Physicochemical Kinetics and Transport at Biointerfaces
IUPAC SERIES ON ANALYTICAL AND PHYSICAL CHEMISTRY OF ENVIRONMENTAL SYSTEMS
Series Editors Jacques Buffle, University of Geneva, Geneva, Switzerland Herman P. van Leeuwen, Wageningen University, Wageningen, The Netherlands Series published within the framework of the activities of the IUPAC Commission on Fundamental Environmental Chemistry, Division of Chemistry and the Environment. INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY (IUPAC) Secretariat, PO Box 13757, 104 T. W. Alexander Drive, Building 19, Research Triangle Park, NC 27709-3757, USA Previously published volumes (Lewis Publishers): Environmental Particles Vol. 1 (1992) ISBN 0-87371-589-6 Edited by Jacques Buffle and Herman P. van Leeuwen Environmental Particles Vol. 2 (1993) ISBN 0-87371-895-X Edited by Jacques Buffle and Herman P. van Leeuwen Previously published volumes (John Wiley & Sons, Ltd): Metal Speciation and Bioavailability in Aquatic Systems Vol. 3 (1995) ISBN 0-471-95830-1 Edited by Andre´ Tessier and David R. Turner Structure and Surface Reactions of Soil Particles Vol. 4 (1998) ISBN 0.471-95936-7 Edited by Pan M. Huang, Nicola Senesi and Jacques Buffle Atmospheric Particles Vol. 5 (1998) ISBN 0-471-95935-9 Edited by Roy M. Harrison and Rene´ E. van Grieken In Situ Monitoring of Aquatic Systems Vol. 6 (2000) ISBN 0-471-48979-4 Edited by Jacques Buffle and George Horvai The Biogeochemistry of Iron in Seawater Vol. 7 (2001) ISBN 0-471-49068-7 Edited by David R. Turner and Keith A. Hunter Interactions between Soil Particles and Microorganisms Vol. 8 (2002) ISBN 0-471-60790-8 Edited by Pan M. Huang, Jean-Marc Bollag and Nicola Senesi
IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Volume 9
Physicochemical Kinetics and Transport at Biointerfaces Edited by HERMAN. P. VAN LEEUWEN Wageningen University, Wageningen, The Netherlands ¨ STER WOLFGANG KO Swiss Federal Institute for Environmental Science and Technology (EAWAG), CH-8600 Du¨bendorf, Switzerland
Copyright ß 2004 by IUPAC Published in 2004 by John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) 1243 779777 Email (for orders and customer service enquiries):
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[email protected], or faxes to (þ44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103–1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02–01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Library of Congress Cataloging-in-Publication Data Physicochemical kinetics and transport at biointerfaces / edited by H.P. van Leeuwen, W. Ko¨ster. p. cm. – (IUPAC series on analytical and physical chemistry of environmental systems; . v.9) Includes bibliographical references and index. ISBN 0-471-49845-9 (ppc : alk. paper) 1. Biological interfaces. 2. Chemical kinetics. 3. Biological transport. I. Leeuwen, H. P. van. II. Ko¨ster, Wolfgang. III. Series. QP517.S87P485 2004 571.6’4–dc22 2003060730 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0471 49845 9 Typeset in 10/12 pt Times by Kolam Information Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by MPG, Bodmin, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.
Contents List of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2
3 4
5
6 7
8 9 10
Physicochemical Kinetics and Transport at the Biointerface: Setting the Stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Ko¨ster and H. P. van Leeuwen Molecular Modelling of Biological Membranes: Structure and Permeation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. A. M. Leermakers and J. M. Kleijn Biointerfaces and Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. P. van Leeuwen and J. Galceran Dynamics of Biouptake Processes: the Role of Transport, Adsorption and Internalisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Galceran and H. P. van Leeuwen Chemical Speciation of Organics and of Metals at Biological Interphases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. I. Escher and L. Sigg Transport of Solutes Across Biological Membranes: Prokaryotes . . . . . W. Ko¨ster Transport of Solutes Across Biological Membranes in Eukaryotes: an Environmental Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. D. Handy and F. B. Eddy Transport of Colloids and Particles Across Biological Membranes . . . . M. G. Taylor and K. Simkiss Mobilisation of Organic Compounds and Iron by Microorganisms . . . . H. Harms and L. Y. Wick Critical Evaluation of the Physicochemical Parameters and Processes for Modelling the Biological Uptake of Trace Metals in Environmental (Aquatic) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. J. Wilkinson and J. Buffle
vii ix xi
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15 113
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205 271
337 357 401
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Herman P. van Leeuwen is an electrochemist who obtained his degree in chemistry at the State University of Utrecht, The Netherlands, in 1969. His thesis was in the field of pulse methods in electrode kinetics, and his Ph.D. degree was awarded cum laude (best 5% in The Netherlands) in 1972. He then joined the Colloid Chemistry and Electrochemistry group of Professor J. Lyklema at Wageningen University, where he became a senior lecturer in 1986. His teaching includes analytical/ inorganic chemistry, electrochemistry and environmental physical chemistry. He was appointed Extraordinary Professor at the University of Geneva in 2000. His current major research interests are twofold: (1) ion dynamics and electrokinetics of colloids, and (2) dynamic speciation and bioavailability of metals in environmental systems. He has published some 140 research papers, reviews and book chapters in these fields. He was chairman of the IUPAC Commission on Fundamental Environmental Chemistry from 1995 to 1999, and Chairman of the Electrochemistry Section of the Royal Dutch Chemical Society from 1993 to 2001. Together with J. Buffle, he edits the IUPAC Book Series on Analytical and Physical Chemistry of Environmental Systems, launched in 1992. Wolfgang Ko¨ster studied biology at the Universities of Bielefeld and Tu¨bingen, Germany. Placing emphasis on biochemistry, plant physiology, genetics and microbiology he was influenced by the work of Professor V. Braun, Professor E. Sander and Professor H. Za¨hner. In 1986, he earned his Ph.D. from the University of Tu¨bingen. With a grant from the German Science Foundation (DFG), in 1988 he became a post-doctoral fellow in the laboratory of Professor R. J. Kadner at the School of Medicine, University of Virginia, USA. He was then promoted to a position equivalent to Assistant Professor and the ‘Habilitation’ in Microbiology at the University of Tu¨bingen. Between 1998 and 1999 he held the position of ‘Visiting Scientist’ (Cantarini Fellowship of the Institut Pasteur and Fellowship of the Centre National de la Recherche Scientific (CNRS), France) in the laboratory of Professor M. Hofnung, Institut Pasteur, Paris, France. In 1999, he joined as a Senior Scientist (leading the group Drinking Water Microbiology) the Swiss Federal Institute for Environmental Science and Technology (EAWAG). He gained teaching experience from the Universities of Tu¨bingen and Hohenheim and ETH Zu¨rich by conducting lectures, seminars and practical courses at undergraduate and graduate levels in the areas of microbiology, genetics, biochemistry, molecular biology and environmental science. His major areas of work and interest comprise: (1) survival strategies and molecular detection methods for bacteria in drinking water and environmental habitats, (2) membrane-associated transport phenomena in microbes, with focus on metal transport in bacteria, and (3) bioavailability and ecotoxicity of metals and hydrophobic organic compounds in green algae.
List of Contributors J. Buffle CABE (Analytical and Biophysical Environmental Chemistry/Chimie Analytique et Biophysicochimie de l’Environnement), 30, quai Ernest Ansermet, Universite´ de Gene`ve, CH-1211 Gene`ve 4, Switzerland F. B. Eddy Environmental and Applied Biology, School of Life Sciences, The University of Dundee, Nethergate, Dundee, DD1 4HN, Scotland, UK B. I. Escher Environmental Microbiology and Molecular Ecotoxicology, Swiss Federal Insti¨ berlandstrasse 133, tute for Environmental Science and Technology (EAWAG), U CH-8600 Du¨bendorf, Switzerland J. Galceran Departament de Quı´mica, Universitat de Lleida, Av. Rovira Roure 191, 25198 Lleida, Spain R. D. Handy School of Biological Sciences, The University of Plymouth, Drake Circus, Plymouth, PL4 8AA, UK H. Harms Swiss Federal Institute of Technology, ENAC–ISTE–LPE, Baˆtiment GR, CH1015 Lausanne (EPFL), Switzerland J. M. Kleijn Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB Wageningen, The Netherlands W. Ko¨ster Environmental Microbiology and Molecular Ecotoxicology, Swiss Federal ¨ berlandstrasse Institute for Environmental Science and Technology (EAWAG), U 133, CH-8600 Du¨bendorf, Switzerland F. A. M. Leermakers Laboratory of Physical Chemistry and Colloid Dreijenplein 6, NL-6703 HB Wageningen, The H. P. van Leeuwen Laboratory of Physical Chemistry and Colloid Dreijenplein 6, NL-6703 HB Wageningen, The
Science, Wageningen University, Netherlands Science, Wageningen University, Netherlands
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LIST OF CONTRIBUTORS
L. Sigg Analytical Chemistry of the Aquatic Environment, Swiss Federal Institute for ¨ berlandstrasse 133, CHEnvironmental Science and Technology (EAWAG), U 8600 Du¨bendorf, Switzerland K. Simkiss School of Animal and Microbial Sciences, University of Reading, Whiteknights, Reading, RG6 6AJ, UK M. G. Taylor School of Animal and Microbial Sciences, University of Reading, Whiteknights, Reading, RG6 6AJ, UK L. Y. Wick Swiss Federal Institute of Technology, ENAC–ISTE–LPE, Baˆtiment GR, CH1015 Lausanne (EPFL), Switzerland K. J. Wilkinson CABE (Analytical and Biophysical Environmental Chemistry/Chimie Analytique et Biophysicochimie de l’Environnement), 30, quai Ernest Ansermet, Universite´ de Gene`ve, CH-1211 Gene`ve 4, Switzerland
Series Preface The main purpose of the IUPAC Series on Analytical and Physical Chemistry of Environmental Systems is to make chemists, biologists, physicists and other scientists aware of the most important biophysicochemical conditions and processes that define the behaviour of environmental systems. The various volumes of the Series thus emphasise the fundamental concepts of environmental processes, taking into account specific aspects such as physical and chemical heterogeneity, and interaction with the biota. Another major goal of the series is to discuss the analytical tools that are available, or should be developed, to study these processes. Indeed, there still seems to be a great need for methodology developed specifically for the field of analytical/physical chemistry of the environment. The present volume of the series focuses on the interplay between organisms and the physical chemistry of the environmental media in which they live. It critically discusses the different physicochemical and biophysical features of the kinetics of processes at the biointerface, with special attention given to aspects such as bioavailability of chemical species, analysis of the necessary mass transfer towards/from the biointerface, routes of transfer through the biomembrane, etc. This volume was realised within the framework of the activities of the former IUPAC Commission on Fundamental Environmental Chemistry of the Division of Chemistry and the Environment. We thank the IUPAC officers responsible, especially the executive director, Dr John Jost, for their support and assistance. We also thank the International Council for Science (ICSU) for financial support of the work of the Commission. This enabled us to organise the discussion meeting of the full team of chapter authors (in Du¨bendorf, Switzerland, 2001) which formed such an essential step in the preparation and harmonisation of the various chapters of this book. The series is indeed being well received, and is growing prosperously. New volumes, on fractal properties of soil particles and physical techniques for micro/nanoparticle characterisation respectively, are in an early stage of preparation. As with all books in the series, these volumes will present critical reviews that reflect the current state of the art and provide guidelines for future research in the field. Jacques Buffle and Herman P. van Leeuwen Series Editors
Preface The idea of broadening the scope of analytical and physical chemistry of environmental systems, to include the interactions of chemical species with living organisms, has been on the priority list of the former IUPAC Commission on Fundamental Environmental Chemistry for some time. It had been recognised that the distribution and transport of chemical components in biotic and abiotic reservoirs is of paramount importance in understanding the effects and fate of organic and inorganic material in environmental systems. The development of mechanistic models for the transport and distribution of chemical components both within and between biotic and abiotic environments requires an integrated approach, with functional links between the various modes of transport of bioactive chemical species and the biophysicochemical processes to which they are subjected. This challenging goal has sparked interest across many fields of research, with the result that much of the key knowledge necessary for progress has become dispersed over several rather poorly interacting disciplines. It is thus timely to integrate these activities, which are all focused essentially on a common broad objective. In doing so, this book will provide the current overall state of the art, as well as highlight key directions for future research. At the end of the 1990s Professor Alex Zehnder (EAWAG/Du¨bendorf CH) and Professor Ronny Blust (Antwerpen University/Belgium) made the first steps towards the creation of this publication. In close cooperation with the chairman of the IUPAC Commission at that time (Dr Herman P. van Leeuwen) it was decided to focus the subject of the book on the physicochemical kinetics of the various processes at the biointerface, and their coupling with the mass transfer of the chemical species involved. This brief would necessarily encompass subtopics such as the structure and permeative properties of biomembranes and their aqueous interfacial layers, the description of diffusive and convective processes at the biointerface, the routes for transport of chemical compounds across membranes, and the biological chemistry of organisms as relevant, for example, for the mobilisation of essential chemicals in the medium. The heart of the book would be on the interphasial region between external medium and organism, and not so much on details of the various chemical conversions inside the organism. A meeting with all prospective chapter authors was hosted by EAWAG/ Du¨bendorf (Switzerland) in early 2001, and the organisation and editing of the book finally came into the hands of the undersigned. It was decided to
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PREFACE
create an opening chapter in order to introduce some of the basic physicochemical features of biointerfaces, e.g. those regarding their characteristic spatial organisation and timescales of transport and chemical reactions. Obviously, chemical speciation and bioavailability are elements of inherent importance in this context. Internalisation of chemical species by organisms is considered in itself, on the level of routes of transfer through the biomembrane, as well as in relation to preceding chemical conversions, stimulated or not by specific reactions of the organism. The elements of the different chapters, that span such apparently disparate topics as the statistical thermodynamics of membrane formation and the endocytosis of colloidal particles, are interlinked as much as possible. The book is concluded by a chapter where experimental biouptake data is critically interpreted in terms of available knowledge, so as to provide an impression of the state of the art. The editors would like to express their gratitude to EAWAG/Du¨bendorf (Switzerland) for hosting the preparatory meeting, to the various external reviewers who carefully looked into the draft chapters and, last but not least, to Dr. Raewyn M. Town (Queen’s University of Belfast) for scrutinising all chapters in terms of their scientific and linguistic qualities and the proper use of IUPAC terminology. Herman P. van Leeuwen and Wolfgang Ko¨ster
1 Physicochemical Kinetics and Transport at the Biointerface: Setting the Stage ¨ STER WOLFGANG KO Microbiology, Swiss Federal Institute for Environmental Science and Technology ¨ berlandstrasse 133, CH-8600 Du¨bendorf, Switzerland (EAWAG), U
HERMAN P. VAN LEEUWEN Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB Wageningen, The Netherlands
Life has developed in media with very diverse chemical compositions and with a variety of physical conditions, including temperature, pressure and their gradients. Evolution actually implies an optimisation in the functioning of organisms in response to these physical and chemical conditions in which they live. It follows that a change in conditions will give rise to a change in the properties of the organism, and this is known in biology as adaptation. The chemical conditions relevant to survival, evolution and adaptation comprise not only the composition and the chemical dynamics of the medium in which the organism is living, but also the availability of the various chemical species. Therefore the distribution and mobilities of inorganic and organic materials in abiotic and biotic media are of paramount importance in understanding their fate and effects in environmental systems. The present book is concerned with the coupling between environmental media and biota, and focuses on the physicochemical features of processes at their interphases.1 Every living cell, whether it be a unicellular organism on its own or a part of a multicellular organisation, is encircled by a biological membrane. In this context, the terms ‘cell membrane’, ‘plasma membrane’, and ‘cytoplasmic membrane’ are used synonymously. Generally, the interphase between an organism and its environment encompasses the elements outlined in Figure 1. The scheme shows that the cell membrane, with its hydrophobic lipid core, has the most 1 Depending on the context, we sometimes prefer the term ‘interphase’ over ‘interface’ because the latter refers to an infinitely sharp dividing plane between two phases. Organisms generally form boundary layers, e.g. the cell wall, that are characterised by a gradual transition from the biological phase to the medium phase, and if we discuss the volume properties of such layers the term ‘interphase’ is more appropriate.
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
2
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Cell membrane O(5 to 10) nm Cell wall layer O(up to 10) nm
O r g a n i s m
Electric double layer O(1 to 10) nm
Medium
Diffusion layer O(10 to 100) µm
Figure 1. Schematic outline of the typical dimensions of the various physically relevant layers at the organism/medium interphase: cell membrane, cell wall layer, electric double layer, diffusive depletion layer
prominent function in separating the hydrophilic aqueous medium from the interior of the cell. The limited and selective permeabilities of the cell membrane towards components of the medium, be they nutrients or toxic species, play a key role in the transport of material from the medium towards the surface of the organism. The lipid bilayer has a very low water content and its core behaves quite hydrophobically, while the cell wall is rather hydrophilic, containing some 80% of water. Physicochemically, the cell wall is particularly relevant because of its high ion binding capacity, and the ensuing impact on the biointerphasial electric double layer. The presence of such an electric double layer ensures that the cell
¨ STER AND H. P. VAN LEEUWEN W. KO
3
wall possesses Donnan partition characteristics, leaving only a limited part of the interphasial potential decay in the diffuse double layer of the adjacent medium. Mass transfer phenomena usually are very effective on distance scales much larger than the dimensions of the cell wall and the double layer dimensions. Thicknesses of steady-state diffusion layers1 in mildly stirred systems are of the order of 105 m. Thus, one may generally adopt a picture where the local interphasial properties define boundary conditions while the actual mass transfer processes take place on a much larger spatial scale. The availability of chemical species to organisms is defined by a number of basic features, including: . their chemical reactivity, as derived from equilibrium distributions of species and their rates of interconversion; . the supply (flux) of these chemical species to the relevant sites at the surface of the organism, as governed by their mass transport properties and the concentration gradients that arise at the interphase as a consequence of the interplay between chemical reactivity and biological affinity; and . the internalisation of the chemical species, governed by an internalisation rate constant, kint , usually accompanied or followed by some bioconversion process. The actual processes of uptake of chemical species by an organism typically encompass transport in the medium, adsorption at extracellular cell wall components, and internalisation by transfer through the cell membrane. Each of these steps constitutes a broad spectrum of physicochemical aspects, including chemical interactions between relevant components, electrostatic interactions, elementary chemical kinetics (in this volume, as pertains to the interface), diffusion limitations of mass transfer processes, etc. Life on Earth in all its diversity could never have evolved without the existence of lipids that are able to spontaneously arrange in aqueous solutions into structures such as micelles or bilayers. Although the composition of biological membranes varies markedly with their various functions, and with the type of organism and environmental conditions, there is a common structural organisation involving lipid, protein, and carbohydrate components. With only a few exceptions, lipids are arranged as bilayers and constitute the basic characteristic architecture for a variety of biomembranes found in different living organisms. The average thickness of biomembranes is approximately 7 nm. The majority of lipids in the plasma membrane of bacteria (prokaryotes) 1 Such layers are frequently denoted as ‘unstirred’ layers. The term ‘unstirred’ however, is physically incorrect [1], since velocity profiles in liquid media are continuous functions which only approach zero at the actual interface. In gel layers the liquid velocity is generally low, but this is due to their high viscosities.
4
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
and eukaryotes are comprised of acyl chains (e.g. palmitic acid, stearic acid, oleic acid), which are ester-linked to glycerol. However, many other more complex lipids that contain additional elements like phosphorus, nitrogen or sulfur, may also be found in biomembranes. In addition, hydrophilic components such as small sugars, choline, serine, or ethanolamine are commonly found. Phospholipids containing a phosphate group are ideal amphipathic components, and represent the largest group of membrane lipids. Sterols like cholesterol are almost exclusively found in eukaryotic membranes, where they can make up to 25% of the total lipids. Archaebacteria can exist in the most extreme conditions, and their membrane composition differs from those of the bacteria and eukaryotes. Some unusual components like hopanoids have been found in this group of organisms. Hopanoids are pentacyclic triterpenoids, biosynthetically derived from the linear molecule squalen, which is formed by joining six isopentenyl units. It is assumed that hopanoids may play a role similar to that of sterols in eukaryotic cells. Another common feature of archaeal membranes are acyl chains derived from repeating units of isoprene (e.g. phytanol) which are ether-linked to glycerol or nonitol. The membranes of hyperthermophilic Archaea living at high temperatures are composed of glycerol di-ethers and glycerol tetra-ethers. Lipids containing biphytanyl chains can form monolayers (resembling somewhat the usually found bilayers) with a hydrophobic milieu inside and hydrophilic surfaces outside. They are very stable under extremely high temperatures. Proteins represent another major group of membrane components. They play structural roles and/or are involved in many cellular processes, which are strictly coupled to membranes. Proteins can be either entirely embedded within the bilayer, or they might be firmly anchored (e.g. by a hydrophobic transmembrane segment composed of hydrophobic amino acid side-chains or as lipoprotein), or they can be just associated with the surface. Carbohydrates related to membranes can be found as lipopolysaccharides or as parts of glycoproteins. Sugars are often characteristic determinants of cell surfaces (see below). The great majority of carbohydrates are found in the outer leaflet of a membrane, resulting in an asymmetrical structure. This is especially true for many plasma membranes and the outer membrane of Gram-negative bacterial cells (see below). The membrane is the regulating barrier for exchange of chemical species between the environmental medium and cell interior. It may be practically impermeable to one type of species and highly permeable to another. In the chain of transport steps from the bulk of the medium to the cell interior, the membrane transfer step may thus vary from fully rate-limiting to apparently fast with respect to transport in the medium. The overall rate of this biouptake process is determined by mass transport either in the medium or through the membrane: the actual rate-limiting step will depend on a large variety of factors. Membrane
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5
transfer rates may be influenced by external chemical conditions, such as pH, ionic strength, presence of surfactants, etc., which alter the permeability features of the membrane, as well as by biological factors like conditioning and adaptation, which may regulate the effectiveness and abundance of transporter functions inside the membrane. An intact and largely undisturbed cytoplasmic membrane or plasma membrane representing the innermost layer enclosing a biological cell is absolutely essential for its vitality. Any major impairment or even a small hole would cause unimpeded exchange of ionic species and thus electrical depolarisation of the membrane, resulting in immediate cell death. This effect can also be generated by certain toxins, which assemble into pores in the membrane. Therefore, channels that are simultaneously open to both sides of the cytoplasmic membrane cannot persist in a living cell. With the help of the atomic force microscopy (AFM) technique, it is possible to obtain three-dimensional images of surface structures at the nanometre scale. Erythrocyte membranes, which are stable during the preparation of an AFM experiment, can be used as a rather basic model with respect to composition and surface structure. This enables a number of details to be visualised, e.g. the deformation of a rhesus monkey erythrocyte membrane caused by an infecting virus (Figure 2). In most cases, however, a given solute approaching the surface of a living cell has to deal with more complex structures, a ‘naked’ membrane surface being highly unusual. In human or animal cells, various glycopeptides and glycosylated proteins are integrated into the lipid bilayer, while most plant (a)
(b)
500 2000 400 1500 300
500
2000
1000 1500 1000
500
400
200 300 200
100
500 0 0
100 0
0
Figure 2. Atomic force microscopy images showing the surface of a rhesus monkey erythrocyte membrane. Damage, such as formation of humps on the peripheral surface and pits in other parts, results from the interaction with virions of the canine parvovirus. (a) edge of erythrocyte; (b) pits on membrane surface. (Source: http://www.ntmdt.ru/ publications/download/211.pdf, Reproduced with permission from Dr Boris N. Zaitser)
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
cells are surrounded by a cell wall composed of polymers of carbohydrates. Bacteria are usually encircled by a sacculus: this peptidoglycan (or murein) sheet contains glycan chains formed by the alternating sugar derivatives N-acetylglucosamine and N-acetylmuramic acid, which are cross-linked by small peptides building a network. Gram-positive bacteria are characterised by multiple layers of peptidoglycan with attached teichoic acids (acidic polysaccharides consisting predominantly of glycerophosphate mannitol phosphate or ribitol phosphate). Alternatively, Gram-negative bacteria possess only a single peptidoglycan layer, but additionally have a second membrane, called the ‘outer membrane’, harbouring various proteins, lipoproteins, and lipopolysaccharides. Moreover, a number of bacterial species produce external capsules or slime layers, while others are capable of building spores that are highly resistant to adverse environmental conditions. This book focuses on the processes that control the transfer of chemicals between environmental media and living organisms. The major driving forces for transport and chemical conversion are contained in the electrochemical potential, that is, the chemical potential difference plus the electrostatic freeenergy change for charged species. Electrical potential differences between the inner and outer boundary of the biological membrane play a crucial role in the various physiological mechanisms. Such potential differences, usually denoted as membrane potentials, derive from differences in permeability of the membrane with respect to ions in the inner and outer media. Common membrane potential expressions, like the Goldman–Hodgkin–Katz equation [2,3] for Naþ , Kþ and Cl , are valid under steady-state conditions of zero net charge transport: X
zi Ji ¼ 0
(1)
i
where i indexes the permeable ionic species, zi is the charge number of i, and Ji is the flux1 of i across the membrane. Fluxes of charged species are generally the result of gradients in concentration, potential and pressure, which are collect~. ively represented by the electrochemical thermodynamic potential m Application of elementary conservation laws leads to formulation of a general expression for Ji , which is often denoted as the Nernst–Planck equation: Ji ¼ Di grad ci (zi =jzi j) ci ui grad C þ ci
(2)
1 For convenience, fluxes from the medium towards the organism will be counted as positive throughout this book. Analysis of mass transport in the medium is usually based on a coordinate system with the origin at the interface and a positive axis going outward, leading to a negative sign in fluxes towards the interface.
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Equation (2), where D denotes diffusion coefficient, c concentration, u mobility, C electric potential, and flow velocity, explicitly shows the diffusion, conduction and flow terms respectively. Within the context of biological systems, transport represented by this Nernst–Planck equation (2) is often referred to as ‘passive’ transport. This qualification is intended to make a distinction from situations where apparently transport takes place in the opposite direction, against a gradient of concentration or potential or pressure. Obviously, such ‘uphill’ or ‘active’ transport requires special conditions, which in biomembrane transport are created by metabolic chemical reactions such as ATP hydrolysis. The coupling of the ionic transport process with the energy providing chemical reaction must be of an asymmetrical nature, in the sense that the production/ consumption of ions at the inner side of the membrane is different from that at the outer side. It has been hypothesised that the asymmetry is in the kinetic features of the interfacial transfer process, in such a way that, in the apparent steady-state, the ratio between influx and efflux is modified. Under such conditions, which essentially are of a nonequilibrium nature, it is possible to realise net uphill ionic transport, and this is the basis of the biologically well-known ionic pumps. The existence of ionic pumps is not in conflict with fundamental transport laws like the Nernst–Planck equation (2): these pumps are generated by the special geometrical and chemically asymmetrical conditions in a biological membrane. In fact, for a rigorous analysis of the pump situation, the Nernst–Planck conservation equation has to be complemented with a chemical source term with a confined spatial distribution. Transport across biological membranes is facilitated by their fluid-like nature. The water content varies strongly from the core to the outer boundary; overall it comes to some 25% by mass. The classical Singer–Nicholson fluid mosaic model represents the biomembrane as a two-dimensional sea of the lipid bilayer, in which proteins and other constituents are floating around. Indeed, most lipid membranes are fluid at physiological temperature, and consequently the lateral mobility of the lipids and proteins is relatively high, whereas the transversal movements (including the flip-flop exchange of lipids between the inner and outer sides) are strongly limited. This feature explains the maintenance of the asymmetry of the membrane with respect to composition and orientation of the ion transporter proteins. As outlined above, this chemical asymmetry is essential for the basic functioning of the biomembrane. Below a certain temperature, the fluid bilayer turns into the crystalline–gel state, in which the lateral mobility of the constituents is greatly diminished. In the fluid state, the lateral diffusion coefficient of lipids in the bilayer structure is O(1013 ) m2 s1 (the symbol ‘O’ is used to indicate order of magnitude). Interestingly, it has been shown that the diffusion coefficients of phospholipids may differ greatly from the inner to the outer leaflet of the biomembrane layer [4,5]. Again, this is related to the differences in chemical
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
composition. Lateral transport of lyophobic species like water and ions in the core of the bilayer is not very relevant, because of their extremely low local concentrations. Mobilities of ions in the interphasial region, even inside the stagnant water layer at the actual interface between the aqueous phase and the lipid bilayer, are on the level of that in the bulk solution [6]. As noted above, biouptake involves a series of elementary processes that take place in the external medium, in the interphasial region, and within the cell itself. One of the most important characteristics of the medium is the chemical speciation of the bioactive element or compound under consideration. Speciation not only includes complexation of metal ions by various types of ligands, but also the distribution over different oxidation states, e.g. Fe(II) and Fe(III), and protonation/deprotonation of organic and inorganic acids of intermediate strength. The relationship between speciation and the direct or indirect bioavailability1 of certain species has received a lot of recent attention. Organisms are able to take advantage of a wide range of nutrients, ranging from trace elements to biopolymers such as proteins, DNA, RNA, starch, lignin, etc. Although they are often present in relatively large amounts, these compounds are not always accessible, as illustrated by the following examples: (1) iron, which is an essential nutrient for most living bacteria (lactobacilli being the only notable exception), is the fourth most abundant metal on Earth. However, iron is not readily bioavailable under ‘normal’ physiological conditions. In the environment it is mainly found as a component of insoluble hydroxides; while in biological systems it is chelated by highaffinity iron-binding proteins (e.g. transferrins, lactoferrins, ferritins) or found as a component of erythrocytes (haem, haemoglobin, haemopexin). As a consequence, organisms have evolved a number of different sequestering strategies for this metal. Under anaerobic conditions, ferrous iron can be transported without the involvement of any chelators. Likewise, at pH 3, ferric iron is soluble enough to support growth of acid-tolerant bacteria. At higher pH values, however, iron is mostly found in insoluble compounds. Therefore, a great variety of low-molar-mass iron ligands, socalled siderophores, which bind Fe3þ with very high affinity, are produced by many bacterial species, certain fungi, and some plant species. These chelators are released in their iron-free forms and subsequently transported back into the organism as ferric-siderophore-complexes. Furthermore, a 1 The notion ‘bioavailability’ is used with different meanings. Environmental chemists understand it in terms of the supplying potential of the medium, whereas (micro)biologists relate it to the assimilation properties of the organism. In the case of metal uptake, for example, a certain complex may be fully labile and thus potentially contribute to the supply of free metal ions. In contact with an organism with a modest affinity towards the metal in question the uptake requirements may be so small that such labile complexes are completely unimportant and their lability irrelevant. In microbiological jargon this complex would be ‘not bioavailable’, whereas a chemist would say that this complex is fully available to the organism.
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number of organisms are able to use haem-bound iron from haemoglobin and similar molecules. Some bacterial species can acquire iron that is released by an as-yet unknown mechanism from transferrins or lactoferrins, whereas vertebrates take up the whole iron–protein complex. All these processes involve specific uptake systems in the cell envelope and in the cytoplasmic membrane. (2) although many biopolymers represent an excellent source of nutrients, they are often too large to be transported into a biological cell. A number of species have developed ways to overcome this problem by the secretion of enzymes, which are able to breakdown polymers into their constituents. Many organisms originating from all kingdoms of life are known to use this strategy. So-called exo-enzymes, which are released from the producing cell, can be classified according to their functions (e.g. proteases, lipases, nucleases). Although in some cases these enzymes only carry out a partial degradation, oligomers (e.g. peptides) up to a certain size become ‘bioavailable’ and can then be transported into the cell. In particular, the kinetics of dissociation reactions as preceding steps in the biouptake of organics and metals from complex media have been extensively studied. It is likely that the gap between the concentration of labile species, as measured by a certain dynamic analytical technique [7], and the effective bioavailability of that species will soon be bridged. The role of the adsorption of bioactive species in the cell wall region becomes important as soon as a mechanistic interpretation of biouptake fluxes beyond their mere values in the ongoing steady-state is sought. Back-extrapolation of fluxes to zero time, or even better, analysis of the initial transient behaviour of the flux, will provide more comprehensive information on the molecular details of the internalisation kinetics. Such means will enable distinction between receptor sites (physiologically active) and mere adsorption sites (physiologically inactive), metal ion buffering action of the adsorption sites in the cell wall region, and true nonconditional rate constants of the actual membrane transfer steps. Comprehensive models for the overall biouptake process range from simplifying schemes like the free-ion activity model (FIAM) [8] and the biotic ligand model (BLM) [9] to more differentiated approaches at the level of the Best equation (i.e. Michaelis–Menten control of the uptake and mass transport limitations in the medium) coupled with homogeneous chemical kinetics of formation of the bioactive species in the medium [10–12]. Clearly, the local speciation in the biological interphase may be very different from that in the bulk phase, and this may have a great impact on the nature and rate of bioaccumulation processes. Thus, with the ionic composition of the medium generally being very different from that inside the organism, ion trapping mechanisms may be essential in facilitating efficient transport across the cell membrane.
10 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
In addition to their function as a permeability barrier to the extracellular environment, membranes also fulfil important tasks inside most eukaryotic cells and in some bacteria. One crucial role is the separation of different cell compartments. A few examples of intracellular membranes may illustrate the large variety of membrane functions: . a special type of membrane represents the so-called ‘tonoplast’ that surrounds the vacuoles characteristic of many plant cells. Vacuoles, which can differ in size, help to maintain the osmotic pressure of the cell, and are used as temporary stores for reserve materials or final storage compartments for waste products of the cell metabolism. The central vacuole of a fully differentiated cell can reach an extensive size, thus constituting the major part of the cell’s volume. . the nuclear envelope consists of an outer and an inner membrane surrounding the nucleus, which harbours most of the genetic information of the eukaryotic cell. The nucleus is the location of, for example, replication, transcription, and RNA processing, and the enzymes involved in these vital functions have to be imported from the cytosol. . an extensive intracellular membrane system, the so-called endoplasmatic reticulum (ER) is directly connected to the nuclear envelope. A significant portion of protein synthesis is associated with the ER. . stacks of membranous cistern-like structures (dictyosomes) as well as derived small vesicles and tubules (Golgi vesicles) form the Golgi apparatus. Dictyosomes and Golgi vesicles are involved in intracellular transport and secretion of macromolecules. Exocytosis describes a process in which such vesicles undergo a fusion with the plasma membrane and consequently release enclosed substances into the external medium. Likewise, this membrane flow can occur as a reverse process: endocytosis. In this case invagination of membrane areas leads to intracellular vesicles containing substances from the external medium. This process of ‘budding’ can also occur in the opposite direction, thus delivering cellular components (or membraneenclosed phage particles) to the external medium. . membranes are essential elements of organelles that are exclusively found in plants – the plastids. Among them, the chloroplasts, typical of green plants and algae, display a complex structure. Surrounded by an envelope composed of outer and inner membrane, a complicated system, the thylakoid membranes (see Figure 3), harbours all elements essential for photosynthesis. . a special compartment, also consisting of an outer and inner membrane, is realised in mitochondria. These organelles contain all components for generating energy in the form of adenosine triphosphate (ATP) via oxidative phosphorylation. The examples mentioned above exclusively apply to eukaryotic cells. In prokaryotic cells, intracellular membranes are the exception. However,
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Figure 3. Example of intracellular membrane organisation: a transmission electron microscopy (TEM) image of a section through the thylakoid stack from a chloroplast. (Source: http://www.ru.ac.za/administrative/emu/gr10p6.htm, Reproduced with permission from Dr. R. Cross)
exceptions are known in a few groups of bacteria where complex intracytoplasmic membrane systems result from the invagination of the plasma membrane. Vesicles, tubuli and thylakoid-like structures are reported. Some of them are present in certain phototrophic bacteria. Extensive intracytoplasmic membrane systems are also found in nonphototrophic nitrifying methane-utilising bacteria. Transport processes across membranes can be divided into several categories: . transport of signals in the form of a signal transduction cascade can be achieved by a series of conformational changes in the components involved, or by consecutive modification events (e.g. phosphorylation–dephosphorylation, methylation–demethylation). These processes enable cells to communicate with their environment, and allow them to respond to changing conditions such as pH, osmolarity, pressure, temperature. . uptake of ions and nutrients (mostly molecules of lower molar mass) and the secretion of metabolites and other smaller molecules (e.g. signalling molecules, siderophores) depend on different types of transport systems, which are either using primary energy sources such as ATP or which are coupled to a gradient like the membrane potential. . transport (import and export) of polymers, including proteins, is also mediated by special transport systems which, in many cases, represent multicomponent systems.
12 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
All these transport processes are of comparable importance for an organism in order to adapt to changing conditions and to exist in a given environment. This book focuses on the mass transfer aspects across biomembranes, involving ions, molecules, and particles. Intact membranes are essential for a great variety of vital functions, such as energy-generating processes taking place in the mitochondria of eukaryotes (see above) or at the cytoplasmic membrane of bacteria. In addition, membranes are indispensable for components involved in electron transport chains, and photosynthesis is strictly coupled to the lipid bilayers. Export machineries for proteins, as well as secretion systems for a variety of substances (such as metabolites, signalling molecules, enzymes and extracellular structures) are located in membranes. Moreover, components involved in cell growth and cell division are specifically associated with membranes. In bacteria, a great variety of extracellular structures are anchored in the membranes, which constitute the envelope. Some structures (e.g. pili, fimbriae) take part in cell–cell interaction, adhesion to surfaces, and biofilm formation, others (e.g. flagellae) allow locomotion and mobility. Since so many functions and processes all occur either within, or coupled to, lipid bilayers, it is easy to realise that a fine-tuned balance of embedded and associated components is highly important for the integrity and functionality of all the different types of biomembranes. Therefore, the design and interpretation of test systems and in vitro assays for studying phenomena related to membranes must consider that both the elimination or overproduction of a single membrane protein (or indeed any set of components) may disturb a fragile system and lead to artificial results. The various aspects mentioned above can be summarised as follows: fluxes and distribution of solutes in aqueous solutions and at (or through) hydrophilic/hydrophobic interphases (see Figure 4) can be modelled by following the rules of physics and physical chemistry. In many cases, equations describing transport phenomena can be rather simple, so long as processes like diffusion and osmosis are dominant, and the shapes and surfaces of particles or cells are not very complicated. However, the complexity of the situation increases greatly once parameters like ‘transport against a concentration gradient’, ‘multicomponent systems’, ‘high or low affinity to a substrate’ or ‘complex structures associated with a cell surface’ have to be taken into account. Thus the development of mechanistic concepts and models for the transport and compartmentalisation of chemicals in bioenvironmental systems requires an integrated approach, which provides functional links between processes at different levels of organisation. Indeed, a thorough understanding of environmental processes can only be achieved if studies of the chemistry (e.g. reaction kinetics and mobilisation) and the biology (e.g. transport near and across biological interphases) are combined. Although it is now widely recognised that integration is the way to proceed, these areas of research have to date been the subjects
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diffusion
MY
kd(MY)
ka(MY) kin(M) Mads
M
ka(ML)
kd(ML)
kin(ML) ML
excretion
Figure 4. Schematic representation of the various processes involved in the transfer of metal ions from a complex medium to an organism. The free metal ion and the lipophilic complexes ML are effectively bioactive. Bioinactive complexes MY, present in the medium, can only contribute to biouptake processes via dissociation into M
of the individual disciplines, with little interaction between them. The present book critically summarises and integrates current knowledge of the physicochemical mechanisms, kinetics, transport and interactions involved in processes at biological interphases in environmental systems. It starts with fundamental chapters on the physical chemistry of the structure and permeation properties of the lipid bilayer membrane (Chapter 2), and the basic features of various chemical gradients at the biological interphase and ensuing mass transport from/towards its environment (Chapter 3). The coupling of transport processes in the medium with the actual transfer of chemical species through the cell
14 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
membrane, whether or not this occurs via an adsorbed intermediate, is analysed in Chapter 4 and the role of the chemical speciation of both organic compounds and metals at the biological interphase is discussed in Chapter 5. The biochemical background of transporter functions for the transfer of chemical species across the biological membrane is highlighted for prokaryotes (Chapter 6) and for eukaryotes (Chapter 7). The particular case of transfer of colloids and particles across the biological membrane, known as endocytosis, is reviewed in Chapter 8. The active mobilisation of components in the medium by specific chemical strategies of organisms, with emphasis on mobilisation of organics, is evaluated in Chapter 9. Finally, a number of elements of the foregoing chapters are integrated in Chapter 10, where experimental data for the biological uptake of trace elements from aquatic media are modelled on the basis of knowledge of the speciation and transport parameters of the medium and the cell membrane.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10.
11. 12.
Levich, V. G. (1962). Physicochemical Hydrodynamics. Prentice Hall, Englewood Cliffs, NJ. Goldman, D. E. (1943). Potential, impedance and rectification in membranes, J. Gen. Physiol., 27, 37–59. Hodgkin, A. L. and Katz, B. (1949). The effect of sodium ions on the electrical activity of the giant axon of the squid, J. Physiol. (Lond.), 108, 37–77. Cevc, G. and Marsh, D. (1987). Phospholipid Bilayers. Wiley-Interscience, New York. van der Wal, A., Minor, M., Norde, W., Zehnder, A. J. B. and Lyklema, J. (1997). Electrokinetic potential of bacterial cells, Langmuir, 13, 165–171. Lyklema, J. (1995). Fundamentals of Interface and Colloid Science. Volume II: Solid–Liquid Interfaces. Academic Press, London. Buffle, J. and Horvai, S. eds. (2000). In Situ Monitoring of Aquatic Systems. Chemical Analysis and Speciation. Vol. 6, IUPAC Series on Analytical and Physical Chemistry of Environmental Systems, Series eds. Buffle, J. and van Leeuwen, H. P., John Wiley & Sons, Ltd, Chichester. Morel, F. M. M. and Hering, J. (1983). Principles and Applications of Aquatic Chemistry. Wiley-Interscience, New York. Playle, R.C. (1998). Modelling metal interactions at fish gills, Sci. Total Environ., 219, 147–163. Whitfield, M. and Turner, D.R. (1979). Critical assessment of the relationship between biological thermodynamic and electrochemical availability. In Chemical Modeling in Aqueous Systems. ed. Jenne, E. A., ACS Symposium Series, Vol. 93, pp. 657–680. Hudson, R. J. M. (1998). Which aqueous species control the rates of trace metal uptake by aquatic biota? Observations and predictions of non-equilibrium effects, Sci. Total Environ., 219, 95–115. van Leeuwen, H. P. (1999). Metal speciation dynamics and bioavailability. Inert and labile complexes, Environ. Sci. Technol., 33, 3743–3748.
2 Molecular Modelling of Biological Membranes: Structure and Permeation Properties FRANS A. M. LEERMAKERS AND J. MIEKE KLEIJN Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB, Wageningen, The Netherlands
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2
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Water and the Hydrophobic Effect . . . . . . . . . . . . . . . . . . . . . 1.2 The Hydrocarbon Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Hydrocarbon–Water Interface . . . . . . . . . . . . . . . . . . . . . 1.4 Surfactants and the Surfactant Packing Parameter . . . . . . . . 1.5 Membrane Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Lipid Phase Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Models of Lipid Bilayers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Ensembles in Molecular Modelling . . . . . . . . . . . . . . . . . . . . . The Molecular Dynamic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Force Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Time and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Dipalmitoylphosphatidylcholine Bilayers . . . . . . . . . . . . . . . . 2.7 Coarse-Grained MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Monte Carlo Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Box and the Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Pragmatic Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Hybrid MC and MD Approaches . . . . . . . . . . . . . . . . . . . . . . 3.5 Typical Monte Carlo Results . . . . . . . . . . . . . . . . . . . . . . . . . . The Self-Consistent-Field Technique . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
16 19 20 20 21 24 26 30 30 31 32 33 33 34 35 35 39 40 44 46 46 47 48 48 49 51 52
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4.2 4.3
The Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 The Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3.1 Lipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3.2 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.3 Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.4 Free Volume and the Pressure . . . . . . . . . . . . . . . . . . . 56 4.4 The Segment Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 The SCF Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 Phosphatidylcholine Bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.7 The Lateral Pressure Profile. . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.8 Comparison of SCF and MD for SOPC Membranes. . . . . . 70 4.9 Case Studies: SCF Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.9.1 Effects of the Length of the Hydrocarbon Tails . . . . 74 4.9.2 Lipid Variations: Charged Lipids in Bilayers . . . . . . 75 4.9.3 The Gel-phase of DPPC Bilayers . . . . . . . . . . . . . . . . 76 4.9.4 Mechanical Parameters of Lipid Bilayers. . . . . . . . . . 78 4.9.5 Membrane–Membrane Interactions . . . . . . . . . . . . . . 83 4.9.6 A DPPC Layer as a Substrate for a Polyelectrolyte Brush . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5 Transport and Permeation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1 Solubility–Diffusion Mechanism . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 Equilibrium Aspects: Partitioning . . . . . . . . . . . . . . . . 88 5.1.2 Dynamic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Pore Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 MD Modelling of Mediated Membrane Transport . . . . . . . 97 6 Summary, Challenges and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 99 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
1
INTRODUCTION
In this review we bring together issues relevant for the structure and permeation properties of biological membranes, from a theoretical, physicochemical perspective. After an introduction concerning the nature of biological membranes, models to evaluate their structural, thermodynamic and mechanical properties will be critically discussed. The input and output of models with molecular detail, in particular the molecular dynamics (MD) and self-consistent-field (SCF) approaches, are analysed. The underlying idea is that all membrane properties should be deducible from that of their constituents. We will pay some attention to the relative importance of intra- and intermolecular forces. Most SCF results that will be presented are updated, i.e. literature results are recomputed using a considerably improved set of interaction parameters. We
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will attempt to cover the complete set of membrane properties that has been considered by SCF modelling to date. The hope is that, eventually, modelling will support measurable system characteristics and predict the unmeasurable ones. With respect to the structure of membranes, we will concentrate on that of phospholipid bilayers, and only briefly mention the work on more complex systems. The review will be biased towards those issues that are relevant to permeation. Our conclusion is that detailed knowledge is available on the structural properties of membranes. The molecularly realistic models mentioned are all in good mutual agreement and indeed complement available experimental data. Structure is only one aspect relevant for transport of molecules across the bilayer; partitioning and dynamics are others. Not surprisingly, significantly less is known about the molecular details that control permeation issues. Indeed, modelling of transport phenomena, especially when specialised molecules are involved, is one of the key challenges for the near future. The fluid mosaic model of the nature of the biological membrane, as put forward by Singer and Nicolson [1] in 1972, is still the starting point for most of the modern work done on biomembranes. The first-order effects described by this model are undisputed. The basis of the biological membrane is a bilayer of lipid molecules, i.e. the lipid bilayer. Computer graphics allow beautiful representations of such a bilayer. The example given in Figure 1 depicts a snapshot of a MD simulation of a bilayer composed of phosphatidylcholine lipids in a slab of water. The fluid mosaic model points further to the role of the lipid molecules. On the one hand, the bilayer forms a barrier to transport for many molecular species that go from one compartment to the other, i.e. the membrane is semi-permeable. The matrix also provides a medium in which protein molecules are incorporated in such a way that they are biologically active. The fluid mosaic model also points correctly to the strongly anisotropic mobility of molecules in this topology. A lipid molecule easily moves around in its ‘own’ monolayer, but it is strongly hindered from flipping from one side to the other (flip-flopping) or jumping out of the bilayer into the aqueous phase. The fluid mosaic model conveniently describes how the constituent molecules are ordered, and it correctly describes, in first order, some of the membrane’s properties. However, it does not give explicit insight into why the biological membrane has a particular structure, and how this depends on the properties of the constituent molecules and the physicochemical conditions surrounding it. For this reason, only qualitative and no quantitative use can be made of this model as it pertains to permeation properties, for example. It is instructive to review the physicochemical principles that are responsible for typical membrane characteristics. In such a survey, it is necessary to discuss simplified cases of self-assembly first, before the complexity of the biological system may be understood. The focus of this quest for principles will therefore be more on the level of the molecular nature of the membrane, rather than viewing a
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Figure 1 (Plate 1). A molecular view of a small section of a flat lipid bilayer generated by molecular dynamics simulations. The bilayers are composed of 1-stearoyl-2-docosahexaenoyl-sn-glycero-3-phosphatidylcholine lipids, i.e. the sn1 chain is 18 C atoms long and the sn2 chain has 22 carbons, including six cis double bonds. The hydrophobic core is in the centre of the picture, and the hydrated head-group regions are both on top and bottom of the view graph. The head group is zwitterionic and no salt has been added. From [102]. Reproduced by permission of the American Physical Society. Copyright (2003)
membrane as a flexible sheet that can be characterised by a number of mechanical parameters. However, the mechanical parameters must also have a molecular origin. Indeed, the large-scale properties of the bilayer (i.e. the lamellar topology) are essential for the compartmentalisation function of membranes. This has a direct link to transport and permeation. For this reason, we will also give this topic some attention. Lipid assemblies of the lamellar type, such as lipid bilayers, can feature a true phase transition in which the topology does not change. Upon cooling, the bilayer goes from the fluid phase to the gel phase. In the fluid phase, the acyl chains are disordered, in the sense that there is enough free volume around the chains to allow for chain conformation variations. In the gel phase, the acyl chains are more densely packed and believed to be ordered in an all-trans (straight) configuration. For very pure systems, at temperatures below this sharp gel-to-liquid phase transition, there are several other states and distinct transitions detectable (pre-transition, ripple phase, etc.). These phases will not be reviewed here. In biomembranes, many type of lipids (and other molecules) occur, and it is known that for this reason the gel-to-liquid phase transition is
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not sharp and occurs over a temperature range of ten degrees or more. In the transition region, there are relatively large density fluctuations, especially when the transition is sharp. These fluctuations lead to pronounced changes in permeation characteristics [2]. As these changes are transient we will not focus on these effects. 1.1
WATER AND THE HYDROPHOBIC EFFECT
Soft condensed matter, e.g. a liquid, is composed of molecules that strongly interact with each other. Without the intermolecular interactions, one only would have gases and, at extreme densities, solids. In all systems that are of interest for biology, water is the main liquid. The water H2 O molecule is very special, and to some extent water has peculiar properties [3]. It has a very low molar mass and, as such, one would expect water to be a gas at ambient conditions, like methane (CH4 ). However, water has the ability to form up to four H-bonds per molecule (each molecule can donate two and accept two Hbonds) and at room temperature is far closer to its freezing temperature than to its critical temperature. Water has a local tetrahedral arrangement dictated by the strongly angular-dependent H-bonds between neighbours. This ‘network’ leaves open voids or free spaces that are equal in size to the water molecule themselves. These voids are stabilised against a collapse by repulsive forces between the molecules, which occur when two water molecules make close contact with each other in an orientation unsuitable for H-bond formation. A realistic picture of water is that H-bonds tend to break and reform easily, and in such a dynamic situation it can be understood that water has a relatively low viscosity. Water is not a particularly good solvent. Not many types of molecules mix with water in all proportions. When a compound has no ability to participate in the H-bonding network, it is likely to be rejected from the water phase. We call these compounds hydrophobic. Hydrocarbon chains are representatives of this class of molecules: hydrocarbons and water strongly segregate. Charged molecules, i.e. ions, are an exception to this rule. The water molecule has a dipole with a fractional negative charge near the oxygen, and a corresponding positive charge halfway between the protons. This dipole will orient in such a way that the molecule is attracted to the ion. As a result, a hydration layer coats the ion. This cluster of molecules dissolves easily, often to very high concentrations, in water. Usually, dissolution of a small amount of one compound in a pure liquid is enthalpically unfavourable and driven by an increase in (mixing) entropy. At room temperature, the opposite is true for the dissolution of a small apolar compound in water. This unexpected behaviour is referred to as the hydrophobic effect [4]. Classically, this effect has been rationalised by ordered water structures around apolar compounds (entropy reduction) and the increase in number
20
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
of H-bonds supposed to form in this layer (enthalpy gain). This interpretation has its problems. It is not very likely that the apolar compounds that prevent the H-bonding network in water from developing can generate structures that overcompensate this loss. We strongly prefer the less-known interpretation of Besseling et al. [5,6]. These authors argue that dissolving a small apolar compound has the effect that the H-bonding network is deformed (entropy loss). The enthalpic gain is not explained by an increase of H-bonds (this remains constant or can even go down when the network is stretched too much), but by the fact that the additives can screen the unfavourable inter-water contacts discussed above. In this picture, the hydrophobic effect and the density maximum of water at 4 8C are strongly correlated phenomena. It is of interest to mention that, at elevated temperature, the usual thermodynamic behaviour is also found for dissolving apolar entities in water. Irrespective of how the free energy of mixing is split up into an enthalpic or entropic part, in effect it remains very difficult to dissolve much of the apolar compound in water. These thermodynamic subtleties are important for explaining the sensitivity for membranes with respect to temperature. We will not do this here. 1.2
THE HYDROCARBON CHAINS
Hydrocarbons can be viewed as semi-flexible chains. The rotation around a C–C bond can be expressed by the rotational isomeric state scheme. The trans configuration is favoured with respect to gauche conformers by an energy difference of about 1 kT. This means that the persistence length – the length along the contour of the chain in which a free-flying chain in a good solvent can ‘remember’ the direction it is going in – is four to five carbon atoms. Short hydrocarbon chains with a chain length of order 16 carbons can thus be viewed as extremely short polymer chains of just about four segments long. In a hydrocarbon melt, the chains are oriented isotropically. This is not the case for hydrocarbon chains at the hydrocarbon–air interface. These chains have anisotropic conformations and are more ordered. Upon cooling a hydrocarbon phase, the freezing of the liquid starts at the hydrocarbon–air interface and at slightly lower temperature the bulk solidifies. This phenomenon is called surface freezing, and has received considerable attention in recent years [7,8]. The slight (order-induced) increase in density of the acyl chains at the boundary is responsible for this effect. Indeed, the shift in freezing temperature is marginal (only one or two degrees), which indicates that the density change of the ordered top layer differs only slightly from that in the bulk. 1.3
THE HYDROCARBON–WATER INTERFACE
When sufficient amounts of hydrocarbon chains are mixed with water, macroscopic phase separation takes place. The saturation value of hydrocarbon
F. A. M. LEERMAKERS AND J. M. KLEIJN
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molecules in water decreases exponentially with the length of the hydrocarbon chain [4,9]. Typically the free energy penalty to transfer a CH2 group from its own environment to water at around room temperature is of order 1 kT (i.e. the thermal energy) [4]. For this reason, the surface tension between a hydrocarbon phase and water is relatively high (ca. 30 mN m1) and the interface between these two phases is sharp, that is only one or two times the size of a water molecule. It is instructive to rationalise the latter result from the point of view of energy transfer of a CH2 group to water. As mentioned above, this value is about 1 kT. This means that the thermal energy suffices to allow the transfer of one or two CH2 groups of the hydrocarbon chain to water, but not many more. This means that the protrusion of hydrocarbon fragments into the water phase is also limited to a few CH2 groups. In other words, the hydrocarbon–water interface is fairly sharp, but it is not a mathematical step-function. The amount of water that can dissolve in an apolar phase is very low (mole fraction of the order of 0.001). The origin of this low value is the loss of H-bond energy when a water molecule enters the hydrophobic phase. This result is only weakly dependent on the acyl chain length. Finally, it is noted that at room temperature all lipid acyl chains are far from their critical points for the water–hydrocarbon unmixing. Therefore the density of the hydrocarbon phase is high. Indeed, the density of a hydrocarbon melt does not differ much from that of water. 1.4 SURFACTANTS AND THE SURFACTANT PACKING PARAMETER There are molecules that dissolve in water (hydrophilic molecules) and molecules that do not (hydrophobic molecules). Of course there are also molecules that chemically combine both entities. These molecules have peculiar mixing properties with water. It is as if these molecules become ‘frustrated’ when mixed with water. They are called amphiphiles or, more frequently, surfactants. Here we are interested in those surfactants that consist of two hydrocarbon chains as the hydrophobic part of the molecule combined with a hydrophilic moiety (usually a charged group), such that these are in a head–tail configuration. These molecules tend to show a solubility limit in water, typically near a value that would have been found for the hydrophobic part alone, but the result of supersaturation is not a macroscopic phase separation. Instead, the aggregation is stopped by the hydrophilic part of the molecules [10]. This process, arrested phase separation, is known as self-assembly. The results of self-assembly of lipid molecules into flat bilayer objects are called membranes. Other geometries exist. The start–stop mechanism of self-assembly dictates a specific rule regarding the type of geometries that can form. At least one of the dimensions of these objects, which are generally called micelles, must remain of the size of the surfactant molecule. As not all dimensions of the micelle can grow without bounds, these micelles can be referred to as being mesoscopic in size.
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It is important to remember that the surfactants in the micelles are in equilibrium with the remaining surfactants in solution. The concentration of free surfactants is known as the critical micellisation concentration (c.m.c.). The molecules in these objects are mobile, and they can exchange with molecules in the bulk. Furthermore, the micelle concentration and the size and shape of the micelles may be a function of the surfactant concentration. Typically, the main effect of increasing the overall surfactant concentration is that the number of micelles increase. The secondary effect is that the free surfactant concentration increases slightly and, because of this, the micelle size adjusts. When the number of surfactants per micelle increases, it can result in some type of packing frustration. When the aggregation number is no longer compatible with the micellar shape, shape changes are implemented. This leads to cylindrical or lamellar-shaped micelles at surfactant concentrations well above the c.m.c. In this review, we are mostly interested in lamellar topologies, i.e. membranes. Israelachvili and co-workers [11] argued that in order to have stable bilayers the averaged shape of the molecule should be a cylinder. This means that the so-called surfactant parameter S ¼ =‘ao (where ‘ is the all-trans length of the hydrophobic tail, ao is the area occupied by the hydrophilic head group, and is the volume occupied by the tails) assumes a value of S 1. Straightforward geometry tells us that cylindrical micelles are favourable when S 1=2, and spherical micelles should be expected when S 1=3. This rule of thumb works very well indeed. It predicts, for example, that surfactants with one tail, for which S 1=3, will tend to form micelles, but surfactants with the same head group, i.e. with same ao , on to which two similar tails of length ‘ are connected will be more likely to form membranes. This is because now is larger by a factor of two and thus S 2=3. Although the following will not add to the proof of the effectiveness of the surfactant parameter approach, it is instructive to mention the implicit assumptions made in it (see also Figure 2). . the first approximation is that the area per molecule a can be defined. There are two counter acting forces that control this area. Firstly, there is the tendency to minimise hydrocarbon–water contacts. The free energy contribution of this is fh (a) ¼ ~ga. Here, ~g is the surface tension between the hydrocarbon phase and the water phase. This tends to decrease the area per molecule. Secondly, pressing head groups near to each other gives repulsive forces that may be of an electrostatic or steric origin. A generic form for this repulsion is fr (a) ¼ k=a, where k is a constant modulating this repulsion (that is, a function of ionic strength, etc.) and can, in principle, weakly depend on the surfactant tail length or micelle shape. The sum of these contributions: ft (a) ¼ fh (a) þ fr (a) may be optimised to give the parea ffiffiffiffiffiffiffiffi per molecule. The optimal area per molecule is found when a ¼ ao ¼ k=~g. Variations in this area are typically small. This means that membranes are laterally rather incompressible.
F. A. M. LEERMAKERS AND J. M. KLEIJN
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:-(
Incompressible
:-(
:-)
:-(
Only small protrusions
:-)
:-(
No interdigitation
:-) :-(
Tails semi-flexible
Figure 2. This figure gives a schematic illustration of various fluctuations that exist in lipid bilayers. From top to bottom: (1) the increase in area and concomitant reduction in membrane thickness is strongly damped. (2) Up and down movements of the lipids are restricted to small amplitudes, i.e. much less than the tail length. (3) Interpenetration of lipids into the opposite monolayer is, in first approximation, forbidden. (4) Conformations of the lipid tails have only few gauche defects, so that the tail is only slightly curved. Reproduced from (58) with permission from the Biophysical Society
. lipid protrusions, i.e. movements of lipids in membranes in the normal direction of the membrane surface, are small as compared with the length of the amphiphile (cf. the width of the oil–water interface discussed above). This allows in principle for a definition of (an intrinsic) membrane thickness and membrane volume (necessary for the estimation of partition coefficients of additives). . in a typical aggregate the chain may have a number of gauche defects which will reduce its effective length. Nevertheless, the thickness of the bilayer is still expected to scale linearly with the extended length of the tails, ‘. . in addition, there is the issue of the volume occupied by the tails in an aggregate. Above, it was argued that hydrocarbons strongly segregate from water. This means that the density of the hydrocarbon phase is high; the acyl chains pack rather densely. Inside a membrane the density is not necessarily
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
equal to the bulk density of the corresponding hydrocarbon phase because the apolar tails in the aggregated amphiphiles are anisotropically oriented. However, as a first approximation, one can equate these densities. This means that the volume occupied by the surfactant tails is under water– hydrocarbon phase segregation control. . chains from one leaflet do not interpenetrate (interdigitate) into the opposite leaflet. Although there is no obvious (free-energy) argument why this is generally the case, it is common belief that it is (in first order) true. . cooperative lateral movements of many lipid molecules, usually called undulations, do not frustrate the packing arguments. This means that these fluctuations are on a length scale that is large as compared with the size of the lipid molecules (or, equivalently, the membrane thickness). A detailed justification of the surfactant parameter approach is still the subject of theoretical investigations, and we will return to several issues below. We mention that the surfactant parameter approach is consistent with the fluid mosaic model of Singer and Nicolson. It tells us that the self-assembly of amphiphiles is driven by the strong segregation of water and hydrocarbon chains, and that packing effects dominate the self-assembly process. All of the above considerations have sometimes led to a too rigid picture of the membrane structure. Of course, the mentioned types of fluctuations (protrusions, fluctuations in area per molecule, chain interdigitations) do exist and will turn out to be important. Without these, the membrane would lack any mechanism to, for example, adjust to the environmental conditions or to accommodate additives. Here we come to the central theme of this review. In order to come to predictive models for permeation in, and transport through bilayers, it is necessary to go beyond the surfactant parameter approach and the fluid mosaic model. Of course it is extremely challenging to build a molecular model for the bilayer which incorporates all of the above properties, and which includes the important fluctuations. From a physicochemical point of view, it is clear that such a model should accurately represent the size and shape of the surfactants. This is necessary because we have seen that the packing effects are important. Such a model should be able to work at very high densities of strongly interacting molecules. The interactions are both of the short-range type (hydrophobic–hydrophilic) as well as of a longer-range type (electrostatics, van der Waals). Below we will discuss only those methods that are powerful enough to do so. 1.5
MEMBRANE TENSION
The combination of the first and second laws of thermodynamics exactly defines the equilibrium of a system. Of course, many biological systems are
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not in equilibrium, and not even in so-called steady-states. It is not known by how much such dynamically evolving bilayers differ structurally from equilibrated ones. In the following sections of this review, we are going to assume that the bilayers have come to a thermal equilibrium. The simple reason for this is that such a bilayer system is unequivocally defined. For obvious reasons, we need to introduce surface contributions in the thermodynamic framework. Typically, in interface thermodynamics, the area in the system, e.g. the area of an air–water interface, is a state variable that can be adjusted by the observer while keeping the intensive variables (such as the temperature, pressure and chemical potentials) fixed. The unique feature in selfassembling systems is that the observer cannot adjust the area of a membrane in the same way, unless the membrane is put in a frame. Systems that have self-assembly characteristics are conveniently handled in a setting of thermodynamics of small systems, developed by Hill [12], and applied to surfactant self-assembly by Hall and Pethica [13]. In this approach, it is not necessary to make assumptions about the structure of the aggregates in order to define exactly the equilibrium conditions. However, for the present purpose, it is convenient to take the bilayer as an example. Let us consider, for example, a flat symmetrical bilayer of which the area is large, so that end-effects can be ignored. Finite size effects are important, and will be discussed in the following section. The membrane is freely floating in solution, i.e. it is not supported by a frame. Combination of the first and second laws of thermodynamics gives for the difference of internal energy dU of a bulk system with membranes with area A: X mi dni þ gdA (1) dU ¼ TdS pdV þ i
where S is the entropy, T the temperature, p the pressure, V the volume, i is the index pointing to molecular components and m the chemical potential. The intensive variable associated with the membrane area is the surface tension g. In principle, the last term seems to be redundant, because the first three terms should already cover any energy change (in a macroscopically homogeneous bulk). In other words, we have introduced extra knowledge of the existence of membranes. The last term is thus some kind of ‘hidden’ contribution. It is often more convenient to control the temperature than to control the entropy, and therefore it is more convenient to switch to the Helmholtz energy F ¼ U TS, for which we can write: X mi dni þ gdA (2) dF ¼ SdT pdV þ i
It will be clear that the membrane is free to adjust its area. Let the system be closed, i.e. let the number ni of molecules in it be fixed. Let the temperature
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
T and volume V also be fixed. In this case the system minimises its Helmholtz energy. This can be done by adjusting its surface area:
qF qA
¼g¼0
(3)
T , V , {ni }
This leads to the equilibrium condition that the membrane tension is zero. For stability reasons it is necessary that q2 F =qA2 0, or equivalently that qg=qA 0. To understand this, it may be convenient to consider an off-equilibrium membrane under tension. A finite tension leads to a bilayer in which the area per molecule is slightly larger than the optimal one. This also means that the thickness is slightly reduced. A finite tension means thermodynamically that it is unfavourable to have a large surface area. As a response, the system will try to reduce its area. When doing so, the area per molecule goes down, the membrane thickness must increase (the loss of material to the bulk can often be ignored) and, as a consequence, the membrane tension is reduced. Exactly the opposite will occur for membranes with a negative surface tension. Sufficiently rigorous models of equilibrated flat bilayers must therefore necessarily have the tension-free state of the bilayer as a constraint. There is a discussion in the literature about the effect of undulation entropy on the equilibrium membrane tension [14,15]. Formally, undulations are included in the surface tension, and thus we need not worry about this. However, if in some model the two are artificially decoupled, one may allow for a very small (positive) surface tension as the equilibrium structure. In other words, the entropy (per unit area) from undulations should compensate for the tension (excess free energy per unit area). 1.6
VESICLES
Flat bilayers only exist in lamellar phases (or in theoretical models). In practice, end-effects are important, and can be eliminated by closing the bilayer into vesicles, known in biology as liposomes. Closing a vesicle introduces curvature energy into the system. It is important to discuss this aspect in some depth. Any curved interface can be described by determining at each point of the surface, two radii of curvature R1 and R2 [16,17]. Typically, we will be interested in a large radius of curvature (R very much larger than the membrane thickness), and therefore it is convenient to define two small parameters: the total curvature J ¼ 1=R1 þ 1=R2 , and the Gaussian curvature K ¼ 1=R1 R2 . Both J and K are invariant upon interchanging the numbering. For a spherical object, R1 ¼ R2 ¼ R and J 2 ¼ 4K ¼ 4=R2 . For cylindrical vesicles one R is infinite and thus K ¼ 0 and J ¼ 1=R.
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Helfrich has shown [18] that the surface tension of a curved interface can be expressed as a Taylor series up to second order in the radius of curvature: 1 K g ¼ go þ kc (J Jo )2 þ k 2 1 1 K ¼ go þ kc Jo2 kc Jo J þ kc J 2 þ k 2 2
(4)
are (phenomenoThe mean bending modulus kc and the saddle splay modulus k logically) mechanical parameters specific for a particular interface. A molecular model is necessary to give an interpretation to these parameters. We will show this below. Equation (4) also allows for a finite so-called spontaneous curvature Jo ¼ 1=R1o þ 1=R2o . For a particular membrane, the Jo (sphere) ¼ 2Jo (cylinder). When a lipid bilayer is composed of just one type of lipid, it must be true that the flat bilayer is stable against small curvature fluctuations. In other words, for reasons of symmetry one should expect that for such a bilayer Jo ¼ 0. In principle, the situation may be fundamentally different for membrane systems which are composed of more than one type of amphiphile. In this case it is, at least in principle, feasible that the flat bilayer is unstable against curvature fluctuations. When a small curvature is imposed, one may witness an uneven partitioning of the various lipids between the inner and outer leaflet of the curved bilayer. Although the two lipids may differ in their surfactant parameter (and thus there may be a thermodynamic driving force for uneven partitioning), entropy strongly counteracts any strong segregation of the lipids. As a result, we must expect that the reshuffling of the lipids is modest and the spontaneous curvature remains strictly zero. However, when entropy cannot counteract a major sorting of the lipids, e.g. when the lipids laterally segregate due to unfavourable lateral interactions, one may expect a nonzero spontaneous curvature. This may be the case when a mixed bilayer membrane is near its gel–liquid phase transition temperature, or when the chemistry of the mixed surfactants is sufficiently different, e.g. when surfactants with fluorinated carbon tails are mixed with hydrogenated ones [19]. Systems of this type, i.e. lipid systems, which are composed of components that laterally segregate, may however, prefer to form two types of vesicles, each composed of one type of lipid only. This again counteracts the nonzero value of Jo for each vesicle. In biological systems, one often observes membrane structures with nonzero spontaneous curvatures, e.g. in mitochondria. This type of bilayer structure is also essential in various transport related processes such as endo- and exocytosis (see Chapter 8 of this volume). These curved membrane systems may be stabilised by protein aggregation in the bilayer, or may be the result of the fact that biological membranes are constantly kept off-equilibrium by lipid transport and/or by (active) transport processes across the bilayer. These interesting
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
subjects are not yet within reach of state-of-the-art molecular modelling techniques. In model systems for bilayers, one typically considers systems which are composed of one type of phospholipid. In these systems, vesicles very often are observed. The size of vesicles may depend on their preparation history, and can vary from approximately 50 nm (small unilamellar vesicles or SUVs) up to many mm (large unilamellar or LUV). Also one may find multilamellar vesicular structures with more, and often many more than, one bilayer separating the inside from the outside. Indeed, usually it is necessary to follow special recipes to obtain unilamellar vesicles. A systematic way to produce such vesicles is to expose the systems to a series of freeze–thaw cycles [20]. In this process, the vesicles are repeatedly broken into fragments when they are deeply frozen to liquid nitrogen temperatures, but reseal to closed vesicles upon thawing. This procedure helps the equilibration process and, because well-defined vesicles form, it is now believed that such vesicles represent (close to) equilibrium structures. If this is the case then we need to understand the physics of thermodynamically stable vesicles. For lipid bilayers, equation (4) can be simplified. Above we have seen that the flat unsupported bilayer is without tension, i.e. g(0, 0) ¼ 0, and therefore the first two terms must cancel: go ¼ 12 kc Jo2 . As argued above, Jo ¼ 0, and thus also the third term drops out. The remaining two terms are proportional to the curvature to the power two. For a cylindrical geometry only, the term proportional to J 2 is present. For spherical vesicles, the two combine into one: )J 2 . The curvature energy of a homogeneously curved bilayer is ( 12 kc þ 14 k found by integrating the surface tension over the available area: ] sphere gAs ¼ 4p [2kc þ k 1 gAs =L ¼ 2p [ kc J] cylinder 2
(5)
In a cylindrically curved bilayer, the area per unit length As =L is of course used. The terms in the square brackets are necessarily positive, and therefore there is > 0. Considering the importance a constraint on spherical vesicles that 2kc þ k of the two bending moduli, it is rather surprising that estimates for these quantities are very rare. The few experiments on phospholipids point to kc values ranging between 10 and 40 kT [21]. To date, there is no experiment which . Again, molecular modelling is needed to leads directly to reliable values for k gain insight into these quantities. controls the memIt is believed that the Gaussian bending modulus k brane topology. In particular, a negative value of this constant is needed for stable bilayers. A positive value will induce nonlamellar topologies, such as is negative for bicontinuous cubic phases. Therefore, it is believed that k membranes.
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). As The total curvature energy of a spherical vesicle is given by 4p(2kc þ k all experimental data on phospholipids indicate that kc is not small, one is inclined to conclude that the vesicles are thermodynamically unstable: the reduction of the number of vesicles, e.g. by vesicle fusion or by Ostwald ripening, will reduce the overall curvature energy. However, such lines of is sufficiently negative to allow the thought overlook the possibility that k overall curvature free energy of vesicles to remain small. is only slightly larger than zero, one can envisage that the When 2kc þ k mixing entropy of the vesicles can compensate for the curvature energy. Accounting for the mixing entropy, which, for dilute solutions, is proportional to lnj of the vesicle solution, equilibrium vesicles are expected when ) ¼ 0. Equivalently, j ¼ exp[ 4p(2kc þ k )=kT]. Fmix ¼ kTlnjy þ 4p(2kc þ k y Safran showed [22] that when the vesicles are thermodynamically stabilised by translational entropy, the vesicle size decreases with decreasing lipid concentration with a power-law R(j) / j0:25 . Experimentally, a smaller power-law exponent is found [23]. The reason for this is that vesicles are not rigid and can assume shape fluctuations, often referred to as undulations. The effect of undulations may be understood from the observation that a membrane may be characterized by a membrane persistence length x. The persistence length is the length along the surface over which the membrane can ‘remember’ its orientation. Orientational information is lost due to the fact that the bilayer is not rigid but semi-flexible. It is known that the persistence length is an exponential function of the mean bending modulus x ‘m exp(kc =kT ) [24,25]. Here ‘m is a molecular length. When we consider two points on the surface that are closer together than x, we know they are on a piece of the bilayer that may be considered approximately flat. However, when the two points are further apart than x, the local orientation of the bilayer on these two points is uncorrelated. This means that when a bilayer closes on length scales larger than x, the curvature energy must vanish. One way to , and thus the curvature energy implement this idea is to allow both kc and k of the vesicle, to depend on the radius of the vesicle. For example, one can write the renormalised mean bending modulus of the form kc (R) ¼ kbare ln( ‘Rm ), c bare is the unnormalised mean bending modulus (all coefficients are where kc ignored for the sake of the argument). When R x, it is clear that the renormalised constant vanishes. Such renormalisation leads to the conclusion that fluctuation-induced stabilisation of vesicles occurs when R x. Usually, one assumes that this argument leads to the prediction that only very large vesicles can exist, because of the very strong dependence of x(kc ) and the relatively high is of the same order of values of kc reported in the literature. However, when k magnitude as kc , the argument may well explain the thermodynamic stability of the vesicle. In conclusion, lipid vesicles that are fully equilibrated (freeze–thaw procedure) may be stabilised by both translational as well as undulational entropy.
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Taking both contributions into account leads to the prediction [26,27] that R(j) / j0:12 . Recent experiments are consistent with this [23]. 1.7
LIPID PHASE BEHAVIOUR
Besides the regulation of the surface area, surfactant (lipid) aggregates have the extra degree of freedom of choice of topology. Above, we have already discussed the formation of curved bilayers or vesicles and mentioned the possibility of formation of (spherical) micelles. Other aggregates are also possible, i.e. in some systems, spherical, cylindrical or lamellar micelles (membranes) may form. These objects can float either randomly in solution or they can be ordered in a liquid-crystalline array, leading to, for example, a cubic-, a hexagonal-, or a lamellar phase. On top of these phases there may be various phases in which saddle-shaped surfaces occur, such as double diamonds (cubic) and sponge phases. A phase diagram is therefore rather crowded and complex. Nomenclature exists to refer to particular phases: for example, phases with lamellar topology are referred to by the letter L. The liquid-crystalline phase of the bilayers is known as the La phase and the gel phase is Lb . Extra labels on the latter may refer to packing variations of the tails in the various gel phases. The sponge phase L3 is locally lamellar, but this phase has ‘handles’ in between the bilayers. Hexagonal phases are referred to using the letter H. Again, indices may refer to cases where the head groups are on the outside or are in the inside (inverted hexagonal) of the cylindrical objects. Isotropic phases, i.e. dilute micellar phases, are assigned the letter I. However, in this review, we will not need this nomenclature extensively, as we are mainly interested in La phases. It is important for the theoretical understanding of the formation of various topologies that these aggregates have entropic contributions on the scale of the objects, i.e. on a much larger scale than set by the molecules. These cooperative entropic effects should be included in the overall Helmholtz energy, and they are essential to describe the full phase behaviour. It is believed that the mech and Jo , control the phase behaviour, anical parameters discussed above: kc , k where it is understood that these quantities may, in principle, depend on the overall surfactant (lipid) concentration, i.e. when the membranes are packed to such a density that they strongly interact. 1.8
COMPLEXITY
It is necessary to elaborate on yet another essential aspect of biological membranes, i.e. their ‘complexity’. This keyword points to the large number of different molecules that are usually found in the biological membrane. First of all, there is a large variety of lipid molecules. The lipid composition of the biological membranes varies from one species to another, and is adapted to meet the needs of organs, cells, organelles, etc. The variations in the head–tail
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overall structure are large. There are differences in hydrocarbon chain length, chain branching, number and position of unsaturated bonds, hydrophilic head group size, charge, etc. There may be other lipid components, such as cholesterol, modulating the physical properties of the bilayers. Most of all, there are proteins embedded into the bilayer or associated with it. The complexity of the biological membranes poses huge problems for the molecular modelling of these objects. Without doubt, the complexity is there for good reasons. For example, the many protein molecules facilitate the chemical reactions and transport events. From a physicochemical point of view, we are still far from a full understanding of complexity; why is there such a variety of lipids and additives (other than proteins) in the biological membrane? Trying to understand model systems is a good way to start, followed by gradually increasing the complexity. This is the route that will be followed in this review. Firstly, the focus will be on ‘simple’ membranes composed of just one type of (saturated) lipids. After that we will consider the effect of number and position of unsaturated bonds, lipid mixtures and the effects of additives. When we arrive at this stage, we can also address permeation issues. 1.9
MODELS OF LIPID BILAYERS
From the above, it will be clear that molecular modelling of lipid bilayers is a challenging task. Models are tailored to give insight into many different aspects. Here we will concentrate mainly, but not exclusively, on models that feature lipid molecules, of which both the hydrophilic as well as the hydrophobic parts are taken into account. The models should reproduce self-assembly, and there should be both a relevant driving force for phase separation and a realistic stopping mechanism. The results should be at least qualitatively consistent with the surfactant parameter approach and the fluid mosaic model. Most importantly, the model should go beyond these approaches by including as many of the aforementioned fluctuations as possible. For such a model, there is hope that it may give predictive insight into the membrane structure and performance, e.g. into permeation issues. It will be rather obvious that models that satisfy these requirements will require the use of a computer to do the detailed work of dealing with all the molecules and the forces between them. Apart from this, there is usually a considerable amount of information that needs to go into the formulation of the problem before one can get something useful out. In this review, we will demonstrate that the ‘break-even point’ has been passed convincingly by several but not all modelling approaches in recent years. To elaborate on this, we will discuss both the in- and output of these models. A further issue is the quality of the results. It will not come as a surprise that the quality of the information gain depends nonlinearly on the computation time that one is willing or able to spend on the problem. There are highly
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
detailed models in which very subtle information (on a particular aspect) is obtained. There are more pragmatic models in which it is rather easy to scan some part of the parameter space and gain qualitative insight. How these models work and why these models work, what the limitations are, etc., will be discussed next. Basically, there are two types of molecular approaches: molecular simulations and self-consistent-field (SCF) methods. We will discuss these two approaches in order. The molecular simulations will include molecular dynamics (MD), and Monte Carlo (MC) simulation. Moreover, we will briefly discuss the recently developed dissipative particle dynamics (DPD) method. Like SCF, the last technique is not exact, because not all excludedvolume (packing) correlations are properly accounted for, but it may be used to obtain dynamical information on much longer timescales than can be reached by MD. 1.10
ENSEMBLES IN MOLECULAR MODELLING
The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29]. Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. . the canonical ensemble: (N, V , A, T) systems, where the number of molecules N, the volume V, the area A, and the temperature T, are fixed, and, as a consequence, the internal energy of the system can fluctuate, . the grand canonical ensemble: (m, V , A, T) system, which allows transfer of molecules from a reservoir with fixed chemical potentials to the system and back, but which restricts the spatial dimension in the system, . the constant-pressure/surface-tension ensemble: (N, p, g, T)-systems, where both the volume and the area of the system may vary (in order to keep the pressure and the surface tension constant), but where both the number of molecules and the temperature remain fixed. The relative importance of fluctuations, e.g. in the membrane area, as is possible in a (N, p, g, T ) ensemble, becomes larger when the system that is under investigation is smaller. For macroscopic systems, the fluctuations in such quantities are negligible, but, for some of the modelling techniques that are restricted to small systems, one cannot avoid dealing with it. Of course, for a (N, p, g, T ) ensemble in which both the volume and the surface area are allowed
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to fluctuate, one can compute the time-average of these quantities. Using these averaged quantities subsequently to specify macroscopic boundary conditions in a canonical ensemble, one typically will recover structural characteristics of the system that match those of the (N, p, g, T ) ensemble. The reverse can also be done. One can perform computations in a canonical ensemble, compute, e.g. the pressure or the surface tension, and subsequently adjust the boundary conditions iteratively until these intensive variables assume the ‘equilibrium’ values of e.g. p ¼ 1 atm and g ¼ 0.
2 2.1
THE MOLECULAR DYNAMIC TECHNIQUE THE STRATEGY
The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. For example, the average kinetic and potential energy in the system is easily obtained from the velocities and the pair interactions. The notoriously difficult part is the evaluation of the entropy. To compute the entropy, it is necessary to know the probability that each distinguishable state can occur in the system. The ‘book-keeping’ to find this probability distribution is not feasible – even more so because in a typical MD run only a part of the possible and allowed configurations of the system is sampled. All known alternatives to compute the entropy are also computationally expensive. Correspondingly, it is very hard to compute, the Helmholtz energy, for example. Fortunately, it is straightforward to obtain differences between Helmholtz energies, upon small changes in the system. Therefore, it is not too difficult to calculate the pressure (tensor) and/or the surface tension, for example. For details we refer to the literature [30].
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
The strong point of molecular dynamic simulations is that, for the particular model, the results are (nearly) exact. In particular, the simulations take all necessary excluded-volume correlations into account. However, still it is not advisable to have blind confidence in the predictions of MD. The simulations typically treat the system classically, many parameters that together define the force field are subject to fine-tuning, and one always should be cautious about the statistical certainty. In passing, we will touch upon some more limitations when we discuss more details of MD simulation of lipid systems. We will not go into all the details here, because the use of MD simulation to study the lipid bilayer has recently been reviewed by other authors already [31,32]. Our idea is to present sufficient information to allow a critical evaluation of the method, and to set the stage for comparison with alternative approaches. 2.2
THE BOX
There are several commercial packages that realise the above strategy for molecularly realistic systems. It is useful to discuss some of the limitations. Ideally, one would like to do simulations on macroscopic systems. However, it is impossible to use a computer to deal with numbers of degrees of freedom on the order of Nav . In lipid systems, where the computations of all the interactions in the system are expensive, a typical system can contain of the order of tens of thousands of particles. Recently, massive systems with up to a million particles have been considered [33]. Even for these large simulations, this still means that the system size is limited to the order of 10 nm. Because of this small size, one refers to this volume as a box, although the system boundaries are typically not box-like. Usually the box has periodic boundary conditions. This implies that molecules that move out of the box on one side will enter the box on the opposite side. In such a way, finite size effects are minimised. In sophisticated simulations, i.e. (N, p, g, T)-ensembles, there are rules defined which allow the box size and shape to vary in such a way that the intensive parameters (p, g) can assume a preset value. The finite size of the box has several important consequences. One of them is that the area of the membrane piece is only of the order of 100 nm2 . It is expected that the membrane is, on this length scale, roughly flat, i.e. the area is small as compared with the persistence length x for the bilayer. Interestingly, however, in recent simulations the first signs of fluctuations away from the flat bilayer structure (undulations) have reportedly been found by MD simulations [33]. The periodic boundary conditions applied in the system have the consequence that one bilayer leaflet can interact in the normal direction with the other leaflet, not only through the contact region in the core of the bilayer, but also through the water phase. To minimise artifacts, one should systematically increase the size of the water phase. However, this is expensive, especially if the main interest is in the behaviour of the lipids. Another solution is to cut off the
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permitted range of interactions, to be smaller than the typical box-size. But then salient features in, for example, spatially varying water densities or water orientation in the normal direction, may remain undetected and have an effect in the simulations. These effects can influence, e.g. the pressure measurements. Physically, it means that a simulation focuses not on isolated membranes but on a stack of membranes as in, for example, a lamellar phase, where the membranes interact with each other. 2.3
THE MOLECULES
Many MD packages contain some basic organic chemistry to accurately define the molecule structure including the positions of H-atoms. For an illustration of the level of detail of how lipids can be represented, we will refer to Figure 3. Many packages can also make use of crystallographic data, such that one can position the molecules to relative coordinates given by the unitcell data characteristics. Then it is rather trivial to build a bilayer of lipid molecules that can serve as a reasonable initial guess configuration for the simulation run. An example is given in Figure 4. The bilayer is positioned in, for example, a pre-equilibrated water phase, by removing the water molecules at the positions of the lipid molecules. Finally, one should introduce ions in the system. These ions are necessary to screen the electrostatic interactions. As very accurate input data are needed for a successful MD run on lipid systems, it is not surprising that most of the simulations done are for a very limited number of systems for which these are available. Phosphatidylcholine (PC) bilayers have been and still are popular [31,33–41], but, nowadays, other types of lipid bilayers are under investigation as well [42–46]. MD studies on lipid mixtures, as well as a lipid bilayer including some protein-like object, give all kinds of additional problems that we will touch upon below. 2.4
THE FORCE FIELD
The full description of the interactions in the system that are included in the simulations is called the force field. A typical potential function of the system features extremely simplified forms (for example, harmonic terms) for the various contributions: ! 2 X1 X Aij Bij X qi qj b 0 k þ þ r b V¼ ij ij 2 ij 4p"0 rij r12 r6ij ij i<j i<j bonds 2 h io X1 X f n þ kijkl 1 þ cos n fijkl f0ijkl kyijk yijk y0ijk þ 2 angles dihedrals
(6)
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Figure 3 (Plate 2). Representation of molecular structure in MD simulations. Shown here is the SOPC lipid, discussed in the text. The numbers at each atom indicate the partial charge on the atom. The space-filling picture on the left gives insight into the van der Waals radii of the various groups, and thus into the shape of the molecule. Reproduced from (58) with permission from the Biophysical Society
Here the atoms in the system are numbered by i, j, k, l ¼ 1, . . . , N. The distance between two atoms i, j is rij , q is the (partial) charge on an atom, y is the angle defined by the coordinates (i, j, k) of three consecutive atoms, and f is the dihedral angle defined by the positions of four consecutive atoms, "o is the dielectric permittivity of vacuum, n is the dihedral multiplicity. The potential function, as given in equation (6), has many parameters that depend on the atoms involved. The first term accounts for Coulombic interactions. The second term is the Lennard–Jones interaction energy. It is composed of a strongly repulsive term and a van der Waals-like attractive term. The form of the repulsive term is chosen ad hoc and has the function of defining the size of the atom. The Aij coefficients are a function of the van der Waals radii of the
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Figure 4 (Plate 3). Typical snapshot of a start configuration in MD simulation. Figure 1 (Plate 1) resulted from the equilibration of this initial guess. From [102]. Reproduced by permission of the American Institute of Physics. Copyright (2003)
two atoms si , sj and interaction energy "ij between the two atoms. The Bij coefficient is also a function of the two radii and the interaction energy "ij . The third term in equation (6) deals with stretching of bonds. The spring constant kbij is only nonzero when i and j share a bond and boij is the equilibrium bond length. The fourth term accounts for the energy penalty of bending of a bond away from the preferred angle yoijk . For small differences, again a harmonic assumption is taken and the constant kyijk is the force constant for the angle deformation, and is only nonzero when i, j and k are consecutive atoms in the molecule along a chain. The final term accounts in a similar way for the rotation around bonds, and depends on four consecutive atoms. We note that modern force fields and simulation packages may use extra and/or more complicated forms of the potential V. The first and second terms in equation (6) are rather expensive in terms of computer power needed. They include the interaction of all atoms with all other
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
atoms. This becomes a problem for large systems. As the interaction energy decreases with the distance between the atoms, it is reasonable to cut them off at some distance. The decay of the van der Waals term of the Lennard–Jones contribution is fast, and the cut-off distance is set to a few times the size, s, of the atoms involved. The electrostatic contribution decays very slowly (1/r), and in the literature there are a number of suggestions to deal with this contribution accurately and economically. One of the problems is that the box size can be small as compared with the decay length. Then the Ewald method [30] can be used to sum the contributions of neighbouring boxes. At present, this methodology is still under development for the modelling of charged membranes. In the integration scheme, not only the potential energy in the system, but also the potential energy felt by a particular atom, is needed. This value is trivially found, similar to an equation as given by equation (6) by selecting only those terms in which atom i occurs. It is instructive to see that the kinetic energy does not occur in equation (6), and that equation (6) only depends on the coordinates of the atoms which are unambiguously known at a given moment in time. The Newton equation F ¼ ma is conveniently cast in the form:
qV q2 ri ¼ mi 2 qt qri
(7)
and of course there is one such equation for each atom i. The number of parameters that occur in equations (6) and (7) is huge, and it should be noted that there is a whole industry in finding acceptable values for a balanced force field. Ideally, one would like to have one correct parameter set which can be used to solve all problems, and one would like to have a theoretical justification for these values, e.g. from quantum mechanics, or unambiguous justification of the parameters from spectroscopic data. However, in practice the situation is not so. Many of the parameters are chosen quite empirically, and if they can be obtained from spectroscopic data or from ab initio quantum mechanics, this still may not guarantee good results in a potential form that contains simplified (harmonic) contributions. Therefore, there is a large number of force fields available. One force field tends to perform better for intermolecular interactions. The other one is tuned to represent intramolecular forces accurately. Therefore the optimum force field may depend on the type of problem at hand. An example of a force field used in lipid systems is the ‘optimised parameters for liquid systems’ (OPLS) [47,48]. In such force fields, polarisable atoms are usually given partial charges (cf. in equation (6) or Figure 3). These partial charges may be found from quantum mechanical analysis, and obviously these values differ per force field. Of course the water molecules should be treated with more than average care. Several water models exist and have been applied to lipid systems [49]. There are, for example, a single point charge model (SPC) [50], an extended single point charge model (SPC/E),
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and a TIP water model, of which the first one is popular because of economic reasons. See, for example, [51] for a critical evaluation of parameters for the modelling of alkanes. It is not unimportant to realise that there are several possible ways to implement the Newton equation in an MD simulation. It is not trivial to make sure that all degrees of freedom, e.g. very fast C–H vibrations, or much slower conformational changes, have their proper share of the internal energy. One MD-‘dialect’ may be superior to the other in, e.g. distributing the energy in such a way that the temperature is constant. It would take too long to discuss all the options, so we will only mention that well-known packages such as GROMOS [52], CHARMM [53] and AMBER [54] are widely available. New developments are still being made, such as the collisional dynamics method [55]. In this method, the atoms experience collisions with ghost particles. These collisions are an efficient way to distribute the energy in the system, and the method has the property that all degrees of freedom are at the same temperature. 2.5
TIME AND LENGTH SCALES
To integrate the equations of motion in a stable and reliable way, it is necessary that the fundamental time step is shorter than the shortest relevant timescale in the problem. The shortest events involving whole atoms are C–H vibrations, and therefore a typical value of the time step is 2 fs (1015 s). This means that there are up to one million time steps necessary to reach (real-time) simulation times in the nanosecond range. The ns range is sufficient for conformational transitions of the lipid molecules. It is also sufficient to allow some lateral diffusion of molecules in the box. As an iteration time step is rather expensive, even a supercomputer will need of the order of 106 s (a week) of CPU time to reach the ns domain. Of course there are many phenomena that equilibrate on the nanosecond timescale. However, the majority of relevant events take much more time. For example, the ns timescale is much too short to allow for the self-assembly of a set of lipids from a homogeneously distributed state to a lamellar topology. This is the reason why it is necessary to start a simulation as close as possible to the expected equilibrated state. Of course, this is a tricky practice and should be considered as one of the inherent problems of MD. Only recently, this issue was addressed by Marrink [56]. Here the homogeneous state of the lipids was used as the start configuration, and at the end of the simulation an intact bilayer was found. Permeation, transport across a bilayer, and partitioning of molecules from the water to the membrane phase typically take also more time than can be dealt with by MD. We will return to this point below. A typical MD simulation usually consists of two parts. The first part is an equilibration time (ca. 1 ns) starting with a reasonably accurate initial configuration (see, for example, Figure 4), which is followed by a production run.
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Considering the relatively short equilibration phase, one should always be alert for possible artifacts that originate from the choice of the initial configuration of the system. In particular, if for equilibration, a considerable adjustment of the area of the membrane is necessary, it may take a very long time before this is completed. An example where this is an issue, is the level of interdigitation of the lipid tails. If the simulations are started in a fully noninterdigitated state, significant chain interdigitations may never be found – not even in the production phase. There can also be hindered equilibration effects that have their origin in finite box size effects. For example, if the number of lipids on each side of the bilayer is taken to be of order 30 to 100, which is reasonable, a fluctuation in this number is not expected to occur in the simulation, because this would be too large an overall perturbation. Cooperative movements of many molecules at the same time, which occasionally may be necessary before equilibration is a fact, may also be very slow. 2.6
DIPALMITOYLPHOSPHATIDYLCHOLINE BILAYERS
From the above, one may be left with the impression that the MD technique has major problems. It is important to realise that there are relatively straightforward ways to systematically improve the method. In the future, the force fields will become more accurate, the computer power will increase and allow larger box sizes and longer (real-time) simulation times. Even today, MD simulations are the closest to this ‘ideal’ situation as compared with other methods. To show this, we will present and discuss some typical results readily available in the literature. Here we will select systems simulated in the constantpressure constant-surface-tension ensemble, i.e. with a temperature, pressure and surface tension control. MD simulations need a sufficiently accurate initial guess for the bilayer structure in order to find relaxation of the structure in the nanosecond range. X-ray diffraction techniques can be used to determine the electron density profile across the bilayer to high precision, and, from these, the required input data can be extracted. However, typically for X-ray diffraction experiments it is necessary to have oriented bilayers, and therefore most results are available for systems with very little water between them. These results are of interest for lipid membranes that strongly interact. Indeed, fully hydrated bilayers are expected to have a slightly different structure. In 1996, Nagle et al. [38] reported the (X-ray) structure of a highly hydrated DPPC bilayer. One problem is that in the fully hydrated state the bilayers do not remain perfectly flat. The shape of the bilayer may be sinusoidally deformed, i.e. it can undulate up and down, and therefore the intrinsic membrane structure is blurred and an ‘averaged’ one is found. Nagle took these undulations into account, and, as a result, there are accurate numbers for the structural parameters of DPPC layers. For example, the area per lipid molecule is a0 ¼ 0:629 0:013 nm2 .
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The number of water molecules per lipid molecule is 29.1 when the equilibrium (centre to centre) distance (swelling limit) between bilayers is 6.72 nm. These data serve also as an accurate test for MD methodology (including the force field). Rand and Parsegian have collected accurate structural parameters for many other lipid systems [57]. From this perspective it is not surprising that most simulations are done on DPPC bilayers, albeit that, from a biological perspective, other lipids, e.g. partially unsaturated lipids, are perhaps more relevant. DPPC bilayers have been extensively reviewed in the literature [31,36], and therefore we will visit this system only briefly. The MD simulation results that easily attract attention are snapshots. From these, it is easy to tell how the molecules are distributed in the box at a particular point of time in the simulation. See Figure 1 for a typical example. These snapshots are important to obtain a good idea of how order parameters and density profiles follow from the positions of the atoms. However, these structural parameters can only reliably be found after sufficient averaging over a large number of such snapshots. Of course, one can find many MD results in the literature, and it is always arbitrary to pick one of those. We choose to present some good results from a simulation by Berger and co-workers [58]. They paid attention to particular values of the force-field parameters, such that the lipid densities were in accordance with experimental observations, and therefore we should interpret the result depicted for the atom distribution profiles as given in Figure 5 to be rather accurate. In Figure 5, the overall water profile, the overall lipid distribution, as well as that of the CH2 , CH3 groups and several head-group fragments, are plotted. It is of interest to discuss some highlights, because we will return to these when discussing corresponding distributions that result from more approximate modelling attempts. The water profile penetrates into the bilayer just up to the glycerol units. Thus, in MD, one does not find any water molecules in the hydrophobic core. The atom density of the lipid shows a dip in the core. The shape of the dip correlates with the distribution of the CH3 groups. These groups have a large volume per atom. As a result of the dip, the profiles suggest that the two monolayers do not overlap greatly (the tails do not interdigitate much). If this were the case, one would expect a bimodal distribution for the CH3 groups. However, there are plenty of chains that cross the symmetry plane and the CH3 is nicely unimodal. The distributions of the head-group fragments are depicted only on the righthand side of Figure 5. Obviously the CO groups that originate from the glycerol backbone are between the PC head group and the hydrocarbon chains. Interestingly, the average position of the P and that of the N are at approximately the same z-coordinate. This means that the average P–N vector is almost parallel to the membrane surface. Close inspection shows that the N profile is significantly wider than that of the phosphate. This reflects the higher mobility
42
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 50 Lipid
r
no. of atoms/V
40 CH2
30 Water
20
PO4
N(CH3)3
10 0
CH3
0
1
2
CO
3 z / nm
4
5
6
Figure 5. Profiles across the bilayer of the total lipid density of DPPC, the water density and the densities of certain lipid groups as obtained from MD simulations by Berger et al. [58]. The profiles are found by taking the time average over the last 300 ps of the simulation. The densities for the lipid head-group components are only shown on one side for clarity. The origin of the z-axis is arbitrarily positioned on the left of the bilayer. On the y-axis, the atom density in atoms per nm3 is given. Redrawn from [58] by permission of the Biophysical Society
of this group, which is expected because it is positioned at the end of the head group. There is a tendency for the distribution of the N group to be bimodal, something which will be discussed below. From these overall profiles, it is not easy to extract conformational properties, other than that it will be clear that the lipid molecules are strongly anisotropically oriented in the bilayer. For this, other characteristics are much more appropriate. It is possible to define an order parameter which indicates how much the lipid tails are oriented normal to the membrane: 1 S ¼ h3 cos2 W 1i 2
(8)
Here the angular brackets mean that the ensemble average is taken, and the angle W can be defined and computed for each bond in the molecule. When a bond is perfectly aligned normal to the bilayer surface, the order parameter assumes a value of unity. In a random orientation, the order parameter is zero. Bonds that are perfectly aligned parallel to the surface gives a negative order parameter of S ¼ 0:5. Such order parameters can also be obtained from deuterium NMR measurements, and therefore one frequently finds predictions for these order parameters in the literature. The order parameter profile that belongs to the tails of the lipid membrane as given in Figure 5 is shown in Figure 6. There is a consensus in the literature about the typical patterns found in the order parameter profile along the chain in the phospholipid bilayer. The order
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0.2 S
0.1
0
0
2
4
6 8 10 12 Carbon atom number
14
16
Figure 6. Variation of the orientational order parameter S along the hydrocarbon chains of the lipids of DMPC lipid bilayers, according to MD simulations of Berger et al. [58]. The line is drawn to guide the eye. The spheres are experimental values obtained by Seelig and Seelig [59] using 2 H-NMR spectroscopy. (Numbering of C-atoms from the head group to the CH3 terminal group). Redrawn from [58] by permission of the Biophysical Society
is relatively high for hydrocarbon chain segments near the glycerol backbone, and slowly decreases toward the ends of the tails, which are more randomly oriented. The absolute value of the order parameter profile is also correct, as can be seen by the satisfactory comparison with data from Seelig and Seeling [59]. It is important to note that the absolute value of the order parameter is of course strongly dependent on the value of the area per lipid in the membrane. In these simulations, the area found is sufficiently close to the experimental data, i.e. a0 ¼ 61 nm2 . Although not an easy task, it is possible to compute, for example, the electrostatic potential profile across the bilayer. In Figure 7, we give an example for DPPC, as reported by Tieleman and Berendsen [49]. In MD many atoms and molecules contribute to the electrostatic potential, because of the partial charges assigned to them. From such an exercise, it turns out that the electrostatic potential assumes very large values of several hundred mV. This is in line with experimental data, where it has been found that the magnitude of the dipole potential of biomembranes is typically several hundreds of millivolts [60] – the hydrocarbon core being positive with respect to the aqueous phases. The surface potential of most biomembranes is of the order of tens of millivolts. As seen in Figure 7, simulations also indicate that the electrostatic potential in the membrane is positive with respect to the aqueous phase. The region 1 < z < 1 is the hydrophobic core of the bilayer, where the effective dielectric permittivity is low (and thus potentials are high). In the aqueous parts of the system, the potential drops rapidly to very low values. This is because the presence of the water molecules, which can orient to compensate for the high potentials (i.e. they have a high dielectric permittivity).
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 0.8
y/ V
0.6
0.4
0.2
0
−3
−2
−1
0 z / nm
1
2
3
Figure 7. The total electrostatic potential profile across the DPPC bilayer as given by Tieleman and Berendson [49]. Redrawn by permission of the American Institute of Physics
Tu et al. [61] analysed the dynamics of water molecules in the vicinity of the bilayer. Water molecules remain bound to the phosphate and carbonyl groups on a 10 ps timescale. The lipid molecules change their conformations more slowly. Only at the hydrocarbon chain ends are there some isomerisation events on the 10 ps timescale. Then, over the period of 100 ps, water molecules can diffuse away from the lipid. Also the centre of mass of the hydrocarbon chains rattles vertically and laterally. On the nanosecond timescale the lipid molecules experience rotational and translational motion (rattling in a cage, no large-scale movement across the box), and the head group can undergo major conformational changes. We need to go to much longer times to see partitioning and transport across the bilayer. It is therefore not possible to use MD to model all relevant processes for transport and permeation. We will return to this in Section 5. 2.7
COARSE-GRAINED MD
Even when the computer ‘power’ will increase significantly, it is not easy to greatly exceed the 10 ns timescales in MD simulations. This is a problem, because many relevant membrane problems in which major lipid rearrangements are necessary have much longer characteristic times. One way of proceeding is to be pragmatic and simplify the models that one is willing to work on. Indeed, in an all-atom MD simulation, most of the simulation time is used to accurately follow the C–H vibrations. This is no longer needed if united atoms are used where a CH2 - or a CH3 group (or in fact several of these groups) are merged together in effective (spherical) units, sometimes called segments. As a result, it is possible to use larger time-steps in the integration of the Newton equations. Of course, for such artificial units, new force-field parameters are
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necessary. The values of the effective interactions between the segments must be chosen such that the physical behaviour on the length scales of the segments is similar to that of the more detailed simulations. The process of ‘integrating’ out the short time-scale events is known as coarse graining. An interesting simulation on coarse-grained lipids is reported by Lipowsky and co-workers [62]. These authors used surfactants that are composed of just a few beads and simplified potentials, similarly to those used in surfactant studies performed by Smit and co-workers [63]. One of the results that was reported is an estimate of the stretching modulus of a model bilayer. We will discuss this point below, where similar properties are reported to have been obtained by the SCF theory. One aspect of MD simulations is that all molecules, including the solvent, are specified in full detail. As detailed above, much of the CPU time in such a simulation is used up by following all the solvent (water) molecules. An alternative to the MD simulations is Brownian dynamics (BD) simulation. In this method, the solvent molecules are removed from the simulations. The effects of the solvent molecules are then reintroduced into the problem in an approximate way. Firstly, of course, the interaction parameters are adjusted, because the interactions should now include the effect of the solvent molecules. Furthermore, it is necessary to include a fluctuating force acting on the beads (atoms). These fluctuations represent the stochastic forces that result from the collisions of solvent molecules with the atoms. We know of no results using this technique on lipid bilayers. Coming from the field of rheology, there is yet another interesting technique to model complex fluids. This technique is called dissipative particle dynamics (DPD) and has recently been applied to (model) lipid bilayers. The fundamental trick here is that the lipids are represented by a set of soft beads, with either hydrophilic or hydrophobic properties that are linked according to the lipid architecture. For soft beads (particles), the appropriate interaction potentials differ fundamentally from the usual Lennard–Jones ones. In DPD, the interaction curve between beads does not have the usual ‘hard-core’ part, and it also lacks an attractive well at small separation. Instead, a weak repulsion, linear with respect to the centre-to-centre distance, is implemented. This potential then allows the molecules to interpenetrate each other to some extent. The weakness of the potential has the big advantage that long time steps can be used in the simulation. This then opens up the possibility of covering significantly longer evolution times than MD. In principle, one should map the DPD potentials to the force fields in atomistically realistic MD simulations. How accurately this can be done is not yet known exactly. A strong point of DPD simulations is that the simulations conform to Navier–Stokes hydrodynamics. Qualitatively, the bilayer structures that result from DPD simulations are reasonable [65]. In the simulation box, it is possible to find a stable bilayer in which the head groups shield the apolar core from the water phase. This means that the model effectively features a start-and-stop mechanism for
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Figure 8 (Plate 4). Typical snapshot of DPD simulation results [64]. The hydrophobic part of mixed bilayers of DPPC-like lipids and up to 0.8 mole-fraction of the non-ionic surfactant C12 E6 (left) and 0.9 (right). The surfactant C12 chains are represented by grey curves, and the lipid C15 chains are black. The hole in the left conformation is transient; on the right they are stable. Reproduced by permission of the Biophysical Society
self-assembly. A recent simulation of a mixture of lipids and surfactants clearly proves that the DPD method has additional value [64]. When a bilayer of such a mixture was kept under a finite lateral tension, it was possible to visualise and analyse the formation of pores or holes. A graphical representation is given in Figure 8. It is clear that bilayers that lose their integrity also lose their biological functions. These holes were attributed to the presence of the surfactants, and were therefore interpreted to be relevant for the toxic effect of surfactants for bacteria. Of course the possibility that there are transient holes or water channels is relevant for transport and permeation. Such spontaneous fluctuations may determine to some extent the permeability of the bilayer for hydrophilic compounds. We will return to these phenomena below.
3 3.1
THE MONTE CARLO TECHNIQUE THE STRATEGY
In many simulations that use a time trajectory to sample membrane properties, it is the equilibrium situation that one is interested in, rather than the dynamics themselves. The dynamics are then just a by-product that is only used to judge the degree of equilibration.
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When dynamic properties are not one’s main interest, one can consider alternatives to obtain solutions for the equilibrium properties of complex molecular systems. As equilibrium fluctuations can be used to estimate the response of a system when it is slightly off equilibrium (Onsager reciprocal relation [66]), it is still of interest to discuss these alternatives in this review. In the MD simulation, the system can only evolve towards equilibrium by a dynamic path (trajectory), through all kind of possible states of the system (phase space). In a Monte Carlo (MC) simulation technique, the trajectory is typically nonphysical. For example, it is possible to suggest and implement conformational changes, or large changes in system properties, from one MC step to the other. There is no limitation at all on the type of changes that can be made in the system before an evaluation of the system’s energy takes place. In order to prevent ad hoc changes in the system from leading to a random mixing of all molecules (this is often the maximum in the entropy of the system), there is a simple rule which can tell whether or not the changes are accepted or rejected. If the potential energy V in the system is decreased, the suggested move is favourable, and the changes are accepted. If, however, the potential energy V increases, which is unfavourable for the system, the move is either rejected or accepted. This is how the choice is made: (1) Let the potential energy increase be DV > 0, then the Boltzmann factor P ¼ exp ( DV =kT) is a number smaller than unity. (2) Next a random number is generated between 0 and 1. (3) If this number is smaller than P, the move is accepted (and implemented). If it is larger than P, then the suggested changes in the system are rejected, and the old state is counted once more (for taking appropriate averages). This strategy, known as the Metropolis scheme, guarantees that the MC trajectory evolves towards equilibrium [30]. Drawing of the random numbers guarantees, for example, that the system can escape from a local minimum, i.e. that it can sample all relevant system configurations. 3.2
THE BOX AND THE MOLECULES
One can apply the MC technique to the same molecular model, as explored in MD. One can use the same box and the same molecules that experience exactly the same potentials, and therefore the results are equally exact for equilibrium membranes. However, MC examples of this type are very rare. One of the reasons for this is that there is no commercial package available in which an MC strategy is combined with sufficient chemistry know-how and tuned force fields. Unlike the MD approach, where the phase-space trajectory is fixed by the equations of motion of the molecules, the optimal walkthrough phase space in an MC run may depend strongly on the system characteristics. In particular, for densely packed layers, it may be very inefficient to withdraw a molecule randomly and to let it reappear somewhere else in
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the system. Excluded-volume constraints will almost certainly reject the move. Snake-like movements (reptation) would be more efficient, as would small conformational changes within a molecule (typical moves that are done in a MD simulation). Also, the water phase poses huge problems for MC simulations. We recall that the interactions between the water molecules (H-bonds) are very strong, and that each water molecule has, on average, four of these ‘contacts’. It is necessary to invent ingenious MC moves in order to prevent a significant reduction of H-bonds in the water network upon a series of random changes in the positional and orientational distribution of the water molecules. Again, the acceptance probability for such system changes is expected to be small. It is the ingenuity of the researcher to implement efficient trajectories through phase space that can make the difference between failure and success. The lack of guidance as how to design the MC trajectories also explains the lack of commercial MC packages. 3.3
PRAGMATIC APPROXIMATIONS
As the dynamics of the system are removed from the model, it is no longer necessary to allow the molecules to ‘live’ in a continuous space. Instead, the use of lattices – discrete sets of coordinates on to which the molecules are restricted – is popular. Digital computers are of course much more efficient with discrete space than with continuum space. The use of a lattice implies that one removes all properties that occur on shorter length scales than the lattice spacing from the model. This is no problem if the main interest is in phenomena that are larger than this length scale. With the Monte Carlo technique, a very large number of membrane problems have been worked on. We have insufficient space to review all the data available. However, the formation of pores is of relevance for permeation. The formation of perforations in a polymeric bilayer has been studied by Mu¨ller by using Monte Carlo simulation [67] within the bond fluctuation model. In this particular MC technique, ‘realistic’ moves are incorporated, such that the number of MC steps can be linked to a simulated time. 3.4
HYBRID MC AND MD APPROACHES
In principle, MC algorithms can be tuned for particular systems and can thus be more efficient than MD for obtaining equilibrium distributions. An interesting idea is to use MC simulations to obtain accurate initial guesses for subsequent MD simulations. Already as early as 1993, Venable and co-workers [68] used a scheme for efficiently sampling configurations of individual lipids in a mean field. These configurations were then used to develop the initial conditions for the molecular dynamic simulations.
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49
TYPICAL MONTE CARLO RESULTS
Monte Carlo may be used to study the lateral distribution of lipid molecules in mixed bilayers. This of course is a very challenging problem, and, to date, the only way to obtain relevant information for this is to reduce the problem to a very simplistic two-dimensional lattice model. In this case, the lipid molecules occupy a given site and can be in one of the predefined number of different states. These pre-assigned states (usually about 10 are taken), are representative conformations of lipids in the gel or in the liquid state. Each state interacts in its own way with the neighbouring molecules (sitting on neighbouring sites). Typically, one is interested in the lateral phase behaviour near the gel-to-liquid phase transition of the bilayer [69,70]. For some recent simulations of mixtures of DMPC and DSPC, see the work of Sugar [71]. Levine and co-workers [72,73] have considered an ensemble of chains end-grafted on a plane and the space is represented by an ingenious (highcoordination lattice suitable for representing saturated as well as unsaturated acyl chains. In the MC simulation, only moves that could realistically take place, i.e. small conformational changes, etc., are incorporated, and, for this reason, one can optimistically connect the MC trajectory with a time-axis. Although this is not completely unrealistic, there always is some ambiguity in how this mapping is done. A number of very interesting predictions that compare favourably to that of MD simulations as well as experimental observations, were drawn. For example, the average order of the chains as a function of the distance away from the grafting plane can be computed. In Figure 9, a typical example is shown for unsaturated alkyl chains. A pronounced dip in the order profile at 0.3 S 0.2
0.1
0
0
3
6
9 t
12
15
18
Figure 9. A comparison of the order parameter profile as found by MC simulations [72] of model 9/10 cis unsaturated chains in a monolayer (the x-line is to guide the eye) with experimental data obtained from NMR experiments () on the same chains incorporated into a biological membrane. Redrawn from [72] by permission of the American Institute of Physics
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 0.8
0.6
S 0.4 N = 10 13
0.2
15
0
6
9
15
12
19
18
t
Figure 10. Calculated order parameters of the CH2 groups of isolated molecules of linear alkanes with chain length N as indicated, starting from the centre of the molecule towards its end. The ranking number along the chain is indicated by the letter t. MC simulations by Rabinovich and co-workers [74–76]. Copyright (1997) by Elsevier
the position of the double bond is a very characteristic feature. This dip shows that the double bond significantly disturbs the local order in the bilayer. Not all MC simulations are done on a lattice. We would like to mention a particularly interesting study by Rabinovich and Ripatti [74–76], who analysed the conformational states of a short acyl chain with and without unsaturated bonds (see Figure 10) The idea of the simulations is to consider isolated chains with realistic short-range potentials. As the chains are not much longer than their persistence length, they cannot return to themselves. For this reason, there is no need to specify interaction potentials other than those that exist for two neighbouring segments (vibrations), three segments in a row (bending), and those found for four consecutive segments (rotation). The usual Lennard–Jones parameters for the nonbonded interactions are not needed. Many conformations were sampled by the usual MC procedure. The result is of course that there is no preferred orientation of the molecule. Each conformation can, however, be characterised by an instantaneous main axis; this is the average direction of the chain. Then this axis is defined as a ‘director’. This director is used to subsequently determine the orientational order parameter along the chain. The order is obviously low at the chain ends, and relatively high in the middle of the chain. It was found that the order profile going from the centre of the molecules towards the tails fell off very similarly to corresponding chains (with half the chain length) in the bilayer membrane. As an example, we reproduce here the results for saturated acyl chains, in Figure 10. The conclusion is that the order of the chains found for acyl tails in the bilayer is dominated by intramolecular interactions. The intermolecular interactions due to the presence of other chains that are densely packed around such a chain,
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determine the orientation of the molecule to be roughly perpendicular to the membrane surface. The level of alignment is high when the area per molecule is low, and vice versa.
4
THE SELF-CONSISTENT-FIELD TECHNIQUE
The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a ‘probe’ molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical
z U(z) for head groups MEAN-FIELD 0 APPROXIMATION
u(z) U(z) for tail segments
Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z ¼ 0. These lines are to guide the eye; the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
CPU times of 100 s or less. The method can additionally provide insight into the mesoscopic behaviour of the bilayer systems, and can deal with polymeric additives. Dynamic information is not directly available, although there are several attempts to reintroduce dynamics in a mean-field framework [77,78]. More so than in ‘exact’ simulations, it is necessary to give details of SCF methods. This is to give an insight into how and when approximations are introduced. Only then it is possible to build a realistic picture of the approach, understand its limitations in terms of conformational detail and loss of information about dynamics, and to interpret the results. 4.1
THE STRATEGY
A central issue in statistical thermodynamic modelling is to solve the best model possible for a system with many interacting molecules. If it is essential to include all excluded-volume correlations, i.e. to account for all the possible ways that the molecules in the system instantaneously interact with each other, it is necessary to do computer simulations as discussed above, because there are no exact (analytical) solutions to the many-body problems. The only analytical models that can be solved are of the mean-field type. The most essential step in a mean-field theory is the reduction of the manybody problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg–Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. Needless to say, a simplified model leads to corresponding thermodynamic quantities, i.e. not all correlations are included. However, the thermodynamic framework itself is fully internally consistent. This is an important observation, because such a model can for this reason be of use to establish the thermodynamic feasibility of ‘what-if ?’ questions. Full control over the absolute deviations from the true thermodynamic behaviour is unfortunately not possible. The approach ignores important (cooperative) fluctuations, and it is expected that especially near phase transitions the approach may give only qualitative results. In particular, comparison of SCF results with experiments or with simulation data can lead to insights into how rigorous the method is. It is of interest to mention that, once particular choices are made concerning how the mean-field interactions are incorporated into the model, the corresponding partition function and thermodynamics follow in a straightforward manner. In particular, there exists a method based upon a variational argument, to formulate the best possible corresponding (mean-field) potential fields. We will not go into these details here, but refer to the variational method, as
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explained in standard textbooks [79]. Typically, the optimal potential fields become a function of the average properties of all test molecules in the system (one for each united-atom type). On the other hand, the average properties of a test molecule depend on the actual values of the potentials. This interconnection of dependencies is solved numerically by some iterative process. As soon as the potentials and the densities are mutually consistent, one can evaluate the partition function, and all thermodynamic and structural characteristics are extracted from it. Such a fixed point is called, for obvious reasons, the selfconsistent-field solution, and the models that make use of this strategy are self-consistent-field models. SCF models typically make use of lattice approximations. As dynamics are not an issue, it is not necessary to specify all the potentials in equal detail. Therefore there are many differences between the SCF and simulation methods. Comparing and contrasting both methods remains of interest, because this will give insight into essential and less essential aspects of membrane formation. 4.2
THE BOX
It is important to realise that the SCF technique differs significantly from the simulation methods, due to the above-mentioned averaging procedure and the use of (average) density-dependent, self-consistent external potentials. This averaging should be done over a large number of molecules, or equivalently, over a sufficiently large region. An averaging procedure is only expected to work if the property over which averaging is done does not fluctuate strongly. Therefore, it makes sense to average in planes parallel to the membrane surface, and account for density-and potential gradients perpendicular to this, i.e. in the z-direction. In this section we will restrict ourselves to this approach. In effect, it is assumed that the system is translationally invariant in the x–y directions. As a consequence, the presentation of the results will also differ from that in a MD or MC box, where a full set of molecules can be depicted (as snapshots). In an SCF model, all properties will be presented in, for example, (average) numbers of molecules per unit area of the membrane, or equivalent, i.e. the (average) densities of molecules as a function of the z-coordinate. The ‘box’ thus consists, if one insists, only of one coordinate. For this reason, we can refer to this method as a one-gradient SCF theory or simply 1D-SCF theory. Extensions towards 2D-SCF are available, where lateral inhomogeneities in the bilayer can also be examined [80]. There are even implementations of 3D SCF-like models, but here the interpretation is somewhat more delicate [78]. For computational reasons, it is necessary to discretise the space in such a way that one considers only a finite number of z-coordinates for the segments of the molecules to take positions. Again, the coarse graining is justified as long as the relevant phenomena do not take place on a smaller length scale. Here we consider lattices with sites that fit united atoms, e.g. CH2 units. The
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
z-coordinates can be numbered arbitrarily, z ¼ M, M þ 1, . . . , 1, 0, 1, . . . , M 1, M, where M is a sufficiently large number such that the membrane can fit into the ‘box’. Typically, we will put the centre of the bilayer on the z ¼ 0 spot. The trick of putting the bilayer in the centre of the coordinate system is a convenient one, in order to remove an uninteresting translational entropy term from consideration. Computational tricks are not included in this review [81]. 4.3
THE MOLECULES
In the SCF technique, there is a straightforward way to account for intramolecular excluded-volume correlations [82], i.e. by generating all possible selfavoiding conformations, albeit a computationally expensive task. Accounting for intermolecular excluded-volume correlations cannot be treated exactly, not even by using up much more CPU time (unless, of course, one is willing to resort to simulations). One of the challenges is to put as many of the effects of intermolecular excluded-volume correlations in to the procedure as possible. From the MC results discussed above, we have learned that for short chains it is sufficient to just account for short-range along-the-chain excluded-volume correlations. For this reason, it is tolerable to generate conformations in a rotational isomeric state scheme (RIS). In this context, we refer to this scheme as a finite memory Markov process: excluded-volume correlations over a distance of four consecutive united atoms are included, i.e. these cannot overlap, but for segments further apart this is not strictly forbidden. This problem is (partially) corrected by a contribution to the segment potential as discussed below. In contrast to the RIS scheme used in MD only discrete angles are included on the lattice typically only those corresponding with the trans (lowest energy) and the two gauche (two local minimums of the energy) conformers. Correspondingly, only the energy difference between the trans and a gauche state becomes a parameter in the model. In calculations discussed below, we will use a value of 0.8 kT for this difference, except when we mention otherwise. 4.3.1
Lipids
Molecules that are restricted with their segments on to lattice sites need a force field with characteristics different from those in MD simulations. For example, restricting atoms on the lattice leads to fixed bond lengths, and the bond length control is not needed. By the same token, the bond angles are fixed and there are no parameters to control this angle. In fact, we will consider models in which all united atoms in the molecule are of equal size, exactly fitting to a lattice site. The number of united atoms, as well as the type of united atoms (polar, apolar, charged, etc.) determines the molecular architecture and the type of interactions felt by the molecule. Instead of Lennard–Jones potentials, the
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square-well potential is popular. This potential assumes a finite value if the segments that interact occupy neighbouring lattice sites. If they are further apart then the interaction is assumed to be zero. In a modern implementation of the SCF approach, GOLIATH [83], detailed architectures of the molecules can be handled. Typically the protons are not explicitly defined, but molecular groups that are large can be split up into a group of beads. In this way, the lipid molecules may have rather realistic structures, i.e. they consist of a glycerol backbone on to which two acyl chains and a head group with accurate composition are linked [84–86], as shown in Figure 12. There are also possibilities for considering organic molecules with various chemical groups and spatial arrangements. Typically, unsaturated bonds in the acyl chains can be mimicked, at first approximation, by forcing a gauche kink in the chain. This way of dealing with lipid unsaturation is, of course, much more approximate than can be done in MD or in MC simulations. 4.3.2
Water
A relevant model of self-assembly of hydrocarbon-based surfactants in aqueous solutions should pay sufficient attention to water. Detailed attention has been given to water in MD simulations [31,36], but much less effort has been spent on water in a realistic SCF model. In many SCF treatments of lipid bilayer systems, the chains are end-grafted on to a surface, and the interaction with water is considered to be so strong that excursions into the water phase are excluded [87–91]. At this limit, the water phase is (almost) completely eliminated from the problem. In this review, we will concentrate on SCF models that include water in the model. Up to very recently, even the most detailed SCF models featured a simplistic water model. Typically, water was a spherical monomer that strongly repelled hydrocarbon segments. The membranes resulting from such a model show large amounts of water in the core. Typically, in the order of 5% of the volume in the core was occupied by water [86]. This value is several orders of magnitude too high. A reasonable model that tackles this problem will be discussed next. The most important aspect of water that should be accounted for in this context is that water and hydrocarbons strongly unmix. In practice, this can be attributed to the strong associative nature of water molecules. This is the rationale of using a model that accounts for lateral (in a lattice layer) association of water monomers forming dimers, trimers, etc. It is assumed that water clusters are in equilibrium with each other, according to the reaction: XN þ X Ð XNþ1 , where X is the concentration of water monomers, and the subscript indicates the cluster size. In fact, each cluster of N-water molecules is given only one amount of entropy of mixing, irrespective of the size of the cluster. The total water density can easily be expressed in terms of the reaction
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
constant K and the concentration of water monomers, if it is assumed that K does not depend on N. A natural consequence worth mentioning is that the cluster size distribution depends very strongly on the local water concentration. We consider a system with K >> 1. At positions where the water density is low (i.e. in the centre of the bilayer), one can have only free monomers, and large clusters do not exist. On the other hand, in the aqueous phase where the water density is high, almost all water molecules are incorporated in (large) clusters. As a consequence, the free monomer density in the bulk remains very low. Of course, the free monomers in the aqueous bulk are in equilibrium with the free monomers in the bilayer in the same way as in the classical model. The larger clusters prefer the bulk and avoid the membrane core. A typical value of K ¼ 100 is sufficient to remove most of the water from the membrane interior. One can argue that the value of K should be related to the strength of an H-bond in water. Then, a much larger K value should be appropriate. However, this is not implemented for computational reasons. 4.3.3
Ions
In biologically relevant systems, the ionic strength is not extremely low, and the charges in the head-group region are largely screened. The screening by positively or negatively charged ions influences the conformational properties of the ionic head groups and, to a lesser extent, that of zwitterionic head groups. The Gouy–Chapman theory [3] describes the distribution of ions near a charged interface. The classical Gouy–Chapman theory, including a Stern-layer, cannot account for relevant features such as penetration of ions in the head-group region, but it gives the distribution of ions in the immediate surroundings of the bilayers. The self-consistent-field model as elaborated on below may be viewed as an extremely sophisticated Gouy–Chapman model. On top of the use of the Poisson–Boltzmann equation for the charged groups, it has the chain statistics and volume constraint effect (e.g. the Stern concept) included automatically. 4.3.4
Free Volume and the Pressure
One usually distinguishes two types of lattice models. The first type may be called lattice-gas models. In this case, the number of molecules in the system is less than the number of available sites. In other words, there are vacant sites. The second type of lattice models may be called lattice fluids. In this case, all lattice sites are filled exactly by the molecular components in the system; the system is considered to be incompressible. It is easily shown that a two-component incompressible lattice–fluid model can be mapped on a one-component lattice gas one. In other words, it is possible to interpret vacant sites to be occupied by a ‘ghost’
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particle. The chemical potentials in the lattice fluids and the chemical potential and pressure in the lattice–gas model can be related to each other. This can be generalised to more components in the system. Below we will see that a sufficiently general model for the lipid bilayer system should account for some free volume (unoccupied lattice sites) in the bilayer core. If this is not done, then the lipid bilayer system will change spontaneously into the gel phase (at all reasonable temperatures). The choice of lattice-gas versus lattice-fluid models also has consequences for the natural choice of the type of parameters used. Let us consider the squarewell potential as an example. In a lattice-gas model where only one type of molecules, A, exists, there is only one relevant parameter accounting for the particle–particle contacts, i.e. UAA . Neither the interaction energy of a particle with a vacant site, nor the interaction between vacant sites, will contribute to the total energy in the system. In a lattice-fluid system there are two molecular species, A and B. As a consequence, there are UAA , UAB and UBB contact energies. However, the total number of contacts in the system is fixed, because all lattice sites are exactly occupied. It can be shown that in this system one can use Archimedes-like parameters. These parameters, known as the Flory– Huggins (FH) interaction parameters, combine the three energies mentioned above. For convenience, the FH parameter is normalised by the thermal energy kT, and, for technical reasons, it is multiplied by the lattice coordinationZ number Z: wAB ¼ 2kT ½2UAB UAA UBB . The reference is the UAA and the UBB contact energies, and therefore the wAA and wBB quantities have a numerical value of zero. A positive w-value means that A and B repel each other. In fact, the difference between the lattice-gas and lattice-fluid parameters is a difference of reference energy only. When, for example, the B-units are interpreted as vacant sites, we have UAB ¼ UBB ¼ 0 and wAB ¼ ZUAA =(2kT). We note that a particular choice for the reference energy is inconsequential for the properties of the system under consideration, and the set of wAB can always be converted into the appropriate U parameters. For historical reasons, the incompressible lattice-fluid system description is used, even if the distribution of one of the components is coupled to the distribution of vacant sites. Constant ‘pressure’ SCF calculations are the same as constant chemical potential calculations for the vacant sites. These conditions are used below. 4.4
THE SEGMENT POTENTIALS
Unless one is willing to become involved in many intricacies, a lattice model with united atoms (segments) features segments which are all of equal size. The price we have to pay for this is that there is no unique way to convert from lattice units to real space coordinates. We will discuss this point in the ‘Result’ sections in more detail.
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In a lattice model, the only way to discriminate between various segment types is by assigning to each segment type a unique set of interactions, e.g. of the nearest-neighbour type parameterised by Flory–Huggins parameters, or of electrostatic origin influenced by the charge and polarisability of the various fragments of the molecule. These segment-type dependent interactions lead to segment-type dependent potentials. Typically, the segment potentials are used in Boltzmann-like weighting factors which influence the molecular distributions. Without going into too much detail, we can note the segment potentials are composed of four terms. For a unit of type A they are given by: uA (z) ¼ u0 (z) þ A ec(z) X 1 1 þ kT wAB ½jB (z 1) þ jB (z) þ jB (z þ 1) jbB "rA E(z)2 3 2 B (9) The first term is a contribution to the segment potential that does not depend on the segment type. The value of this term is chosen such that the predicted density of molecules in the system is consistent with the incompressibility constraint: X
jx (z) ¼ 1
(10)
x
which applies in each layer z. The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and c(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson–Boltzmann (Gouy–Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. The third term is the one that accounts for the short-range nearest-neighbour contact energies, parameterised by the Flory–Huggins parameter introduced above. The summation is over all segment types in the system. Here the approach is used that nearest-neighbour contacts can be made with segments that reside in the same – or in one of the adjacent – layers. This implies an a priori weighting of contacts of l1 ¼ 1=3. We note that we here assume that the lattice cell is not isotropic, i.e. the size in the x and y direction is not the same as the size in the z-direction. We will discuss this below. The volume fraction jB (z) is equal to the probability of finding a segment of type B in layer z. The quantity jbB is the probability of finding this segment type in the homogeneous bulk with whichPthe system is in equilibrium. In the bulk all densities add up to unity as well: x jbx ¼ 1.
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The fourth term is a polarisation term. Here E(z) ¼ qc=qz is the electric field at position z. In previously published SCF results for charged bilayers, this last term is typically absent. It can be shown that the polarisation term is necessary to obtain accurate thermodynamic data. We note that all qualitative results of previous calculations remain valid and that, for example, properties such as the equilibrium membrane thickness are not affected significantly. The polarisation term represents relatively straightforward physics. If a (united) atom with a finite polarisability of "rA is introduced from the bulk where the potential is zero to the coordinate z where a finite electric field exists, it will be polarised. The dipole that forms is proportional to the electric field and the relative dielectric permittivity of the (united) atom. The energy gain due to this is also proportional to the electric field, hence this term is proportional to the square of the electric field. The polarisation of the molecule also has an entropic consequence. It can be shown that the free energy effect for the polarisation, which should be included in the segment potential, is just half this value –12 "rA E 2 . If for each segment in the system the proper segment potential (according to equation (9)) is computed, one can evaluate easily the overall potential energy in the system. This result is comparable with the first two terms of equation (6) for MD. As discussed above, the bond length is fixed and the RIS scheme is used to control the chain flexibility. For a particular conformation c of a molecule, the positions of all (united) atoms in space as well as the chain conformers are known. The potential energy of this conformation is therefore just the sum of the contributions, as given by equation (9) for all the united atoms and a particular energy quantity per gauche bond in the chain. The statistical weight for this conformation is proportional to the Boltzmann factor containing this segment potential: c
Pci / eui =kT
(11)
The next step is to generate all possible and allowed conformations, which leads to the full probability distribution Pci . The normalisation of this distribution gives the number of molecules of type i in conformation c, and from this it is trivial to extract the volume fraction profiles for all the molecules in the system. With these density distributions, one can subsequently compute the distribution of charges in the system. The charges should be consistent with the electrostatic potentials, according to the Poisson equation: q"(z)E(z) ¼ q(z) qz
(12)
where "(z) is the local dielectric constant that is dependent on the local composition. From equation (12), the electrostatic potential is extracted, and, together
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
with the density profile information, equation (12) is recomputed. What follows has been explained above. The potentials and the densities should be consistent with each other, and these solutions are found routinely by computer. Of course, such a solution should also obey the compressibility conditions. The formalism sketched above has been used in the literature in more or less the same detail by many authors [87–92]. The predicted membrane structure that follows from this strategy has one essential problem: the main gel-to-liquid phase transition known to occur in lipid membranes is not recovered. It is interesting to note that one of the first computer models of the bilayer membrane by Marcˇelja [93] did feature a first-order phase transition. This author included nematic-like interactions between the acyl tail, similar to that used in liquid crystals. This approach was abandoned for modelling membranes in later studies, because this transition was (unfortunately) lost when the molecules were described in more detail [87]. The reason for the failure of such an SCF model to predict a gel-to-liquid phase transition is known. The coupling of the single-chain behaviour to the external field is insufficient to induce some cooperative phenomenon necessary for this transition to occur. In the Marcˇelja approach, this coupling was made through a Mayer–Saupe (thermotropic) potential. In this case, the interaction energy is made more attractive for the tails when they are aligned normal to the membrane than when they do not line up. This is reasonable for the molecule as a whole, as may be judged from conformational phase transitions that take place in polymer brushes in the presence of Mayer–Saupe interactions [94], but the ordering efficiency becomes too weak when it is applied on the CH2 CH2 level. It can be shown that in order to recover a cooperative gel-to-liquid phase transition, one should include conformational entropy effects (lyotropic effects). Indeed, it was shown that anisotropic lateral interactions are not necessary for the transition. The solution for the problem was found by introducing weighting factors for placing the bonds of the molecule in the system. It may be of interest to qualitatively explain the origin of the effect. In the derivation of the mean-field partition function, it is necessary to know the probability for inserting a chain molecule in a given conformation into the system. The classical way to compute this quantity is by approximating it by a product of the local volume fractions of an unoccupied site (averaged over lattice layers). It was realised that, besides the density information, information on the bond distributions is also available. The bond distribution gives information on the average local order. Using this information, it becomes possible to more accurately access the vacancy probability. For example, if many chains are perfectly aligned, e.g. in the z-direction, the probability of the test chain also placing its segments in the z-direction becomes strongly enhanced. The oriented molecules leave an open corridor in the direction of the director. In terms of the vacancy probability, it is an event that it is enhanced with respect to the classical mean-field prediction. As a result, the test
F. A. M. LEERMAKERS AND J. M. KLEIJN
61
molecule is likely to follow the average orientation of its neighbours. For very densely packed chains, this effect may introduce a cooperative phase transition. The SCF theory that accounts for the anisotropic placing of segment bonds is called SCAF (self-consistent anisotropic field) theory [95]. It is now realised that in a SCAF analysis the chain statistics are done in a quasi-chemical approximation (on a pair-level). Typically, the interactions are still implemented on the Bragg–Williams level, as outlined above (but this is not a matter of principle). Theories that combine the SCAF approach (lyotropic effects) with anisotropic Mayer–Saupe (thermotropic) interactions do not exist. Such a model in combination with a realistic model for water (based upon the quasi-chemical approximation), may well become a next generation of SCF modelling. 4.5
THE SCF SOLUTION
In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of Z N number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excludedvolume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. There exists a less-dangerous method which has the additional advantage that it is computationally extremely cheap. In an RIS scheme, it suffices to know the exact positions of four consecutive segments. Ignoring any longerrange correlations along the chain allows the use of a Markov scheme to generate conformations. In this scheme the combined statistical weight of the full set of conformations is evaluated using a computation time that is linearly proportional to the number of segments in the molecule. This trick is the basis for the huge gain in CPU time. Again, the MC simulations have indicated that such a Markov process is legitimate. The SCF solution, i.e. the condition that the segment densities and the segment potentials are consistent with each other, is found for a canonical ensemble. This means that the number of molecules of each molecule type is fixed. As explained above, membranes should be modelled in a (N, p, g, T), i.e.
62
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
in a fixed surface tension (g ¼ 0) and fixed pressure ensemble. Both the pressure and the surface tension can be evaluated as a function of the composition, that is, the number of lipid molecules per unit area in the system. Typically, an extra iteration of the composition is needed to evolve to an equilibrium tension-free membrane. Unless otherwise mentioned, we will present results for membranes that are free of tension. With respect to SCF models that focus on the tail properties only (typically densely packed layers of end-grafted chains), the molecularly realistic SCF model exemplified in this review needs many interaction parameters. These parameters are necessary to obtain colloid-chemically stable free-floating bilayers. A historical note of interest is that it was only after the first SCF results [92] showed that it was not necessary to graft the lipid tails to a plane, that MD simulations with head-and-tail properties were performed. In the early MD simulations (i.e. before 1983) the chains were grafted (by a spring) to a plane; it was believed that without the grafting constraints the molecules would diffuse away and the membrane would disintegrate. Of course, the MD simulations that include the full head-and-tails problem feature many more interactions than the early ones. The set of parameters, i.e. the force-field parameters used in the SCF calculations, are listed in Table 1. We will not discuss all of them. The most important one is the repulsion between water and hydrocarbon. The value of this FH parameter is set to wH2 O, C ¼ 0:8. One should remember however that in Table 1. Parameters used in (most of) the SCF calculations as presented in this chapter H2 O
V
C
CH3
N
P
O
Na
Cl
"r
80 0
1 0
2 0
2 0
10 1
10 –0.2
10 0
10 1
10 –1
w
H2 O
V
C
CH3
N
P
O
Na
Cl
0 2.5 0.8 0.8 0 0.5 0 0 0
2.5 0 2 1 2.5 2.5 2.5 2.5 2.5
0.8 2 0 0.5 2.6 2.6 2 2.6 2.6
0.8 1 0.5 0 2.6 2.6 2 2.6 2.6
0 2.5 2.6 2.6 0 0 0 0 0
0.5 2.5 2.6 2.6 0 0 0 0 0
0 2.5 2 2 0 0 0 0 0
0 2.5 2.6 2.6 0 0 0 0 0
0 2.5 2.6 2.6 0 0 0 0 0
H2 O V C CH3 N P O Na Cl
In the first row the relative dielectric constant for the compound is given. In the second row the valency of the unit is given. The other rows give the values for the various FH parameters. Remaining parameters: the characteristic size of a lattice site 0.3 nm; the equilibrium constant for water association: K ¼ 100; the energy difference for a local gauche conformation with respect to a local trans energy: DU tg ¼ 0:8 kT; the volume fraction in the bulk (pressure control) of free volume was fixed to jbV ¼ 0:042575
F. A. M. LEERMAKERS AND J. M. KLEIJN
63
the calculations the water–cluster approximation has been implemented. Consistent with this model is a fairly low repulsion between water and hydrocarbon. The parameter has been chosen in order to find the predicted critical micellisation concentrations for surfactants that match experimental ones [97]. In Table 1, the interaction parameters of the unoccupied sites V with the real segments in the system are all positive and relatively high. Again, the reason for having interaction parameters for these ‘contacts’ originates from the Archimedes-like FH parameters. The effect of the relatively high values is that the vapour phase is a bad solvent for all components in the system. The parameters shown in Table 1 must still be considered preliminary. For example, several interactions between two polar compounds are assigned the athermal w ¼ 0 value. This ideal value is chosen in order to keep the set as simple as possible. We also kept the system exactly symmetrical with respect to the interactions of the Na and Cl ions. Again this is a simplification of the real system. One undoubtedly has an asymmetry with respect to the interaction with the hydrophobic core for these two ions. This in itself will generate an electrostatic potential at the membrane–water interface. These ion-specific effects may be studied systematically and are relevant for permeation and transport phenomena. 4.6
PHOSPHATIDYLCHOLINE BILAYERS
A selection of the predictions of the equilibrium structure of DPPC bilayers as found by numerical self-consistent-field calculations is given in the following figures. In a series of articles, the SCF predictions for such membranes were published, starting in the late 1980s. As discussed above, we will update these early predictions for the theory outlined above with updated parameter sets. The calculations are very inexpensive with respect to the CPU time, and thus variations of the parameter-set will also provide deeper insight into the various subtle balances that eventually determine the bilayer structure – the mechanical properties as well as the thermodynamic properties. The saturated PC molecule differs (apart from the length of the tails) from the SOPC molecule depicted in Figure 12 only with respect to the double bond, which is replaced by a single one. In this figure, each unit that is included in the SCF model is drawn. The corresponding parameters are collected in Table 1. From this it is clear that a united atom approach is implemented. This means that the protons are not explicitly included. Note that, in contrast to the recent work of Meijer [84–86], the end groups of the acyl chains now have double the volume as compared with CH2 groups. This is realised by a branching point at the CH2 group on the last but one position in the chain. Furthermore, the CH3 group is given some small repulsion with the CH2 groups to specify that these units differ significantly. The overall density profiles across the DMPC bilayer are given in Figure 13 (see also refs. [84–86,92,95,98]). Together with these profiles, the water-density
64
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES NC 3
+
PO 4
–
C=C 9 10
5
i ii
1
iii
1
sn2 sn1 10
5
15
Figure 12. A typical example of the level of detail used to represent molecules in the SCF calculations: sn1-stearoyl-2-oleoyl-phosphatidylcholine. The open squares are the CH2 groups, the light grey squares on the end of the tails represent the CH3 groups. The C C unsaturation (cis and on position 9) along the acyl chain is indicated. The mid-grey w squares represent oxygen. The darkest grey square is the phosphate. The nitrogen atom as well as the three surrounding carbons in the choline group (CH3 ) are striped. The numbering of the carbons along the sn1 and sn2 tail (1–18) and the carbons in the glycerol backbone (i, ii, iii) are indicated. The positive charge is located at the nitrogen, and the negative charge is distributed over the five units of the phosphate group
Water
DMPC 0.8
j Head
0.4
CH3 0 −20
−10
0 z
Free volume 10
20
Figure 13. The overall density (volume fraction) profile for DMPC bilayers is shown here. Apart from the distribution of the overall DMPC molecules, the density distribution of the head-group units (including the choline group, the phosphate group and the oxygens of the glycerol unit), and the end groups of the lipid tails are also indicated. In addition, the free-volume profile and the water profile are depicted
profile is given. In line with data from the MD simulations, the SCF method predicts that the water molecules only penetrate into the head-group region, and that the water density in the core is reached already in the region of the glycerol backbone. In the model there is also a component which is interpreted as free volume. The parameters were chosen such that the amount of free volume in the core of the bilayer is slightly higher than in the water phase. The free volume density in water is fixed to the equilibrium value found for the water–vapour coexistence calculations: jbv ¼ 0:042575. This value was fixed, which implies, from a lattice-gas perspective, that the pressure was fixed to some reference value, i.e. p ¼ 1.
F. A. M. LEERMAKERS AND J. M. KLEIJN
65
In the interfacial region between the hydrophobic core and the water phase, there is always a small increase in free volume. The increase is not particularly large, and we are not aware of attempts in MD simulations to measure this small drop in local density. As can be seen from Figure 13 the distribution of head groups is almost as wide as the width of the hydrophobic core. On the one hand, this is because we have collected several head-group units in this overall head-group distribution. On the other hand, there are a number of protrusion fluctuations that are automatically ‘excited’ in the computations. The end groups of the hydrocarbon tails are depicted separately in Figure 13. In line with data from MD simulations, we see a significant spread of the end groups throughout the core. This means that there are considerable fluctuations in conformations of the tails. The atom density dip, as found above in the MD simulation, correlates with the distribution profile of the CH3 units. In reality, the CH3 groups are rather voluminous, and thus conversion of the MD data to volume fractions, or the SCF data to atom densities, leads to the conclusion that the SCF results are at least qualitatively consistent with the MD predictions. The membrane thickness can be defined in many ways. The z-axis is in units of a lattice-cell dimension. As can be seen, the thickness is 20 to 30 lattice units. In principle, the size of a lattice unit can still be chosen. When this value is scaled to the size of a water molecule, i.e. 0.3 nm, a bilayer thickness of the order of 6 nm is obtained. However, a C–C bond in the hydrocarbon chain is significantly shorter, and taking this (i.e. 0.15 nm) as a measure for the lattice size, a P–P distance of approximately 20 lattice units corresponds with a size of order 3 nm. Of course some average is more appropriate. To dissolve the conversion problem from lattice units to real space coordinates it is necessary to compare the results to accurate simulations or experiments. Comparison of MD results, for example as presented in Figure 5, and the SCF result shown in Figure 13 suggests that a more accurate scaling of the size of a lattice site should be around d ¼ 0:2 nm. This value is indeed between the size of a water molecule and that of a C–C bond. We will use this value below to convert from lattice units to real coordinates. Using this value, the bilayer thickness given by the cross-membrane P–P distance for the membrane given in Figure 13 is approximately 4 nm. A more detailed investigation of the DMPC bilayer is given in Figure 14. In this figure, the distributions are split up into two groups, depending on which side the choline group of the lipid is positioned. In this way it is possible to illustrate how much the chains from opposite monolayers interact with each other. The average position of the CH3 groups of the chains is approximately at the centre of the bilayer, albeit that the end groups belonging to chains that have the head group on the positive side have a positive average position. Surprisingly, however, the maximum in the end-point distribution is positioned slightly on the opposite side.
66
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
CH2
0.6 j
0.4
0.2 CH3
P 0 −20
−10
0 z
N 10
20
Figure 14. Volume-fraction profiles of parts of the DMPC molecules for lipids that have the head group at positive coordinates (continuous lines) and at negative coordinates (dashed lines). The centre of the bilayer is positioned at z ¼ 0. The phosphate group, the nitrogen of the choline group and the CH3 groups of the tail ends, as well as the other hydrocarbon units, are indicated
Now we can discuss the interdigitation issue in more depth. The key point is that there is undoubtedly some interdigitation, but that this remains relatively limited. The two monolayers are thus not completely independent. A limited interpenetration should also be expected from entropic considerations. Below we will see that the chain ends are not particularly highly ordered. Thus the chains do not have a clear memory as to which side of the bilayer they ‘belong’. In previous SCF calculations, the interpenetration is likely to have been overestimated. This is because the interpenetration is sensitive to the detailed properties of the CH3 groups. These groups have a larger volume and are expected to be more hydrophobic than CH2 groups. These properties are represented in the parameter set discussed above (and not present in previous ones). The quantitative results depend to some extent on the details of the parameter set. However, it is virtually impossible to completely remove all the overlap between opposing monolayers. This is of course an important observation for the ability of the bilayer to absorb additives in the core. We will return to this issue below. The orientation of the head group can also be deduced from Figure 14. In line with results published in SCF results of Meijer et al. [85], and also in line with all MD simulations of PC layers, the head group is laying flat. This flat orientation is very much promoted by the zwitterionic nature of the head group. The acyl groups connected to the nitrogen group may even push the choline group towards the membrane core. However, entropically it is expected that the end group can be positioned more freely and thus will have a wider distribution. For this reason, the choline group can be found on both sides of
F. A. M. LEERMAKERS AND J. M. KLEIJN
y
67
0.05
0.006
0.025
0.003
0
0
−0.025
−0.05 −30
q
−0.003
−20
−10
−0.006 0 z
10
20
30
Figure 15. The electrostatic potential (V) profile (dashed line; left ordinate) and the overall charge (in units of elementary charges) profile (continuous line; right ordinate) across the DMPC bilayer, as found by SCF calculations
the phosphate group. Meijer [84] has shown that with increasing ionic strength the bimodal character of the N profile becomes more pronounced. This result is also in accordance with experimental observations. It has been reported that in PC crystals both conformations of the head group are present [99]. The charge distribution and the electrostatic potential profile across the DMPC bilayer are much more complex that one would intuitively expect. The electrostatic potential profile is presented in Figure 15. It is slightly positive on the outer side of the bilayer as well as in the membrane core. A region with negative electrostatic potential is found at the position of the phosphate group. The electrostatic potential profile must be zero throughout the bilayer, as long as the phosphate and the choline group have identical z-positions. These identical positions do not occur, and thus there is a potential profile and a charge distribution which remains finite, albeit that the local charge density as well as electrostatic potentials are small. The differences between the electrostatic potential profiles as found by SCF calculations and that of MD simulations (cf. Figure 7) are large. First of all, there is a huge difference in absolute value. Furthermore, the SCF results are more detailed. It is instructive to discuss these disparities. For this we need to return to the force fields used in both methods. In MD, many segments have partial charges, and therefore they all contribute to the electrostatic potential. In SCF, these partial charges only occur on the groups which carry an effective charge. All the other contributions are ‘hidden’ in the Flory–Huggins parameters and the local dielectric permittivities, and as such do not show up in the charge distribution and the mean-field electrostatic potential. For this reason, it is not a surprise that the two methods give different results. Perhaps it remains worthwhile to note that both methods indicate that the electrostatic potential in
68
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 0.0015 j
Cl 0.001 Na
0.0005
0 −30
Figure 16.
−20
−10
0 z
10
20
30
The distribution of the ions across the DMPC bilayer
the core is positive with respect to that in the water phase, in accordance with experimental data. The distributions of the ions (Na and Cl) follow the electrostatic potential profile accordingly, as is seen from Figure 16. Of course the ions have a strong preference for the aqueous phase. Nevertheless, they have to partition inside the membrane to some extent. Because the electrostatic potential is positive inside, the negative ions have a small preference for the membrane interior over the positive ions. It should be noted that this preference is solely the result of the electrostatic potential, because the size and the nonelectrostatic interactions were chosen to be identical for the two ions. A more realistic model can discriminate between the ions with respect to the intrinsic affinities for oil and water, respectively. The electrostatic potential profile in the water phase is of course well understood, and follows exactly the Gouy–Chapman theory [3]. For sufficiently low electrostatic potentials (smaller than 25 mV), there is an pffiffiffiffi exponential decay with a decay length given by the Debye length kD 3= js where js is the volume fraction of ions that is proportional to the concentration of ions. The order parameter is directly available from the calculations and the SCF results are given in Figure 17. The absolute values of the order parameter are a strong function of head-group area. Unlike in most SCF models, we do not use this as an input value; it comes out as a result of the calculations. As such, it is somewhat of a function of the parameter choice. The qualitative trends of how the order distributes along the contour of the tails are rather more generic, i.e. independent of the exact values of the interaction parameters. The result in Figure 17 is consistent with the simulation results, as well as with the available experimental data. The order drops off to a low value at the very end of the tails. There is a semi-plateau in the order parameter for positions t ¼ 6 14,
F. A. M. LEERMAKERS AND J. M. KLEIJN 0.4
69
sn2
S
sn1 0.3
0.2
0.1
0 0
2
4
6
8 t
10
12
14
16
Figure 17. Order parameter profile for the two tails of DMPC bilayers. The chain closest to the head group is named sn1 and the other one sn2. Segments closest to the glycerol backbone are numbered t ¼ 1, and the chain ends are t ¼ 16. The lines are drawn to guide the eye
and there is a maximum in the order parameter at chain segments near the glycerol backbone. Together with the known positions of the chain segments and the order along the chain, we can deduce that the central region in the bilayer is slightly more disordered than the outer part of the core. Such a result is relevant for the mechanism of incorporation of foreign objects in the bilayer. The two chains in the lipid bilayer are not identical, because they are positioned asymmetrically with respect to the PC head. It turns out that the tail closest to the head group is buried less deep in the bilayer than the other chain [98]. The difference is not very large; it amounts to about half a segment size. From this difference, we can rationalise the disparities in order between the two tails. The chain that is pulled out will be stretched most, and the order tends to be higher than the other chain. Below we will see that the differences in the behaviour of the tails can become larger if there are differences between the tails, e.g. with respect to the degree of unsaturation. 4.7
THE LATERAL PRESSURE PROFILE
Recently, the lateral pressure profile in a bilayer has been discussed in the context of partitioning of proteins in bilayers [100]. It is argued that this pressure profile can be used to rationalise the effects of additives on the membrane properties. Here a note of caution is necessary. It is not possible to define the lateral pressure profile through a bilayer unambiguously. The reason for this problem is that the local pressure not only has contributions that come from the local densities (this property is uniquely defined), but also from
70
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
interaction between molecules. As molecules typically do not occupy the same coordinate, the method of keeping track of binary, tertiary, etc., interactions becomes somewhat arbitrary. All choices that one can make to do this are equally good. For example, all results of thermodynamic observables that result from moments over the lateral pressure profile and moments over the derivatives of the lateral pressure profile, such as the surface tension and the mechanical parameters of the bilayer, can be shown to be independent of this choice [16,101]. The fact that the lateral pressure profile is ill defined indicates that this property is not a useful one to be monitored or to be discussed in the context of partitioning of additives in membranes. For this reason, we will neither discuss the lateral pressure profile further, nor give an example of the lateral pressure profiles. For one and the same membrane we can generate many strongly varying lateral pressure profiles [16,101]. 4.8
COMPARISON OF SCF AND MD FOR SOPC MEMBRANES
In biological membranes, the class of fully saturated PC lipids is just one of many. Indeed, variations in the lipid architecture are frequently encountered. There are variations with respect to the head-group structure as well as variations in the tail architecture. For example, in the tails, single or multiple unsaturated bonds are very common. There is relatively little knowledge about the role of lipid unsaturation in biomembranes. There are relatively few MD simulations on bilayers composed of lipids with unsaturated bonds. Nevertheless, single unsaturation as well as multiple unsaturated chains have been of some interest. For a recent review see reference [102]. We will now present a selection of the results. We will also take this opportunity to compare the results from the MD simulations with those from the SCF calculations. In Figure 1 we showed a snapshot of a typical MD result for a membrane composed of lipids of the type 1-stearoyl-2-docosahexaenoyl-sn-glycero-3phosphatidylcholine, which has six unsaturated bonds. From a snapshot, it is not possible to understand the role of unsaturation in any detail. However, the averaged density profiles are more informative. In Figure 18, various profiles as found by SCF modelling are compared with corresponding MD results for SOPC bilayers in the hydrated state (see also reference [103]). Before this figure can be discussed, one should realise that in MD simulations the results are typically presented in terms of the density in g cm3 versus the distance from the centre of the bilayer in nm units. The SCF results are available in terms of dimensionless densities (volume fractions) and the dimensionless distance to the centre of the bilayer (i.e. in lattice units). Implementing the above suggested conversion of 0.2 nm leads to a good match in membrane thickness. The problems with the conversion of volume fractions to densities are not easily resolved, and therefore we need to concentrate on the shape of the distributions and not on the heights of the distributions.
(a)
F. A. 1M. LEERMAKERS AND J. M. KLEIJN Tails
(b)
0.8
1.2
Water
1
0.6
0.8
0.4
Head
r/g cm−3
j
71
Water
Gly 0.2
CH3 V
Tails 0.6
Head
0.4
Gly 0.2
0.015
0.2
Na
0.01
Cl
0.005
r/g cm−3
j
0 P
0.075
j
P
0.1
0.05 N
0.025 −20
−15
−10
−5
0
z
5
10
15
N
0 20
0
−4
−3
−2
−1
0
1
2
3
4
z /nm
Figure 18. Molecular modelling results for hydrated bilayers of 1-stearoyl-2-oleoyl-snglycero-3-phosphatidylcholine PC (SOPC). (a) SCF results. The volume fraction profiles for tails, head group, glycerol backbone, CH3 end-groups and water (top panel), Na and Cl (middle panel) and the P and N atoms of the head group (bottom panel) are presented. Note that the volume fractions are dimensionless densities and that the z coordinate is made dimensionless by the lattice spacing. (b) MD results. Mass density distributions, g cm3, for the tails, head-group segments, glycerol backbone units and water (top panel), the P and N atoms (bottom panel). In the MD simulation systems there were 48 times two lipids per cell, 24 H2 O molecules per lipid, and altogether there were 20 544 (SOPC) atoms. Mean cross-sectional area per lipid molecule was 0.6624 nm2. The MD trajectories of 1018 ps were computed at T ¼ 303 K. The zero point (z ¼ 0) is the centre of the bilayer: it was calculated during the MD simulations as the middle point between the centres of P–N vectors of the two monolayers (the centre of the P–N vector is the middle of P–N distance). The ratio between the CPU times (MD/SCF) needed to obtain these results is in the order of 105
First of all it is seen that the SCF results are free of any noise, whereas there is plenty of noise in the MD profiles (note, however, that the density profiles on both halves of the bilayer are in this case not averaged; the close resemblance between the profiles on both halves thus indicates that the membranes are well equilibrated). Apart from this, inspection of Figure 18 shows a remarkable resemblance between the two set of predictions. Many details are in semiquantitative agreement. Moreover, many of the features of membranes composed of SOPC resemble those of DMPC discussed above. For example, the width of the membrane–water interface is about 1 nm, i.e. the size of just two to three water molecules. This width is consistent with the scaling arguments mentioned at the beginning of this chapter. A more accurate comparison
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
between the SCF and MD results shows that even the shapes of the head-group profiles match accurately and the distributions of the P and N atoms have the same features. The fact that these two distributions are strongly overlapping means that the PC head group is roughly oriented parallel to the membrane surface, as is the case for DMPC. Of course there are differences. For example, in the SCF calculations a 1:1 electrolyte solution was used, whereas in the MD simulations no salt ions were added. We note that the densities of the two ions, as shown in Figure 18a, are almost the same in the centre of the bilayer. This means that in this case the electrostatic potential in the centre of the bilayer is virtually zero. Other features of the ion profiles are the same as in Figure 16 for DMPC. The noted difference is not attributed to a fundamental difference between DMPC and SOPC bilayers. The difference is caused by the fact that the ionic strength used in the SOPC system (Figure 18a) is 10 times higher than in the DMPC system (Figure 16). The higher ionic strength leads to a more efficient screening of the electrostatic potential. A minor effect already noticed by Meijer and co-workers [84] is that the head-group properties slightly depend on the ionic strength, even in the case of a zwitterionic head group. From the density profiles one cannot really judge the effect of the double bonds; the density profiles for membranes of saturated lipids are very similar to those of unsaturated ones. Therefore it is necessary to consider some of the conformational characteristics of the tails. It is possible to compute the order parameter profile for both the saturated and the unsaturated chains. The order parameter profile for the saturated chain closely follows the results presented in Figure 17 for DMPC membranes for both the SCF and the MD predictions. The order parameter profiles for the unsaturated chain closely follows the MC predictions, as discussed in Figure 9. A pronounced dip is found near the cis double bond. For this reason, we choose here to present complementary data about the conformational properties of the acyl chains. In Figure 19 the root-mean-square positions (fluctuations) of all segments of both tails as a function of the average positions are shown for both modelling techniques. These results are significantly more informative than order parameter profiles. From Figure 19 it is easily seen that towards the tail end the fluctuations increase and that the segments close to the glycerol backbone have the sharpest distributions. The average positions of the first segments of the two tails differ slightly; the sn1 tail is (similarly to in DPPC bilayers) buried slightly deeper in the bilayer than the sn2 tail, as can be judged from the hzi positions of the first segments. In lipid bilayers with saturated tails, the two tails differ only with respect to this small shift. However, in SOPC the unsaturated chain is buried significantly less deep in the bilayer than the saturated one; the unsaturated chain behaves effectively as a shorter chain [104]. The fluctuations tend to increase relatively fast for the free chain ends, especially for the unsaturated tail, i.e. left of the arrow (which indicates the position of the double bond) in Figure 19. The fact that the average end position (CH3 groups) of the saturated
F. A. M. LEERMAKERS AND J. M. KLEIJN
73
5 1.4
(a)
(b)
18 18
4
18
18
RMSZ /nm
RMSZ
1.2 18:1
3 18:0
18:0
1 2
1
18:1
2
2 2
1
0.8 2 1
1
1 0
2
4
6
8
0.6 0
0.5
1
1.5
2
< z >/nm
Figure 19. The root-mean-square (RMS) position of each segment of both acyl chains of SOPC lipids is plotted as a function of the average position of the segment. The sn1 tail is given by the closed symbols, and the sn2 tail is given by the open symbols. Various numbers of the tail segments and of the backbone segments are indicated. The lines are drawn to guide the eye. The arrow points to the position of the unsaturated bond. (a) SCF results (conversion from dimensionless units to real units is approximately a factor of 0.2 nm), (b) MD results (the average over the sides of the bilayer is taken)
tail is near the centre of the bilayer, and because the fluctuations are large, it is again true that there is a significant interdigitation between tails of opposite monolayers. The interpenetration is significantly less for the unsaturated chain. In both the SCF calculations, as well as for the MD results, the two curves in Figure 19 cross each other. The local growth of the fluctuations at the position of the double bond and suppression of fluctuations just around the double bond are both very characteristic of a cis double bond. Similar results are also discussed in some depth in a paper where statistical mechanical calculations are compared with corresponding MD simulations [105]. The mean-field model that is used in the latter study is one where chains of equimolar amounts of saturated and unsaturated bonds are grafted at a density which corresponds with the experimental data. Not all studies agree on the effect of unsaturation. In an MD study made by Heller [106] it was found that around the unsaturated bond there is a region with increased fluctuations, whereas above these regions there were just a few less fluctuations. One can argue that this might be due to an insufficient equilibration of the system. From the modelling results for bilayers composed of unsaturated lipids one can begin to speculate about the various roles unsaturated lipids play in biomembranes. One very well-known effect is that unsaturated bonds suppress the gel-to-liquid phase transition temperature. Unsaturated lipids also modulate the lateral mobility of molecules in the membrane matrix. The results discussed above suggest that in biomembranes the average interpenetration depth of lipid tails into opposite monolayers can be tuned by using unsaturated lipids. Rabinovich and co-workers have shown that the end-to-end distance of multiple unsaturated acyl chains was significantly less sensitive to the temperature than that of saturated acyls. They suggested from this that unsaturated
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
lipids may have a role in providing a temperature-insensitive lipid jacket around protein molecules [102]. Whether it is generally true that membrane proteins are preferentially surrounded by unsaturated lipids remains to be proven. 4.9
CASE STUDIES: SCF RESULTS
We have seen from the above that the simulations and the calculations give a consistent picture of the PC bilayers. The strong and weak points of both methods are now well illustrated, and we can proceed by presenting results for gradually more complex problems obtained by the SCF method. 4.9.1
Effects of the Length of the Hydrocarbon Tails
It is of interest to elaborate on the effect of the tail lengths of the PC lipids on various (overall) membrane parameters. Here we will concentrate on the area per molecule and the compressibility modulus in the range C12 to C16 . Relevant data can easily be obtained from SCF calculations, and the predictions of this theory are presented in Figure 20. In line with experimental data, the area per molecule is almost independent of the length of the tails. Only a very small linear correction is found, which indicates that the longer the tail length the larger is the head group area. As a consequence, the thickness of the bilayer is to a good approximation proportional to the length of the chains. It is necessary to convert the dimensionless area as given in Figure 20 to real areas per molecule. For this, we need to know the cross-sectional area of a lattice site. The first-order approximation of this is d 2 . However, this is only correct for isotropic lattice sites. It can be shown that, consistent with the weighting of the interactions in equation (9) (l1 ¼ 1=3), the area is a factor of three higher: 0.9
(a)
5.4
0.2
1.3
(b)
C16PC
1.1
g
5.3
0.85
0.1 0.9
0.8 12
13
14 t
15
5.2 16
0 0
0.05
0.1 (a − a0)/a0
0.15
d ln a / d g
d ln a/d g
a0
0.7 0.2
Figure 20. (a) The (dimensionless) lateral compressibility (dilatational modulus, elastic area expansion modulus) (left ordinate) and the dimensionless area per molecule (right ordinate) as a function of the tail length (t) of the PC lipids in equilibrium bilayer membranes. The conversion to real compressibilities and areas per molecule is discussed in the text. (b) The (dimensionless) surface tension and the (dimensionless) lateral compressibility as a function of the relative expansion for the C16 PC lipid
F. A. M. LEERMAKERS AND J. M. KLEIJN
75
as ¼ 3d 2 . Again, when we use d ¼ 0:2 nm, as explained above, it is possible to convert the prediction to the area per molecule. One simply needs to multiply the dimensionless areas by 0.12 to obtain the area in units of nm2 . The correspondence with experimental data is satisfactory. The lateral compressibility, i.e. the relative area change upon an imposed membrane tension, decreases slightly more than linearly with the chain length. This means that it is more difficult to expand the membrane surface area of a long-chained lipid than a shorter one. In Figure 20 dimensionless units are used, which means that the surface tension is given in units kT=as . Again, using a lateral dimension of a site, d ¼ 0:2 nm, and the lattice site area as ¼ 3d 2 , means that g ¼ 1 corresponds with about 33 mN m1 lateral tension. In other words, one needs to apply a lateral tension of order 40 mN m1 to double the membrane area. This prediction seems to be a factor of about six lower than estimates that were recently reported by Evans and co-workers [107]. These authors use micropipettes to pressurise giant vesicles and obtain values of the order KA ¼ qg=q ln a ¼ 230 mN m1 . There are also some data on the compressibility modulus, as found by MD simulations on primitive surfactants [62] KA ¼ 400 mN m1 . In a molecular detailed simulation study on DPPC lipids, Feller and Pastor [40] report a KA value of about 140 mN m1 . In Figure 20 the surface tension of the bilayer is given as a function of the relative expansion of the bilayer. Of course, when the surface area is increased, the surface tension invariably goes up. The slope of this curve decreases slightly with increasing relative expansion. From this, it is seen that the membrane compressibility increases when the membrane is stretched. 4.9.2
Lipid Variations: Charged Lipids in Bilayers
An important ingredient at the disposal of nature to vary the membrane properties in subtle ways is to make changes in the head-group structure. A PC lipid is zwitterionic and, as such, it is rather inert with respect to variations in ionic strength. We note that the insensitivity with respect to the membrane structure does not imply that the colloidal stability is not affected by the ionic strength. Meijer [84] has shown that it is expected that PC bilayers are only colloid-chemically stable at an intermediate ionic strength; then the electrostatic repulsion is largest. Both at high and low ionic strength attractive contributions may dominate over repulsive interactions. Charged lipids have not frequently been modelled. In simulations, the charges impose significant problems because it is extremely expensive to account for long-range interactions. Also by SCF techniques there has been little work done. Meijer et al. [84] reported on some properties of phosphatidylserine-type lipids. We connect here to these results and present a short survey. In an SCF model the PS lipids are parameterised on the same level of detail as the PC lipids. The extra carboxyl group is modelled with one negative charge with similar properties as a negative ion, on to which
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24 5.8 dP−P
ao PS
20 5.4 PC
16 5 0
0.2
0.4
0.6
0.8
1
fPS
Figure 21. The area per molecule (left ordinate) and the distance between the phosphate groups on opposite sides of the bilayer of the PC and PS lipids as indicated (right ordinate) as a function of the fraction of PS molecules in equilibrated bilayers composed of mixtures of C16 PC and C16 PS lipids. Reproduced from ref (85) with permission from the American Chemical Society
two oxygens are coupled. For PS, there are two negative charges and one positive charge in the head group. A net negative charge on the head group gives the head group a more hydrophilic nature. This leads to a more hydrated head group that tends to extend more towards the water phase than the PC group. The average phosphate – phosphate distance across the bilayer is a convenient measure for the thickness of the bilayer. This property is presented in Figure 21 for bilayers composed of mixtures of PS and PC lipids. The phosphate of the PS head group is extended more outwards than the phosphate of the PC group. At the same time, the area per molecule increases with an increase of the PS in the bilayer. These effects are shown in Figure 21. It is anticipated that the membranes composed of charged lipids are more sensitive to the ionic strength. Very often in such cases the Debye length is the controlling parameter. As the Debye length is proportional to the square root of the ionic strength, we present in Figure 22 the area as well as the hP Pi distance as a square root of the ionic strength. The number of molecules per unit area (the inverse of the area per molecule) is inversely proportional to the Debye length (i.e. ðjs Þ1=2 ). When the area per molecule goes down, the molecules must pack more closely, and thus it is natural to find that the membranes become thicker. This is reflected by the increase in hP Pi, increasing again with the square root of the ionic strength. 4.9.3
The Gel-Phase of DPPC Bilayers
At sufficiently low temperature, the liquid state of the bilayer is not stable and the membrane abruptly ‘freezes’ into the so-called gel state. The structure of a
F. A. M. LEERMAKERS AND J. M. KLEIJN
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5.9
dP−P
a0
20 5.7
19
5.5
5.3
18 0.02
0.04
0.06
0.08
0.1
js
Figure 22. The average dimensionless area per molecule (left ordinate) and the phosphate – phosphate distance in lattice units across the bilayer (right ordinate) of phosphatidyl serine bilayers as a function of the square root of the ionic strength
bilayer membrane in the gel state has recently been modelled by MD simulations [35,106,108]. To obtain good results, it is necessary to have a constant pressure rather than a constant-area simulation. In the gel phase, the molecules are almost in an all-trans configuration. Only approximately 0.1 gauche torsion was found per acyl chain. The acyl chains can be ordered in several ways with particular tilting angles. The SCF modelling of gel-state membranes has also some history. There are some early works on the gel-to-liquid phase transition [109,110], where the main interest is in the cooperative ordering of the acyl chains. The full problem of thermodynamically relaxed bilayers composed of lecithin-like molecules has been reported by Leermakers et al. more than a decade ago [95]. Although the SCAF model makes use of lattice approximations which become rather restrictive when the molecules go towards the fully ordered state, it was shown that it is possible to generate tensionless membranes in the gel state. Both fully interdigitated gel phases and noninterdigitated ones could be found which depended on the degree of hydration of the bilayers. We will not present these results in this review in more detail. In passing, we note that Schick and co-workers also modelled lipid monolayers using a SCAF-like technique [111]. They go essentially off-lattice by allowing steps in many different directions. By doing so, they found some cooperative tilt direction of the lipid tails as a function of the headgroup area. Such an approach should be applied to the bilayer problem as well. Then gel phases in which the molecules are cooperatively tilted with respect to the membrane normal can be analysed. To our knowledge this has not yet been done.
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4.9.4
PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Mechanical Parameters of Lipid Bilayers
In principle, one can analyse the equilibrium fluctuations as found from an MD simulation around the perfectly flat configuration of a bilayer to obtain information on the mechanical properties of the bilayer [14,15]. The finite size of the box is a serious limitation to do this accurately, and results of this type can best be reached using coarse-grained MD simulations. The alternative to learning about the mechanical parameters of the bilayer is to do molecular realistic simulations on nonlamellar geometries (i.e. curved bilayers). Periodic boundary conditions dictate that the (average) shape of the bilayers in MD simulations is flat. This is why such results do not (yet) exist in the literature. The authors are aware of some preliminary MD simulations on primitive surfactants (coarsegrained MD) in which the molecules follow the equations of motion in the presence of some type of bias force. This force is chosen such that it maintains a well-defined local curvature in the bilayer [112]. The strength of the bias gives information on the mechanical resistance of the bilayer against shape changes. Some information on the behaviour of lipids in curved geometries for a molecularly simplistic model is available from MC simulations [113]. Much more work has been done using SCF theory. Above, we argued that a portion of a finite-sized membrane can close upon itself to remove edge effects. In this way, vesicles are formed. The thermodynamic stability of vesicles is still a topic of hot debate in the literature, primarily because there are so many scenarios. The SCF analysis of vesicles leads to information on the mechanical parameters for a particular membrane system. The first point to be made is that the structure of the bilayers changes due to an imposed curvature. These curvature effects are easily monitored in an SCF analysis. Unless one is willing to do a three-dimensional analysis, the method is restricted to homogeneously curved bilayers, i.e. cylindrical or spherically shaped vesicles. Curvature-induced structural changes are best observable when the curvature is extremely high. Curved vesicles composed of lipids with molecular detail were first performed by Leermakers and Scheutjens [114]. The results presented in Figure 23 were obtained with the same parameter-set as discussed in Table 1, but the anisotropic bond-weighting factors were omitted. This means that the bilayers are slightly thinner than given in Figure 13. In Figure 23 the centre of the vesicle is at r ¼ 0. The head groups on the inner leaflet, positioned at r ¼ 20 have a higher local density than the head groups in the outer leaflet. The tails, excluding the CH3 end-groups, show a clear maximum on the outer side of the core region. The CH3 groups, on the other hand accumulate slightly more on the inner side. This shows that two halves of the curved bilayer contain chains that have different conformational properties. The total contribution of the tails, i.e. including the CH2 and CH3 groups, still reaches a maximum in the
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1 Tails
0.8 j
0.6
0.4 Head
0.2
0
CH3
0
10
20
30
40
50
r
Figure 23. Radial segment density profile through a cross-section of a highly curved spherical vesicle. The origin is at r ¼ 0, and the vesicle radius is very small, i.e. approximately r ¼ 25 (in units of segment sizes). The head-group units, the hydrocarbons of the tails and the ends of the hydrocarbon tails are indicated. Calculations were done on a slightly more simplified system of DPPC molecules in the RIS scheme method (thirdorder Markov approximation), i.e. without the anisotropic field contributions
outer region of the core. These curvature-induced changes in the membrane structure are the locus of some additional free energy in the system. By means of a thermodynamic analysis, one can extract these changes in free energy and obtain information on the mechanical properties of the bilayers. The thermodynamic analysis of curved bilayers is typically done on very weakly curved objects (much less curved then the example shown in Figure 23). This is to guarantee that the free energy of curvature remains quadratic in J and K. The bending modulus that controls the undulation fluctuations kc is known to depend strongly on the length of the hydrocarbon chains. The classical theory of elastic deformation [115] already indicates that the energy needed to bend a plate scales with the third power of its thickness. Such a strong increase of the bending modulus with lipid chain length appears to be consistent with both theoretical SCF computations [116] and with experimental findings [107]. It is also believed that for stable bilayers the Gaussian bending modulus is negative [25]. There is no experimental technique available that points directly to this quantity. It is believed that theoretical predictions may help to understand these matters in more detail. A typical outcome of the Helfrich analysis for charged bilayer systems is presented in Figure 24b–d in combination with experimental results for the dependence of the equilibrium radius of vesicles composed of DOPG and DOPC as a function of the ionic strength in Figure 24a. The vesicle solution was equilibrated using the freeze–thaw method [20], and the size was measured
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 140
(a)
(b) 6.5
R / nm
120
kc/kT 100
DOPG
cW-ion = −2
5.5
80 cW-ion = 0
DOPC 60 0
100
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300 cNaBr /mM
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400 (c)
_ k/kT
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−15 0
4.5 0
0.01
0.02
0.03
fs
0.04
0
0
0.01
0.02
0.03
0.04
fs
Figure 24. (a) Experimental results for the radius of vesicles composed of DOPG and DOPC lipids as a function of the concentration of NaBr. The vesicles were prepared by the freeze–thaw method mentioned above, and the size R was found from dynamic lightscattering analysis. (b) The mean bending modulus as found by SCF calculations for DOPG-like lipids as a function of the ionic strength of 1:1 electrolyte. (c) the Gaussian bending modulus as found by SCF calculations for DOPG-like surfactants as a function ) as of the ionic strength of 1:1 electrolyte. (d) The total curvature energy em ¼ 4p(2kc þ k a function of the ionic strength of 1:1 electrolyte. In b, c and d the two curves illustrate the effect of the solvent quality for the ions (as indicated). Reproduced from (132) with permission from the American Institute of Physics
using dynamic light scattering [23]. The experimental data show that the vesicle size R goes through a minimum for DOPG, and is essentially an increasing function for DOPC as a function of the ionic strength. For computational reasons, the corresponding SCF analysis has been done on a relatively primitive model. The chain statistics is set to the lowest order (i.e. a first-order Markov approximation instead of the RIS scheme). The lipid molecules were modelled as linear chains with a charged head group in the middle of two C18 tails, i.e. with a Gemini-like architecture: C18 M2 C2 M2 C18 . The charge on the head-group units was set to vM ¼ 0:25 mimicking DOPG, so that the overall charge was –1. The ionic strength is given by the volume fraction of ions in the bulk, which was varied from 104 to 0.04. Interaction parameters were chosen to be as simple as possible (still reflecting the essential amphiphile characteristics): w ¼ 1:6 for water–C as well as ion (M, Na, Cl)–C contacts. In the calculations the affinity of ions for the water phase was varied
F. A. M. LEERMAKERS AND J. M. KLEIJN
81
to show the effect of changes with respect to the ionic properties. Below, results are presented for the athermal interactions, i.e. wionwater ¼ 0 and for attractive interactions between ions and water wionwater ¼ 2. The interaction parameter for the M–water contacts was fixed at zero. Lekkerkerker [117] examined the curvature dependence of the free energy of the electric double layer for fixed surface charge density. From this, the electrostatic part of the mean and Gaussian bending moduli of charged bilayers was predicted as a function of the ionic strength. Both the mean bending modulus and the Gaussian bending modulus were found to depend relatively strongly on the ionic strength. With increasing ionic strength, the absolute value a of these quantities decreases as a power law (kc / ca salt and k / csalt ). The power-law exponent is predicted to be a function of the surface charge density. At high surface charge density, a ¼ 0:5, whereas at low surface charge density a value of a ¼ 1:5 was reported. One problem with these early predictions is that the surface charge density is fixed for all values of the imposed curvature J and K and the value of the ionic strength. Thus the model does not allow for curvature-dependent head-group areas and local charge compensation mechanisms such as the penetration of the ions in between the charged head groups. Indeed, for lipid membranes, one should expect that the head-group area, and thus the effective charge density of the bilayer, is a function of the ionic strength. These restrictions do not apply for the SCF analysis. In these calculations, the surface charge density adjusts itself automatically. In the SCF analysis of curved bilayers, it was found that all results could be fitted with the Helfrich equation, without the need to invoke a nonzero J0 . This means that the vesicles are typically stabilised by translational and undulational entropic contributions. This result is consistent with results by Leermakers [114] for uncharged lipid bilayers, and can be rationalised by symmetry arguments as discussed above. In Figure 24b and 24c, the mean and Gaussian bending moduli are plotted as a function of the ionic strength for charged lipid vesicles in a 1:1 electrolyte ) is presolution. For convenience, the total curvature energy em ¼ 4p(2kc þ k sented in Figure 24d. Results are presented for two values of the ion–water interactions. Let us first discuss the ideal case where the ions interact athermally with the solvent (closed spheres). In this case, the predictions of Lekkerkerker [117] are qualitatively recovered: with increasing ionic strength the Gaussian bending modulus becomes less negative, the mean bending modulus is reduced and the overall curvature energy is an increasing function of the ionic strength. Forcing a power law through the data of Figures 24b,c (solid points), however, gives a much lower ionic strength dependence than predicted by Lekkerkerker [117] (a 0:1). The reason for this weak ionic strength dependence is that with increasing ionic strength the area per molecule goes down, which tends to increase the absolute values of the bending moduli.
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One may argue that the ideal interactions of all ions with water are not physically realistic. For this reason, we have repeated the full analysis with better than athermal solvent conditions. The results of this exercise are given by the open circles in Figure 24b,c,d. Indeed, for this case, the mean bending modulus goes through a minimum as a function of the ionic strength. For the other properties presented in Figure 24, the qualitative behaviour is not affected. We may argue that, with increasing ionic strength, the solvent (water þ ions) becomes a worse solvent for the apolar tails. As a consequence, the membrane thickness increases just slightly more with increasing ionic strength than in the athermal case (not shown). At high ionic strength, the change of the bare bending modulus due to the increase in thickness of the bilayer, dominates over the change of the electrostatic part of the bending modulus. Above we have argued that the equilibrium radius of a vesicle may be related to the persistence length of the bilayer. This implies that the radius of the vesicle kc ). The strong correlation between the curves for DOPG in R x / exp ( kT Figures 24a and 24b (open circles) strongly indicates that this line of reasoning makes some sense. Using this approach, one can extract from Figure 24a how the bending modulus depends on the ionic strength. Consistent with the SCF results, a very weak ionic strength dependence is found (a 0:06). It is of significant interest to discuss the overall curvature energy (as shown in Figure 24d) as a function of the ionic strength, in somewhat more detail. Consistent with predictions based upon the electrostatic part of the bending moduli [117], it is found that the overall curvature energy goes down with decreasing ionic strength. When the overall curvature energy drops below zero, the vesicles become unstable. The SCF thus predicts that at very low ionic strength, a micellar phase, not the vesicles, becomes the more favourable aggregation state of these lipids. This is consistent with the surfactant parameter arguments mentioned in the introduction. With decreasing ionic strength, the effective size of the head group increases. Therefore, the effective shape of the surfactant becomes less like a cylinder and more like a cone. In other words, the surfactant parameter drops to such a low value that bilayers are not stable. Interestingly, at the point where the curvature energy vanishes, the size of the vesicles is very large, because kc increases with decreasing ionic strength. The transition from the solution with vesicles to a micellar solution is predicted to be a jump-like (first-order) phase transition. Going in the opposite direction, i.e. when we consider the membrane stability towards zero. Going with increasing ionic strength, we notice the approach of k towards this value, the tendency of the bilayers to form saddle-shaped connections (also called ‘stalks’) between bilayers increases. Saddle-shaped membrane structures also occur in processes like vesicle fusion, endo and exocytosis. The SCF predictions thus indicate that these events will occur with more ease at high ionic strength than at very low ionic strength.
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4.9.5
83
Membrane–Membrane Interactions
Invariably, the thermodynamic stability of lipid vesicles in the colloid–chemical sense is an important issue. In this context, it is very important to know how a given bilayer interacts with other bilayers of its kind. In MD simulations, it is virtually impossible to do predictions for isolated, noninteracting bilayers. Typically, one uses experimental data on the equilibrium membrane–membrane spacing when a simulation box is created. Systematic thermodynamic analysis of the effect of the membrane–membrane spacing is not available from simulations. For this, we again have to turn to more approximate SCF modelling. In an SCF technique the partition function is accurately available, and therefore it is straightforward to examine interacting membranes. In Figure 25 we show a prediction of Meijer et al. [85] for DMPC membranes interacting with each other as a function of an applied lateral tension (see reference [85] for the details of the values of interaction parameters used). Membranes under lateral tension have an increased area per molecule, and thus are thinner than equilibrium membranes. As a result, the number of water– hydrocarbon tail contacts is increased, and also the head-group conformation may be altered. As a result of this, one would expect that membranes under tension become more attractive to each other than when they are free of tensions. Basically, this is also what can be extracted from Figure 25. The repulsion at high membrane spacing has an electrostatic origin. When increasing tension is applied to the membrane, the repulsion goes down. The attraction in the curve is due to conformational changes in the head-group region when membranes are in very close proximity. The depth of this minimum is not 0.0008
F int γ(∞)= 0.160 0.0004 0.0 0.087
0
5
7
9
11
D
Figure 25. The free energy of interaction between two DMPC membranes at various degrees of surface tension as indicated. We note that the indicated surface tension is the tension at large membrane–membrane spacing. The salt bulk volume fraction was js ¼ 0:002. Redrawn from [85] by permission of the American Chemical Society
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
affected greatly by the lateral tension. The position of the minimum is a function of the applied tension, as must be expected from the reduced dimensions of the bilayers. The steep repulsion at very close proximity is due to a compression of the bilayers, which is, as explained above, energetically unfavourable. Stress-induced adhesion is also found in experimental systems, as discussed by Helfrich [118]. The typical interpretation of stress-induced adhesion differs significantly from the one given above. It is generally believed that the main effect of an applied tension is to reduce undulations. In the absence of undulations, the corresponding repulsion vanishes and the membranes can attract each other by, for example, van der Waals forces. It should be noted that the interaction free energy of the bilayers reported in Figure 25 is an intrinsic interaction. For a complete picture, one needs to add the van der Waals contribution, but also, more importantly, the contributions due to the undulation. To some extent, these two contributions cancel each other out, and it is expected that the intrinsic effects as discussed above are relevant for the interaction of bilayers at rather close proximity. 4.9.6
A DPPC Layer as a Substrate for a Polyelectrolyte Brush
In many biological systems the biological membrane is a type of surface on which hydrophilic molecules can be attached. Then a microenvironment is created in which the ionic composition can be tuned in a controlled way. Such a fluffy polymer layer is sometimes called a slimy layer. Here we report on the first attempt to generate a realistic slimy layer around the bilayer. This is done by grafting a polyelectrolyte chain on the end of a PC lipid molecule. When doing so, it was found that the density in which one can pack such a polyelectrolyte layer depends on the size of the hydrophobic anchor. For this reason, we used stearoyl C18 tails. The results of such a calculation are given in Figure 26. Because the amount of matter in the form of DPPC lipids in these calculations is about 25 times higher than for the polyelectrolyte modified lipids, it is necessary to rescale the latter densities such that one can understand the interesting aspects. Therefore there are different scales on left and right ordinates in the figures. In Figure 26a, the overall density profiles of both lipids are shown. The DPPC lipids are not much affected by the intrusion of the PEmodified ones. The overall membrane thickness due to the DPPC chains is about equal to the thickness of an unperturbed DPPC bilayer. The conformations of the PE chain are understood from polymer brush theory. In particular, the work on polyelectrolyte brushes has advanced in recent years [119,120]. The distribution of small counterions in the polyelectrolyte layer is such that the charge is almost exactly compensated locally. It is relevant to mention that the charge per segment on the PE chain was set to ¼ 0:25 and
F. A. M. LEERMAKERS AND J. M. KLEIJN DPPC
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(b)
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C PC-P
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N
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10
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Figure 26. The structural properties of a mixture of C16 PC (DPPC) and stearoyl C18 PC lipids of which there is a polyelectrolyte chain attached to the N of the choline group. The polyelectrolyte (PE) chain is a string of hydrophilic segments, each of which has a charge of ¼ 0:25. The amount of lipid is y ¼ 10:1, and that of the PE chain is y ¼ 0:4. In all views graph profiles are shown for only half the bilayer, i.e. starting from the centre of the bilayer (at z ¼ 0). (a) The overall profiles for C16 PC and water are solid lines, left ordinate, and C18 PCP200 , Na and Cl all dashed lines, right ordinate. (b) The charge distribution, q, left ordinate and electrostatic potential profile, c, right ordinate. (c) Detailed distribution of the PE-modified lipid C18 PCP200 . The C units and the PE chain are dashed (left ordinate), and the P and N distributions are solid lines (right ordinate). (d) The P and N distributions for the C16 PC lipids, broken lines (left ordinate) and that for the PE-modified lipid, solid lines (right ordinate)
that on the small mobile ions is unity. Therefore the density of polymer is about a factor of four higher than the difference between the density of positive ions and negative mobile ions. This local electroneutrality is illustrated in Figure 26b, where the charge density profile q(z) is plotted. The electrostatic potential profile is rather complex. In the hydrophobic region, the features are similar to those discussed above for the pure DPPC layer cf. Figure 15. The electrostatic potential profile in the PE layer is parabolic, and outside the PE layer the potential is very low, but decays according to the classical Gouy–Chapmann theory, i.e. exponential decay towards zero. In Figure 26c and 26d some more details are plotted for the PE-modified lipid. On the PE chain there are some features deviating from the parabolic profile near the lipid–water interface. We should realise that in the membrane
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there is a positive electrostatic potential, whereas the electrostatic potential in the brush is negative. The relatively high brush density near z ¼ 20 is therefore attributed to electrostatically driven ‘adsorption’. One would intuitively argue that grafting a long hydrophilic tail on the end of the head group would cause the P–N vector to orient towards the membrane normal. This is indeed the case. The distribution of the N is shifted to higher z values than the P group, as is best seen in Figure 26d. In this part of the graph it is shown that the head-group orientation of the unmodified DPPC is surprisingly unperturbed by the presence of the PE chains near the membrane.
5
TRANSPORT AND PERMEATION
To facilitate specific uptake of nutrients and secretion of products, as well as to maintain the ionic concentration gradients and membrane potentials essential for life processes, nature has developed different kinds of more or less specialized transport systems incorporated in the lipid bilayer. However, some types of molecules and ions can pass biomembranes by themselves. Examples are water, oxygen and carbon dioxide (which need to move rapidly into and out of cells), nitrous oxide, and various toxins and drugs. For the explanation of this so-called nonmediated transport, two alternative mechanisms are commonly used. In the first, it is proposed that molecules diffuse across the membrane (solubility–diffusion mechanism), in the other the transport is assumed to take place along water wires or stochastic pores in the lipid bilayer (pore mechanism). In the next subsections we will go into some detail with respect to these two mechanisms, and will discuss the molecular modelling performed in this direction using MD simulations and SCF calculations. In Section 5.3 we will pay attention to the modelling of mediated membrane transport, i.e. the transport through peptide or protein channels, or otherwise catalysed by carriers or transport proteins. 5.1
SOLUBILITY–DIFFUSION MECHANISM
Travel across the lipid bilayer must be an exciting undertaking. The molecular traveller has to go through a complex inhomogeneous soft material. In doing so, the molecule first passes a crowded head-group region that is still rather aqueous. Then, on a length scale of about a nanometre (as soon as the glycerol backbone is passed), the polarity drops to a value that is close to that found in hydrocarbon liquids. Subsequently, a region with a relatively high orientational order is encountered. Near the centre of the bilayer the order gradually diminishes somewhat. After this, the molecule has to move out again through the other half of the bilayer, encountering the similar regions in reverse order. In short, there are gradients in polarity, in electrostatic potential, in packing
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density and orientational order. Not many molecules can readily cope with all these gradients. In 1899, Overton [121], who, of course, at that time had no conception of the structural complexity of biological membranes, recognised that these provide a barrier to the free diffusion of solutes. From numerous investigations on the osmotic properties of both plant and animal cells [121,122], he found that the permeabilities of membranes to different solutes correlate with the water– octanol partition coefficients of the solutes. The resulting Overton rule, implying that solutes that dissolve well in organic solvents (such as hexane or octanol) can penetrate membranes more rapidly than solutes that are less soluble in such solvents, has been for a long time the most important guideline in physiological studies concerning membrane permeability and related issues (see Figure 27). In more recent years it was found, however, that for small nonelectrolytes (e.g. water, ammonia, formamide) the Overton rule does not hold. Their permeabilities are much higher than predicted from the trends seen for larger molecules, and are inversely proportional to their size [123,124]. The ideas of Overton are reflected in the classical solubility–diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick’s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst–Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the −1 −2
log P (P in m s−1)
−3 −4 −5 −6 −7 −8 −7
−6
−5
−4
−3
−2
−1
0
log K
Figure 27. Relationship between water–hexadecane partition coefficients and membrane permeabilities for a broad selection of solutes. (Data collected by Walter and Gutknecht [124]. Reproduced with permission from the American Chemical Society)
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concentration profile of the permeant molecule in the membrane is taken to be linear, and the concentration drop going from one side to the other is related to the concentration differences in the solutions at both sides of the membrane, according to the equation: Dcmembrane ¼ KDcw
(13)
with K the water/membrane partition coefficient of the molecule. If any interfacial barriers for the entrance and exit of the membrane are neglected, the permeability P of a solute over the membrane is given by: P¼
KD d
(14)
Here, D is the diffusion coefficient of the solute in the membrane, and d is the thickness of the membrane. In all its simplicity, the above equation shows that both equilibrium and dynamic information is needed for predicting the permeability of biological membranes for particular solutes, and therefore has added value with respect to the Overton rule. The permeation of molecules across membranes, and their distribution in the lipid bilayer are closely related phenomena, but, to describe membrane permeability, knowledge of the diffusive behaviour of the solute in the bilayer must be available as well. Of course, with increasing knowledge of the structure of lipid bilayers and biological membranes, the classical solubility–diffusion model has been refined. The most obvious improvement is to divide the bilayer into different regions (e.g. an inner hydrophobic region and two more hydrophilic outer regions with dielectric properties different to those of the surrounding electrolyte solutions, and with different mobilities for the permeating solutes [127]). Meanwhile, computational methods have reached a level of sophistication that makes them an important complement to experimental work. These methods take into account the inhomogeneities of the bilayer, and present molecular details contrary to the continuum models like the classical solubility– diffusion model. The first solutes for which permeation through (polymeric) membranes was described using MD simulations were small molecules like methane and helium [128]. Soon after this, the passage of biologically more interesting molecules like water and protons [129,130] and sodium and chloride ions [131] over lipid membranes was considered. We will come back to this later in this section.
5.1.1
Equilibrium Aspects: Partitioning
The equilibrium modelling of additives in bilayers is already a challenging task. For example, in MD simulations it is difficult to consider very low loading
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because one can consider only very small systems. Above a certain (still low) concentration of foreign molecules in the bilayer, one should expect serious perturbations of the membrane properties, and the possibility exists that this loading threshold is exceeded by just putting a single foreign molecule in the box. In SCF calculations, one can both investigate the infinite dilution case and study the effects that occur upon increasing the loading. In the former case, the equilibrium membrane properties are not affected, but in the latter one they are. However, for the prediction of partition coefficients from first principles it is necessary that all parameters are known. This situation has not been reached yet. For this reason we can only rely on predictions of trends in partition coefficients for a series of additives in which some parameter is varied systematically. As indicated above, in MD the infinite dilution case cannot be investigated. For hydrophobic small inert units like noble gases, however, it is assumed that their distribution in the bilayer follows the distribution of free volume. An extensive analysis of free-volume-related properties (free-volume fraction, hole size distribution, local order parameters) of the lipid membrane has been given by Marrink et al. [132]. On the basis of MD simulations of a DPPC bilayer, the membrane was divided into four distinct regions with very different free-volume properties, shown in Figure 28. The first region contains the adjacent water layer, the second is the head-group layer, the third comprises the first half of the hydrophobic tails, and the fourth is the centre of the membrane, i.e. the second half of the tails. The largest free-volume fraction is found in the centre of the membrane, with large cavities in which even larger molecules with a diameter of 0.6 nm, like urea, could fit without disturbing the structure. The third region, where the order in the lipid tails is high (cf. the order
Accessible free-volume fraction
0.6
1
2
3
0.5 0.4
4 0.0
3
2
1
0.1
0.3
0.2 0.2
0.28 0.34 0.4 0.6
0.1 0
−2
−1
0
1
2
z / nm
Figure 28. Accessible free-volume fraction profile across a DPPC bilayer for solutes of different sizes (diameters indicated in nm; 0.0 corresponds with the total free volume). Results of Marrink et al. [132]. Redrawn by permission of the American Institute of Physics
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parameter profile for a DMPC bilayer given in Figure 17), has clearly the lowest free-volume fraction, and is therefore considered to represent the largest resistance to the permeation of small solutes. This view is supported by MD simulations [133,134]. Xiang and Anderson [133] investigated the processes to insert noble gases into the bilayer interior. The unfavourable free energy to generate a cavity in the bilayer was analysed by scaled-particle theory, and the interaction energy between the inserted particle and the lipids in the molecule was determined. They found that it was much more difficult to create a cavity in a membrane with oriented lipids than in an isotropic alkane (dodecane), which suggests a model with an anisotropic surface tension at the noble gas–lipid interface. In these MD simulations the alkyl chains were grafted with a harmonic potential to a plane. In a more recent paper [134], they evaluated, from a series of MD simulations on a DPPC bilayer, the contributions of functional groups like CH3 , Cl, OH and COOH to the free energy for transfer of solutes from water to the ordered-chain region of the bilayer. In an experimental study, the same authors [135] systematically explored the effects of permeant size and shape, using as permeants seven monocarboxylic acids with different chain length and chain branching. Deviations of the permeability coefficients found for permeation across DPPC bilayers from the values predicted by solubility– diffusion theory, were found to correlate with the minimum cross-sectional area of the permeants, which was explained as resulting from lipid-chain ordering. The pronounced size-dependency of bilayer permeation by small solutes has also been analysed in terms of the structure and ordering of the lipid-chain region of bilayers by Mitragotri et al. [136], using scaled-particle theory. The partitioning of polar molecules is restricted to the head-group region and typically remains limited. Since the partitioning of lipophilic molecules in the interior of the bilayer is expected to be relatively high, it can be studied in MD simulations with some confidence. For example, in animal cells the cholesterol/ lipid ratio is relatively high and therefore suitable for MD studies. Indeed, there are quite a number of simulation studies on the effect of cholesterol on lipid bilayers (see the review by Ohvo-Rekila¨ et al. [137] and references therein). The rigid steroid ring system of the cholesterol molecules affects the conformations of the acyl chains, and tends to increase the population of trans conformations and decrease the free-volume fraction. Cholesterol contains one hydroxyl group, which gives this otherwise hydrophobic compound its hydrophilic character and ability to partition in hydrogen-bonding, with water molecules or with the lipid carbonyl and phosphate groups [138]. Simulations further show that the introduction of cholesterol allows for a broader distribution of P–N angles in the lipid head groups. This is caused by the relatively low position of cholesterol in the lipid bilayer, leaving holes in the bilayer surface, which are filled by choline N-groups from neighbouring lipids [139]. This leads to rearrangement of water molecules and changes the electrical properties of the
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bilayer/water interface in the sense that the membrane dipole potential increases. An increase in dipole potential with increasing cholesterol content has experimentally been found for egg phosphatidylcholine monolayers [140]. Furthermore, cholesterol decreases the fluidity of membranes by slowing down large-scale rearrangements and reorientational motions. The experimental observation that the permeability of bilayers for small permeants like water decreases with increasing cholesterol content [141], is probably connected with both the tighter packing of the lipid chains, as well as with the slower dynamics. In the literature, one can find many more interesting MD studies concerning lipid bilayers with additives. In particular, a wealth of MD simulations of such systems is in the field of anaesthetics (for a review see [142]). Many anaesthetics tend to accumulate at the membrane/water interface, implying that their potencies are not related to their ability to cross the membrane. Instead, it seems to be more likely that their functioning is via binding to membrane receptors. Generally, they have an effect opposite to that of cholesterol, i.e. they increase the membrane fluidity and permeability. In Section 4.6 we discussed the partitioning of ions in the bilayer as found from SCF calculations. Bilayers possess an internal dipole potential that favours the partitioning of anions over cations (see Figure 16). This is due to the orientation of molecular dipoles at the bilayer interface, lipid head groups and bound water, rendering the hydrocarbon core positive with respect to either aqueous phase. Apolar molecules may preferentially partition in the core of the bilayer, and then the saturation level is potentially high, especially when the additive is able to separate the two halves of the bilayer. Amphiphiles partition in the outer regions of lipid membranes. The effect of charged surfactants on the orientation of the head group of DMPC as determined by SCF calculations has been reported by Meijer et al. [85]. It should be noted, by the way, that strong partitioning of lipophiles or amphiphiles in the core of the bilayer or the bilayer/solution interface, respectively, slow down trans-bilayer transport. Experimentally determined rates of transport of drug molecules have been analysed in terms of the amphiphilicity and lipophilicity of these molecules, by Bala´z˘ [143]. In Figure 29a we show the well-known result that for alkanes the partition coefficient is a strongly increasing function of the molecular weight. (It should be noted, as mentioned earlier, that SCF calculations can only give trends in partition coefficients.) This result can be explained by realising that by partitioning in the bilayer the alkane gains interaction energy, which scales linearly with the chain length DU ¼ NwC, water . However, the transfer of an additive from the water phase to the membrane phase implies a loss of translational entropy. This loss is given by DS ¼ k log K. The free-energy balance DF ¼ DU T DS ¼ 0 gives log K / N. This law is followed as shown in Figure 29a. In Figure 29b three typical absorption isotherms are given near the saturation levels of the alkanes in the DPPC membrane. The results presented in this graph
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 8 (a)
(b)
n excN
7
6
log K 5
4
C8 3
2
C24 C4
1 0
5
10
15 N
20
25
0 −0.5
−0.4
−0.3
−0.2
∆m
−0.1
0
0.1
Figure 29. (a) The log of the partition coefficient as a function of the chain length of alkanes CH3 (CH2 )N2 CH3 (at infinite dilution) in DPPC membranes. The CH3 units are modelled as a double bead, similarly to the DPPC lipids. (b) The absorption isotherm of three alkanes in equilibrated DPPC membranes. The excess number of molecules (multiplied by the chain length) is plotted as a function of the chemical potential of the alkane minus the chemical potential of alkane at saturation (binodal) in water
are highly nontrivial. They show that the absorption of both low- and highmolecular-weight alkanes is limited, whereas, for intermediate chain lengths, the amount of alkanes is unbounded at the bulk saturation conditions. For C8 – an intermediate case – the isotherm is smooth. With an increasing concentration of octane, i.e. with increasing chemical potential, the absorbed amount increases. It is expected that long-range van der Waals interactions, not included in the model calculations, will keep the absorbed amount finite, also at coexistence. For C4 the absorbed amount remains very low. The arrow in Figure 29b points to this low value. In fact, this result is rather surprising, because one may argue that short-chain alkanes can easily penetrate between the tails and expand the bilayer, this behaviour being predicted from polymer brush theory [24]. However, butane has two CH3 groups, which are given small, unfavourable interactions with CH2 groups, see Table 1. For this reason, the incorporation of low-molecular-weight alkanes is limited for enthalpic reasons. This mechanism is expected to limit the concentration of additives in membranes in most, if not all, cases. This is because there are always chemical differences between the additives and the acyl chains. The finite absorbed amount for very large alkanes C24 is expected, because the long chains experience a reduction in conformational entropy when they partition inside the finite-sized bilayer. This is an old result already predicted by Gruen [144]. It is also known that a melt of long chains does not penetrate a brush of short chains (at high grafting densities) [24]. The absorption isotherm is nontrivial in this particular case. Before the finite absorbed amount of the C24 alkane at the binodal, indicated by the arrow in Figure 29b, is found, there is a stepwise growth of the absorbed amount. This jump-wise growth means that there is a first-order phase transition taking place in the layer. Such a stepwise growth of absorbed amount may be expected when
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the transition is from a finite absorbed amount to an infinite absorbed amount, which may be generated by changing the temperature, for example, in a firstorder phase transition. Then, the C24 case is just close to such a transition. The unexpected effect is that the step in the isotherm is found for the case that the absorbed amount at coexistence remains finite. This must indicate that there are two mechanisms active on two length scales that should be considered for the control of additive uptake. One of these is the hydrophobic compatibility. The other one is likely to be the ability of the additive to be compatible with the local order in the bilayer. These data have been collected as an example for this review. More work is needed to understand the physics involved here. The authors do not know sufficiently accurate experimental data on absorption isotherms to test the above predictions. In Figure 30, the partition coefficients are given for a series of molecules with equal overall composition, but which differ in architecture. This graph is reproduced from the work of Meijer and co-workers [86]. The key point here is that the partition coefficient is largely determined by the chain composition, but in addition to this there is a weak contribution to how the various groups in the molecule are linked to each other. The additive, which is just designed to show the potential application of the model, has a positive charged N on one side and a negatively charged P group on the other. In addition, there is a C6 group somewhere along the chain, splitting the chain into two fragments. These fragments can be positioned on the chain in an ortho, a meta or a para position, as depicted in Figure 30. All these variables have their effect on the partition coefficient. The partition coefficient is largest when the C6 group is positioned about halfway up the chain, and in particular when the two chain fragments are
1.0⫻107
o m 8.0⫻106
p K 6.0⫻106 +
o
N
1
4.0⫻106
Cn+1
m
Cm+1 −
P
4
p
2.0⫻106
0
2
4
6
8
10
12
14
n
Figure 30. Partition coefficients for a series of molecules that all have the same overall composition, but in which the architecture is changed systematically. The second chain can either be positioned on the ortho, meta or para position of the C6 -ring
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PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
in the ortho position. This can be understood, because in this way the C6 group can be positioned near the centre of the bilayer, where the order is lowest, while the two charges can still reach the membrane–water surface. The asymmetry with respect to the C6 group being near the positive or negative chain is interesting. As the positive charge in the PC head group is more mobile, it can adjust to accommodate the negative charge on the ring, irrespective of whether it sits in an ortho, meta or para position. However, the restricted phosphate cannot do the same when the positive charge is positioned at the ring. Then the ortho position is particularly unfavourable. 5.1.2
Dynamic Aspects
Study of the dynamic aspects of permeation processes across lipid bilayers from molecular modelling is still in its infancy. As yet, SCF theory has not been developed to deal with dynamics (modelling of steady-state processes is just starting), whereas the MD technique cannot cover the relevant time frame to simulate the full dynamic process of a particular molecule crossing the lipid bilayer. Despite this limitation, permeation processes may be investigated with MD, using indirect methods. For example, Marrink and Berendsen [129] deduced the rate of permeation of water across DPPC membranes via the computation of the free-energy profile and diffusion rate profile across the bilayer. For the head-group region and adjacent water layer, the diffusion rate profile was obtained by determination of the mean squared displacement of the water molecules in a short time interval (1 to 5 ps trajectories). For the interior part of the bilayer, a force-correlation method was used, in which the autocorrelation function of random forces acting on a water molecule (constrained or inserted as a ghost particle, i.e. without disturbing the configuration) is converted into a local friction coefficient x. This friction coefficient is related to the local diffusion coefficient by Einstein’s well-known equation D ¼ kT=x. From the results (depicted in Figure 31), it is concluded that the rate-limiting step is the permeation of the water molecule through the dense, ordered region of the lipid tails. In the centre of the membrane, where the hydrocarbon tails are disordered (comparable with the situation in liquid alkanes), diffusion is fastest. MD simulations of Bassolino-Klimas et al. [145,146] suggest that cavities play a role in the permeation of small solutes across membranes. In these simulations, occasionally large and rapid displacements of a benzene molecule are observed. The benzene molecule is trapped in a cavity, where it undergoes rattling motions, until a pathway opens to another, nearby cavity. This usually requires a rearrangement of the lipid chains, often involving a gauche–trans transition. Jumps from one cavity to another seem to be of particular importance for diffusion through the best-ordered region of the lipid tails, i.e. near the head-group region. Hopping between voids was already proposed by Lieb and Stein [123] on the basis of the size-dependent permeability of small solutes.
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⫻10−5 20
1
2
3
4
3
2
1
D/cm2 s–1
15
10
5
0
−2
−1
0
1
2
z / nm
Figure 31. Diffusion coefficient of water in a DPPC bilayer as a function of position. Results of MD simulations of Marrink and Berendsen [129]. Redrawn by permission of the American Chemical Society
Obviously, the availability of large enough cavities decreases strongly with the size of the solute. We have already discussed another explanation of the relationship between permeation rate and solute size, based completely on solubility, i.e. the size-selectivity of solute partitioning in the ordered lipidchain region [133–136]. 5.2
PORE MECHANISM
Experimental results have shown that the solubility–diffusion model is satisfactory for many permeants, but for some solutes it clearly fails. One of the features that will frustrate its applicability is the possibility that thermal fluctuations give rise to the formation of transient water-channels or hydrated pores. The smallest type of short-lived hydrophilic channels may be single strands of hydrogen-bonded water molecules or water threads. When such a channel is somewhat larger, the head groups may also cover the water–tail interface inside the pore. Such a mesoscopic pore can grow to semi-macroscopic size, especially when the membrane is under tension [147]. This mechanism may be considered when the failure of the bilayer under tension is studied. Earlier we discussed the example of such a (stressed) membrane as simulated by DPD [64], where it was shown that surfactants stabilise long-lived pores, giving a mechanism for the toxicity in microorganisms. Mesoscopic pores may in turn provide a mechanism for lipid flip-flop events. Related to the pore mechanism is the process of ion transport over bilayers, as suggested by molecular dynamics simulations by Wilson and Pohorille [131]. These simulations show that permeation of ions into a glycerol 1-monooleate (GMO) bilayer is accompanied by the formation of deep thinning defects in the bilayer, whereby the lipid head groups and water penetrate into the bilayer interior. As the ion passes the mid-plane of the
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membrane, the deformation switches to the other side, and, while the initial defect slowly relaxes, a defect forms in the outgoing side of the bilayer. During the process, the ion remains hydrated. As discussed earlier, lipid bilayers possess an internal dipole potential, which favours the partitioning of anions over cations. Therefore, the solubility– diffusion model predicts lower permeabilities for cations compared with anions [148]. However, in particular for small cations and relatively thin bilayers, the permeability is much higher than predicted on the basis of this model. Back in 1969, Parsegian [149] calculated the free energy of a small cation located in an aqueous pore, and showed that this is significantly lower than in the hydrophobic part of the bilayer, implying a lower diffusion barrier for crossing the bilayer along a water pore. If the pore-mechanism applies, the rate of permeation should be related to the probability at which pores of sufficient size and depth appear in the bilayer. The correlation is given by the semi-empirical model of Hamilton and Kaler [150], which predicts a much stronger dependence on the thickness d of the membrane than the solubility–diffusion model (proportional to exp(–d) instead of the 1/d dependence given in equation (14)). This has been confirmed for potassium by experiments with bilayers composed of lipids with different hydrocarbon chain lengths [148]. The sensitivity to the solute size, however, is in the model of Hamilton and Kaler much less pronounced than in the solubility-diffusion model. A number of MD studies have been published on permeation along water wires, all devoted to proton permeation through bilayers [130,151,152]. This process is a very special case. Experimental studies [153–155] have shown that protons penetrate bilayers at rates five orders of magnitude faster than other monovalent cations. In contrast to the nonmediated transport of other ions, for protons the rate of this process is so high that it is of physiological significance. Protons can diffuse by a special mechanism not applicable for other cations: hopping along hydrogen-bonded strands of water molecules (see, for example, ref. [156] and references therein). One of the first MD studies in which the hypothesis of proton transport through water pores via hopping over a single strand of water molecules was tested, was performed by Marrink et al. [130]. As the observation of spontaneously formed water channels is still far from accessible in MD studies, a strand of water molecules was gradually induced across a DPPC bilayer (64 DPPC and 736 water molecules in the box). From the forces needed to constrain the pore in the bilayer, its free energy of formation was calculated. After lifting the constraints, it was found that the lifetime of the pore before being dispersed is in the order of a few picoseconds, which would be – according to the authors – just long enough to accommodate the transport of a single proton. However, Venable and Pastor [151] argued that this lifetime is too short to demonstrate the feasibility of proton transport along water wires. Their MD simulations on
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a DPPC bilayer, as well as on a crude model system for a lipid bilayer (water/ octane/water sandwich model), suggest much longer lifetimes of water wires, on the 50–100 ps timescale. In both systems a water wire was inserted, preformed in vacuum under cylindrical constraints. Subsequently, the systems were energy-minimised and subjected to 10 ps of MD with restraints to keep the water molecules in the wire. After this, simulations were performed without restraints until the wire dissipated. In a similar way Zahn and Brickmann [152] studied the breaking of water wires in a DLPE bilayer, and they found an average lifetime of about 90 ns. These authors give a rough calculation of the minimum lifetime to allow proton transport over the hydrophobic core of a DLPE bilayer (6.7 ps), based upon the rate of proton transfer from one water molecule to another (1.2 (ps)1) and the number of water molecules in a singlefile water pore long enough to bridge this hydrophobic core (8). The obvious conclusion is that water chains are sufficiently stable to allow proton permeation. Conduction along water wires may as well be the dominant mechanism in the permeation of protons in channels; an MD study of proton transport through a gramicidin channel can be found in, for example, [156]. 5.3
MD MODELLING OF MEDIATED MEMBRANE TRANSPORT
The Holy Grail of membrane modelling is without doubt the modelling of molecular realistic biological membranes, including membrane transport systems or systems with other functionalities (like the photosynthetic reaction centre, the respiratory chain and molecular complexes for recognition). Membrane transport systems comprise pores and channels, which only facilitate downhill or passive transport (i.e. movement towards equilibrium), and carriers and pumps, which are able to perform transport uphill. Pores select their transported substrates according to size, while channels are more selective and characteristically transport ions [157,158]. Ion-selective transmembrane protein channels form dynamic structures that rearrange their structure in response to outer influences, such as changes in ion gradients or electric potential across the membrane, and ligand binding. If the different conformational states are characterised by different ion permeabilities (open and closed states), this is called channel gating. Generally, gating is a complex process, involving various intermediate states [159,160]. A well-known example is the potentialdependent opening and closing of Naþ channels in nerve membranes. Ionselective channels can also be formed by molecules other than proteins. These have very specific properties, like a flexible conformation (so that conformational changes connected to ion transport can occur readily), sufficient length, and a hydrophobic exterior (except at the ends of the channel). A well-known example is gramicidine, a pentadecapeptide, of which two molecules in a b-helix conformation form, head-to-head, a transmembrane channel [161]. Cyclic
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oligopeptides with a hydrophobic exterior, like the antibiotic valinomycin, are characteristic ion carriers. A general difficulty that is encountered in MD simulations of mediated transmembrane transport is, again, the timescale of relevant processes. For example, the mediated transport of a single ion across a membrane takes tens to hundreds of nanoseconds, and gating transitions several microseconds. Coarse-grained MD simulations do not provide insight into the delicate balance of forces and interactions involved in the transport. Furthermore, to construct a good starting conformation is especially difficult for the simulation of protein– membrane complexes, since structural information on membrane proteins is still sparse [162]. On the nanosecond timescale, simulation results are still sensitive to details of the starting structure of the proteins. Nevertheless, there are many recent papers concerning MD simulations of lipid bilayers containing biomolecules, among these various articles on proteins and oligopeptides forming transmembrane ion channels, reviewed by, amongst others, Biggin and Sansom [163], Feller [164] and Roux [165,166]. These simulations focus on different aspects, like the testing of channel structures, the effect of the lipid bilayer on the structure and vice versa, the dynamics of ion or water transport, and the identification of structurally relevant features that affect this process: e.g. multiple occupancy and gating. The above-mentioned problems have been dealt with (as fully as possible) by using somewhat simpler molecules to model the functioning of membrane proteins, special techniques, or combining MD simulations with other methods (see the review by Feller [164]). For example, helix bundles of (synthetic) channel-forming peptides have been studied as a model for the bilayer-spanning part of membrane proteins [163,167]. We end this section with an example of a recent MD study of Tieleman and co-workers [168] on the selectivity of a bacterial Kþ channel, KcsA, which is formed by a protein. Experimentally, it has been found that the channel is selective for Kþ , but under certain conditions it is possible for Naþ to pass through it. Furthermore, intracellular Naþ can block the channel. In Figure 32, the structure of the channel is depicted. The starting conformation was built using X-ray diffraction data (which have a relatively low resolution of 0.32 nm), and the missing KcsA side-chain coordinates were added by building stereochemically preferred conformers. A number of amino acid residues at the N and C terminal regions were not included in the simulations. The KscA protein was embedded in a preequilibrated POPC bilayer (116 POPCs in the extracellular leaflet, 127 in the intracellular one), hydrated with 9821 water molecules. All ionisable side-chains were in their default ionisation state, and Cl ions were added to have a net charge of zero in the system. Van der Waals and electrostatic interactions were cut off at 1.0 and 1.7 nm, respectively. The duration of the simulations (ca. 2 ns, taking about 20 days of CPU time on eight processors) was an order of
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Figure 32. (a) Overview of the simulation system for the bacterial potassium channel KcsA, as used by Shrivastava et al. [168]. The channel consists of four peptide chains (for clarity in the picture only two are given), and is embedded in a lipid bilayer. The structure can be thought of as being made up of a selectivity filter, a central cavity, and a gate at the inner side of the bilayer. (b) Water molecules and Kþ ions in the selectivity filter; SO represents the extracellular mouth. From [168]. Reproduced by permission of the Biophysical Society
magnitude shorter than the estimated time for an ion to pass through the channel. Therefore, only short time- and length-scale motions can be observed, and the study of ion transport concentrates on the passage through the selectivity-filter part of the channel. Another limitation is that the parameters for ion–filter and ion–water interactions are not accurately known, but the authors checked that the results are not dependent on small changes in these parameters. It was found that the structure of the selectivity filter is somewhat flexible and is distorted by the presence of ions; Kþ ions and water translocate between the binding sites in the filter (in either direction) on a subnanosecond timescale, while Naþ remains bound. The distortion of the filter is significantly larger when the smaller Naþ ion is present. The results clearly correlate with the experimentally observed selectivity of the channel. 6
SUMMARY, CHALLENGES AND OUTLOOK
In this review, we have presented a molecular picture of the lipid bilayer system in relation to its permeation properties, as it appears from simulations and from self-consistent field calculations. Of course, it was not possible to go into all the details. In fact, the practicalities are always much more complicated, and the fine details are only of interest to the experts. The level of detail in this review is sufficient to evaluate the state of the art, at least to some extent. We hope that the previous pages have shown that the molecularly realistic models that have been elaborated on by the various
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computational tools start to give a consistent and transparent picture of the lipid bilayer membranes. The computations can help to interpret experimental findings, and the MD snapshots will further guide our intuition. The MD simulations of molecular detail can give all the information that can be known for a small part of an ideal bilayer for fast processes (on the order of 10 ns). The dynamic aspects that take place on longer timescales can be investigated by coarse-grained methods, or with DPD or quasi-dynamic MC simulations. Due to continuously increasing computer power, the field of MD is gradually moving into larger systems and longer (real-time) scales on the one hand, and more complex, biologically relevant systems, e.g. including additives and functional molecules, on the other. Equilibrium properties are surprisingly accurately predicted by molecularlevel SCF calculations. MC simulations help us to understand why the SCF theory works so well for these densely packed layers. In effect, the high density screens the correlations for chain packing and chain conformation effects to such a large extent that the properties of a single chain in an external field are rather accurate. Cooperative fluctuations, such as undulations, are not included in the SCF approach. Also, undulations cannot easily develop in an MD box. To see undulations, one needs to perform molecularly realistic simulations on very large membrane systems, which are extremely expensive in terms of computation time. One can formulate many challenges for the near future. One of these is to include dynamics in SCF calculations in a convincing way. The advantage of SCF is that the technique is computationally extremely efficient. Therefore it is more easy to sample large parts of the parameter space, and, by doing so, one can understand the regularities of particular problems. Some first initiatives to take the approach into the world of dynamics have already given interesting results, but these approaches have not yet been applied to lipid bilayers. There is also some work under way to analyse off-equilibrium so-called steady-state distributions of molecules that are exposed to some chemical potential gradients. This development will help us understand many aspects that have to do with the semi-permeable properties of the bilayer membranes. The link from lipid properties to mechanical properties of the bilayers is now feasible within the SCF approach. The next step is to understand the phase behaviour of the lipid systems. It is likely that large-scale (3D) SCF-type calculations are needed to prove the conjectures in the field that particular values of the Helfrich parameters are needed for processes like vesicle fusion, etc. In this context, it may also be extremely interesting to see what happens with the mechanical parameters when the system is molecularly complex (i.e. when the system contains many different types of molecules). Then there will be some hope that novel and deep insights may be obtained into the very basic questions behind nature’s choice for the enormous molecular complexity in membrane systems.
F. A. M. LEERMAKERS AND J. M. KLEIJN
GLOSSARY a a A As BD c.m.c. d DPPC DLPE DMPC DPD DSPC e E f F F int J J0 k kc k K K K l L m M MC MD ni N N Nav p P P PC PG q
Area per molecule, ao equilibrium area (m2 ) Acceleration (m s2 ) Surface area of the system (m2 ) Surface area of membrane (m2 ) Brownian dynamics Critical micellisation concentration (mol m3 ) Thickness of the membrane (m) Dipalmitoylphosphatidylcholine Dilauroylphosphatidylethanol Dimyristoylphosphatidylcholine Dissipative particle dynamics Distearoylphosphatidylcholine Elementary charge 1:6 1019 (C) Field strength (V m1 ) Free-energy contribution (J) Helmholtz energy (J) Free energy of interaction (J) Total or mean curvature (m1 ) Spontaneous curvature (m1 ) Boltzmann’s constant 1:38 1023 (J K1 ) Mean bending modulus (J) Saddle splay modulus (J) Gaussian curvature (m2 ) Cluster constant used in water model Water/membrane partition coefficient Length of surfactant tail (m) Length of cylinder (m) Mass of particle (kg) Last layer in lattice used in SCF calculation Monte Carlo Molecular dynamics Number of molecules of type i Number of molecules in system Number of segments in chain Avogadro’s number 6:02 1023 (mol1 ) Pressure (N m2 ) Probability Permeability (m s1 ) Phosphatidylcholine (zwitterionic) Phosphatidylglycerol (negatively charged) Charge (C)
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r rij ri R1 , R2 R S S S SCF SOPC t t T U u vA v V V z Z g e0 erA em kD l x ji js c w
Dimensionless radial coordinate Distance between atoms i and j (m) Position of particle number i (m) Radii of curvature (m) Radius of vesicle (m) Surfactant parameter Entropy of the system (J K1 ) Order parameter Self-consistent-field 1-stearoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine Time (s) Ranking number of tail segments Absolute temperature (K) Internal energy of the system (J) Self-consistent (segment) potential (J) Valency of component A Volume of surfactant tail(s) (m3 ) Volume of the system (m3 ) Potential energy in MD simulation (J) Layer number Lattice coordination number Surface tension (J m2 ¼ N m1 ) Dielectric permittivity of vacuum 8:85 1012 (C V1 m1 ) Dielectric permittivity times segment volume of A (C V1 m2 ) Total curvature energy of vesicle (J) Debye length (m) A priori step probability in lattice Membrane persistence length (m) Volume fraction of component i Volume fraction of salt Electrostatic potential (V) Flory–Huggins interaction parameter
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the liquid-crystal phase and comparison to self-consistent field modeling, Phys. Rev. E, 67, 011909. Leermakers, F. A. M., Rabinovich, A. L. and Balabaev, N. K. (2003). Selfconsistent field modeling of hydrated unsaturated lipid bilayers in the liquidcrystal phase and comparison to molecular dynamics simulations, Phys. Rev. E, 67, 011910. Rabinovich A. L. and Ripatti, P. O. (1991). On the conformational, physical properties and functions of polyunsaturated acyl chains, Biochim. Biophys. Acta-Lipid. Metabol., 1085, 53–62. Fattal, D. R. and Ben-Shaul, A. (1994). Mean-field calculations of chain packing and conformational statistics in lipid bilayers: comparison with experiments and molecular dynamics studies, Biophys. J., 67, 983–995. Heller, H., Schaefer, M. and Schulten, K. (1993). Molecular dynamics simulation of a bilayer of 200 lipids in the gel and in the liquid-crystal phases, J. Phys. Chem., 97, 8343–8359. Rawicz, W., Olbrich, K. C., McIntosh, T., Needham, D. and Evans, E. (2000). Effect of chain length and unsaturation on elasticity of lipid bilayers, Biophys. J., 79, 328–339. Tu, K., Tobias, D. J., Blasie, J. K. and Klein, M. L. (1996). Molecular dynamics investigation of the structure of a fully hydrated gel-phase dipalmitoylphosphatidylcholine bilayer, Biophys. J., 70, 595–608. Nagle, J. F. (1980). Theory of the main lipid bilayer phase transition, Ann. Rev. Phys. Chem., 31, 157–195. Marsh, D. (1991). Analysis of the chainlength dependence of lipid phase transition temperatures: main and pretransitions of phosphatidylcholines; main and non-lamellar transitions of phosphatidylethanolamines, Biochem. Biophys. Acta– Biomembranes, 1062, 1–6. Schmid, F. and Schick, M. (1995). Liquid phases of Langmuir monolayers, J. Chem. Phys., 102, 2080–2091. Den Otter, W. K. and Briels, W. J. (2003). The bending rigidity of an amphiphilic bilayer from equilibrium and nonequilibrium molecular dynamics, J. Chem. Phys., 118, 4712–4720. Baumgartner, A. (1994). Asymmetries of a curved bilayer model membrane, J. Chem. Phys., 101, 9060–9062. Leermakers, F. A. M. and Scheutjens, J. M. H. M. (1989). Statistical thermodynamics of association colloids. II. Lipid vesicles, J. Chem. Phys., 93, 7417–7426. Love, A. E. H. (1907). Lehrbuch der Elastizita¨t. Teubner Verlag, Wiesbaden, Germany. Szleifer, I., Kramer, D., Ben-Shaul, A., Roux, D. and Gelbart, W. M. (1988). Curvature elasticity of pure and mixed surfactant films, Phys. Rev. Lett., 60, 1966–1969. Lekkerkerker, H. N. W. (1990). The electric contribution to the curvature elastic moduli of charged fluid interfaces, Physica A, 167, 384–394. Helfrich, W. (1995). Handbook of Biological Physics. Tension-Induced Mutual Adhesion and a Conjectured Superstructure of Lipid Membranes. Chapter 14. Elsevier, Amsterdam. Zhulina, E. B., Borisov, O. V., van Male, J. and Leermakers, F. A. M. (2001). Adsorption of tethered polyelectrolytes onto oppositely charged solid–liquid interfaces, Langmuir, 17, 1277–1293.
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120. Borisov, O. V., Leermakers, F. A. M., Fleer, G. J. and Zhulina, E. B. (2001). Polyelectrolytes tethered to a similarly charged surface, J. Chem. Phys., 114, 7700–7712. ¨ ber die allgemeinen osmotischen Eigenschaften der Zell, ihre 121. Overton, E. (1899). U vermutliche Ursachen und ihre Bedeutung fu¨r die Physiologie, Vierteljahrsschr. Naturforsch. Ges. Zuerich, 44, 88–114. ¨ ber die osmotischen Eigenschaften der lebenden 122. Overton, E. (1895). U Pflanzen und Tierzelles, Vierteljahrsschr. Naturforsch. Ges. Zuerich, 40, 159–201, 1895. 123. Lieb, W. R. and Stein, W. D. (1969). Biological membranes behave as non-porous polymer sheets with the respect to the diffusion of non-electrolytes, Nature, 224, 240–243. 124. Walter, A. and Gutknecht, J. (1986). Permeability of small nonelectrolytes through lipid bilayer membranes, J. Membr. Biol., 90, 207–218. 125. Hanai, T. and Dayton, D. A. (1966). The permeability to water of bimolecular lipid membranes, J. Theor. Biol., 11, 1370. 126. Finkelstein, A. and Cass, A. (1967). Effect of cholesterol on the water permeability of thin lipid membranes, Nature, 216, 717–718. 127. Aguilella, V., Belaya, M. and Levadny, V. (1996). Ion permeability of a membrane with soft polar interfaces. 1. The hydrophobic layer as the rate-determining step, Langmuir, 12, 4817–4827. 128. Sok, R. M., Berendsen, H. J. C. and van Gunsteren, W. F. (1992). Molecular dynamics simulation of the transport of small molecules across a polymer membrane, J. Chem. Phys., 96, 4699–4704. 129. Marrink, S. J. and Berendsen, H. J. C. (1994). Simulation of water transport through a lipid membrane, J. Phys. Chem., 98, 4155–4168. 130. Marrink, S. J., Ja¨hnig, F. and Berendsen, H. J. C. (1996). Proton transport across transient single-file water pores in a lipid membrane studied by molecular dynamics simulations, Biophys. J., 71, 632–647. 131. Wilson, M. A. and Pohorille, A. (1996). Mechanism of unassisted ion transport across membrane bilayers, J. Am. Chem. Soc., 118, 6580–6587. 132. Marrink, S. J., Sok, R. M. and Berendsen, H. J. C. (1996). Free volume properties of a simulated membrane, J. Chem. Phys., 104, 9090–9099. 133. Xiang, T. X. and Anderson, B. D. (1999). Molecular dissolution processes in lipid bilayers: a molecular dynamics simulation, J. Chem. Phys., 110, 1807–1818. 134. Xiang, T. X. and Anderson, B. D. (2002). A computer simulation of functional group contributions to free energy in water and a DPPC lipid bilayer, Biophys. J., 82, 2052–2066. 135. Xiang, T. X. and Anderson, B. D. (1998). Influence of chain ordering on the selectivity of dipalmitoylphosphatidylcholine bilayer membranes for permeant size and shape, Biophys. J., 75, 2658–2671. 136. Mitragotri, S., Johnson, M. E., Blankschtein, D. and Langer, R. (1999). An analysis of size selectivity of solute partitioning, diffusion, and permeation across lipid bilayers, Biophys. J., 77, 1268–1283. 137. Ohvo-Rekila¨, H., Ramstedt, B., Leppima¨ki, P. and Slotte, J. P. (2002). Cholesterol interactions with phospholipids in membranes, Progr. Lipid Res., 41, 66–97. 138. Robinson, A. J., Richards, W. G., Thomas, P. J. and Hann, M. M. (1995). Behavior of cholesterol and its effect on head group and chain conformations in lipid bilayers: a molecular dynamics study, Biophys. J., 68, 164–170.
110 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 139. Tu, K., Klein, M. L. and Tobias, D. J. (1998). Constant-pressure molecular dynamics investigation of cholesterol effects in a dipalmitoylphosphatidylcholine bilayer, Biophys. J., 75, 2147–2156. 140. McIntosh, T. J., Magid, A. D. and Simon, S. A. (1989). Cholesterol modifies the short-range repulsive interactions between phosphatidylcholine membranes, Biochemistry, 28, 17–25. 141. Fettiplace, R. and Haydon, D. A. (1980). Water permeability of lipid membranes, Physiol. Rev., 60, 510–550. 142. Pohorille, A., New, M. H., Schweighofer, K. and Wilson, M. A. (1999). Insights from computer simulations into the interactions of small molecules with lipid bilayers. In Membrane Permeability, Vol. 48: 100 Years Since Ernest Overton. eds. Deamer, D. W., Kleinzeller, A. and Fambrough, D. M., Academic Press, San Diego pp. 50–76. 143. Bala´zˇ, S. (2000). Lipophilicity in trans-bilayer transport and subcellular pharmacokinetics, Persp. Drug Discov. Des., 19, 157–177. 144. Gruen, D. W. R. and Haydon, D. A. (1981). A mean-field model of the alkanesaturated lipid bilayer above its phase transition. II. Results and comparison with experiment, Biophys. J., 33, 167–188. 145. Bassolino-Klimas, D., Alper, H. E. and Stouch, T. R. (1993). Solute diffusion in lipid bilayer membranes: an atomic level study by molecular dynamics simulation, Biochemistry, 32, 12 624–12 637. 146. Bassolino-Klimas, D., Alper, H. E. and Stouch, T. R. (1995). Mechanism of solute diffusion through lipid bilayer membranes by molecular dynamics simulation, J. Am. Chem. Soc., 117, 4118–4129. 147. Sandre, O., Moreaux, L. and Brochard-Wyart, F. (1999). Dynamics of transient pores in stretched vesicles, Proc. Natl Acad. Sci. USA, 96, 10 591–10 596. 148. Paula, S., Volkov, A. G., van Hoek, A. N., Haines, T. H. and Deamer, D. W. (1996). Permeation of protons, potassium ions, and small polar solutes through phospholipid bilayers as a function of membrane thickness, Biophys. J., 70, 339–348. 149. Parsegian, A. (1969). Energy of an ion crossing a low dielectric membrane: solutions to four relevant electrostatic problems, Nature, 221, 844–846. 150. Hamilton, R. T. and Kaler, E. W. (1990). Alkali metal ion transport through thin bilayers, J. Phys. Chem., 94, 2560–2566. 151. Venable, R. M., and Pastor, R. W. (2002). Molecular dynamics simulations of water wires in a lipid bilayer and water/octane model systems, J. Chem. Phys., 116, 2663–2664. 152. Zahn, D. and Brickmann, R. (2002). Molecular dynamics study of water pores in a phospholipid bilayer, Chem. Phys. Lett., 352, 441–446. 153. Nichols, J. W. and Deamer, D. W. (1980). Net proton hydroxyl permeability of large unilamellar liposomes measured by an acid–base titration technique, Proc. Natl Acad. Sci. USA, 77, 2038–2042. 154. Elamrami, K. and Blume, A. (1983). Effect of lipid phase transition on the kinetics of Hþ/OH diffusion across phosphatidic acid bilayers, Biochim. Biophys. Acta, 727, 22–30. 155. Perkins, W. R. and Cafiso, D. S. (1986). An electrical and structural characterization of Hþ/OH currents in phospholipid vesicles, Biochemistry, 25, 2270–2276. 156. Pome`s, R. and Roux, B. (2002). Molecular mechanism of Hþ conduction in the single-file water chain of the gramicidin channel, Biophys. J., 82, 2304–2316.
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157. Post, R. L. (1999). Active transport and pumps. In Membrane Permeability: Vol. 48 100 Years Since Ernest Overton. eds. Deamer, D. W., Kleinzeller, A. and Fambrough, D. M. Academic Press, San Diego pp. 397–417. 158. Eisenberg, B. (1998). Ionic channels in biological membranes: natural nanotubes, Acc. Chem. Res., 31, 117–123. 159. Honig, B. H., Hubbell, W. L. and Flewelling, R. F. (1986). Electrostatic interactions in membranes and proteins, Ann. Rev. Biophys. Biochem., 15, 163–193. 160. Schulze, K. D. (1994). Investigations of the channel gating influence on the dynamics of biomembranes, Z. Phys. Chem., 186, 47–63. 161. Arseniev, A. S., Barsukov, I. L., Bystrov, V. F., Lomize, A. L. and Ovchinnikov, Y. A. (1985). 1H-NMR study of gramicidin-A transmembrane ion channel: headto-head right handed single stranded helices, FEBS Lett., 186, 168–174. 162. Werten, P. J. L., Re´migy, H. W., de Groot, B. L., Fotiadis, D., Philippsen, A., Stahlberg, H., Grubmu¨ller, H. and Engel, A. (2002). Progress in the analysis of membrane protein structure and function, FEBS Lett., 529, 65–72. 163. Biggin, P. C. and Sansom, M. S. P. (1999). Interactions of a-helices with lipid bilayers: a review of simulation studies, Biophys. Chem., 76, 161–183. 164. Feller, S. E. (2000). Molecular dynamics simulations of lipid bilayers, Curr. Opin. Coll. Interf. Sci., 5, 217–223. 165. Roux, B. (2002). Theoretical and computational models of ion channels, Curr. Opin. Struct. Biol., 12, 182–189. 166. Roux, B. (2002). Computational studies of the gramicidin channel, Acc. Chem. Res., 35, 366–375. 167. Law, R. J., Tieleman, D. P. and Sansom, M. S. P. (2003). Pores formed by the nicotinic receptor M2d peptide: a molecular dynamics simulation study, Biophys. J., 84, 14–27. 168. Shrivastava, I. H., Tieleman, D. P., Biggin, P. C. and Sansom, M. S. P. (2003). Kþ versus Naþ ions in a K channel selectivity filter: a simulation study, Biophys. J., 83, 633–645.
3 Biointerfaces and Mass Transfer HERMAN P. VAN LEEUWEN Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB Wageningen, The Netherlands
JOSEP GALCERAN Departament de Quı´mica, Universitat de Lleida, Av. Rovira Roure 191, 25198 Lleida, Spain
1 2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Organism–Medium Interphase . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Physicochemical Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ionic Distributions at Biological Interphases . . . . . . . . . . . . . 2.3 Electrodynamics of Biological Interphases . . . . . . . . . . . . . . . 3 General Mass Transfer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conservation Laws and the Nernst–Planck Equation. . . . . . 3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Impact of Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Transient and Steady-State Conditions. . . . . . . . . . . . . . . . . . 3.5 Diffusion in Composite Media . . . . . . . . . . . . . . . . . . . . . . . . . 4 Convective Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Convective Diffusion from an Unbounded Laminarly Flowing Liquid to a Planar Surface . . . . . . . . . . . . . . . . . . . . 4.2 Convective Diffusion from a Channelled Laminarly Flowing Liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Convective Diffusion from a Liquid to a Moving Spherical Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Latin Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
114 115 115 117 120 122 122 124 125 125 127 129 130 135 137 141 141 141 142 142 143 143
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1
INTRODUCTION
Life has evolved in media with colourful chemical composition, under a variety of physical conditions, which include temperature, pressure and their gradients (see Chapter 1 of this volume). Evolution implies an optimisation of the functioning of organisms with respect to the physical and chemical conditions in which they live. Likewise, a change of conditions will give rise to a change in the properties of the organism, and this is known as adaptation. The chemical conditions relevant to evolution, adaptation and survival do not only comprise the composition and the chemical dynamics of the medium in which the organism is living. More generally, it is the operational availability of the various chemical species that determines their effect. In general terms, the availability of chemicals is defined by a number of basic features: . the chemical availability as derived from equilibrium distributions of species and their rates of interconversion; . the supply of these chemical species to the relevant sites at the surface of the organism; and . the actual internalisation of the chemical species by the organism, usually followed by some bioconversion process. The second feature is largely concerned with mass transfer of chemicals from the medium to the biosurface or back. It generally involves diffusion (transport due to a concentration gradient), flow (transport due to a pressure gradient), and sometimes also conduction (transport due to an electric potential gradient). Organisms may be actively involved in the efficient organisation of transport processes. This may be realised, for example, by the creation of flow via their own motility, or by tuning the physical structure of their active uptake surface to the transport conditions. This chapter deals with the basic principles of mass transfer, and their application to the specific case of transport to/from biological surfaces. It will do so after an appropriate characterisation of the physicochemical features of the biological interphase between an organism and an aqueous medium. The origin and nature of chemical gradients and ionic distributions near the biointerphase will be outlined, with special emphasis on electric double layers, diffusion layers, and their dimensional and dynamic properties. The mathematical formulation of mass transport processes is derived from basic conservation laws, which include conservation of mass and conservation of momentum. Their application to the mass transport situation, with simultaneous gradients in pressure, concentration and electric potential leads to the well-known Nernst–Planck equation, which comprises individual recognisable terms for diffusion, flow and conduction. The elaboration of this fundamental equation to practical biouptake situations calls for explicit consideration of such aspects
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as the geometrical conditions and the distinction between transient and steadystate transport situations. The combination of flow and diffusion is especially important in many practical transport situations, so, by way of illustration and implementation of the basic equations, some cases of convective diffusion are dealt with, such as diffusion towards planar active surfaces or moving spherical organisms. More pertinent elaboration on the resulting biouptake fluxes can be found in Chapters 4 and 10 of this volume.
2
THE ORGANISM–MEDIUM INTERPHASE
Quite generally, the interphase between an organism and its environment encompasses the elements outlined in Figure 1 of Chapter 1. The scheme shows that the cell membrane, with its hydrophobic lipid bilayer core, has the most prominent function in separating the external aqueous medium from the interior of the cell. The limited and selective permeabilities of the cell membrane towards components of the medium – nutrients as well as toxic species – play a governing role in the transport of material from the medium towards the surface of the organism. While the lipid bilayer has a very low water content, and therefore behaves quite hydrophobically, especially in its core (see Chapter 2 of this volume), the cell wall is rather hydrophilic, with some 90% of water. Physicochemically, the cell wall is particularly relevant because of its high ion binding capacity and the ensuing impact on the biointerphasial electric double layer. Due to the presence of such an electric double layer, the cell wall possesses Donnan-like features, leaving only a limited part of the interphasial potential decay in the diffuse double layer in the adjacent medium. For a detailed outline, the reader is referred to recent overviews of the subject matter [1,2]. Mass transport phenomena usually are effective on distance scales much larger than cell wall and double layer dimensions. Thicknesses of steady-state diffusion layers in mildly stirred systems are in the order of 105 m. Thus one may generally adopt a picture where the local interphasial properties define the boundary conditions, while the actual mass transfer processes take place on a much larger spatial scale. 2.1
PHYSICOCHEMICAL CHARACTERISTICS
The chemical and structural features of the membrane and cell wall are extensively discussed elsewhere in this volume (see Chapters 2, 6, 7 and 10). They usually contain numerous charged groups, which, as far as they are not internally compensated by counterions, give rise to the formation of an electric double layer at the interphase. The net charge of membrane surface plus cell wall is counterbalanced by a diffuse charge with opposite sign. This so-called diffuse
116 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
double layer extends into the medium over a distance depending on the electrolyte concentration. In aqueous systems of low salt concentration, for example, rivers and lakes, the diffuse countercharge around the cell wall of a microorganism may extend over distances of the order of 10 nm, whereas, in more saline systems, such as seawater and most body fluids, the compensating charge will be situated at distances less than a nanometre. The charge on bacterial cell walls mostly originates from carboxylic, phosphate and amino groups [1,3,4]. The degree of protonation of these anionic and cationic groups is determined by the pH and the ionic composition of the medium. At neutral pH, almost all bacterial cells are negatively charged, because the number of deprotonated carboxylate and phosphate groups is generally higher than that of the protonated amino groups. The compensating charge mainly consists of (positive) counterions that penetrate the porous wall, and, to a minor extent, of (negative) co-ions that tend to be expelled from it. The thickness of bacterial cell walls is in the range of several tens of nanometres [4]. Electrostatic fields associated with biomembranes arise from several sources. The net charge on the membrane generates a potential at the surface relative to the bulk aqueous phase (the surface or double layer potential). This charge resides in the outer parts of the membrane and arises from charged head groups of phospholipids, adsorbed and penetrated ions, and proteins. It is dependent on the pH and ionic composition of the medium, and should be considered in conjunction with the charges in the cell-wall layer. In the hydrophobic core of the membrane, the net charge density is essentially zero. In most biological systems the double layer potential of membranes is negative, owing to the predominance of negatively charged lipids. These typically constitute 10–20% of the effective lipid area of the membrane (about 0:02 C m2 [5–7], corresponding with about one elementary charge per 10 nm2 ). Furthermore, most of the membrane proteins have isoelectric points below pH 7. Altogether, this results in z-potentials (i.e. potentials in the electrokinetic slip plane, that is, the outer boundary of the hydrodynamic stagnant layer at the surface, usually some tenths of a nm thick) of 8 to 30 mV [5,7]. Internal membrane potentials are due to the inhomogeneous distribution of charges within the membrane. Lipid bilayers possess a substantial dipole potential arising from the structural organisation of dipolar groups and molecules. These groups are oriented in such a way that the hydrocarbon phase is positive with respect to the outer membrane regions. The magnitude of the dipolar potential is usually large, typically several hundreds of millivolts [5,7,8]. Other internal potentials arise from membrane asymmetry, including differences in adsorption at the two sides of the membrane, and from differences in penetration of ions (see Chapter 2 of this volume). Finally, separation of charge across the membrane gives rise to a transmembrane potential. The transmembrane potential is defined as the electric potential
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difference between the bulk phases at the two sides of the membrane, and results from selective transport of charged species across the membrane. In biological systems it is typically of the order of 10 to 100 mV [6,7]. As a rule, the potential at the cytoplasm side of cell membranes is negative relative to the extracellular physiological solution. The issues of ion permeation and transmembrane potential recur in other chapters of this volume (2, 6 and 10) Typical values for the dimensions of the various layers are included in Figure 1 of Chapter 1. Diffusion layer thicknesses depend on the timescale and hydrodynamic conditions; they will be dealt with in detail in Sections 3 and 4. 2.2
IONIC DISTRIBUTIONS AT BIOLOGICAL INTERPHASES
In the electrochemical literature one finds the Gouy–Chapman (GC) and Gouy–Chapman–Stern (GCS) approaches as standard models for the electric double layer [9,10]. According to the Gouy–Chapman model, the thickness of the diffuse countercharge atmosphere in the medium (diffuse double layer) is characterised by the Debye length k1 , which depends on the electrostatic properties of the medium: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ¼ ee0 RT =2F 2 z2 c (1) where F stands for the Faraday constant, c is the concentration of the (supposedly symmetrical) electrolyte, z the ionic charge number (z ¼ zþ ¼ z ) and ee0 the permittivity. Typical values for k1 are 1 nm for 0:1 mol dm3 1–1 electrolyte and 10 nm for 103 mol dm3 1–1 electrolyte. In the diffuse part of the double layer, the profile of the electrical Volta potential c [10] is given by an exponential decay function, with k as the mathematical argument: c ¼ c0 exp ( kx)
(2)
where c0 is the electrical potential at the surface. The quantity ee0 k is the capacitance Cd of the diffuse double layer. The applicability of GC and GCS models to biological systems is limited because they completely ignore the interphasial structure with its spatial distribution of charged sites, counterions and co-ions. Nevertheless, in a number of cases, GCS adequately describes the dependence of potential on distance at a lipid membrane surface [11–13]. An obvious first attempt to take into account the three-dimensional distribution of charges in the interphase is to extend the double layer model with a Donnan layer. A Donnan layer contains a number of fixed charges and is accessible for water and dissolved ions. For a biological interphase, the Donnan layer would encompass the head-group charges of
118 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
the membrane lipids, the ionisable sites in the cell-wall layer, and part of the countercharge. The countercharge is generally distributed over the Donnan layer and the diffuse layer, the distribution ratio being dependent on the electrolyte concentration in the medium. A theoretical derivation of the potential in a system with a combination of a Donnan layer and a diffuse layer has been given by Ohshima and Kondo [14–17]. The basic step is to write Poisson’s equation in terms of all relevant charge carriers. For the Donnan layer, it reads: d2 c r þ re ¼ w eD e0 dx2
(3)
where rw represents the space charge density due to the lipid head groups and the charged groups of the cell wall, re that of the electrolyte ions and eD is the relative permittivity of the Donnan layer phase (see Figure 1). For the adjacent diffuse layer in the aqueous phase, the GC approach is maintained. The potential profile can be derived if rw is taken as constant for any x and the distribution of ions that generates re follows a simplified solution of the Poisson–Boltzmann equation [14,17]. Using the continuity of potential and potential gradient across the interphase and taking eD as identical to the relative permittivity e in the surrounding medium, Ohshima and Kondo [17] found: cDm ¼ cD
RT zF cD tanh zF 2RT
(4)
where cDm is the potential at the boundary between the Donnan layer and the medium, and cD is the Donnan potential, corresponding with the limiting situation of the pure Donnan profile (no diffuse layer at all): cD ¼
RT r sinh1 w 2zFc zF
(5)
The expression for the potential variation with x within the Donnan phase is approximated by [17]: h i1=4 2 x c(x < 0) ¼ cD þ ðcDm cD Þ exp k 1 þ ðrw =2zFcÞ
(6)
where k1 is the Debye length given by equation (1). Figure 1 shows a set of calculated potential profiles for different electrolyte concentrations. For each curve, the asymptotic value on the left is cD , while the intercept with ordinates is cDm . For large c, the profile approaches the step functionality of the pure Donnan model. Note that for low potentials, the ratio
H. P. VAN LEEUWEN AND J. GALCERAN −10
−5
119
x / nm 0
0
5
c = 0.2 mol dm−3 −2
c = 0.1 mol dm−3 c = 0.05 mol dm−3
−4
y / mV
−6
−8 c = 0.02 mol dm−3 −10
−12 c = 0.01 mol dm−3 −14
Figure 1. Potential profile c(x) across the cell wall (x < 0) according to the simplified expression (6) from the Ohshima and Kondo model [15]. The profile within the solution (x > 0) is assumed to follow the usual GC decay expression c ¼ cDm exp ( kx) (equation (2)). Parameters: T ¼ 298 K, z ¼ þ1, e ¼ 78:5, thickness of the Donnan layer ¼ 10 nm, rw ¼ 1447 C dm3 . Curves correspond with c ¼ 0:2, 0:1, 0:05, 0:02 and 0:01 mol dm3
cDm =cD tends to 1/2 according to equation (4). The diffuse double layer potential is lower than in the pure GC model, because part of the ‘fixed’ charge is already compensated within the Donnan layer phase. In the Ohshima-modified Donnan approach it is assumed that the distribution of fixed charges in the Donnan layer, rw , is homogeneous and that the thickness of this layer is larger than k1 , resulting in an exponential variation of the potential in the interphase. This feature makes the Ohshima-modified Donnan approach primarily suitable for describing the double layer at the cell wall–solution interface, and less appropriate for the lipid bilayer membrane. For the construction of the full electric potential profile of a biological interphase, the preceding approach is a good starting point. It divides the interphase into different regions, each with its own thickness, permittivity, and fixed charge density, and applies a Poisson–Boltzmann type of approach. At all boundaries between distinguishable layers, we have the Gaussian conditions of continuity, taking into account the differences in permittivity for the various layers cþ ¼ c ; eþ (dc=dx)þ ¼ e (dc=dx) . For the diffuse part of the double layer in the solution, the GC model may be used again. More detailed reviews concerning the electric double layer at biological interphases are given in references [5,11,18,19].
120 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
2.3
ELECTRODYNAMICS OF BIOLOGICAL INTERPHASES
How fast does an electric double layer in a biological interphase and the adjacent solution build up or adjust itself to changing conditions? In other words, what are the characteristic time constants for formation of the interphasial double layer? The time constant, td , for relaxation of the diffuse part of the double layer is determined by bulk properties of the medium: td ¼
ee0 K
(7)
where K is the specific conductivity. In terms of electric equivalent circuitry, the relaxation process corresponds with the discharge of the capacitor with capacitance ee0 (i.e. the dielectric formed by the bulk electrolyte solution) across the solution of resistivity K 1 . In electrical jargon, td represents the RC time constant of the solution, and this indeed governs the rate of interphasial double layer formation. Equation (7) can be rewritten in terms of the ionic diffusion coefficients Di because: K ¼F
X
jzi jui ci ¼
i
F2 X 2 z D i ci RT i i
(8)
in which ui represents the ionic mobilities. For a single symmetrical electrolyte with Dcation ¼ Danion ¼ D, and using equation (1), it follows that: td ¼
1 Dk2
(9)
Thus, the time constant td is directly related to the time necessary for ions to migrate over a distance equal to the Debye length k1 . For example, 1 for a 103 mol dm3 aqueous electrolyte solution with D ¼ 109 m2 s , and 8 1 7 k ¼ 10 m , equation (9) yields a td value of 10 s. Note that k is proportional to the square root of the electrolyte concentration, (see equation (1) ), so that td is inversely proportional to c. Thus, for electrolyte concentrations of the order of 0:1 mol dm3 , td reaches values as low as 109 s. Lipid bilayer membranes, with their apolar cores, generally have extremely poor conductivities. K can be as low as 106 V1 m1 [20], and also depends on the type of ions transferring the charge across the membrane. The time constant tm (as given by equation (7) with em e0 and Km representing the membrane permittivity and conductivity, respectively) to relax from electric polarisation by conduction through the bilayer is correspondingly large. For a membrane with a relative permittivity of the order of 10 and a thickness of about 108 m, the geometrical capacitance is 102 F m2 . Thus, the corresponding time
H. P. VAN LEEUWEN AND J. GALCERAN
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constant tm is 104 s. This means that on the relevant timescales of biological processes, the double layer at the solution side adjusts instantaneously (td typically 109 to 107 s), whereas the apolar core of the lipid bilayer may behave as a dielectric that does not allow appreciable passage of charge on timescales below 104 s. In real biological systems, this is generally overruled by specific ionselective permeation processes (e.g. see Chapters 5, 6 and 10). When the bilayer contains ion-transport channels or other transport mediators, the individual channels have conductivities typically of the order 1011 V1 m1 [21]. Realising that such a channel occupies a part of the membrane surface area of the order of 10 nm2 , this corresponds with a local K of about 106 V1 m1 . Thus, locally, the relaxation time constant comes down to around 108 s, which logically is again in the range of that for electrolyte solutions. Lateral transfer of ionic species through the biointerphasial double layer has only recently received attention. Yet it is a subject of significant relevance, because it may play a crucial role in the interactions of organisms with their surroundings, for example in bacterial adhesion processes, biofilm formation (and removal), etc. As an example, we may refer to a study of bacterial cells of the genus Corynebacterium. The cells are more or less spherical, with diameters of 1.1 and 0:8 mm for the longer and shorter axes, and can be prepared as fairly homodisperse suspensions. Electrophoretic mobilities of Corynebacterium cells are displayed as a function of pH in Figure 2. Via the z-potential and other double layer features, the mobility represents the particle charge (see Chapter 4 in [10]). Thus, the mobility increases with increasing charge in the wall, and with decreasing electrolyte concentration. For the Corynebacterium cells, the mobility versus charge function levels off and becomes quite insensitive to the electrolyte concentrations at higher pH, i.e. at increasing (negative) charge. This behaviour points to surface conduction effects, i.e. tangential displacement of counterions inside the double layer. A comprehensive analysis of the electrodynamic features is accessible by combining mobility data with the dielectric spectra of the dispersed cells [22,23]. As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer d that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance: the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities
122 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 3
108 u/m2 V-1 s-1
2
1
0
−1 −2 −3 0
2
4
6 pH
8
10
12
Figure 2. Electrophoretic mobility of Corynebacterium versus pH. The general trend of the markers for each KNO3 concentration (D ¼ 0:001; O ¼ 0:01 and ¼ 0:1mol dm3 ) is indicated with an auxiliary line. Redrawn from reference [22]
of charged species (different from their values just outside the double layer by some type of Boltzmann factor). These aspects are of primary importance in defining the boundary conditions (see Section 3.2).
3
GENERAL MASS TRANSFER EQUATIONS
3.1 CONSERVATION LAWS AND THE NERNST–PLANCK EQUATION Basic elements of transport equations are the laws expressing conservation of mass and conservation of momentum. The former is self-evident; the latter derives from Newton’s second law (stating that the sum of forces acting on a system equals the rate of production of momentum in that system). For details on the basic premises and features, refer to the specialised literature [24–29]. Let us consider the very general situation where a liquid system is subject to: . a pressure gradient, so that there is a flow of the liquid with velocity v, . a concentration gradient for a certain component i, so that there is diffusion of i, and
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. an electric potential gradient, so that the charged components of the system are subjected to conduction (often denoted as ‘migration’ in the electrochemical literature). For this set of stimuli, the general linear flux equation for component i can be formulated as Ji ¼Di grad ci
|fflfflfflfflffl{zfflfflfflfflffl} diffusion term
zi ci ui grad c þ ci v jzi j
|fflfflfflffl{zfflfflfflffl}
|fflfflfflffl{zfflfflfflffl}
conduction term
flow term
(10)
where subscript i denotes that the magnitude (J for flux, c for concentration, u for mobility and z for charge) corresponds with the species i, and v is the velocity of the solution at the point where the flux is measured. For transport in one dimension, equation (10) reduces to: Ji ¼ Di
dci zi dc ci ui þ ci v dx jzi j dx
(11)
Equation (10) is known as the Nernst–Planck equation. This equation can be given in all kinds of formulations. Another common one is: Ji ¼
Di ~i þ ci v ci grad m RT
(12)
where the diffusion and conduction terms are contained in the gradient of the ~i . electrochemical potential m Mass conservation for component i implies that at a certain point in space, the time dependence of ci is related to the divergence of the flux of i. It is easily understood that a finite positive divergence in the flux will lead to depletion, i.e. to a lowering of the concentration. This can be expressed as: qci ¼ div Ji qt
(13)
where div denotes the divergence, i.e. div J ¼ qJx =qx þ qJy =qy þ qJz =qz in Cartesian coordinates, which reduces to dJ/dx in the one-dimensional case. We shall also encounter situations where a chemical reaction influences the concentrations. For a volume reaction of the type j Ð i, the reaction rate for the production of i is given by kf cj (if we assume it to be an elementary reaction), where kf is the forward reaction rate constant, and the rate of consumption of i by a term kb ci , where kb is the rate constant of the backward
124 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
reaction. Then the conservation equation (13) has to be extended with the chemical source terms: qci ¼ div Ji þ kf cj kb ci qt
(14)
The relative importance of reaction with respect to diffusion can be described in terms of the nondimensional (second) Damko¨hler number [30–36] (also called Thiele modulus), in terms of the reaction layer thickness [37,38] or in terms of lability criteria [39,40]. 3.2
BOUNDARY CONDITIONS
For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a ‘boundary value problem’. The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism: . fixed concentration at a boundary. If an isolated organism is immersed in a huge volume, then the bulk concentration of the species taken up is constant and its fixed concentration (either at finite or infinite distance) can be used as a boundary condition. In the particular case of fixed zero concentration at the organism surface, we would have the maximum (‘limiting’) flux allowed by the diffusion supply. . fixed flux at a boundary. If a biological internalisation flux can be considered constant along a certain period, then a fixed-flux condition holds at the interfacial boundary. If the species cannot cross a surface (e.g. a ligand of the species taken up not being able to cross a cell membrane), then a zeroflux condition for that species applies. The nonflux condition would also be relevant for the case of an organism in a finite medium, as the internalised species in the medium is not replenished [41]. Obviously, many combinations arise: different boundary conditions for different species on the same surface, different boundary conditions on different adjacent surfaces (mixed boundary conditions) for the same species, prescribed combinations of flux and concentrations at a certain surface, etc.
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3.3
125
THE IMPACT OF GEOMETRY
For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v ¼ 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator r2 , also frequently denoted as D. Thus, for a case of diffusion and flow, equation (10) becomes: qci ¼ Di r2 ci div(ci v) qt
(15)
which, performing a local balance of matter, is a continuity equation. Like the divergence or the gradient, the Laplacian operator has different forms for different geometrical situations [42]; see Table 1.
3.4
TRANSIENT AND STEADY-STATE CONDITIONS
The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of ci as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function ci (x, y, z) is essentially independent of time t, i.e. in the steadystate. Then equation (15) reduces to: Di r2 ci divðci vÞ ¼ 0
(16)
It is useful to point out here that we frequently encounter partial steadystates. An important example is the case where the diffusion process is much faster than a surface process, and thus a quasi-steady-state is reached for the diffusion concentration profile at each changing concentration of the surface. This distinction between different timescales of the processes can lead to a significant simplification of complex problems, see end of Section 4.3 or Chapter 4 in this volume. The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which – for practical purposes – can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di ¼ 109 m2 s1 towards a spherical organism of radius r0 ¼ 1 mm is practically attained at t r20 =Di ¼ 1 ms.
Variable
x r r r, z
r, y, j
x, y, z
System of coordinates
Planar (1D) Spherical (1D) Cylindrical (1D) Axisymmetrical (2D)
Spherical polar (3D)
Cartesian (3D)
q2 S=qx2 þ q2 S=qy2 þ q2 S=qz2
(qS=qx)ex þ (qS=qy)ey þ (qS=qz)ez
(qS=qx)ex (qS=qr)er (qS=qr)er (qS=qr)er þ (qS=qz)ez qS 1 qS 1 qS er þ ey þ ej r sin y qj r qy qr
r2 S q2 S=qx2 q2 S=qr2 þ (2=r)(qS=qr) q2 S=qr2 þ (1=r)(qS=qr) q2 S=qr2 þ (1=r)(qS=qr) þ q2 S=qz2
q2 S 1 qS 1 q qS 1 q2 S þ þ sin y þ qr2 r qr r2 sin y qy qy r2 sin2 y qj2
grad S
Table 1. Gradient and Laplacian operators for different geometries. S stands for a scalar and ei stands for a unit vector associated to coordinate i
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127
Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. As steady-state can only be maintained by the concurrence of some pumping energy, it has been associated with many life processes of no change. Conversely, equilibrium would correspond with a no-change situation of ‘death’.
3.5
DIFFUSION IN COMPOSITE MEDIA
For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as s. The first boundary condition follows from the continuity of the material flux: Ji, s ¼ Ji, sþ
(17)
where sþ and s indicate opposite sides of the separating surface. The second boundary condition arises from the continuity of chemical potential [44], which implies – under ideally dilute conditions – a fixed ratio, the so-called (Nernst) distribution or partition coefficient, KN , between the concentrations at the adjacent positions of both media: c i , s ¼ KN c i , s þ
(18)
The transient solution can be obtained, for planar or spherical geometry, using Laplace transforms [26,45], but, for simplicity, we restrict ourselves to the steady-state solutions. We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are Di,1 and Di,2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance di (diffusion layer thickness) from the surface where ci ¼ 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst–Planck equation (10), it can be seen that the flux at any surface is:
128 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Medium 2
Medium 1
Di,2
Di,1
s ci,s−
* ci,s+ ci (di) = c i,2
ci (0) = 0
xs di
Figure 3. Outline of the concentration profile of species i due to steady-state diffusion in a composite region with two media
Ji ¼
c i, 2 xs = Di,1 KN þ ðdi xs Þ=Di,2
(19)
where xs denotes the position of the separating plane s. Expression (19) can be physically interpreted as due to the ‘resistance’ to flow generated by each medium due to its length and diffusion coefficient. Although Di,1 is expected to be lower than Di,2 , the fact that xs is di can result in the practical neglect of the retardation due to the thin wall layer. Expressions analogous to (19) have been used in interpreting analytical techniques with composite media [46]. In the case of semi-infinite diffusion towards a spherical organism of radius r0 , with the surface of separation (concentric with the organism’s surface) at radius rs , the flux at the membrane surface is [26]: Di,1 KN ci,2 Ji ¼ ð1=rs Þ ð1=r0 Þ Di,1 KN =rs Di,2 r20
(20)
If rs and r0 are very similar, the previous expression practically reverts to the diffusion in only one medium. If the transported species is volatile, it is convenient to relate the concentration to the pressure via the solubility (or Henry law coefficient), because the steady-state solution does not depend on the particular values of the diffusion
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coefficients and of the solubility parameters of each medium [26,44], but on their product, which is known as the permeability of this medium. Other geometries can be substantially more difficult to solve [45], mostly when the one-dimensional approach is not appropriate. The steady-state analytical solution through a multilayer medium towards a disc surface is available [47], but for most problems (especially transient ones) only numerical simulation is feasible.
4
CONVECTIVE DIFFUSION
Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration ci . Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion–advection mass transport [48,49]. The relative contribution of advection to total transport is characterised by the nondimensional Pe´clet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh – 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. Transport of i due to the flow of the surrounding liquid travelling at a velocity v amounts to ci v so that the total flux Ji resulting from diffusion and flow comes to: Ji ¼ Di grad ci þ ci v
(21)
which is nothing other than the Nernst–Planck equation (10) for the case without the conduction term. We note that for not too high concentrations of i, the diffusion coefficient Di can be considered as a (temperature-dependent) constant. In many biological works, which have tended to focus on the organism rather than on the surrounding medium, the flux coming from the medium and crossing the surface is taken as positive (see Chapter 1). Following this sign convention, and considering the usual absence of flow of the fluid across the surface, the flux reads: Ji (diffusion) ¼ þDi grad ci
(22)
which is just the first term from the Nernst–Planck equation (10). Taking into account that the liquid acting as a medium for the microorganism is incompressible (div v ¼ 0), equation (15) becomes:
130 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Di
q2 c i q2 c i q2 c i þ þ qx2 qy2 qz2
!
qci qci qci vx ¼0 þ vy þ vz qx qy qz
(23)
which is the general equation of steady-state convective diffusion. Mathematically, it is a second-order partial differential equation, which can be solved (either analytically or more frequently numerically) for a given a set of boundary conditions that are characteristic for the problem under consideration. In most cases, the velocity profile is known or can be solved prior to (and independently of) the determination of the diffusion profile. Unfortunately, the rigorous solution of most realistic hydrodynamics problems is rather involved (for instance, due to the presence of turbulence) and sometimes it is reasonable to use empirical expressions for the velocity profile. One particular simplification arises when the scale of the turbulence [48] is much larger than the size of the microorganism, which effectively results in a locally linear shear (velocity gradient) [49,51] or, for that matter, a laminar sublayer close to the biological surface [25,52]. Below we shall treat three relevant, although simplified, examples. 4.1 CONVECTIVE DIFFUSION FROM AN UNBOUNDED LAMINARLY FLOWING LIQUID TO A PLANAR SURFACE We consider a plane with length l and width w, which is exposed to a flowing solution with a dissolved component i at a bulk concentration ci . The direction of the bulk stream of flow is parallel to the plane in the direction of the y-axis; the x-axis is perpendicular to the plane, as shown in Figure 4. The bulk velocity of the flow far from the plane is denoted as v. The z-axis is considered immaterial, due to a sufficiently large value of w compared with the dimensions of the velocity and concentration perturbations. l is assumed to be much larger than the diffusion layer thickness (di ). The first step in solving convective diffusion problems is the derivation of the velocity profile. In this case, the flow arriving with velocity v is modified by y x
w
l
Direction of flow
z
Figure 4. Schematic representation of the convective–diffusion problem for an active plane parallel to the direction of flow dealt with in Section 4.1. The liquid flow extends up to x ¼ 1, where its free velocity is v in the direction of increasing y. The leading edge of the plane is the segment x ¼ 0, y ¼ 0, 0 z w
H. P. VAN LEEUWEN AND J. GALCERAN
131
the leading edge of the plane (y ¼ 0) and develops a velocity profile for which the rigorous solution, attributed to Blasius in 1908 (p. 20 in [25], p. 396 in [53], Chapter 4 in [54]), can be approximated by: vy 0:332v3=2 x=
pffiffiffiffiffi ~y
(24)
vx 0:083ðv=yÞ3=2 x2 ~1=2
(25)
where ~ is the kinematic viscosity of the solution (the quotient between the viscosity and the density, which for aqueous solutions at room temperature is about 106 m2 s1 ). Notice that vx and vy depend on both x and y, but vx =vy (usually small except close to the leading edge, where the Blasius solution is not accurate [50,55]) is independent of v. Exact (continuous line) and approximate (dotted line, given by equation (24) ) profiles of the velocity are depicted in Figure 5. An important concept in fluid mechanics is the hydrodynamic boundary layer (also known as Prandtl layer) or region where the effective disturbance 16 di
ci /ci*
vapprox
0.8
vexact
0.6
d0 8
ci /ci*
108v/m s−1
12
1.0
0.4 4 0.2
0 0
0.1
0.2
0.3 x/mm
0.4
0.5
0.0 0.6
Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y ¼ 105 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters: Di ¼ 109 m2 s1 , v ¼ 103 m s1 , and ~ ¼ 106 m2 s1 . The hydrodynamic boundary layer thickness (d0 ¼ 5 104 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ci =ci , the linear profile of the diffusion layer approach (continuous line) and its thickness (di ¼ 3 105 m, equation (34) ) have been added. Notice that the linearisation of the exact velocity profile requires that di d0
132 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
of the main flow takes place, with d0 labelling its thickness as seen in Figure 5. It is good to emphasise that the diffusion layer and the hydrodynamic boundary layer are principally different. The former denotes the layer of solution where the concentration of diffusing species is depleted, whereas the latter refers to the layer of solution where the liquid velocity is reduced with respect to the bulk velocity. The d0 (assumed to be much smaller than l or w) can be quantified as the distance from the plane where vy 0:99v, using the exact (numerical) solution. In this problem, d0 can be taken as [53,55,56]: pffiffiffiffiffiffiffiffiffiffi d0 5 ~ny=v
(26)
With the expressions for the velocities, i.e. equations (24) and (25), at hand, one can turn to the diffusion problem and seek to solve equation (23). Apart from dropping the terms with derivatives in the z-direction, it can be shown that the diffusive term q2 ci =qy2 is also relatively small in comparison with q2 ci =qx2 (see p. 87 in [25]). Thus, equation (23) boils down to:
q2 c i qci qci ¼0 þ vy Di 2 v x qx qx qy
(27)
The leading boundary conditions correspond with semi-infinite diffusion with an instantaneous sink of the diffusing species at the plane x ¼ 0: ci (x, y) ¼ 0 ci (x, y) ¼
x¼0
ci
x!1
8y
(28)
8y
(29)
with ci (0, 0) well behaved. Substitution of equations (24) and (25) into equation (27) and further manipulations (see p. 88 in [25] for details), leads to the solution: 3=2 3 1=2 ci =ci ¼ 1 0:373G 1=3, 0:0277D1 x ~n i ðv=yÞ
(30)
where G stands for the incomplete gamma function: G(a, z) ¼
Z
1
ta1 expt dt
(31)
z
It can be seen that the profile of the concentration (see Figure 6 and dashed line in Figure 5) is largely linear over a wide range of x-values away from the plane. On the other hand, the change in ci with y, at a given x, is of a parabolic nature (see Figure 7). With the complete concentration profile ci (x, y) at hand, it is straightforward to compute the magnitude of the flux Ji towards the plane:
H. P. VAN LEEUWEN AND J. GALCERAN
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1.0
y = 30 µm
0.8
ci /mol m−3
y = 20 µm 0.6
y = 10 µm
0.4
0.2
0.0 0
4
8
12
16
20
x/µm
Figure 6. Concentration profiles of species i (equation (30)) as a function of the distance from the surface (x) in the case of laminar flow parallel to an active plane (Section 4.1). 1 Parameters: Di ¼ 109 m2 s1 , v ¼ 0:012 m s1 , ci ¼ 1 mol m3 , and ~ ¼ 106 m2 s
1.0
ci /mol m−3
0.8
0.6
x = 30 µm
0.4
x = 20 µm 0.2 x = 10 µm 0.0 0
250
500 y/µm
750
1000
Figure 7. Concentration profiles of species i (equation (30) ) as a function of the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (Section 4.1). Other parameters as in Figure 6
134 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 2=3
Ji ¼ Di ðqci =qxÞx¼0 ¼ 0:339ci Di
pffiffiffiffiffiffiffi 1=6 v=y~
(32)
which shows that the diffusive flux of solute i towards a plane in a laminarly flowing solution is proportional to its diffusion coefficient Di to the power 2/3 [48]. Thus, changes in the exponent of Di (sometimes on empirical basis) are linked to the hydrodynamic regime and the contribution of advection. The magnitude of the flux given by equation (32) decreases with increasing distance from the upstream edge of the plane and with decreasing velocity of the liquid according to square root dependencies. Formally, the flux can also be written: Ji ¼ Di ci =di
(33)
where di is the thickness of the diffusion layer of species i. Comparison with equation (32), yields di for the present case of laminar flow along a plane: 1=3
di ¼ 2:95Di
pffiffiffiffiffiffiffi 1=6 y=v~
(34)
Physically, di represents the distance from the plane where bulk concentration would be recovered if a concentration profile were simply linear with a slope given by the tangent of the true concentration profile at x ¼ 0 (see continuous line in Figure 5). Notice that the boundary (equation (26) ) and diffusion (equation (34) ) layers, in this example, extend as the square root of the distance from the leading edge, i.e. the steady-state thicknesses of the hydrodynamic boundary layer and the diffusion layer increase with increasing distance (y) from the upstream edge, according to parabolic relationships. The flux of i at x ¼ 0, being inversely proportional to di , decreases with increasing y. Figure 8 summarises this typical behaviour. It should be noted that the ratio di =d0 is independent of the coordinate y and the liquid velocity, and that the solution for convective diffusion given by (30) and following equations requires that di =d0 is well below unity, to allow for the approximate expressions of the velocities, equations (24) and (25), to apply. One easily verifies that for small ions and molecules in aqueous systems with D O(109 ) m2 s1 and ~ O(106 ) m2 s1 , di =d0 is small enough for the approximate treatment developed here to be acceptable. In passing, it is good to emphasise that the above analysis illustrates the limitations of the widely used Nernst diffusion layer concept. This concept assumes that there is a certain thin layer of static liquid adjacent to the solid plane under consideration at x ¼ 0. Inside this layer, diffusion is supposed to be the sole mechanism of transport, and, outside the layer, the concentration of the diffusing component is constant, as a result of the convection in the liquid. We have seen that, in contradiction with this oversimplified picture, molecular diffusion and liquid motion are not spatially separated, and that the thickness
H. P. VAN LEEUWEN AND J. GALCERAN
135 6.0
d0
0.2
4.0
2.0
0.1
104 J/mol m−2 s−1
Boundary or diffusion layer thickness/mm
0.3
Ji di 0.0 0.0
0.5
1.0 y/mm
1.5
0.0 2.0
Figure 8. Variation of the hydrodynamic boundary layer thickness (d0 , equation (26), continuous line), the diffusion layer thickness (di , equation (34), dotted line) and the ensuing local flux (J i , equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a 1 sink for species i). Parameters: Di ¼ 109 m2 s , v ¼ 103 m s1 , ci ¼ 1 mol m3 , and 6 2 1 ~ ¼ 10 m s . Notice that di d0 (as required for the derivation of the flux equation (32) ), and that the flux decreases when di increases
of the diffusion layer depends on the properties of the diffusing molecules as well as on hydrodynamic characteristics. Needless to say, a formal flux expression like equation (33) becomes meaningful only after combination with the appropriate formulation of the values of di . 4.2 CONVECTIVE DIFFUSION FROM A CHANNELLED LAMINARLY FLOWING LIQUID In some biological cases, liquid is forced to flow inside a channel (e.g. a tube inside an animal, as in gills, for example). The channel can be modelled as a long pipe or cylinder of radius R. The active surface corresponds with r ¼ R (see Figure 9). As usual, we must first solve for the velocity profile. The flow can be taken as laminar for low Reynolds numbers, vR=~ < 2500, where v is the maximum velocity of the flow reached at the centre of the channel (r ¼ 0) [52]. For simplicity, we will consider the particular case where the Poiseuille velocity profile has been fully developed when the solution carrying the active species
136 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Parabolic velocity profile
r
v R
x
y
Figure 9. Model of convective diffusion inside a tube channel (radius R) towards the walls of the tube considered as perfect active surfaces. A Poiseuille profile for the velocity (equation (35) ) is also schematically shown with arrows on the right-hand side. The maximum velocity v is reached at the centre of the tube (r ¼ 0)
reaches the part of the pipe with the active surface. Thus, the main difference with respect to the previous case is that the velocity profile does not change along the direction y (under steady-state conditions, the incompressible fluid is forced to flow at the same velocity all along y), because no end effects distort the velocity profile. The parabolic Poiseuille profile (including the no-slip condition on the surface at r ¼ R) is: vy ¼ v 1 r2 =R2
(35)
Via a simple integration of this parabolic profile, one can find a relationship between v and the effective flow rate vf (volume of liquid crossing any section of the channel per unit of time): vf ¼ pR2 v=2
(36)
A further simplification (known as the Le´veˆque approximation [57]), linearises the profile in the vicinity of the surface (where a local coordinate x, perpendicular to the surface, is taken): vy 2vx=R
(37)
Considering also that in the y-direction the rate of mass transport due to convection is much higher than that due to diffusion [58], equation (23) reduces to: Di
q2 c i qx2
!
2vx qci R qy
¼0
(38)
Through a change of variable (see p. 114 in [25]), the concentration profile, for the limiting situation where the active surface is a perfect sink, can be expressed (similarly to equation (30)) as:
H. P. VAN LEEUWEN AND J. GALCERAN
ci =ci ¼ 1 0:373G 1=3, 2vx3 =(9Di Ry)
137
(39)
Then, we obtain for the flux: 1=3 Ji ¼ 0:678 ci D2i v=ðRyÞ
(40)
As physically expected, the flux diminishes as we move downstream (increasing y) due to the previous depletion at lower y. No direct relation to the viscosity is present, as the flow is forced to go with a constant v (along different y), but the viscosity of the liquid has an indirect effect via Di . The diffusion layer thickness can be computed as: di ¼ 1:474 ðDi Ry=vÞ1=3
(41)
Obviously, this approximate treatment fails when di > R, as then the Le´veˆque linearisation is clearly unapplicable. For an approximation to account for lateral effects of nonactive walls in box-like channels, see ref. [46].
4.3 CONVECTIVE DIFFUSION FROM A LIQUID TO A MOVING SPHERICAL BODY Another case of great practical importance in the bioenvironmental context is the convective–diffusive transport of solutes towards the surface of suspended bodies (or particles) in a liquid medium. We will consider the situation where a spherical body of radius r0 moves with respect to the liquid, which, of course, is mathematically equivalent to the situation of a stagnant particle in a flowing liquid [48]. Figure 10 shows a planar cross-section of the situation and the polar coordinate system relevant to the analysis. As the starting velocity profile (for vr and vy ), we take the simplified regime corresponding with ‘creeping’ flow around the sphere [24], which limits the applicability of the results to parameters corresponding with low Reynolds numbers, i.e. cases where turbulences do not come into play: < 0:1 vr0 =~
(42)
The field of velocities corresponding with such a regime (which includes ‘no-slip’ of the fluid at the sphere surface) is indicated by the arrows in Figure 10, where the length of each arrow is proportional to the local velocity of the fluid. The perturbation of the bulk velocity extends to quite large distances; for instance, it is necessary to move 75 times the radius r0 from the centre of the sphere normal to the free stream direction of flow in order to obtain 99% of v.
138 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
r
r0 q
Figure 10. Velocity field of creeping flow around a sphere [24]. Each arrow represents the velocity at the origin of the depicted vector. The length of the arrow corresponding with the free velocity v would equal the radius of the sphere
In the steady-state situation qci =qt ¼ 0, and we can use equation (23) as the appropriate starting equation. For the case of diffusion towards a sphere in polar coordinates (see Table 1), this comes to: Di
q2 ci 2 qci þ qr2 r qr
! vr
qci qci vy ¼0 qr qy
(43)
where r and y are the two relevant variables. Due to axisymmetry, the elevational angle j is immaterial (not depicted in Figure 10), so that we are left with two essential spatial variables, i.e. r and y. For clarity, we choose again the boundary conditions corresponding with limiting transport (maximum flux): ci (r, y) ¼ 0
r ¼ r0
8y
ci (r, y) ¼ ci
r!1
8y
(44)
H. P. VAN LEEUWEN AND J. GALCERAN
139
The second expression in equation (44) implies that the bulk concentration in the medium is not affected by the consumption of i towards the particle, i.e. the overall depletion is insignificant. In case of the presence of an ensemble of bodies (or particles), this means that the distance between different bodies (or particles) is sufficiently large compared with the steady-state diffusion layer (i.e. the dispersion should be sufficiently diluted). Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer di is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is:
1=3 di 1:29 Di 1 ¼ y sin 2y r0 sin y vr0 2
(45)
The value of the product v r0 becomes critical as, on one hand (see condition (42)), it must be low, but on the other it must be high enough to satisfy di r0 [48]. For instance, Figure 11 (where the end of the diffusion layer is depicted with a discontinuous line) just fulfils both conditions, except in the rear down1 stream flow if ~ < 106 m2 s . Inspection of equations (33) and (45) confirms that the flux Ji is maximal for y ¼ 0. It decreases with increasing y, and tends to zero for y ¼ p, where the depletion layer thickness is no longer small compared to r0 . This result is not practically important, since the fluxes for y close to p are relatively small and hardly count in the total transport rate. It may also be noted that, as a consequence of the approximations in the derivation, the limit v ! 0 of equation (45) does not approach the purely diffusional value given by r0 . The total transport rate (in mol s1 ) to the whole body (or particle) is found 1 by integration of the flux (in mol m2 s ) over the total surface area: ð
Ji dA ¼ 2pr20
ðp Ji sin ydy
(46)
0
which, after substitution of equations (33) and (45), yields the average flux: Ji ¼ 0:65(D2i v=r20 )1=3 ci
(47)
It should be highlighted that equation (47) holds for solid particles. In the case of liquid particles, e.g. with emulsions, the convective diffusion process is very different due to interfacial momentum transfer which gives rise to a different velocity profile. Consequently, convective diffusion to/from a liquid particle is more effective than that for a solid particle. Starting again from equation (43),
140 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 0.4
x/mm
0.3
Flow 0.2
0.1
di r0 0.0 −0.4
−0.3
−0.2
−0.1 y/mm
0
0.1
0.2
Figure 11. Position of the diffusion layer limit (dashed line; computed with equation (45) ) for a species diffusing towards a spherical surface (continuous line) when creeping flow coming from the right-hand side is considered. Notice that the ensuing flux will be maximum at positions facing the flow. Parameters: r0 ¼ 104 m, Di ¼ 109 m2 s1 , and v ¼ 103 m s1
the following equivalent of equation (47) for a moving particle can be obtained (see page 404 in [25]): 1=2 1=2 1=2 Ji ¼ 0:65Di ci r0 v0
(48)
Z v v0 ¼ m 2 Zm þ Zp
(49)
with:
where Zm and Zp are the viscosities of the medium and the particle, respectively. Literature data seem to suggest that the behaviour of vesicles with a lipid bilayer skin is more close to that of rigid particles [2,12,23]. In many interfacial conversion processes, certainly those at biological interphases, the diffusion situation is complicated by the fact that the concentration at the organism surface is not constant with time (c0i (t) not constant). However, in most cases of steady-state convective diffusion, the changes in the surface
H. P. VAN LEEUWEN AND J. GALCERAN
141
concentration are slow compared to the characteristic time for reaching a steady-state diffusional situation (typically of the order of r20 =D [59] in the spherical case). This means that the thickness of the diffusion layer and the shape of the concentration profile have reached their steady-state properties. Then, it is allowed to apply the flux equations above together with the appropriate expression for di , withci replaced by the relevant concentration change over the diffusion layer, i.e. ci c0i (t) . Applying this, by way of example, to the spherical body case, we may write the flux as [9]: Ji ¼ ms ci c0i (t) (50) where ms is shorthand notation for the steady-state mass transfer coefficient, which for the case of diffusion towards a rigid spherical body in creeping flow (see equation (47)) is: 1=3 ms ¼ 0:65 D2i v=r20
(51)
Thus, with a variable degree of empiricity, mass transfer coefficients for a number of biologically relevant cases have been described: phytoplankton uptake [60], periphyton uptake [61], coral-reef supply of nutrients [62–64], fixation of carbon at leaf surfaces [65], etc. Due to the difficulties in having rigorous analytical expressions for the flux at any given geometry and flow conditions, in many instances it is convenient to include all the characteristics of the supply in the mass transfer coefficient ms and use expression (50). It must be pointed out that expression (50), stating a linearity between the flux and the difference between bulk and surface concentrations, cannot be – in general – valid for nonlinear processes, such as coupled complexation of the species i with any other species (see Chapters 4 and 10 for a more detailed discussion).
GLOSSARY GC GCS RC
Gouy–Chapman Gouy–Chapman–Stern Resistance–Capacitance
LIST OF SYMBOLS SUPERSCRIPTS * 0
bulk conditions (spherical) surface of the organism
142 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
SUBSCRIPTS D Dm e i m m r, y w x,y,z s þ – 1,2 0
Donnan phase Donnan–medium interphase electrolyte ions charge species i medium membrane polar components of a vector (fixed) wall phase charge Cartesian components of a vector separation surface between different media limit approaching from the right limit approaching from the left labels of media spherical surface of the organism
LATIN SYMBOLS Symbol
Name
Units
Equation
A c D div F grad J K KN k k f , kd l ms
Area Concentration Diffusion coefficient Divergence Faraday constant Gradient Flux Conductivity Nernst distribution coefficient Boltzmann constant Kinetic constants Length of plane Steady-state mass transfer coefficient Gas constant Radius of the cylindrical channel Radius, radial coordinate Temperature Time Ionic mobility Fluid velocity Flow rate Width of adsorbing plane
m2 mol m3 1 m2 s Operator C mol1 Operator 1 mol m2 s 1 1 V m None J K1 s1 m m s1
(46) (1) (8) (13) (1) (10); see Table 1 (10), (22) (7) (18) (1) (14) Figure 4 (50)
R R r T t u v vf w
1
J K1 mol m m K s 1 m2 V1 s 1 ms 1 m3 s m
(4) (35), (36) (20) (1) (13) (8), (10) (10) (36) Figure 4
H. P. VAN LEEUWEN AND J. GALCERAN
x y z z
Coordinate normal to the surface Coordinate of main flow Ionic charge Cartesian coordinate
143
m m None m
(3) Figure 4 (1), (6), (10) Figure 4
None m m None F m1 V N s m2 rad m1 1 m2 s J mol1 C m3 s V
(31) (33) (26) (1), (3) (1)
GREEK SYMBOLS G di d0 e e0 z Zm , Zp y k ~ ~ m r td c
Incomplete gamma function Diffusion layer thickness Boundary layer thickness Relative permittivity Absolute permittivity of vacuum Electrokinetic potential Medium and particle viscosities Polar coordinate Reciprocal of Debye length Kinematic viscosity Electrochemical potential Density of charge Double layer relaxation time Electrical potential
(49) (43) (1) (24) (12) (3) (7), (9) (3), (4)
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4 Dynamics of Biouptake Processes: the Role of Transport, Adsorption and Internalisation JOSEP GALCERAN Departament de Quı´mica, Universitat de Lleida, Av. Rovira Roure 191, 25198 Lleida, Spain
HERMAN P. VAN LEEUWEN Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB Wageningen The Netherlands
1 2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncomplicated Mass Transfer of the Species Being Taken up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A Transient Model for Two Parallel Internalisation Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 General Model with Two Types of Langmuirian Adsorption Sites Coupled to First-Order Internalisation Processes . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Typical Results for the Case of Two Internalisation Routes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 A Particular Case: One Internalisation Route and One Adsorption-Only Route. . . . . . . . . . . 2.1.3.1 Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.2 Steady-State. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Steady-State Limit for Two Parallel Internalisation Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Linear Adsorption Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Analytical Solution for Steady-State . . . . . . . . . . . . . . 2.3.3 Analytical Solution for the Transient . . . . . . . . . . . . . 2.3.4 The Impact of the Radius . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 The Approach to Steady-State After a Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
149 150 150
150 153 155 155 155 156 160 160 160 161 162 163
148 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
2.3.6
How Long Does it Take to Reach Steady-State? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Discriminating the Individual Parameters of the Product kKH Through the Cumulative Plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Diffusional Steady-State (dSS) Approach . . . . . . . . . . . 2.4.1 The Essence of the dSS Approach . . . . . . . . . . . . . . . 2.4.2 Linear Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Langmuirian Adsorption . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.1 One Site Adsorbing and Internalising . . . . . 2.4.3.2 Two Sites: Site 1 ¼ Adsorption þ Internalisation, Site 2 ¼ Adsorption Only . . . . . . . . . . . . . . . . . . . . . . 3 Biouptake When Mass Transfer is Coupled With Chemical Reaction (Complex Media) . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Totally Inert Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fully Labile Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Partially Labile Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Limiting Supply Flux in the Steady-State Uptake Considering Homogeneous Kinetics . . . . . . . 3.4.2 The Degree of Lability and Lability Criteria . . . . . . . 3.4.3 Combining Supply and Internalisation. . . . . . . . . . . . 3.5 Relationship with the FIAM . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Key Factors and Challenges For Future Research in Biouptake Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Refinements Based on Mass Transport Factors . . . . . . . . . . 4.1.1 Finite Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Nonstagnant Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Complex Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Refinements Based on Adsorption Processes. . . . . . . . . . . . . 4.2.1 Adsorption Isotherm and Kinetics . . . . . . . . . . . . . . . 4.2.2 Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Refinements Based on Internalisation Factors . . . . . . . . . . . 4.3.1 Internalisation Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Efflux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Latin Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165
168 170 170 172 175 175
176 178 178 180 180 182 182 183 184 186 190 190 190 192 192 193 193 193 194 194 194 194 195 195 196 196 197 198
J. GALCERAN AND H. P. VAN LEEUWEN
1
149
INTRODUCTION
Biological systems are open to the exchange of matter with their environment. If species from the medium (either nutrients or pollutants) travel towards the membrane and a net increase in their concentration arises in the cell, then an uptake process is occurring. Apart from the obvious importance for the organism itself, there is an impact on the medium (e.g. regulating the fate of pollutants in the environment). Modelling biouptake processes helps in the understanding of the key factors involved and their interconnection [1]. In this chapter, uptake is considered in a general sense, without distinction between nutrition or toxicity, in which several elementary processes come together, and among which we highlight diffusion, adsorption and internalisation [2–4]. We show how the combination of the equations corresponding with a few elementary physical laws leads to a complex behaviour which can be physically relevant. Some reviews on the subject, from different perspectives, are available in the literature [2,5–7]. Here we discuss in some detail a few recent models for biouptake, highlighting their physicochemical basis, especially the diffusive transport involved. Numerical examples are usually taken from the biouptake literature on bacteria and algae, because the uptake occurs across a single membrane and the application of the modelling is simpler; thus no exploration is done of more ‘macroscale’ models based such as those based on compartments [4,8]. The goal is to help the reader in the building up of a coherent picture of the various processes and their mathematical modelling, taking into account their relative importance and the specific equations describing them (e.g. Henry or Langmuir adsorption isotherm). Special attention is paid to analytical solutions – even if their applicability requires simplifying assumptions – as they can provide primary understanding for the main trends in global behaviour. We shall consider spherical microorganisms [9] (see ref. [10] for spheroidal shapes), but the treatment applies equally well to cases of larger organisms where the uptake takes place in individual, well-separated microregions. The planar case can be found by letting r0 tend to infinity. We recall from Chapter 3 (this volume) that planar semi-infinite diffusion without any convection cannot sustain steady-state, which is attainable in practice only for sufficiently small radii. We will first deal with the uptake of a species that diffuses towards the biosurface without coupled chemical reactions in the medium (Section 2) and then move to the case of a coupled reaction (Section 3). Dynamic aspects are then highlighted, because it might not be realistic to assume that the species being taken up is in equilibrium in the medium surrounding the microorganism [11]. The idea of kinetics possibly being relevant was pioneered by Jackson and Morgan [12] and Whitfield and Turner [9], and then experimentally proved by Hudson and Morel [13] with regard to Fe
150 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
uptake. The consideration of further physical phenomena is discussed in Section 4. A list of the symbols used is given in Appendix A. 2 UNCOMPLICATED MASS TRANSFER OF THE SPECIES BEING TAKEN UP 2.1 A TRANSIENT MODEL FOR TWO PARALLEL INTERNALISATION ROUTES 2.1.1 General Model with Two Types of Langmuirian Adsorption Sites Coupled to First-Order Internalisation Processes Let us consider the uptake of a given species, either a nutrient or a pollutant heavy metal or an organic (macro)molecule, etc., which will be referred to as M. M is present in the bulk of the medium at a concentration, cM , and we assume that the only relevant mode of transport from the medium to the organism’s surface is diffusion. The internalisation sites are taken to be located on the spherical surface of the microorganism or in a semi-spherical surface of a specialised region of the organism with radius r0 (see Figure 1). Thus, diffusion prescribes: " # qcM (r, t) q2 cM (r, t) 2 qcM (r, t) ¼ DM þ qt qr2 r qr
(1)
In the most general case considered here, M is adsorbed on two kinds of sites, labelled ‘1’ and ‘2’. Each adsorption process is assumed to be fast enough (when
Medium
Organism
r0
Mads,1
Minter
M
M
Mads,2
Figure 1. Outline of the uptake model showing the spherical diffusion of species M through the medium towards two different sites where adsorption is followed by internalisation. The radius of the organism is taken as r0
J. GALCERAN AND H. P. VAN LEEUWEN
151
compared with diffusion or the internalisation process) as to be described by a Langmuir (equilibrium) isotherm relating the coverage fraction of each kind of site (y1 and y2 , respectively) and the local concentration of M at the organism surface: cM (r0 , t): y1 (t) ¼
G1 (t) cM ðr0 , tÞ G2 (t) cM ðr0 , tÞ ; y2 (t) ¼ ¼ ¼ Gmax,1 KM,1 þ cM ðr0 , tÞ Gmax,2 KM,2 þ cM ðr0 , tÞ
(2)
where Gj stands for the surface concentration associated with sites of type j, and KM,1 and KM,2 are a type of adsorption constant. The adsorption process can thus be considered as the coordination of M with a finite number of transport sites on the surface of the microorganism [3,11,14,15]. Once adsorbed, we assume that M is internalised following a first-order kinetic process in each of the sites, with internalisation rate constants k1 and k2 respectively [9,16–18]. For each kind of adsorption site, we have an uptake flux given by: Ju, j (t) ¼ kj Gmax , j
cM (r0 , t) KM, j þ cM (r0 , t)
j ¼ 1, 2
(3)
which can be seen as a Michaelis–Menten-like expression, with KM, j called the half-saturation constant or bioaffinity parameter [13,14,19]. Within the hypothesis of adsorption equilibrium, the usual Michelis–Menten coefficient KM, j ¼ kdesorption, j þ kj =kadsorption, j kdesorption, j =kadsorption, j (usually kdesorption, j > kj , due to the fast adsorption process), thus becomes the inverse of the usual Langmuir coefficient [4,13,15] (which is indicative of the stability of the surface complex). For a constant ‘cell quota’ (as amount of M per unit biomass) [20], Ju can be related to the rate of growth of biomass, and equation (3) has been described in the uptake context as a Monod expression [16,21]. Thus, the fundamental boundary condition arising from the flux balance at r ¼ r0 can be written as: dG1 (t) dG2 (t) qcM (r, t) þ ¼ DM k1 G1 (t) k2 G2 (t) dt dt qr r¼r0
(4)
simply expressing that the change in adsorbed amounts follows from the difference between the diffusive supply flux and the internalisation rates. The remaining boundary condition for the diffusion equation (1) is semiinfinite diffusion:
152 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
cM (r, t) ¼ cM
r!1
t0
(5)
and the initial distribution of M is assumed to be homogeneous: cM (r, t) ¼ cM
r r0
t¼0
(6)
As expected from the presence of nonlinear terms in the boundary conditions, no analytical solution for the problem defined by equations (1)–(6) is available to our knowledge, so a numerical strategy is applied here. As seen in ref. [22], the problem given by the differential equation (1) with boundary conditions (4)–(6) can be recast in the form of an integral equation for cM (r0 , t): ðt DM 1 cM ðr0 , tÞ cM pffiffiffiffiffiffiffiffiffiffi dt cM t pffiffiffiffiffiffiffiffiffiffiffi r0 pDM 0 r0 t t ðt DM cM ðr0 , tÞ dt k1 G1 (t) þ k2 G2 (t) þ r0 0
G1 (t) þ G2 (t) ¼
(7)
and the numerical solution of this equation can be obtained by discretising the unknown cM ðr0 , tÞ. Once the value of cM (r0 , t) (which for simplicity will be denoted as c0M from now on) is known, any physical quantity of the system can be computed. In particular, the incoming diffusive (or mass transport) flux: Jm (t) ¼ DM
qcM (r, t) qr r¼r0
(8)
has been selected in this work as a relevant response function. For the general case considered here, equation (4) can be reorganised as: Jm (t) ¼
dG1 (t) dG2 (t) þ þ k1 G1 (t) þ k2 G2 (t) dt dt
(9)
where fluxes towards the organism are counted positively (see Chapters 1 and 3 of this volume). The bioaccumulated amount, Fu , is the time integral of the uptake flux: Fu (t)
ðt Ju (t)dt
(10)
0
representing the amount of matter that has been internalised per unit of surface area. The product 4pr20 Fu is the number of moles taken up per individual cell from the beginning of the process, i.e. the cell quota [23]. On the other hand, the total supply from the medium can be defined as:
J. GALCERAN AND H. P. VAN LEEUWEN
Fm (t)
153
ðt Jm (t)dt
(11)
0
The relationship between the accumulated fluxes can be easily found by integrating the boundary condition (4) as: Fm (t) ¼ Fu (t) þ G1 (t) þ G2 (t)
(12)
which is nothing else than mass conservation of M. 2.1.2
Typical Results for the Case of Two Internalisation Routes
Figure 2 plots the evolution of the incoming fluxes Jm and Ju with time for some typical values from the literature [24,25] and references therein. As expected, the diffusive flux Jm decreases with time and tends towards a steady-state value when converging with Ju . It is noticeable that, in the initial transient, the internalisation flux Ju is much closer to its eventual steady-state 1.00
1.6
0.90
1.4
0.70 1.0
0.60 0.50
0.8 Ju
q
J /10−10 mol m−2 s−1
0.80
q1
Jm
1.2
0.40
0.6
0.30 0.4 0.20
q2
0.2
0.10 0.00
0.0 0
1
2
3
4
5 t/s
6
7
8
9
10
Figure 2. Evolution of diffusive (Jm , continuous line) and internalisation (Ju , circles) fluxes with time for a system with two internalisation sites (Section 2.1.2). Fraction of coverages of each site type, y1 and y2 , are indicated with dashed lines. Parameters: DM ¼ 109 m2 s1 , cM ¼ 104 mol m3 , r0 ¼ 104 m, KM,1 ¼ 105 mol m3 , KM,2 ¼ 103 mol m3 , Gmax,1 ¼ 108 mol m2 , Gmax,2 ¼ 1011 mol m2 , k1 ¼ 102 s1 and k2 ¼ 1 s1
154 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
value than Jm , whose values are huge for short t. Plots on suitable scales establish that Ju goes through a maximum (of around 9:1735 1011 mol m2 s1 ) for t 148 s, which is slightly larger than the steady-state flux of around 9:1734 1011 mol m2 s1 . Obviously, such a small maximum is irrelevant for practical purposes, but other maxima – for other sets of parameters – could appear to be more prominent. For more detailed discussion on these maxima (for the simpler case of linear adsorption), see Section 2.3.5. It is also seen in Figure 2 that there is no saturation of either of the two kinds of sites (the largest coverage is about 90% for sites of type 1), because cSS M is too small. The shape of the various Ju versus t plots reflects the impact of different maximum coverages for each of the site types. Figure 3 shows a system with the same parameters as in Figure 2, but with Gmax,2 a factor of 1000 higher. At around 0.2 s, sites 1 (with high affinity and same Gmax ) depart from the linear regime of the isotherm; a smaller fraction of the supply accumulates on G1 ; c0M increases more steeply and so does the flux Ju,2 and consequently Ju . Sites of type 2 are always in the linear regime; the curvature of Ju,2 (e.g. at 1.2 s) being due to the progressive achievement of the steady-state concentration
10
2.0 Ju
9
Ju,2
8
1.8 Jm
1.4
7
1.2
6
1.0
5
0.8
4
0.6
3
0.4
2
0.2 0.0 0.0
1
Ju,1
0.5
1.0
1.5
2.0
Ju /10−10 mol m−2 s−1
Jm /10−8 mol m−2 s−1
1.6
2.5
0 3.0
t /s
Figure 3. Plot of fluxes (solving integral equation (7) numerically) for parameters in Figure 2, but Gmax,2 ¼ 108 mol m2 . The individual uptake fluxes for sites of type 1 (Ju,1 , þ ) and sites of type 2 (Ju,2 , ) are added to provide the total flux Ju (s). The inflexions in Jm arise from the saturation of sites of type 2 and from the approach to the steady-state
J. GALCERAN AND H. P. VAN LEEUWEN
155
c0M cSS M KM,2 . When the inflexions in (the increasing) Ju occur, there are also the inflexions of (the decreasing) Jm , because Ju increases with c0M , but Jm decreases with c0M . For a detailed discussion on the effects of the parameters on the fluxes, see ref. [22]. 2.1.3 A Particular Case: One Internalisation Route and One Adsorption-Only Route We deal now with two parallel Langmuir adsorption steps, only one of which is followed by internalisation. Parameters of the site with both adsorption and internalisation will be denoted with subscript 1 (physiologically active site), while the other, with no internalisation, will be denoted with subscript 2 (physiologically inactive site) [2]. 2.1.3.1 Transient The numerical solution of this case can be obtained by setting k2 ¼ 0 in the expressions for the general model of two routes (or ‘mouths’) described in Section 2.1.1. Now, material adsorbed on site 2 simply acts as a reservoir, buffering c0M . 2.1.3.2 Steady-State If one defines the limiting biouptake flux for each site (which would appear for cSS M much larger than each KM, j ) as [26]: Ju, j (t) ¼ kj Gmax, j
j ¼ 1, 2
(13)
the internalisation flux Ju (for steady-state) can be written as: Ju ¼ k1 Gmax,1
cSS cSS M M ¼ Ju,1 SS K M, 1 þ c M KM,1 þ cSS M
(14)
which depends only on the parameters of site 1 because, for steady-state fluxes, the adsorption-only route is irrelevant. So, results in this section also apply when there are only internalising sites present. SS can be written: The supply flux Jm c cSS DM cM cSS cSS SS M M Jm ¼ J (15) ¼ DM M ¼ 1 M 1 m r0 r0 cM cM where: Jm DM cM =r0
(16)
is the limiting supply flux, namely the product of bulk concentration and the mass transfer coefficient DM =r0 (see Chapter 3).
156 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
The result of equating the steady-state fluxes, equations (14) and (15), is sometimes known as the Best equation [9,13,16,27–31] which we normalise to its most elementary parameters [26] (other normalisations have also been suggested [21]): ( 1=2 ) (1 þ a þ b) 4b 1 1 J~ ¼ 2b (1 þ a þ b)2
(17)
where J~ is the normalised flux: SS J~ Jm =Ju
(18)
(with Ju ¼ Ju,1 , if there is a second noninternalising site) a is the normalised bioaffinity parameter: a ¼ KM =cM
(19)
(with KM ¼ KM,1 , if there is a second noninternalising site) and b the limiting flux ratio: b ¼ Ju =Jm
(20)
Equation (17) shows quite elegantly that the biouptake flux is governed by the two fundamental parameters a and b. A set values of J~ is easily of limiting derived by using that (1 x)1=2 approaches 1 12 x for x 1. If we consider, by way of example, an organism with a relatively low affinity for M (a 1) in a medium with a relatively high transport flux (b 1), then equation (17) reduces to: ( 1=2 ) a 4b 1 1 2 J~ ¼ 2b a
(21)
which, since for this case 4b=a2 is much smaller than unity, approaches J~ ¼ 1=a or
SS Jm ¼ Ju cM =KM
(22)
2.2 THE STEADY-STATE LIMIT FOR TWO PARALLEL INTERNALISATION ROUTES Let us now consider the simple case of steady-state, towards which the transient solution tends when t ! 1 but which, for sufficiently small r0 , is practically
J. GALCERAN AND H. P. VAN LEEUWEN
157
reached within a very short time. In this case, the balance of flux, equation (4), can be written as [18]: DM
cM cSS cSS cSS M M M ¼ k1 Gmax,1 þ k2 Gmax,2 SS r0 K M, 1 þ c M KM,2 þ cSS M
(23)
This is a cubic equation of the unknown quantity cSS M which is the eventual concentration at the surface of the organism. Then, the resulting steady-state flux can be computed using either side of equation (23). Some insight on the effect of the parameters on the mathematical solution can be gained through a graphical procedure. The basic idea is to plot the uptake and diffusive fluxes as functions of a variable concentration on the surface c0M (i.e. cM (r0 )) and seek their intersection. It is therefore convenient dSS to introduce the ‘diffusive steady-state’ (dSS, see Section 2.4 below) flux, Jm , or flux corresponding to the diffusion profile conforming to the steady-state situation for a given surface concentration c0M : dSS Jm DM
cM c0M r0
(24)
14 12 10 8 6 4
Ju −0.0011
2
−0.0008
−0.0005 cM(r0)/mol m−3
−0.0002
Ju
0 0.0001 −2
J /10−10 mol m−2 s−1
JmdSS
−4 −6
Figure 4. Graphical determination of the three real solutions of the cubic equation (23) corresponding with the steady-state case. The diffusive flux JmdSS (a straight line with negative slope) and the uptake flux Ju are plotted as a function of the variable cM (r0 ) (including physically meaningless negative values). Ju is the addition of two hyperbolae dSS (for Ju,1 and Ju,2 ) with its asymptotes plotted as dashed lines. The intersections of Jm and Ju yield the three solutions, from which one is positive (the physically relevant) and the remaining two real and negative. Parameters as in Figure 2, except r0 ¼ 103 m, Gmax,1 ¼ 5 109 mol m2 , Gmax,2 ¼ 1012 mol m2 and k2 ¼ 10 s1
158 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
As depicted in Figure 4, all the solutions for cSS M can be found graphically at dSS (which is a straight line of slope DM =r0 ) with the each intersection of Jm curve for Ju ¼ Ju,1 þ Ju,2 (which is the sum of two hyperbolae with their corresponding vertical asymptotes at c0M ¼ KM,1 and at c0M ¼ KM,2 ). Due to the positive character of all the physical constants, one concludes that there is only one positive (physically meaningful) solution of equation (23). The above graphical method, where the fluxes are plotted in terms of c0M (in the region c0M > 0), can help in rationalising the impact of several parameters on the steady-state flux (as well as in the graphical solution of the cubic equation with a spreadsheet). Each diagonal line in Figure 5a represents the diffusive flux dSS versus c0M for different cM values (given by the intercept with the abscissas). Jm As seen in Figure 5a, at low cM , a small change in cM implies a large shift SS (upwards) of the ordinate of the intersection (a large change in Jm ) within the linear region of the Ju curve. On the other hand, at large cM , the impact of changing cM on the ordinate of the intersection point is low, because in this region the uptake flux approaches its maximum value: Ju ¼ Ju,1 þ Ju,2 ¼ k1 Gmax,1 þ k2 Gmax,2
(25)
SS Figure 5b shows the resulting steady-state flux Jm (obtained as the positive SS solution for cM from equation (23)) for a range of cM values. At low cM values (usually associated with low cSS M values), there is a linear dependence between SS and cM , as expected from the linearisation of the Langmuir isotherms (see Jm equation (31), below). At large cM values, the usual Michaelis–Menten saturating effect of cM is also seen. From inspection of equation (23), it follows that the effects of r0 and DM dSS are opposed. Three Jm versus cM (r0 ) plots for different ratios DM =r0 are shown as straight lines with different slopes in Figure 5c, converging at a common point at c0M ¼ cM . Thus, the steady-state flux increases with DM =r0 up to an asymptotic value given by the maximum (at this cM ) uptake flux c c SS ! Ju,1 KM,1Mþc þ Ju,2 KM,2Mþc < Ju . The overall effect of changing organism Jm M M SS size (i.e. r0 ) on the Jm can be seen in Figure 5d, which exhibits a sigmoidal c c shape (falling practically from Ju,1 KM,1Mþc þ Ju,2 KM,2Mþc to 0) when the ratio M M spans over a large interval, but could be experimentally seen as practically linear even for certain regular variation ranges of r0 . The same analysis can be performed on the effect of the similar parameters SS increases when any of them k1 , Gmax ,1 , k2 and Gmax,2 . The steady-state flux Jm increases. As seen in Figure 5f for the particular case of Gmax,1 variation, there SS are two asymptotic limits for Jm given by the fixed Ju,2 (at low Gmax,1 ) and by the fixed limiting diffusion value (see equation (16) and Figure 5e).
J. GALCERAN AND H. P. VAN LEEUWEN 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
159
1.2
(a)
(b)
J /10−10 mol m−2 s−1
J /10−10 mol m−2 s−1
1.0
Ju
cM* =
10−4 mol m−3
cM* = 5⫻10−5 mol m−3 0
0.00005
cM* = 2⫻10−4 mol m−3
0.00015
* Ju,1
0.6 0.4
* Ju,2
JmSS
* cM * KM,1 + cM * cM
* KM,2 + cM
0.2
cM* = 1.5⫻10−4 mol m−3
0.0001
0.8
0.0
0.0002
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(c)
5.0
DM −6 −1 r0 = 2.5⫻10 m s
1.0
DM −6 −1 r0 = 10 m s
Ju
J /10−10 mol m−2 s−1
J /10−10 mol m−2 s−1
3.0 2.0
0.0 0
0.00002
0.00004
0.00006
0.00008
0.0001
J*u
1.00 0.80
J*u,1
0.60 0.40
J*u,2
* cM
* KM,1 + cM
JmSS
* cM
* KM,2 + cM
0.20 0.00 −6
−5
−4
−3
cM(r0)/mol m−3
−2
−1
0
log (r0/ m)
(e)
2.0
J /10−10 mol m−2 s−1
1.5
Gmax,1=2×10−8 mol m−2 Gmax,1=5×10−9 mol m−2
0.5
0.0 0
0.00002
0.00004
0.00006
cM(r0)/mol m−3
0.00008
0.0001
J /10−10 mol m−2 s−1
Gmax,1=2×10−8 mol m−2
1.0
1
(d)
1.20 DM −6 −1 r0 = 5⫻10 m s
4.0
0.9
* /10−3 mol m−3 cM
cM(r0)/mol m−3
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
(f)
J*m
JmSS
Ju,2
0.5
1.0
J *u,2 1.5
2.0
2.5
3.0
Gmax,1 /10 −8 mol m−2
SS Figure 5. Effect of the model parameters on the steady-state flux Jm illustrated by the graphical procedure (left column), and the corresponding outcome (right column) with rest of parameters as in Figure 4, except Gmax,1 ¼ 5 108 mol m2 for (a)–(d) and DM ¼ 109 m2 s1 for (d). SS dSS (a) Impact of cM on Jm . Straight lines with negative slope indicate the diffusive flux Jm for different cM . Circles stand for the uptake flux Ju , sum of two terms like equation (3). Each intersection between a straight line and the quasi-hyperbolic succession of circles SS corresponding with that cM value. yields the Jm SS on cM . The maximum values expected for Ju, j at each bulk (b) Dependence of Jm concentration are also depicted. SS dSS . Straight lines indicate Jm at different (c) Impact of the relationship DM =r0 on Jm DM =r0 values.
160 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
2.3 2.3.1
THE LINEAR ADSORPTION LIMIT Model Formulation
For sufficiently low coverages (such as those expected for low bulk concentrations, e.g. trace pollutants), the adsorption isotherms, equation (2), revert to linear ones: G1 (t) cM (r0 , t) ; Gmax,1 K M, 1
G2 (t) cM (r0 , t) Gmax,2 K M, 2
(26)
then equation (4) can be written as: KH
dcM (r0 ) qcM (r) kKH cM (r0 ) ¼ DM dt qr r¼r0
(27)
with an effective linear (Henry) adsorption coefficient: KH ¼
Gmax,1 Gmax,2 þ KM,1 KM,2
(28)
and an effective internalisation rate constant: k1 Gmax,1 k2 Gmax,2 Gmax,1 Gmax,2 k¼ þ þ KM,1 K M, 2 K M, 1 K M, 2 k1 Gmax,1 KM,2 þ k2 Gmax,2 KM,1 ¼ Gmax,1 KM,2 þ Gmax,2 KM,1
2.3.2
(29)
Analytical Solution for Steady-State
The steady-state solution associated with equation (27) is: (d) Dependence of JmSS on the logarithm of the radius. Notice that for r0 ! 0, c c SS ! Ju,1 KM,1Mþc þ Ju,2 KM, 2Mþc < Ju as diffusion is no longer limiting. For r0 ! 1, Jm M M SS Jm vanishes. SS . The straight line indicates JmdSS . Each succession of (e) Impact of Gmax,1 change on Jm circles indicates the uptake flux Ju at a given Gmax,1 (5 109 mol m2 , 108 mol m2 and 2 108 mol m2 ). SS on Gmax,1 (solid line). Notice that for Gmax,1 ! 0, JmSS ! (f) Dependence of Jm SS Ju,2 (cSS M ) (O) and that Ju,2 (cM ) < Ju,2 ¼ k2 Gmax,2 (dashed line). For Gmax,1 ! 1, SS ^ Jm ! Jm ( )
J. GALCERAN AND H. P. VAN LEEUWEN
cSS M ¼
cM 1 þ ðkKH r0 =DMÞ
161
(30)
from which the steady-state diffusion flux can be computed as [11,21]: SS Jm ¼
DM cM cM ¼ r0 þ (DM =kKH ) (r0 =DM ) þ (1=kKH )
(31)
This expression has been interpreted as the total ‘resistance’, being the sum of the diffusion ðr0 =DM Þ and adsorption þ internalisation (1=kKH ) resistances [11,32] or a combination of permeabilities [19]. If the couple: adsorption þ internalisation is much more effective than diffusion (kKH DM =r0 ), then cSS M ! 0 and we recover the steady-state maximum uptake flux for a spherical organism Jm . 2.3.3
Analytical Solution for the Transient
It can be shown (see [33–35]), that the expression for the flux can be expressed in terms of the auxiliary parameters a and b: pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DM r0 þ DM r0 4 DM KH þ kKH2 r0 a pffiffiffiffi 2KH r0 pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DM r0 DM r0 4 DM KH þ kKH2 r0 b pffiffiffiffi 2KH r0
(32)
(33)
type of combination of parameter values Thus, if a ¼ b, that is, for a particular
DM r0 ¼ 4 DM KH þ kKH2 r0 : rffiffiffiffiffiffiffiffiffi SS Jm Jm 1 DM t 1 2 2 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi þ ¼ þ DM cM DM cM p KH r0 pDM t " # pffiffiffiffiffiffiffiffiffi DM t 4KH 1 1 2 2 DM t F þ 2 2KH KH KH r0 2KH r0 where: F(x) exp x2 erfc(x) For the more general case with a 6¼ b:
(34)
162 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES SS Jm Jm 1 ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi DM cM DM cM pDM t pffiffiffiffiffiffiffiffi pffiffi 2 DM DM a þ F(a t) 2 KH r0 (a b) KH r0 a(a b) KH (a b) pffiffiffiffi pffiffi 2 DM DM b F(b t) 2 KH r0 (a b) KH r0 b(a b) KH (a b)
(35)
From the previous expressions, the accumulated uptakes (Fu and Fm , see equation (11)) follow by simple integration [35]. 2.3.4
The Impact of the Radius
Figure 6 shows the evolution of Jm with time, for three different radii. As expected from the enhanced diffusion efficiency, the smaller the radius, the sooner steady-state values are approached. For a large radius of 1 mm, after SS (in practice, with such long 200 s, the value of Jm is still approximately twice Jm times, convection will usually overrule the pure diffusion conditions). It can also be seen in Figure 6 that larger radii yield larger fluxes at intermediSS ate times, while Jm decreases with increasing r0 . Plots at very short times (not 3.00
Jm /10−5 mol m−2 s−1
2.50
2.00
1.50
1.00
0.50
0.00 0
0.1
0.2
0.3
0.4
0.5 t /s
0.6
0.7
0.8
0.9
1
Figure 6. Impact of the microorganism radius, r0 , on the time evolution of the diffusive flux Jm (given by equations (34) and (35)). Three curves are depicted: u (r0 ¼ 10 mm), n (r0 ¼ 100 mm) and þ (r0 ¼ 1 mm). Parameters: KH ¼ 2 105 m, k ¼ 5 105 s1 , DM ¼ 109 m2 s1 , cM ¼ 1 mol m3
J. GALCERAN AND H. P. VAN LEEUWEN
163
shown here) indicate larger fluxes for smaller radii, as expected from the enhanced diffusion transport towards microbodies. The inversion of the order in the fluxes at intermediate times can be explained as larger radii producing more gradually decreasing fluxes than smaller radii. 2.3.5
The Approach to Steady-State After a Maximum
As seen in Figure 7, the surface concentration c0M attains a maximum, overshooting the steady-state concentration value. It has been shown theoretically [35] that the maximum appears for any combination of parameters’ values. Experimental evidence of the appearance of transient uptake rate maxima (which might be totally or partially related to the maximum predicted by the present uptake model) has already been reported [36–39]. The maximum can be physically understood as a result of the simultaneous settling of two processes: adsorption governing short times (see Section 3.2, below) and internalisation ruling longer times. If only adsorption were present (i.e. k ¼ 0), the final cM ðr0 , tÞ value would be cM . So, at relatively short times, for the combined processes, the surface concentration ‘aims at’ a value closer to cM than the eventual cSS M value, overshooting it. After the maximum (longer times), cM ðr0 , tÞ is relatively large, and the corresponding Ju cannot match the
0.25
* cM(r0,t)/cM
0.20 k = 10−2 s−1
0.15
0.10
k = 10−3 s−1
0.05 k = 10−4 s−1 0.00 0
500
1000
1500
2000 t /s
2500
3000
3500
4000
Figure 7. Evolution of cM ðr0 , tÞ=cM for three different combinations of KH and k yielding a common product 105 m s1 . Curves for k ¼ 102 s1 , k ¼ 103 s1 and 9 2 1 m s k ¼ 104 s1 converge towards cSS M =cM ¼ 0:09. Other parameters: DM ¼ 10 and r0 ¼ 1 mm
164 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
‘exhausted’ diffusive flux Jm (due to the depleted surroundings of the surface). Then, cM ðr0 , tÞ begins to decrease towards cSS M . Such a type of overshoot, a typical transient phenomenon, was already seen for adsorption–internalisation in planar geometry [40] and for the electrodic conversion affected by a slow preceding chemical reaction [41]. Figure 8 shows the fluxes’ evolution in terms of c0M for a given set of parameters. When c0M is low (corresponding with short times), the diffusive dSS flux Jm is much larger than Ju . For reference, Jm (diffusive steady-state flux 0 for the same cM ) has been included in the plot. The intercept corresponding dSS SS with Jm ¼ Ju ¼ Jm yields Jm , as seen in the plots of fluxes versus c0M discussed above in the steady-state case. Although time is not explicitly represented in Figure 8, we can follow the transient evolution towards steady-state on this plot. At t ¼ 0, c0M and Ju are zero and the uptake flux starts from the origin of coordinates; then, with increasing (but still short) t, Ju increases. At short t, Jm is extremely large (in comparison with Ju ), but decreasing until both fluxes (Ju and Jm ) eventually meet at a t which is easily identified as tmax (the time when c0M reaches its maximum value). Indeed, if Ju ¼ Jm , the boundary condition equation (27) prescribes dcM ðr0 , tÞ=dt ¼ 0, which is the condition for the maximum. Thus, while t < tmax , Jm > Ju , with their difference indicating the rate of accumulation 3.00
2.50
J/10−6 mol m−2 s−1
JmdSS 2.00 Jm 1.50
1.00 Ju 0.50
0.00
0
0.1
0.2
0.3
0.4 cM(r0
0.5
0.6
0.7
0.8
0.9
1
,t )/mol s−1
Figure 8. Plots of the fluxes versus cM ðr0 , tÞ. n (Jm ), s (Ju ) and u (JmdSS ). Parameters: cM ¼ 1 mol m3 , KH ¼ 2 105 m, k ¼ 0:05 s1 , DM ¼ 109 m2 s1 and r0 ¼ 0:1 mm. 3 For t ! 1, the three fluxes converge at the coordinates cSS M 0:909 mol m , SS Jm 9:09 107 mol m2 s1
J. GALCERAN AND H. P. VAN LEEUWEN
165
2.00 1.80
J /10−9 mol m−2 s−1
1.60 Jm
1.40
Ju
1.20 1.00 0.80
JmSS
Ju
Jm
0.60 0.40 0.20 0.00 0
2000
4000
6000
8000 10 000 12 000 14 000 16 000 18 000 20 000 t/s
Figure 9. Plot of the fluxes showing that proximity to steady-state can require huge times. Parameters: KH ¼ 2 105 m, k ¼ 5 105 s1 , DM ¼ 1011 m2 s1 , cM ¼ 1 mol m3 and r0 ¼ 1 cm
on the organism surface as adsorbate. In the plot, Jm first moves to the right and downwards, but, after cM ðr0 Þ has reached its maximum, moves left towards SS . In the overshoot region (t > 25 s), Jm < Ju (not seen the steady-state value Jm in Figure 8, but obvious in Figure 9 for t > 10 000 s), as the amount of species adsorbed on the surface is decreasing. dSS Further computations show that at very short times, Jm Jm . For the dSS parameters in Figure 8, Jm > Jm at any time, but this relationship can be inverted at intermediate times for other parameters (such as those of curve KH ¼ 2 105 m in Figure 10). 2.3.6
How Long Does it Take to Reach Steady-State?
Mathematically, steady-state is never reached within a finite time. For practical purposes, however, one can compute the time necessary to reach steady-state by imposing the condition that a given transient magnitude (concentration or flux) differs from the steady-state value in a reasonably low relative proportion [42]. For calculating the proximity to steady-state, the diffusive flux Jm is more convenient than the internalisation flux Ju , because of the continuously decreasing behaviour with time of the former. The approach to steady-state can be extremely slow (see Figure 9), and would lead to considerable error in the determination of the characteristic parameters of the system if an inappropriate transient value is taken as the final
166 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 4
Jm KH = 2⫻10−4 m
J /10−9 mol m−2 s−1
3
2
KH = 2⫻10−5 m KH = 2⫻10−6 m
1 Ju 0 0
10
20
30
40
50 t/s
60
70
80
90
100
Figure 10. Evolution of fluxes with t. Upper curves: diffusive fluxes Jm for three different combinations of KH and k, yielding a common product 109 m s1 . Lines with k ¼ 5 104 s1 , k ¼ 5 105 s1 and k ¼ 5 106 s1 converge towards SS Jm ¼ 109 mol m2 s1 . Lower curve with markers s: internalisation flux Ju for KH ¼ 2 105 m and k ¼ 5 105 s1 . Other parameters: DM ¼ 109 m2 s1 , cM ¼ 1 mol m3 and r0 ¼ 10 mm
steady-state value. For instance, in Figure 9 the transient flux at t ¼ 104 s is still twice the true steady-state value; this would lead to a 100 % error in the determination of the product kKH via application of equation (31) (see Section 2.3.2 above). The same relative error appears in the case with the smallest k in SS . Figure 10 if the transient flux at t ¼ 80 is taken as Jm If, at large time intervals, flux data do not change within the experimental error, the steady-state hypothesis is the simplest option. A possible check on that hypothesis would be to analyse the data in comparison with the transient behaviour as derived from the general expressions given above for given values of k and KH . The availability of analytical expressions for the transient diffusive flux Jm allows computation of the time necessary for a given proximity to the steadySS state. Results for the particular case of Jm being 10 % greater than Jm are given as a contour plot in Figure 11. The plot can be used for any set of parameters following the model, because it can be demonstrated that three suitable dimensionless variables suffice to describe the model [33]. We have selected the following dimensionless parameters: DM t=r20 (which could be called a dimensionless time), KH =r0 (a dimensionless adsorption parameter) and kr20 =DM (a dimensionless kinetic parameter). The logarithm of the dimensionless adsorption and kinetic parameters have been used as coordinates in Figure 11. Each
J. GALCERAN AND H. P. VAN LEEUWEN
167
4 3 2
log (kr02/DM)
1 −3
−2
0
−1 0
−1
1
−2
2
−3
3
−4
4
−5 −6 −4
5 6 −3
−2
−1
0
1 log (KH/r0)
2
3
4
5
6
Figure 11. Contour plot of the dimensionless time DM t=r20 needed for Jm to reach SS 1.1Jm in terms of the logarithm of the dimensionless adsorption parameter ðKH =r0 Þ, in abscissas, and the logarithm of the dimensionless internalisation parameter kr20 =DM , in ordinates. The number on each curve indicates the value of log DM t=r20 . The diagonal (dashed line) corresponds with kKH r0 ¼ DM
curve connects systems sharing the same dimensionless time to reach the SS condition Jm ¼ 1:1Jm . Let us illustrate how Figure 11 could be used with the particular DM and r0 data of curve KH ¼ 2 105 m in Figure 10: as the logarithms of the dimensionless parameters for that curve are 0.30 (adsorption) and 5:30 (kinetics), we can roughly read log (DM t=r20 ) ¼ 2:5 in Figure 11 (the point is in between the iso-curves labelled ‘2’ and ‘3’), which implies that approximately 30 s are needed to reach the prescribed proximity to steady-state. If the individual values of k 9 1 and KH were not known, but just their product (say 10 m s ), we would not kr2
have a single point in the diagram but the line log Kr0H þ log DM0 ¼ 5, which crosses contour lines with increasing values of the dimensionless times as the individual value of KH is increased. Inspection of Figure 11 suggests that, in general, increasing KH delays the achievement of the steady-state. This can be physically understood as larger KH implying larger amounts of matter that must be slowly transported (by diffusion) towards the organism surface. The pattern of the iso-curves suggests two large regions roughly separated by the main diagonal (defined by kKH ¼ 1 and plotted as a dashed line in Figure 11): the lower left region (weak adsorption and kinetics) and upper right region (strong adsorption and kinetics), thus indicating the critical role played by the product kKH .
168 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
2.3.7 Discriminating the Individual Parameters of the Product kKH Through the Cumulative Plot SS The knowledge of the steady-state value Jm , allows one to isolate the value of the product kKH from equation (31) – if DM and r0 are known – but not the individual values of k and KH . However, these individual values do affect the transient evolution of the flux. Indeed, it is seen in Figure 10 that by keeping a fixed product for kKH ¼ 109 m s1 , the transient fluxes dramatically increase with the individual value of KH increasing from KH ¼ 2 106 m to KH ¼ 2 104 m. As the two first terms in the expansion of cM ðr0 , tÞ at short t (see ref. [35]) do not depend on k and rapidly decrease when KH increases, a larger KH produces a lower cM ðr0 , tÞ, which in turn produces a larger Jm . Using a different set of parameters, Figure 7 shows very different evolutions for cM ðr0 , tÞ, despite a common product kKH ¼ 105 m s1 : the larger the k, the faster the concentration moves closer to the steady-state value. In conclusion, the individual values of k and KH could be found by fitting the transient behaviour of fluxes (or surface concentrations). Figure 12 (which is similar to reported experimental transient curves [13,36,38,43,44]) shows the cumulative uptakes corresponding with fluxes depicted in Figure 10 with KH ¼ 2 105 m. It can be seen that for quite long times, the adsorbed amount is much larger than the internalised amount. After the building
Fm 2.00
GM
Uptake/10−5 mol m−2
1.50
1.00
0.50 Fu
0.00 0
50
100
150
200 t/s
250
300
350
400
Figure 12. Time evolution of the internalised cumulated uptake Fu (see equation (10) ), total cumulated uptake Fm (see equation (12) ) and surface concentration (G(t) ¼ KH cM ðr0 , tÞ). Same parameters as in Figure 10
J. GALCERAN AND H. P. VAN LEEUWEN
169
up of GM over some 30 s, the system practically reaches a steady-state regime, reflected by the linear nature of the evolution of Fm and Fu for longer times. This linearity suggests one procedure to find the individual values of the parameters in the product kKH , if experimental access to very short times is not feasible, and it is not possible to distinguish between adsorbed and internalised amounts. Figure 13 shows a detail of Figure 12, together with the line corresponding with: SS Fm KH cSS M þ Jm t
(36)
which asymptotically tends to Fm (t) for not too small t. This straight line (dashdotted in Figure 13) can be interpreted as the result of an instantaneous building of the surface concentration up to the steady-state value, followed immediately by the steady-state regime flux. In the case of Figure 13, this approximation is quite reasonable, due to the fast fulfilment of the adsorption process discussed above. Thus, measured steady-state Fm values could be fitted to a straight line, SS the slope of which would yield Jm and the intercept of which would yield SS SS KH cM ¼ Jm =k, from which k (and KH ) can be isolated. This method based on equation (36) can be labelled ‘the instantaneous steady-state approximation’ (ISSA). As expected, if the system is still far from steady-state, the method will yield erroneous values for KH . With another set of parameters, the plot of the 2.01
Uptake/10−8 mol m−2
KHcMSS + JmSS t
2.00 Fm
GM
SS KHcM
1.99
1.98 0
5
10
15
20
25 t/s
30
35
40
45
50
Figure 13. Detail of Figure 12, showing the asymptotic behaviour of the line SS KH cSS M þ Jm t (dash-dotted line) with respect to the total cumulated uptake Fm (see equation (12) ). The dashed line corresponds with the steady-state surface concentration KH cSS M . The evolution of the surface concentration G(t) ¼ KH cM ðr0 , tÞ is also shown
170 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 3.00 Fm
Uptake/10−5 mol m−2
2.50
2.00
1.50 SS KHcM
1.00
0.50
0.00 0
1000
2000
3000
4000
5000 t/s
6000
7000
8000
9000 10 000
Figure 14. Plot of the cumulative flux Fm with the same parameters as Figure 9, showing that a linear fit of relatively long time values of Fm can lead to erroneous determinations of KH . The dashed line corresponds with the steady-state surface concentration KH cSS M
cumulated fluxes (see Figure 14) clearly shows that the intercept is not a measure of GM . From the slope and the intercept of the straight-line fitting of the rightmost part of Fm in Figure 14, a value of KH ¼ 4:3 103 m would be recovered, which is about 200–fold greater than the true value. The error could be easily discovered if the (theoretical) transient with this fitted parameters was plotted: steady-state would not even be reached at 20 000 s. Even simpler would be just to use Figure 11 (the coordinates can be ascribed to the flat region with dimensionless time 100=p), from which one would estimate that at least 109 =p s are needed to begin to consider steady-state as being approached. In fact with the true parameters, 109 =p s is also the estimate for achievement of near steadystate conditions. The plot of the ‘initial’ (t < 20 000 s) fluxes for such a true system (see Figure 9) again highlights that an observed constancy of the flux can hide a extremely sluggish tendency towards steady-state. 2.4 2.4.1
THE DIFFUSIONAL STEADY-STATE (dSS) APPROACH The Essence of the dSS Approach
We have seen that purely diffusion-controlled biouptake fluxes may require time spans of O(103 ) s to decay to their eventual steady-state values (see Section 2.3.6). In reality the situation of pure diffusion as the mode of mass transfer in
J. GALCERAN AND H. P. VAN LEEUWEN
171
the medium, is not generally maintained over such long timescales (see Chapter 3 in this volume). Even in unstirred systems, natural convection overrules mere diffusion after times of O(10103 ) s. In biouptake systems with flowing liquids (e.g. fish gills) or with mobile organisms, the effects of convection are obviously larger. Typically, in situations of mild movement of the medium with respect to the active uptake area, the transition from pure diffusion with a time-dependent diffusion layer thickness dM ( ¼ (pDt)1=2 for the planar case) to convective diffusion with a fixed dM of O(10102 ) mm (see Chapter 3 in this volume for details) is at times of O(101 to 100 ) s. Thus, it seems very appropriate to extend the dynamic flux analysis, as given before for semi-infinite diffusion, with two main simplifying hypotheses: (1) there is a diffusion layer thickness dM with a time-independent value, defined by the hydrodynamic conditions of the experiment, indicating the distance from the surface where bulk conditions are restored: cM (r0 þ dM , t) ¼ cM ; and (2) the steady-state concentration profile between the biological surface and r ¼ r0 þ dM is rapidly adjusted to changes of the surface concentration c0M (t) cM (r0 , t), i.e. the characteristic time for setting up or modifying the steady-state concentration profile is much smaller than the timescale of changes in the surface concentration. We refer to the set of both simplifications as the ‘diffusional steady-state’ approximation (dSS). Apparently, the most important consequence of the dSS approach is the simplification of the expressions for the flux Jm , as compared with the semiinfinite diffusion case. Indeed, for a given c0M , the steady-state flux in spherical geometry is [45]: dSS Jm (t) ¼ DM
1 1 c c0M (t) cM c0M (t) ¼ DM M þ r0 dM d
(37)
where an ‘effective diffusion layer shell thickness’, d, has been introduced: 1 1 1 þ d r 0 dM
(38)
which includes the planar case: r0 ! 1, d ! dM . This effective diffusion layer shell thickness can be seen as just the inverse of the mass transfer coefficient (usually denoted km or b [45]). In many instances, d can be considered as an unknown free parameter to be fitted from the data. For very small microorganisms, with radii typically below 10 mm, dM r0 and then d r0 ; as the time for steady-state spherical diffusion to settle can be estimated with tSS d2M =pDM (see Chapter 3), hypothesis (2) above is so reasonable that the dSS approach suffices without the need to consider semi-infinite diffusion. The key consequence of the dSS approach is that the diffusion problem is greatly simplified. Although c0M still varies with time, the complete profile
172 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
cM (r, t) is defined if c0M is known. Thus the flux-balance equation (4) can be written as: dG1 (t) dG2 (t) c cM (r0 , t) þ ¼ DM M k1 G1 (t) k2 G2 (t) dt dt d
(39)
The relationship between G1 , G2 and c0M has to be specified via the adsorption isotherm. Let us first proceed with the simplest case of the linear Henry regime. 2.4.2
Linear Adsorption
In the range of linear adsorption behaviour, whatever the number of site types (see Section 2.3.1 for the merging of parameters of two sites), the surface concentration G is related to c0M via an effective linear coefficient, KH , while the first-order internalisation processes can also be described by an effective first-order constant, k. Thus, equation (39) can be recast, for instance, in terms of G as: dG(t) c G(t)=KH kG(t) ¼ DM M d dt
(40)
This equation can be integrated straightforwardly [46]. If we take again the case of G(t ¼ 0) ¼ 0, the result is: G(t) ¼
DM t tdSS cM 1 exp d tdSS
(41)
where the characteristic time constant for this case of first-order biouptake kinetics coupled with steady-state diffusion (tdSS ) is defined by: tdSS ¼
DM þk dKH
1 (42)
The concentration of M just outside the adsorbed layer, c0M (t), follows from G(t), via G(t) ¼ KH c0M (t) and then the flux Jm can be computed with equation (37). For example, the flux of M towards the interface is: DM cM DM tdSS t=tdSS 1e 1 Jm (t) ¼ d dKH
(43)
Intuitively [46], the dSS approximation is more likely to hold for times greater than the time to practically reach SS (tSS d2M =pDM ). Thus, the approximation agrees with the rigorous solution (transient diffusion with the bulk value restored at r ¼ r0 þ dM ) for most of the range seen in Figure 15, but not for
J. GALCERAN AND H. P. VAN LEEUWEN 10
173
(a)
Jm /10−4 mol m−2 s−1
8
6
4
2
0 0.0
0.2
0.4
0.6
0.8
1.0
t/s (b)
35
Ju /10−8 mol m−2 s−1
30 25 20 15 10 5 0 0
2
4
6
t/s
Figure 15. Comparison of curves Jm (a) and Ju (b) versus t predicted by different submodels for a system with linear adsorption. Continuous line: rigorous solution of transientwithboundaryconditioncM (r0 þ dM , t) ¼ cM 8t;dashedline:rigorous(transient) solution with semi-infinite diffusion (solving integral equation (7) ); s: dSS approximation (given by equation (43) ). Parameters: cM ¼ 1 mol m3 , DM ¼ 8 1010 m2 s1 , KH ¼ 7:24 104 m, k ¼ 5 104 s1 , r0 ¼ 1:8 106 m, r0 þ dM ¼ 105 m
that of Figure 16. As expected, the dSS approach for Jm (given by equation (43)) is less reliable for short t. Otherwise, the agreement is so good that dSS might be the simplest alternative for semi-infinite diffusion for microorganisms. Note that Ju starts at zero for t ¼ 0, and that it increases proportionally with G which increases with [1 expðt=tdSS Þ according to equation (41) (see Figure 15b). The dSS approximation is even better for Ju than for Jm . For the
174 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Jm /10−8 mol m−2 s−1
1.5
1.0
0.5
0.0 0
2000
4000
6000
8000
10 000
t/s
Figure 16. Jm versus t predicted by the same submodels as in Figure 15, but with parameters: cM ¼ 1 mol m3 , DM ¼ 1011 m2 s1 , KH ¼ 2 105 m, k ¼ 5 105 s1 , r0 ¼ 102 m, r0 þ dM ¼ 1:1 102 m
case where sites 1 are followed by internalisation whilst sites 2 adsorption are not, typical parameter values are [38]: DM ¼ O(109 ) m2 s1 , dM ¼ O(104 ) m, which, together with r0 ¼ 1:8 106 m, renders d r0 ; the combined KH (see equation (28)) is O(104 ) m and the combined k is O(102 ) s1 . For this set of data we compute tdSS to be O(0.1) s, which in this particular case is mostly governed by the settling of the adsorptive steady-state (first term between square brackets in equation (42)). The steady-state flux (common for the dSS approximation and for the rigorous solution with bulk concentrations restored at r ¼ r0 þ dM ) can be written: SS Jm
DM cM c0M cM ¼ ¼ kKH c0M ¼ ð1=kKH þ d=DM Þ d
(44)
and, in the example, yields 109 mol m2 s1 for a bulk concentration of 3 108 mol dm . The numerical example given above also shows that, for certain sets of parameter values, the assumption of linear adsorption may be violated. Since the sites for adsorption plus internalisation have a higher affinity, they are preferentially filled; because they are much lower in number, the nonlinear regime can be surpassed. For the above example, cSS M cM , which combined with KH,2 (for the noninternalisation sites) of O(105 ) m, yields 2 O(1010 ) mol m of occupied sites. If the maximum number of such sites is
J. GALCERAN AND H. P. VAN LEEUWEN
175
2
O(109 ) mol m , one can still accept the linear regime hypothesis (10% occupancy), but an increase of the bulk concentration can overrule this assumption. For such situations, a more detailed analysis with Langmuirian adsorption is required. 2.4.3
Langmuirian Adsorption
If, on the timescale of observation, the degree of coverage of any site type becomes appreciable, the precise nature of the relationship between G1 , G2 and coM has to be taken into account. For the case of a Langmuirian isotherm (implying sufficiently fast kinetics of adsorption/desorption) this means that equations in (2) are applicable. Two particular cases are described here: 2.4.3.1 One Site Adsorbing and Internalising The balance of fluxes (39) can be written in terms of the surface concentration G as: dG(t) DM cM KM DM G(t) kG(t) ¼ d dt dðGmax G(t) Þ
(45)
This equation can be easily integrated and reorganised in terms of coverages as:
e
ktðySS yII Þ
1ySS 1 y=ySS ¼ 1yII 1 y=yII
(46)
where ySS and yII stand for the solutions of the steady-state equation associated with equation (45), yII being the secondary solution (in itself physically meaningless) corresponding with a negative concentration. In a plot of the fluxes versus c0M , those solutions would correspond with the two intersections of the dSS (see Figure 4). hyperbola for Ju and the straight line for Jm Figure 17 compares the internalisation fluxes Ju predicted by the linear and Langmuir isotherms using the dSS approximation. As expected, at short t, they converge as the Langmuir isotherm tends to the Henry isotherm for low surface concentrations c0M . Due to the saturation effect limiting the G value for the Langmuir isotherm, the linear isotherm can yield a larger steady-state value for Ju . Figure 17 also allows comparison of the Langmuir isotherm for two different diffusion regimes: dSS (continuous line) and semi-infinite (dotted line). At short t, dSS underestimates the fluxes because it ignores the initial transient. At intermediate times, the relationships reverse: dSS yields larger fluxes because of the slow supply of matter in the semi-infinite case. With the parameters of the Figure, for both Langmuirian curves there is a practically common steady-state Ju value (reached at relatively short t), because dM r0 .
176 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Ju /10−12 mol m−2 s−1
1.5
1.0
0.5
0.0 0
0.02
0.04
0.06
0.08
0.1
t/s
Figure 17. Comparison of Ju versus t plots predicted by different submodels for a system with one type of site (adsorbing and internalising): linear isotherm with dSS approximation (s) applying equation (43) with KH ¼ 5:2 106 m; Langmuirian isotherm with dSS approximation (continuous line) applying equation (46); Langmuirian isotherm with semi-infinite diffusion (dotted line) by numerically solving integral equation (7) ). Other parameters: cM ¼ 5 104 mol m3 , DM ¼ 8 1010 m2 s1 , KH ¼ 2 105 m, k ¼ 5 104 s1 , r0 ¼ 1:8 106 m, r0 þ dM ¼ 105 m, KM ¼ 2:88 103 mol m3 , Gmax ¼ 1:5 108 mol m2
2.4.3.2 Two Sites: Site 1 = Adsorption + Internalisation, Site 2 = Adsorption Only. The balances of fluxes can be written: Gmax,2 c0M Gmax,1 c0M d Gmax,1 c0M cM c0M þ k ¼ D M 1 d dt KM,1 þ c0M KM,2 þ c0M KM,1 þ c0M
(47)
which can be integrated to [46]: SS 1yII1 II SS 1y1 k1 t ySS ln 1 y1 =yII þ 1 y1 ¼ ln 1 y1 =y1 1 " 1ySS KM,2 Gmax,2 SS SS 2 1 y2 =y1 ln 1y2 =ySS 2 KM,1 Gmax,1 1yII1 II 2 yII ln 1y2 =yII þ 2 =y1 2 # II 2 KM,1 KM,1 KM,2 ySS 1 y1 y2 KM,2 KM,2 þ KM,1 KM,2 yII KM,2 þ KM,1 KM,2 ySS 1 1 (48)
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II where ySS 1 and y1 stand for the solutions of the steady-state equation associated with equation (45). As expected, and already commented in section 2.1.3.2, II those steady-state solutions (ySS 1 and y1 ) do not involve any parameter associated with site 2. Coverages in equation (48) referred to site 2 (e.g. ySS 2 ) can be obtained from site 1 through the relationship:
y2 ¼
K y M, 1 1 KM,2 þ KM,1 KM,2 y1
(49)
Equation (48) reverts to (46) if Gmax,2 ¼ 0. A similar reduction can be obtained when KM,1 ¼ KM,2 , showing that then the time to reach a certain y1 multiplies by 1 þ Gmax,2 =Gmax,1 (with respect to the case when Gmax,2 ¼ 0) [46]. The impact of changing KM,2 on Ju (using equation (48)) is seen in Figure 18. Because KM,1 cM , no saturation effect of the internalising site 1 can be expected. When KM,2 is larger or similar to KM,1 (curves (a) and (b)) the approach of Ju to steady-state follows the usual ‘parabolic’ behaviour. For low KM,2 (curve (c)) the supplied M is mostly adsorbed on to site 2 (because Gmax,2 Gmax,1 ), with adsorption still following practically linear isotherms for both sites with the 12 (a) (b)
Ju /10−12 mol m−2 s−1
10
8 (c) 6 (d) 4
2
0 0
20
40 t/s
60
80
Figure 18. Impact of bioaffinity of the non-internalising sites (KM,2 ) on Ju versus t plots, with dSS approximation for two Langmuirian adsorption isotherms (i.e. applying equation (48) ). Other parameters: d ¼ 1:48 106 m, cM ¼ 5 104 mol m3 , DM ¼ 8 1010 m2 s1 , Gmax,1 ¼ 1:5 107 mol m2 , Gmax,2 ¼ 8:85 106 mol m2 , KM,1 ¼ 2:88 103 mol m3 , k ¼ 5 104 s1 . Curves: (a) KM,2 ¼ 0:05 mol m3 , (b) KM,2 ¼ 0:005 mol m3 , (c) KM,2 ¼ 2:5 104 mol m3 , (d) KM,2 ¼ 5 105 mol m3
178 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
effect of just reducing the slope of Ju versus t. For much lower KM,2 (curve (d)), saturation of site 2 introduces a qualitative change in the shape of Ju versus t. 3 BIOUPTAKE WHEN MASS TRANSFER IS COUPLED WITH CHEMICAL REACTION (COMPLEX MEDIA) 3.1
MATHEMATICAL FORMULATION
Due to the usual diversity of components in the medium, there will be a need to consider that the species taken up interacts with other species while diffusing towards the organism surface (see Figure 19). In some cases (as in the aquatic prokaryotes that exudate Fe chelators called siderophores to improve the availability of Fe; see Chapter 9 in this volume), the medium is modified on purpose by the organisms [11,47–49]. A simple model for this interaction assumes the complexation of M with a ligand, with elementary interconversion kinetics between the free and complexed forms: ka M þ L Ð ML kd
(50)
where ka and kd denote the association rate constant and the dissociation rate constant, respectively. In spherical geometry, the continuity equations for the species are: ! qcM (r, t) q2 cM (r, t) 2 qcM (r, t) þ þ kd cML (r, t) ka cM (r, t)cL (r, t) (51) ¼ DM qt qr2 r qr ! qcL (r, t) q2 cL (r, t) 2 qcL (r, t) þ þ kd cML (r, t) ka cM (r, t)cL (r, t) (52) ¼ DL qt qr2 r qr Organism
ML
r0
Minter
DML
ML
ka kd
Mads
M + L
K DM
M + L
Figure 19. Schematic representation of the coupled diffusion of M (species taken up) and ML (bioinactive complex), with their interconversion kinetics involving the bioinactive ligand L
J. GALCERAN AND H. P. VAN LEEUWEN
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and ! qcML (r,t) q2 cML (r,t) 2 qcML (r,t) ¼ DML þ kd cML (r,t)þka cM (r,t)cL (r,t) (53) qt qr2 r qr Under conditions of complexation equilibrium, such as those assumed in the bulk of medium, we have: K¼
ka c ¼ ML kd cM cL
(54)
where K is the stability constant. For bioinactive complexes ML, the boundary condition at the biosurface is: qcML ¼0 qr r¼r0
(55)
This hypothesis excludes lipophilic complexes [19,50] and complexes subject to accidental uptake via membrane permeases [51,52], for which the analysis would be basically different [5,18]. In this sense, we also disregard here any adsorption of M in the form of its complex ML. Since many environmentally relevant complexes diffuse much slower than M (see Chapter 3), the distinction between DM and DML is important. Their ratio is usually labelled: e
DML DM
(56)
It is apparent that the kinetics of the homogeneous reaction can have a dramatic impact on the overall uptake process by controlling the ratio of complexed to free M, which affects the velocity of transport towards the organism surface. Therefore, kinetics do matter and all the dynamic effects must be properly taken into account. The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53–55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as
180 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Fe in marine water), known as the excess of ligand case, which allows linearisation of the problem via: cL (r, t) cL
(57)
The use of the excess ligand condition, equation (57), spares the need to consider the continuity equation (52) for the ligand. Then, two limiting cases of kinetic behaviour are particularly simple: the inert case and the fully labile case. As we will see, these cases can be treated with the expressions (for transient and steady-state biouptake) developed in Section 2, and they provide clear boundaries for the general kinetic case, which will be considered in Section 3.4. 3.2
TOTALLY INERT COMPLEXES
If the rate constants for interconversion between M and ML are infinitesimally small (on the effective timescale of the experimental conditions), the complex does not contribute significantly to the supply of metal to the biosurface. The equilibrium equation (50) behaves as if frozen. In a biouptake process, the complex ML then does not contribute to the supply of metal towards the biosurface, and all the expressions given in Section 2 apply, with the only noteworthy point that the value of cM to be used differs from the total metal concentration. In this case, the complexed metal is not bioavailable on the timescale considered, as metal in the complex species is absent from any process affecting the uptake. 3.3
FULLY LABILE COMPLEXES
We now turn to the dynamic limit where the rates of association/dissociation of ML are infinitely fast. The complex system will maintain a transport situation governed by the coupled diffusion of M and ML. In the case of excess of ligand conditions, equation (57), the full lability condition implies the maintenance of equilibrium on any relevant spatial scale: cML (r, t) ¼ KcM (r, t)cL
8r, t
(58)
Then, equations (51) and (53) can be summed to cancel out the kinetic terms and provide a simple diffusion equation for the total metal: cT, M (r, t) cM (r, t) þ cML (r, t)
(59)
which reads: qcT, M (r, t) q2 cT, M (r, t) 2 qcT, M (r, t) ¼D þ r qt qr2 qr
! (60)
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181
where D, the average diffusion coefficient, given by: D
DM cM þ DML cML DM þ KcL DML 1 þ eKcL ¼ ¼ D M cM þ cML 1 þ KcL 1 þ KcL
(61)
can be seen as resulting from the frequent flip-flops of the species M from its free state M to its complex state ML, and vice versa [41]. Boundary condition (55) is overruled by lability condition (58), and the usual boundary condition at the biological surface will derive from the coupling of transport with internal isation (concentration and flux at the interphase). Using cM ¼ cT, M = 1 þ KcL , the supply flux becomes: Jm (t) ¼ D
qcM (r, t) qcT,M (r, t) ¼ DM þ DML KcL qr qr r¼r0 r¼r0
(62)
instead of equation (8), for the noncomplex case. Once the transport step is defined, it is necessary to take into account which is the adsorption isotherm preceding the internalisation (the latter is always assumed to be first order). By extending results from Section 2, analytical solutions can be written for steady-state for both linear and Langmuir isotherms, as well as for the transient case with the linear isotherm. Langmuirian isotherms require numerical approaches for the transient case. As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing DM , cT, M replacSS ing cM (including the substitution of cT, M by cM and cSS T, M by cM ) and KH (defined as G=cMðr0 , tÞ in both cases, i.e. with or without the presence of L) by KH = 1 þ KcL . Other cases with analytical solutions arise from the steadystate situation. The supply flux under semi-infinite steady-state diffusion is [57]: SS Jm
DcM cSS M ¼ 1 þ KcL 1 r0 cM
(63)
If instead of semi-infinite diffusion, some distance dM acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where r0 is substituted by 1=ð1=r0 þ 1=dM Þ (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58]. Combining the flux given by equation (63) with a Michaelis–Menten-like expression for Ju , one recovers the modified Best equation (17), where the bioconversion capacity parameter b is now related to the total concentration of free and labile species of M:
182 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
b¼
Ju J r0 ¼ u Jm DcT, M
(64)
The bioaffinity parameter a basically reflects the free metal ion concentration, whereas the limiting flux ratio b reflects the total labile metal species concentration. Due to the complexation, the ratio a/b thus changes by a factor qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ eKcL in spherical geometry, while the factor 1 þ eKcL 1 þ KcL is required for planar geometry [26]. As mentioned in Section 3.1, an analytical solution can be provided for the steady-state of fully labile complexes, without needing to resort to the excess of ligand approximation: SS ¼ DM Jm
cM cSS c cSS M ML þ DML ML r0 r0
(65)
which, as expected, reverts to equation (63) under an excess of ligand. For any relationship between DL and DML , assuming no complex adsorption, the steady-state complex concentration at the microorganism surface needed in equation (65) is based on the equality (except for sign) of the fluxes of L and ML, leading to: cSS ML
K DL cL þ DML cML cSS M ¼ DL þ DML KcSS M
(66)
In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DM cM þ DML cML , the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [11] or Chapter 3 in this volume). 3.4
PARTIALLY LABILE COMPLEXES
3.4.1 The Limiting Supply Flux in the Steady-State Uptake Considering Homogeneous Kinetics The linear steady-state equations associated with (51) and (53), under excess ligand conditions, can be solved analytically [59] in terms of the concentration of M at the surface cSS M . The resulting supply flux is:
J. GALCERAN AND H. P. VAN LEEUWEN
SS Jm ¼
DM cM
r0
cSS M
183
1 pffiffiffi ka C B A @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi eKcL 1 þ eKcL þ ka 0
eKcL
(67)
where: ka
ka cL r20 DM
(68)
can be seen as a dimensionless complex formation rate constant, sometimes denoted as the Damko¨hler number [60,61] (see Chapter 3 in this volume). ka compares the diffusional timescale (r20 =DM ) with the reaction timescale (1=ka cL ) 1=2 or, alternatively, ka compares r0 with the reaction layer thickness pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ DM =ka cL . Equation (67) can be written [26]: SS Jm
¼
, kin Jm
cSS M 1 cM
(69)
where the (homogeneous) kinetic characteristics of the system are included in , kin the limiting supply flux Jm , which is a normalised mass transfer coefficient (see Chapter 3): 1 pffiffiffi ka C B A @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi eKcL 1 þ eKcL þ ka 0 , kin ¼ Jm
DM cM r0
eKcL
(70)
An extension to any number of ligands under any geometry is discussed in ref. [62]. 3.4.2
The Degree of Lability and Lability Criteria
In order to assess the impact of the homogeneous kinetics on the supply flux, the degree of lability can be defined – when only one complex is formed – as: pffiffiffi , kin , inert Jm Jm ka ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x , labile pffiffiffi , inert Jm Jm eKcL 1 þ eKcL þ ka
(71)
Then, equation (67) can be rewritten as: SS Jm ¼
cSS DM cM 1 M 1 þ eKcL x cM r0
(72)
184 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
which clearly shows that an increase in the degree of lability (with other parameters in the expression kept constant) implies an increase in the supply flux. As physically expected, it can be shown [55,59] that the higher the degree of lability, the stronger the depletion of ML at the microorganism surface: x¼
1 cML ðr ¼ r0 Þ=cML 1 cSS M =cM
(73)
A condition for the complex having a predominantly labile behaviour could follow from x 1=2, given that x ranges between 0 (for totally inert) and 1 (for fully labile): ka eKcL 1 þ eKcL
(74)
This expression could be regarded as a general lability criterion for the steadystate supply of M in spherical geometry. For the particular case that eKcL 1, one can recover the condition [57]: sffiffiffiffiffiffiffiffiffiffi DM DML cML c kd ka cL ML r0
(75)
which can be seen as the comparison between the contribution of the kinetic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi flux (dissociation of ML within the reaction layer m ¼ DM =ka cL , also called reacto-diffusive length [11,61]) and the limiting diffusive flux of ML. Similar lability criteria have also been derived for planar geometry and transient regimes [26]. 3.4.3
Combining Supply and Internalisation
Supply and internalisation are coupled through the value of cSS M and the flux of M crossing the interface. For the linear adsorption regime: cSS M ¼
c M
1 þ kKH r0 = DM 1 þ eKcL x
(76)
Similarly, for the steady-state situation with one Michaelis–Menten type of uptake site, the Best equation (17) still applies, now with a bioconversion capacity given by: b¼
Ju J r u 0 ¼ Jm DM cM 1 þ eKcL x
(77)
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185
which also encompasses the totally inert (x ¼ 0, see equation (16)) and fully labile (x ¼ 1, see equation (64)) limits. Let us consider now the influence of the radius of the microorganism. On one hand, it is well known that with decreasing radius, there is an enhancement of the diffusive flux due to the convergent nature of the spherical geometry. This can be seen, for instance, in equation (16) for the case without complexation or in equation (63) for fully labile complexes. For the case of a complexing medium with kinetics, there is also an impact of the radius on the lability: the smaller the radius, the lower the degree of lability [59], because when r0 ! 0, the dimensionless kinetic parameter ka ! 0 (see equation (68)), and the degree of lability x tends to 0 (see equation (71)). Thus, from a physical point of view, a change in r0 has two opposite effects on the supply flux, as can be seen in equation (70). However, by combining the expression for ka (equation (68)) , kin given by equation (70), it can be seen that the convergence enhancewith Jm ment dominates over the lability loss. It has been argued that the search for an improved diffusion efficiency has facilitated the evolution of some small organisms [63–65]. , kin due to changing r0 is bound to affect the This change in the supply flux Jm , kin uptake flux. A decrease in r0 implies an increase in Jm and a decrease in b (see equation (77)) which results in an increase of the normalised flux J~ (see equation (17)). However, it must be pointed out that the internalisation step can become rate determining, and then the flux will not change any more below some r0 value. As an illustration, we analyse the concrete situation for Pb2þ , which has a ka of 107 mol1 m3 s1 [66]. This ka value is a high one, so labile behaviour is expected over a relatively large range of parameters. From Figure 20 we can see that, for the chosen values of the parameters, the degree of lability x is quite high for radii larger than 10 mm. Thus the normalised uptake flux J~ for r0 > 10 mm is much closer to that obtained assuming fully labile rather than inert behaviour. For intermediate radii around 1 mm, the system experiences a dramatic loss in lability, and the uptake is also affected. In this case, ignoring the kinetic effects (e.g. assuming fully labile behaviour) in the interpretation of the measured flux would lead to the determination of quite incorrect parameters. However, if the bioconversion capacity (due to changes in Ju and cL ) is low, the free M by itself is able to sustain the uptake flux. This can be seen in Figure 21, where J~ (computed whilst taking into account ka ) coincides with the fully labile case. When lability is lost (r0 below 10 mm), J~ has already reached the unity value for the inert case (mass transport is no longer a limiting step). Thus, in this latter case, ignoring the kinetics is not relevant. See ref. [57] or [11] for a more detailed discussion and practical cases. For a single microorganism with a given r0 , different metals may have remarkably different behaviour, due to the metal-specific association and dissociation kinetics of the complexes involved. Take, for example, an organism of
186 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 1.0 0.9
~ Jlabile
Normalised flux or x
0.8
x
0.7 0.6
~ J
0.5 0.4 0.3 0.2
~ Jinert
0.1 0.0 −7.0
− 6.5
−6.0
−5.5 log (r0 /m)
−5.0
− 4.5
− 4.0
Figure 20. Normalised flux J~ (equation (17) ) versus log r0 for Pb2þ taking ka ¼ 107 mol1 m3 s1 (solid curve) compared with the labile (x ¼ 1, u) and inert (x ¼ 0, ) limits in equation (77) for parameters: KM ¼ 109 mol m3 , Ju ¼ 5 108 mol m2 s1 , DM ¼ 109 m2 s1 , DML ¼ 1010 m2 s1 , K ¼ 104 mol1 m3 , cL ¼ 101 mol m3 , cT, M ¼ 103 mol m3 . Markers correspond with the degree of lability x of this complexation of Pb2þ at different radii (see equation (71))
2þ
r0 ¼ 105 m which is simultaneously taking up Pb2þ and Ni (the latter metal having relatively slow complex association/dissociation kinetics [14]), and which has a similar affinity a towards both metals. The Pb(II) species are labile and their flux is controlled by their coupled diffusion, whereas the Ni(II) complexes are inert and the flux is determined by diffusion of free Ni(II). However, for similar organisms with decreasing r0 , the Pb-complexes are losing lability, resulting in a strong dimensional dependence of the relative uptake characteristics for the two metals. Again, the main conclusion is that radial transport influences the biouptake in two different ways: it not only generates changes in the magnitude of the diffusional fluxes towards the surface, but it also affects the labilities and hence the bioavailabilities of complex species. 3.5
RELATIONSHIP WITH THE FIAM
The most widely used model in environmental studies is the free ion activity model (FIAM or FIM) which postulates that the uptake is dependent on the bulk activity of free M (i.e. cM as a practical simplification) [2,67], rather than to the total metal concentration [2,5,66,68,69]. This has led to recognition of the
J. GALCERAN AND H. P. VAN LEEUWEN
187
1.0
Normalised flux or x
0.9 0.8
~ Jlabile
0.7
~ J
0.6 0.5 0.4 0.3 x
~ Jinert
0.2 0.1 0.0 −7.0
−6.0
−5.0
−4.0 log (r0 /m)
−3.0
−2.0
−1.0
Figure 21. Figure analogous to Figure 20, showing a case where there is no kinetic impact on the normalised flux J~ . Same parameters as in Figure 20, except Ju ¼ 5 1010 mol m2 s1 and cL ¼ 102 mol m3
importance of the speciation of M in the medium on the biological effects (see [2,9,70] and references therein). The FIAM can be physically interpreted as a model in which mass transport is not the limiting step, so that the surface and bulk concentrations are practically identical. Due to this identification between bulk and surface concentrations, the FIAM has been classified as an equilibrium or thermodynamic model [11]. If Michaelis–Menten internalisation kinetics hold (this is not necessary for the FIAM to apply), the equivalence between the FIAM and no diffusion transport can be shown mathematically [22]. Therefore, strictly speaking, the FIAM could only apply when diffusion (coupled with dissociation of complexes) is able to maintain the surface concentration of the free ion equal to its bulk value [13], although, in practice, small differences between the concentrations might be negligible. It is worth noticing that transport limitation (and failure of the FIAM) is more likely to happen when M is alone in the solution (i.e. ‘unsupported flux of M’) than when non-inert complexes are supporting the flux via dissociation. Figures 22 and 23 illustrate that the FIAM can apply due to dissociative support of M, but can fail when this support disappears. For a fixed amount of free metal ion, at large concentrations of ligand c0M ! cM SS (Figure 22), whilst Jm ! Ju (Figure 23). Moving towards lower concentra0 tions, cM starts to diverge from cM indicating the increasing significance of the concentration gradient. If dissociation of complexes does not counterbalance
188 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 1.00
SS/c * cM M
0.80
0.60
0.40
0.20
0.00 0
0.05
0.1 c L*/mol m−3
0.15
0.2
Figure 22. Plot of normalised surface concentration versus cL to illustrate that the lack of supporting dissociation by complexes can limit the application of the FIAM. Parameters as in Figure 20, but the total metal concentration cT, M is varied to keep a constant free-metal concentration cM ¼ 106 mol m3
1.00
0.80
~ J
0.60
0.40
0.20
0.00
0
0.005
0.01
0.015
0.02
c L*/mol m−3 SS Figure 23. Plot of normalised flux Jm =Ju versus cL to illustrate one limit of application of the FIAM. Parameters as in Figure 22
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189
SS the development of the gradient of cM , then Jm will decrease against the FIAM predictions based on cM . See Chapter 10 of this volume for a detailed discussion on the FIAM and BLM (biotic ligand model, [7,71]). The FIAM has been shown to apply to a large number of cases [2,69,72]. This indicates that the mass transfer step can likely be ignored in many cases [60], probably due to a sufficiently small radius (see equation (77)) and/or relatively low bioaffinity. In any case, whatever the microorganism size, the diffusion step needs to be considered for the interpretation of transient data as they become increasingly available [4,11,19,36,38,39,43,44,47,50,73–77]. If the FIAM apSS plies, i.e. transport is not limiting, Jm does not change with time (as long as there is no medium depletion, see Section 4.1.1 below). For instance, in the case of two sites, of which only one internalises M following the Langmuir adsorption, equation (14) boils down to: SS ¼ k1 Gmax,1 Jm
cM KM þ cM
(78)
which implies that the internalised amount will just increase linearly with time. Notice that lability effects are irrelevant as long as the FIAM applies, because diffusion is not flux-determining. In this context, it is also clear that any increase in the parameters enhancing the actual biouptake rate increases the possibility of kinetic control of the flux [9]. In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the cT, M or cML (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free cM . Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter (‘ligand exchange’) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion due to the labile complexation of M with a coefficients (perhaps arising from D large macromolecule or a colloid particle, see Section 3.3). Against simplistic views of the FIAM, it is necessary to stress that the model does not imply that the free metal ion is the only species available to the microorganism [2,14]. Indeed, the internalisation flux (i.e. the rate of acquisition) depends on the free metal ion concentration at the biological interphase (which in the FIAM is practically cM ), but metal bound to a ligand in the solution can dissociate, can diffuse (under a negligible gradient according to the FIAM), and can eventually be taken up.
190 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES SS In principle, the FIAM does not imply that the measured flux Jm should be linear with the metal ion concentration. The linear relationship holds under submodels assuming a linear (Henry) isotherm and first-order internalisation kinetics [2,5,66], but other nonlinear functional dependencies with cM for adsorption (e.g. Langmuir isotherm [11,52,79]) and internalisation (e.g. secondorder kinetics) are compatible with the fact that the resulting uptake is a function (not necessarily linear) of the bulk free ion concentration cM , as long as these functional dependencies do not include parameters corresponding with the speciation of the medium (such as cL or K [11]).
4 KEY FACTORS AND CHALLENGES FOR FUTURE RESEARCH IN BIOUPTAKE MODELLING The preceding sections have demonstrated the considerable quantitative understanding of biouptake that can be attained by models with a sound theoretical basis. We have shown solutions for a range of conditions, ranging from relatively simple limiting cases to more involved situations involving kinetically limited metal complex dissociation fluxes. In this section, we highlight key points that should be considered in future refinements of biouptake models. 4.1 4.1.1
REFINEMENTS BASED ON MASS TRANSPORT FACTORS Finite Media
In many practical situations, the picture of an isolated microorganism in an infinitely large medium is rather too crude. The combined action of a collection of similar organisms can have a strong impact on the surrounding medium. For instance, the depletion of essential trace elements in the photic zones of lakes and oceans is well known [14]. A simple model for the depletion of the medium can be obtained by combining previous results in this chapter, for the case of excess of ligand and any degree of lability [80]. We assume that each microorganism takes up M from a finite spherical volume of radius rf (at least five times larger than r0 ). In order to obtain an analytical expression, it is convenient to consider two very different timescales [81]: the diffusive steady-state is reached much faster than the depletion process, so that cM (t) cM rf , t and cML (t) cML rf , t are taken as bulk values for the semi-infinite steady-state diffusion, while they are time dependent for the depletion process. The total amount of M (free or complexed) in the volume of the medium per microorganism at a given time, O(t), can be computed by integrating the steady-state profiles associated with equation (67): O(t) ¼ f1 cSS M (t) þ f2 cM (t)
(79)
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191
where f1 and f2 are known functions of the medium parameters (i.e. not including KH or k; see reference [80]). If a linear adsorption isotherm (instead of the Langmuirian case considered in Section 3.4.3) holds, then equation (76) can be used to write cSS M in terms of cM . Equation (67), expression for the flux SS Jm , can also be reformulated as: SS Jm (t) ¼
cM (t) 1 r 0 þ kKH DM 1 þ eKcL x
(80)
Assuming that the time to re-adjust to steady-state is negligible in comparison with the depletion timescale, we can write: SS (t)dt dO(t) ¼ 4pr20 Jm
(81)
SS which, due to the linear relationships of O and Jm with cM , integrates to:
cM (t) ¼ cM (0)exp( t=tdepl )
(82)
thus predicting an exponential decay for the concentrations, with a time decay constant tdepl which is defined by the physicochemical parameters of the model (i.e. also on the kinetics of the complexation if cSS M 6¼ cM , see Section 3.5). For the particular case of no coupled reactions, the approximate expression (82) has been shown to agree reasonably well with the exact solution of the rigorous case of no flux across r ¼ rf . We also note that the total amount of metal taken up, Fm , can be measured from the difference between the original amount in the medium and that at a given time (associated with cM (t)). KH could then be found – following the line of the ISSA described in Section 2.3.7 – by approximating: Fm (t) KH cSS M ðr0 , tÞ þ
ðt 0
SS 0 Jm (t )dt0
(83)
SS where cSS M and Jm (t) can be written as functions of time through cM (t) (see equations (80) and (82)) with known medium parameters ( f1 and f2 ), and the unknown product kKH (or measured tdepl ). Thus, this allows for discrimination between KH and k values. In principle, any kind of limitation of the medium (e.g. due to some kind of clustering in a zone) tends to diminish the individual uptake rate [31]. From the point of view of modelling, the breaking of the symmetry rapidly complicates the problem (see Chapter 3 in this volume). As an exception to the general rule of decreased uptake due to inter-cell competition, it has been shown [49] that biouptake through siderophore excretion is only viable for nonisolated cells.
192 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Thus, the community effect can have a positive impact on the biouptake of certain trace nutrients. The high degree of packing of the organisms within a volume can lead to the formation of flocs (suspended aggregates), where millions of cells cluster to form particles with dimensions in the order of millimetres [29]. Models for uptake by such ecosystems also assume sphericity, and start from a continuity equation accounting for the consumption of the species throughout the floc: qcM (r, t) q2 cM (r, t) 2 qcM (r, t) þ ¼ DM qt qr2 r qr
! kfloc cM (r, t)
(84)
where r is now measured from the centre of the floc, and kfloc represents the first-order kinetic constant for the uptake of M by the organisms that are considered to be smeared out within the floc. Equation (84) (and similar equations) are usually solved for the steady-state case with boundary conditions of no flux of M at r ¼ 0 and cM fixed at the surface of the floc [29]. Of course, many other refinements can also be considered [82–84]. 4.1.2
Nonstagnant Media
In many natural environments, the assumption of a stagnant medium for the organism is too crude [85]. Either the organism will move in the medium or the solution will present some flow or turbulence that disturbs the stagnant regime. In both situations, there is a need to take into account not only diffusion (which is always the final step in the mass transport towards the biological surface [86– 88]), but also convection. The basic concepts of the convective diffusion problem, which is often simplified by introducing an effective diffusion layer thickness, d, are given in Chapter 3 of this volume or refs. [58,81]. Whitfield and Turner [9] estimated such a diffusion layer thickness d for gravitational sinking, convective water movements and swimming of motile phytoplankton cells and concluded that a lower boundary would be of the order of 10 mm. Mass transfer, taking into account a number of the nonstagnant regimes, has been combined with internalisation processes – under steady-state conditions – [10,16,21,30,31,49,89–92] to produce what can be seen as variants of the Best equation (17). The influence of the shape of the organism (or particle) has also been analysed [10,90,91]. 4.1.3
Complex Media
As commented on above, in general, if the complexation process of M is not linear (i.e. an excess of ligand cannot be assumed) only numerical approaches
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will be possible for the transient evolution. In some cases, the medium is far more complex and requires consideration of more than just a single ML species [61]. Then, we need to take into account the competition between several ligands for the same M, or various mixtures of different M and different L. The case of iron is paradigmatic: specific chelators exuded by the microorganisms to enhance iron uptake (siderophores) [49], colloidal suspension of iron oxides [93], etc. Obviously, in such cases, development of numerical approaches is then really almost the only way to obtain quantitative information, especially from a transient model.
4.2 4.2.1
REFINEMENTS BASED ON ADSORPTION PROCESSES Adsorption Isotherm and Kinetics
More complex isotherms than the Henry and Langmuir ones used here could be necessary to take into account interaction between adsorbed species, blocking effects, etc. The finite kinetics of the adsorption/desorption steps at the interface have been extensively studied by Hudson and Morel [13,15]. A wealth of literature is available on dealing with such interfacial processes [94–96] and its inclusion in the biouptake model should be implemented when experimental evidence of its necessity arises. 4.2.2
Competition
If other ions affect the internalisation process via competition for the transport sites [3,5,14,15,37,52,69,93,97–99], then a reasonable starting point is to modify the Langmuir isotherms (3) to: JuM, j (t) ¼ kM j Gmax , j
1 KM c (0, t) P M1 1 þ Ka ca (0, t)
j ¼ 1, 2 . . . a ¼ M, . . .
(85)
a
where now the superscript M is added to distinguish the uptake of this species from the others, and a is the general index that runs over the whole set of species ( j is the general index that runs over the types of internalisation sites). For the sake of simplicity, a common Gmax , j for all the species being taken up has been used. Obviously, analytical solutions become cumbersome or impossible, but the numerical approaches remain essentially the same (provided that the simultaneous fitting of a large number of parameters is not attempted). In many instances, the influence of pH has been dealt with as another competition process [3].
194 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
4.3
REFINEMENTS BASED ON INTERNALISATION FACTORS
4.3.1
Internalisation Kinetics
For simplicity, up to now, first-order kinetics have been assumed, but obviously other rate laws may apply. Further complications can be generated by the presence of multiple paths for M on a variety of sites exhibiting different kinetics [5,11] or sequential enzymatic processes [100]. Some complexes, labelled as ‘lipophilics’, have been shown to cross the membrane without the need for specific pre-adsorption sites [5,11,18,19,50]; see also Chapters 5, 6 and 10 in this volume. Fortin and Campbell [76] have recently reported the ‘accidental’ uptake of Agþ induced by thiosulfate ligand. Maintenance requirements can be included in the uptake flux expressions by the addition of a constant term [30]. This term expresses the minimum flux necessary for the organism to survive. 4.3.2
Efflux
Another factor to take into account in biouptake studies is the possibility that the organism develops strategies of eliminating toxic species by means of efflux [38,52,101]. As a first approach, the efflux rate can be set proportional to the amount of species taken up that has been internalised, thus converting the boundary condition of flux balances for two sites, equation (4), into: dG1 (t) dG2 (t) qcM (r, t) k1 G1 (t) k2 G2 (t) þ kout Fu (t) þ ¼ DM dt dt qr r¼r0
5
(86)
CONCLUDING REMARKS
Modelling biouptake requires the judicious consideration and selection of the underlying physical phenomena responsible for the experimental observations. We have seen that three fundamental phenomena may play a key role in biouptake: mass transfer, adsorption, and internalisation. The inclusion of additional phenomena or refinements (such as nonexcess ligand complexation, non-first-order kinetics, nonlinear isotherms, etc.) may be essential to describe certain cases, but they have handicaps, such as: . analytical solutions are rarely available; . it is difficult to obtain reliable parameters that can adequately describe such additional phenomena;
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. any improvements provided by a more refined theoretical model may be so small that they are indistinguishable from experimental error. Furthermore, there is presently a paucity of experimental data under appropriate conditions to provide both the driving force and the supporting evidence for development of more sophisticated models. On the other hand, if the really relevant phenomena are overlooked, then this could lead to incorrect interpretation of the fitted parameters, and, consequently, invalid predictions, e.g. if they form the basis of a risk assessment. As illustrative examples, consider two cases that highlight the range of convenience of a refinement: (1) most transient effects cannot be seen for microorganisms with very small radii, but the influence of the transient regime can be relevant in the description of accumulation data; and (2) if there is transport limitation (i.e. the FIAM assumption does not hold), the lability of the complexation becomes very relevant for both the flux and the depletion of the medium. A decision about which phenomena to keep and which to neglect – for the specific biological system under consideration and the specific measured quantity – can only be made on the basis of a close interaction between theoretical and experimental studies.
ACKNOWLEDGEMENTS The authors are most grateful to Jacques Buffle and Kevin Wilkinson (University of Geneva), Jaume Puy and Josep Monne´ (University of Lleida), and Raewyn M. Town (Queen’s University of Belfast) for their suggestions and assistance. Part of the preparation of this chapter was performed within the framework of the BIOSPEC project funded by the European Commission (contract EVK1-CT-2001-00086). This chapter is developed from [35], copyright Elsevier, 2003.
GLOSSARY BLM dSS FIAM SS * kin ISSA
Biotic ligand model Diffusive steady-state Free ion activity model Steady-state Bulk (superindex) Kinetic (superindex) Instantaneous steady-state approximation
196 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
APPENDIX A: NOTATION LATIN SYMBOLS Symbol
Name
a
Normalised bioaffinity parameter Ratio of limiting fluxes Concentration of species i (M: taken up; L: ligand; ML: complex) Steady-state concentration of species i at the surface (r ¼ r0 ) Bulk concentration of the species taken up (M). Concentration at the surface organism cM (r0 , t) Total concentration of M (free plus complexed) Average diffusion coefficient Diffusion coefficient of species i Functions of the depleted medium Diffusive flux Steady-state flux Diffusive steady-state flux Uptake (internalisation) flux due to sites of type j. Maximum uptake flux (due to sites j) Total uptake flux Normalised steady-state flux Linearised internalisation kinetic constant. Internalisation kinetic constant for sites of type j Association and dissociation rate constant for complexation Kinetic constant within a homogeneous floc Efflux kinetic constant
b ci cSS i cM c0M c T, M D Di f1 , f2 Jm SS Jm dSS Jm J u, j Ju Ju, j Ju J~ k kj ka kd kfloc kout
Units
Equations
None None
(19) (20),(64),(77)
mol m3
(1),(52),(53)
mol m3
(30), (66)
mol m3
(5)
mol m3
(37)
mol m3 m2 s1
(59) (61)
m2 s1
(1),(52),(53)
m3 mol m2 s1 mol m2 s1 mol m2 s1
(79) (8) (15) (24)
mol m2 s1
(3)
mol m2 s1 mol m2 s1 None
(13), (18), (25) (14) (18)
s1
(29)
s1 mol1 m3 s1 and s1
(3), (4)
s1 s1
(84) (86)
(54)
J. GALCERAN AND H. P. VAN LEEUWEN
K KH K M, j r r0 rf t
Equilibrium constant for complexation Linear adsorption coefficient Bioaffinity parameter for sites of type j. Radial coordinate Radius of the organism Radius of the region depleted by one organism Time
197
mol1 m3 m
(54) (28)
mol m3 m m
(2) (1),(52),(53) (7), Figure 1
m s
Section 4.1.1 (1)
Units
Equations
s1=2
(32), (33)
mol m2
(2)
mol m2 m
(2) (37)
m
(38)
None
(56)
None
(2)
None
(46) (48)
None
(46) (48)
None None s s
(68) (71), (73) (7) (41), (42)
s
(82)
GREEK SYMBOLS Symbol
Name
a, b
Auxiliary parameters for the transient solution Surface concentration of M at sites of type j Maximum surface concentration of M at sites of type j Diffusion layer thickness Effective diffusion layer shell thickness Normalised diffusion coefficient Fraction of coverage of species i Primary solution of coverage under steady-state (for site j, if subindex) Secondary solution of coverage under steady-state (for site j, if subindex) Dimensionless complex formation rate constant Degree of lability Dummy integration time Characteristic time constant Characteristic time for depletion of the medium
Gj Gmax , j
dM d e yi ySS ySS j yII yII j ka x t tdSS tdepl
198 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Fu Fm O
Bioaccumulated amount Total supply from the mass transport Overall amount of M in the depletion region
mol m2
(10)
mol m2
(11)
mol
(79)
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5 Chemical Speciation of Organics and of Metals at Biological Interphases BEATE I. ESCHER AND LAURA SIGG Environmental Microbiology and Molecular Ecotoxicology, and Analytical Chemistry of the Aquatic Environment (EAWAG), Swiss Federal Institute for Environmental ¨ berlandstrasse 133, CH-8600 Du¨bendorf, Switzerland Science and Technology U
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Chemical Speciation of Organic Chemicals . . . . . . . . . . . . . . 1.2 Metal Speciation and Biological Interactions. . . . . . . . . . . . . 2 Speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Speciation of Organic Compounds . . . . . . . . . . . . . . . . . . . . . 2.1.1 Hydrophobic Ionogenic Organic Compounds (HIOCs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Examples of Environmentally Relevant HIOCs . . . . 2.1.3 Redox Speciation of Organic Compounds . . . . . . . . . 2.2 Metal Speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Inorganic Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Organic Complexes: Hydrophilic Complexes . . . . . . . 2.2.3 Organic Complexes: Hydrophobic Complexes. . . . . . 2.2.4 Organometallic Species . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Redox Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Solubility of Solid Phases and Binding to Colloids and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Kinetics of Complexation Reactions . . . . . . . . . . . . . . 3 Membranes and Surrogates or Membrane Models. . . . . . . . . . . . . 3.1 Octanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Other Solvent–Water Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Liposomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Biological Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interactions of HIOCs with Biological Interphases . . . . . . . . . . . . 4.1 Partitioning and Sorption Models . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Octanol–Water Partitioning . . . . . . . . . . . . . . . . . . . . .
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
206 207 207 208 208 208 209 211 211 212 212 215 215 216 216 217 217 217 218 218 219 220 220 220
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4.1.2
Membrane–Water Partitioning: General Derivation of Membrane–Water Partition Coefficients of a Charged or Neutral Compound or Species . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Thermodynamics of Membrane–Water Partitioning . . . . . . 4.3 pH-Dependence of Membrane–Water Partitioning . . . . . . . 4.4 Ion Pair Formation at the Membrane Interphase . . . . . . . . . 4.5 Speciation in the Membrane: Interfacial Acidity Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Sorption of HIOCs to Charged Membranes Vesicles and Biological Membranes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Site of Interaction of HIOCs with Membranes . . . . . . . . . . . 4.8 Effects of Speciation of HIOCs at Biological Interphases . . 4.8.1 Ion Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Bioaccumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Toxicity: Uncoupling Effect . . . . . . . . . . . . . . . . . . . . . 5 Interaction of Metal Species with Biological Interphases . . . . . . . 5.1 Binding to Biological Ligands and Free Ion Activity Model (FIAM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Uptake of Specific Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Interactions of FA and HA with Biological Interphases . . . 6 Interaction of Hydrophobic Metal Complexes and Organometallic Compounds with Biological Interphases . . . . . . . 6.1 Hydrophobic Metal Complexes . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Organometallic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Organolead Compounds. . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Organotin Compounds . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions and Recommendations for Further Research . . . . . . List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
223 226 227 231 232 233 236 238 238 239 239 241 241 244 245 245 245 248 248 248 248 251 252 255
INTRODUCTION
The objective of this chapter is to discuss the role of speciation for interactions and uptake at biological interphases. Both organic chemicals and metals will be considered in order to emphasise common mechanisms, as well as fundamental differences. Acid–base reactions, redox reactions and complexation between metallic ions and various types of ligands have to be considered in regard to speciation. Particular attention will be paid to the role of interactions between metals and organic chemicals. The role of speciation regarding the various types of possible interactions at biological interphases (partitioning into the lipid
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bilayer, specific binding to carrier ligands and unspecific binding to functional groups, polar and electrostatic interactions) will be discussed. A related topic, the role of speciation for interaction with organic matter, is beyond the scope of this chapter. 1.1
CHEMICAL SPECIATION OF ORGANIC CHEMICALS
Historically, organic environmental pollutants were hydrophobic, often persistent, neutral compounds. As a consequence, these substances were readily sorbed by particles and soluble in lipids. In modern times, efforts have been made to make xenobiotics more hydrophilic – often by including ionisable substituents. Presumably, these functional groups would render the compound less bioaccumulative. In particular, many pesticides and pharmaceuticals contain acidic or basic functions. However, studies on the fate and effect of organic environmental pollutants focus mainly on the neutral species [1]. In the past, uptake into cells and sorption to biological membranes were often assumed to be only dependent on the neutral species. More recent studies that are reviewed in this chapter show that the ionic organic species play a role both for toxic effects and sorption of compounds to membranes. Speciation of hydrophobic ionogenic organic compounds (HIOCs) may influence their bioavailability and bioaccumulation, as well as sorption to organic matter and particles. For the charged species, specific sorption processes may play a role, while neutral organic compounds only undergo hydrophobic partitioning via van der Waals plus hydrogen donor/acceptor interactions. Finally, the toxic effect of HIOCs, in particular their specific toxic effect of uncoupling the electron transport from the ATP synthesis, is strongly influenced by their speciation [2]. To date, there have been only very few studies on the sorption of environmentally relevant HIOCs to membranes. In contrast, there exists a large body of literature on ionogenic drug partitioning (see, e.g. [3–5]), and the partitioning of hydrophobic ions [6], and the use of fluorescent membrane probes [7]. This literature review attempts to relate the findings from these related scientific fields to environmental chemistry. 1.2
METAL SPECIATION AND BIOLOGICAL INTERACTIONS
The role of metal speciation in the uptake of metals by biological systems has long been recognised, in particular with respect to aquatic organisms [8], and also in soil systems [9,10]. It is clear that metal ions cannot simply diffuse through the hydrophobic core of membranes. Various biological mechanisms of metal transport across biological membranes are known [11]. These mechanisms include permeation through ion channels, transport by carrier ligands (proteins), binding of metals to various components of the membrane,
208 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
partitioning of hydrophobic metal complexes, and in some cases transport of specific complexes [11,12]. In the case of ion channels and carrier ligands, interactions of metal ions with specific membrane components have to occur. The chemical species, in which the metal ions occur outside the cells and in which they are transported to the biological interphase, are thus of key significance for subsequent specific interactions at the membrane. For any mechanism involving binding of metal ions to biological components, ligand-exchange reactions of metals between the external ligands and the biological ligands have to occur. In the simplest case, such reactions involve the exchange of coordinated water molecules of a hydrated metal ion with a biological ligand. Other complexes may exchange by dissociation of the complexes to the hydrated metal ion, or by other mechanisms. The stability of the metal species outside the membranes, as well as their dissociation kinetics, are thus of significance with regard to binding to biological ligands ([13,14], and Chapter 3 in this volume). On the other hand, the presence of hydrophobic complexes is a prerequisite for partitioning and diffusion of metals into the lipid bilayer. In the following paragraphs, various types of metal complexes will be discussed, which are relevant to the interactions of metals in aquatic systems. The role of these various types of metal complexes with respect to interactions at the biological interphases will be systematically examined.
2
SPECIATION
2.1 2.1.1
SPECIATION OF ORGANIC COMPOUNDS Hydrophobic Ionogenic Organic Compounds (HIOCs)
HIOC is a term that refers to organic compounds that can be ionised in the environment and thus are present in two or more species. Basically, the group of HIOCs includes weak organic acids and bases and organometallic compounds, e.g. organotins. This latter group will be treated separately (see Section 2.2.4). Organic acids dissociate according to: HAw Ð Hþ w þ Aw
(1)
where HAw and A w are the acid and the conjugate base in the aqueous phase, and Hþ represents the proton. The corresponding equilibrium constant, i.e. the w acidity constant Kaw , is defined by: K aw ¼
aAw aHw aHAw
(2)
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where aHAw and aAw are the activities of species HA and A in the aqueous phase and aHw is the hydrogen ion activity ðpH ¼ log aHw Þ. The basicity constant can be defined in an analogous way to the acidity constant. To simplify matters, one can also speak of the acidity constant of the conjugate acid of a base, HBþ . Compounds with an acidity constant, pKaw , in the range of 4 to 10, i.e. weak organic acids or bases, are present in two species forms at ambient pH. This pKaw range includes aromatic alcohols and thiols, carboxylic acids, aromatic amines and heterocyclic amines [15]. Conversely, alkyl-H and saturated alcohols do not undergo protonation/deprotonation in water ðpKaw 14Þ. 2.1.2
Examples of Environmentally Relevant HIOCs
Many pesticides are moderate to weak acids. Strong acid pollutants are fully ionised at ambient pH. Examples include trifluoroacetic and chloroacetic acids, whose use as herbicides has been banned but which still occur as solvent degradation products [16], or the pesticide 2,4,5-trichlorophenoxyacetic acid ðpKaw 2:83Þ. Then there are a number of pesticides, e.g. the phenolic herbicide dinoseb and the fungicide pentachlorophenol, whose speciation varies strongly in the environmental pH-range. For this reason, one has to consider the pKaw when estimating their environmental fate. Structures of the compounds discussed in this section are depicted in Table 1, together with a listing of their pKaw and octanol–water partition coefficients, Kow , of the neutral species (unless otherwise indicated). Typical basic pollutants include the industrial chemicals aniline and N,N-dimethylaniline. Recently, attention has turned to drugs and personal care products, many of which are acids and bases. More than 30 000 ionisable compounds are listed in the 1999 world drug index, which corresponds with 63 % of all drugs [17]. Interestingly, some of the pharmaceuticals are structurally very similar to known environmental pollutants: e.g. the pesticide mecoprop and the hydrolysis product of the lipid regulator clofibrate, clofibric acid, are isomeric molecules that exhibit similar behaviour in the environment [18]. Drugs and personal care products are increasingly detected in the effluents of wastewater treatment plants, rivers, lakes, and the marine environment [19–23]. Prominent examples are the anti-inflammatory drug ibuprofen [24] and disinfectant agents such as triclosan or tetrabromocresol [19]. Ciprofloxazin, a zwitterionic antibiotic (at pH 7) was identified to be the agent responsible for mutagenicity in hospital wastewater [25]. Another example is oxalinic acid, a frequently used antibiotic in fish farming [26]. Since the membrane–water partition coefficient is a relevant parameter for the pharmacokinetic behaviour, i.e. drug uptake, many more studies on the speciation of drugs at membrane interphases can be found in the
210 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES Table 1. Examples of environmentally relevant HIOCs Compound name
Structure
Aniline
NH2
Pentachlorophenol
Cl
Cl
Cl
Application
pKaw
log Kow
Industrial intermediate
4.63a
2.35a
Fungicide
4.75b
5.24b
Pesticide
4.62b
3.56b
Herbicide
3.78c
1.26c at pH 7
Metabolite of pharmaceutical compound (blood lipid regulator)
4.46d
2.57
anti-inflammatory drug
4.45/ 5.7e
3.5e
Antiseptic used in 8.05 f personal care products
5.4 f
OH Cl
Cl
Dinoseb
O2N
OH NO2
Mecoprop
CH3 Cl
O H3C
COOH H
Clofibric acid Cl
O H3 C
Ibuprofen
H3 C
Triclosan
Cl
COOH
OH O Cl Cl
COOH CH3
B. I. ESCHER AND L. SIGG Ciprofloxacin
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O OOC
F
N
Antibiotic in human medicine
6.08 and 8.58g
0.074 at pH 7.4h
Veterinary antibiotic
6.9i
0.68i
N N H H
Oxalinic acid
CH3 O
N
O
COOH O
a
[15]. b [116]. c [245]. d,e [22]. f Own measurements (unpublished). g [246]. h [247]. i [26].
literature than reported for environmental pollutants. Since most of the conclusions are generally valid and since drugs have been identified as emerging environmental pollutants, this review also covers work from the pharmaceutical sector. 2.1.3
Redox Speciation of Organic Compounds
Redox conditions have a minor impact on organic compounds as compared with metals. Most organic chemicals that pose a toxicological hazard due to their chemical reactivity are not hydrophobic enough to create a toxicological problem in a hydrophobic membrane environment. Exceptions include quinones, some of which can be reduced to hydroquinones through enzymatic and nonenzymatic redox cycling with their corresponding semiquinone radicals [27]. Redox cycling may also produce reactive oxygen species that – apart from reaction with other biomolecules – may damage the membrane lipids and membrane-bound proteins. Protection against cell damage due to oxidative stress is provided, amongst others, by glutathione (GSH), a cellular tripeptide with a thiol function in a cysteine residue. GSH is deprotonated to GS , which is a scavenger for electrophilic compounds and is reduced to GS–SG in defense of reactive oxygen species [28]. 2.2
METAL SPECIATION
Metal speciation is discussed here from the perspective of the speciation occurring in natural freshwater environments. This speciation is relevant for interactions of metals with aquatic organisms.
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2.2.1
Inorganic Complexes
Inorganic ligands in aqueous solutions, and in particular in natural freshwaters, include, in addition to H2 O and OH , the major ions carbonate and bicarbonate, chloride, sulfate and also phosphate [29]. The distribution of metal ions between these ligands depends on pH and on the relative concentrations of the ligands. The pH is a master variable with regard to the occurrence of hydrolysed species and to the formation of carbonate and bicarbonate complexes. Stability constants of metal complexes with inorganic ligands are generally well known [30]. Inorganic speciation of metals can therefore easily be evaluated by thermodynamic calculations if the composition of the solution is known [29,31]. Under typical freshwater conditions, at pH 7–9 and in presence of millimolar concentrations of carbonate, most transition metals in solution (Cu(II), Zn(II), Ni(II), Co(II), Cd(II), Fe(III), etc.) occur predominantly as hydroxo or carbonato complexes. For a few metals, chloro complexes may be predominant (Ag(I), Hg(II) ), if chloride is in the range 104 103 mol dm3 or higher. Alkali and alkali-earth cations occur predominantly as free aquo metal ions [29]. At lower pH values, the fraction of free aquo metal ions generally increases. Strong sulfide complexes of several transition metals have recently been shown to occur even under oxic conditions [32,33].
2.2.2
Organic Complexes: Hydrophilic Complexes
Organic ligands that may significantly influence the speciation of metals in natural waters (and in other aquatic media) comprise a large range of compounds. Complexation of metal ions by organic ligands cannot be modelled as simply as the inorganic speciation, because the composition of natural organic matter is not known in detail and the binding characteristics of the macromolecular ligands require sophisticated models. Simple organic molecules such as small carboxylic acids (oxalate, acetate, malonate, citrate, etc.), amino acids and phenols are all ligands for metals. Such compounds may all occur as degradation products of organic matter in natural waters. The complexes formed are typically charged hydrophilic complexes. The stability of the metal complexes with these ligands is, however, moderate in most cases. Model calculations including such compounds at realistic concentrations indicate that their effects on speciation are relatively small [29]. A further category of ligands includes the synthetic chelating polycarboxylate ligands such as EDTA (ethylenediaminetetraacetate) and NTA (nitrilotriacetate). These ligands form very stable complexes with a number of metals. They occur at low concentrations in natural waters, due to inputs from sewage treatment plants [34]. They have also been widely used as complexing ligands in culture media for investigation of interactions between metals and organ-
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isms, especially for algae, in order to control the free aquo metal ion concentrations [35,36]. If present in excess of the trace metals, as it is the case in culture media, EDTA and NTA dominate the speciation of trace metals. These charged hydrophilic complexes are very soluble and increase the overall solubility of trace metals. Fulvic acids (FA) and humic acids (HA) play a very important role in the complexation of metals in natural waters [37–39]. FA and HA are polymeric ligands with molecular weights typically in the range 500–2000 g mol1 for FA and 2000–5000 g mol1 for HA. Recent spectroscopic work has started to reveal more clearly the structures of these polymeric molecules [40–43]. FA and HA comprise both hydrophobic and hydrophilic domains. Phenolic and carboxylic groups bound to various aromatic and aliphatic structures are the most important complexing functional groups in FA and HA. To describe the complexation properties of FA and HA, the heterogeneity of the various binding sites and the polyelectrolyte nature of metal ion binding have to be taken into account [37]. The various models for complexation of metals with FA and HA cannot be discussed here; for details see, for example, [37–39, 44–47]. It is clear that the stability of metal complexes with FA and HA depends on the ratio of metal to ligand, because the strongest binding sites are occupied at low metal concentrations. Increasing evidence for the presence of unknown strong organic ligands for trace metals has been shown by a number of studies in seawater [48–57,58,59] and in freshwater [60–64]. This evidence is based on indirect methods, which indicate strong complexation of the metals, but do not identify the structure of the ligands involved. A controversial discussion is ongoing in the literature on the question whether specific strong ligands can be distinguished from strong binding by humic and fulvic acids [65–67]. Specific strong chelators may be produced biologically and either excreted during growth of the organisms (algae, cyanobacteria, bacteria) or released upon partial decomposition of the biomass [68–71]. They may include biological ligands such as glutathione, phytochelatins and siderophores [59,71,72]. Very stable complexes may be due to the occurrence of N- or S-functional groups in chelating ligands [5,72,73]. In a complex solution the total metal concentration in solution cM, t can be represented by the following general equation: cM, t ¼ cM þ
X i
cMLi inorg þ
X
cMLi org
(3)
i
P where cM represents the free metal ion concentration, cMLi inorg the sum of P i cMLi org the sum of organic complexes. inorganic complexes, and i
The relationship between free metal ion activity (or concentration, depending on the conventions used for the stability constants) and cM, t is given by:
214 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
cM ¼ cM, t
1þ
X
bi cLi inorg
i
þ
X
!1 bi cLi org
(4)
i
where bi represents the overall thermodynamic stability constant of the complex MLi , cLi inorg the concentration of inorganic ligand Linorg and cLi org the concentration of organic ligand Lorg . If all stability constants and ligand concentrations are known, the free metal ion concentration at equilibrium can be calculated by equation (4). Complexation by natural organic matter cannot be reliably represented in terms of simple discrete ligands and requires sophisticated models. Figure 1 illustrates the speciation of Cu in the presence of various inorg
oxal
cit
FA/HA
L1/L2
EDTA
CuCO03
CuCO03
CuOHcit
CuFA
CuL1
CuEDTA
−4
−6
log[Cu-species]
−8 Cu2+
CuOx Cu2+
−10
−12
−14
Cu(OH)02
Cu(OH)02
CuCO03
CuHA CuL2 CuCO03
Cucit Cu2+
Cu2+
Cu(OH)02
Cu(OH)02
CuCO03
CuCO03
Cu2+
Cu2+
Cu(OH)02
Cu(OH)02
−16
Figure 1. Calculated speciation of Cu in presence of inorganic and organic ligands. In all cases, C Cu, t is 2 108 mol dm3 (dashed line), pH 8, total carbonate species 2 103 mol dm3 . Stability constants are from [30], unless otherwise indicated. Inorg: speciation in the presence of the inorganic ligands carbonate and hydroxide only; oxal: 1 106 mol dm3 oxalate; cit: 1 106 mol dm3 citrate; FA/HA: 3 mg dm3 FA and 0:3 mg dm3 HA, according to the WHAM model [44]; L1/L2: lakewater ligands as determined in [60], 8 108 mol dm3 L1 (log K ¼ 14:0, conditional for pH 8) and 5 107 mol dm3 L2 (log K ¼ 12:0, conditional for pH 8); EDTA: 2 105 mol dm3 EDTA in a culture medium also containing Ca and other trace elements. Only the most abundant Cu species are shown
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ligands, namely of inorganic ligands (carbonate and hydroxide), of oxalate, of citrate, of FA and HA according to the WHAM model [44], of strong lakewater ligands (L1/L2) [60], and of EDTA in a culture medium. This figure illustrates how the concentration of the free aquo ion Cu2þ decreases in the presence of strong ligands. Copper is over 90% organically complexed in the presence of citrate, FA and HA, EDTA and lakewater ligands under the conditions considered in Figure 1. 2.2.3
Organic Complexes: Hydrophobic Complexes
A number of organic ligands forming hydrophobic complexes are known in analytical chemistry, in which they have been used as carriers for metal extraction in organic solvents (e.g. dithiocarbamates, oxine [74]) and as ligands in carrier-based ion-selective electrodes (e.g. macrocyclic compounds [75,76]). The occurrence of ligands forming organic hydrophobic complexes in natural waters has not been clearly demonstrated, but is likely. Dithiocarbamate metal complexes are used in fungicides and may therefore occur in natural waters [77]. The significance of other organic pollutants in forming hydrophobic metal complexes has not yet been investigated in detail. Some natural compounds may also form uncharged hydrophobic complexes, such as e.g. catechol. Chlorocatechols form complexes with different hydrophobic character [78]. Furthermore, some uncharged inorganic species (e.g. HgCl02 , HgS0 , AgCl0 ) also behave as hydrophobic complexes [79–81]. The same relationships between free metal ions and total metal concentrations hold as above. Equation (4) may include the concentration of hydrophobic ligands. However, the occurrence of hydrophobic complexes is particularly relevant to the interactions with membranes. 2.2.4
Organometallic Species
The occurrence of organometallic species (compounds with at least one carbon– metal bond) is a key factor for the biological effects of a number of metallic elements. Of particular environmental concern are methylmercury and other organomercury compounds, organotin compounds (e.g. tributyltin) and alkylated lead compounds [82]. Methylmercury has been observed in many aquatic systems in varying proportions to total mercury [83,84–86]. Methylmercury is formed in natural environments by bacterial methylation [81,87]. Methylmercury forms inorganic complexes, which may be uncharged, in particular CH3 HgCl0 with hydrophobic properties [79]. Methylmercury may also strongly bind to humic acids [88]. Triorganotin compounds, in particular tributyltin (TBT) and triphenyltin (TPT) have been introduced directly into the environment as pesticides and
216 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
antifouling agents in boat paint. The organotin compounds form uncharged hydroxo and chloro complexes (R3 SnOH0 , R3 SnCl0 ), which also exhibit hydrophobic properties [89]. At higher pH, triorganotins are present as hydroxides R3 SnOH0 . At low pH, they become protonated to the cation R3 Snþ with a pKaw of 6.25 for TBT and 5.2 for TPT. The cation forms strong complexes with Cl , Br , and NO 3 , and more labile complexes with ClO4 . 2.2.5
Redox Reactions
A number of trace metals may occur in different redox states (Mn(II/IV), Fe(II/III), Cu(I/II), Cr(III/VI), Hg(0/II) etc.). The various oxidation states of an element differ in their coordination properties, and thus in their speciation. For example, Cr(VI) occurs as an anionic species (CrO2 4 ), whereas Cr(III) forms hydrolysed species and has limited solubility as an (hydr)oxide. In oxygenated water, the oxidised species of these elements predominate at equilibrium, but the reduced species may be present at low concentrations as reactive intermediates (e.g. Fe(II), Mn(II)). Low concentrations of Fe(II) have been observed in oxygenated seawater and fresh water under the influence of light [90–93]. Reduction or oxidation reactions may also occur at the biological interphases [94,95]. Binding to biological ligands is also strongly dependent on the redox states of a metal. 2.2.6
Solubility of Solid Phases and Binding to Colloids and Particles
Under relevant conditions (neutral pH, presence of carbonate) the concentrations of dissolved trace metals are limited by the solubility of their solid hydroxides or carbonates. A prominent example is the solubility of Fe(III), which is very low (around 1010 mol dm3 ) under relevant conditions of fresh water or seawater at neutral pH [96]. Iron(III) occurs primarily as solid or colloidal iron hydroxide at neutral pH. Even if the solubility limit of relevant solid phases is not exceeded, a large fraction of trace metals in natural waters often occurs bound to particles of various size ranges (nanometres to millimetres). An important process for binding of metals to particles is adsorption to mineral surfaces (often oxides or hydroxides) [29]. The colloidal size range is generally defined as the range of a few nanometres to less than 1 mm, respectively the range of molar mass larger than 100010 000 g mol1 [97]. This size range comprises small mineral particles, as well as organic macromolecules such as large HA. Recent studies have indicated the significance of the colloidal fraction for the speciation of trace metals in natural waters [98–101]. Metals in this size range may be either organically complexed to macromolecules, adsorbed to mineral surfaces, or (co)-precipitated as solid phases.
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217
Kinetics of Complexation Reactions
The kinetics of ligand-exchange reactions must be considered to evaluate exchange of metals between external ligands and biological ligands. The kinetics of complex formation of a metal with a ligand depend on the water exchange rates of metal ions [102,103]. For most metal ions, the water-exchange rates are fast, and complex formation reactions are generally fast (on timescales of less than seconds). Noteworthy exceptions are complex formation reactions of Ni(II), Cr(III), Fe(III), and Al(III), which are much slower than reactions of other cations and may occur on time scales of hours to days. If the metals bound in complexes exchange with biological ligands, the dissociation kinetics of these complexes, the ligand-exchange kinetics and the association kinetics with the biological ligands must be considered. Simple dissociation kinetics of complexes are related to their thermodynamic stability constants by the relationship: M þ L Ð ML
(5)
K ¼ kf =kd
(6)
where K is the thermodynamic equilibrium constant of reaction (5), kf is the rate constant for formation of the complex ML, and kd is the rate constant for dissociation of the complex ML. Dissociation kinetics of strong complexes are thus typically slow. The role of complexation kinetics for the biouptake processes is discussed in Chapters 3 and 10 in this volume and in ref. [14].
3 3.1
MEMBRANES AND SURROGATES OR MEMBRANE MODELS OCTANOL
The octanol–water partition coefficient, Kow , is the most widely used descriptor of hydrophobicity in quantitative structure–activity relationships (QSAR), which are used to describe sorption to organic matter, soil, and sediments [15], bioaccumulation [104], and toxicity [105–107]. Octanol is an amphiphilic 1 bulk solvent with a molar volume of 0:12 dm3 mol when saturated with water. In the octanol–water system, octanol contains 2:3 mol dm3 of water (one molecule of water per four molecules of octanol) and water is saturated with 4:5 103 mol dm3 octanol. Octanol is more suitable than any other solvent system (for) mimicking biological membranes and organic matter properties, because it contains an aliphatic alkyl chain for pure van der Waals interactions plus the alcohol group, which can act as a hydrogen donor and acceptor.
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3.2
OTHER SOLVENT–WATER SYSTEMS
The octanol–water partition coefficient appears to correlate better with biological activity than partition coefficients in other solvent–water mixtures as, for example, hexane–water, because the amphiphilic nature of octanol can accommodate a greater variety of more or less hydrophobic molecules. In pharmaceutical science, the notion was taken up that no single solvent– water system is able to mimic the complex properties of a biological membrane. Leahy et al. proposed the so-called ‘critical quartet’ of organic solvent–water systems [108]. The original quartet was composed of an alkane, representing a solvent, which can merely interact via van der Waals forces, octanol as amphiprotic solvent, chloroform as a hydrogen donor, and dibutylether or propylene glycol dipelargonate as a hydrogen acceptor. The idea behind this choice was that uptake in biological membranes is a composite process influenced by the different types of interactions listed above. Chiou chose glyceryl trioleate (triolein) as model lipid because of its similarity to triglycerides which are abundant in organisms [109]. Triolein is also a bulk lipid and the good correlation with the bioconcentration factor is restricted to neutral compounds of moderate hydrophobicity. No attempts were made to measure partitioning of ionogenic compounds with the glyceryl trioleate–water partition system.
3.3
LIPOSOMES
All model systems mentioned in Sections 3.1 and 3.2 have the disadvantage of being bulk phases. The simplest model system that mimicks the anisotropic properties and ordered structure of biological membranes are liposomes. Liposomes are artificial lipid bilayer vesicles of known lipid composition and of controlled size [110,111]. They can be prepared either in the gel state or in the liquid crystalline state (as in biological membranes), depending on temperature and the phospholipid composition. Alternatively, planar black lipid bilayers have been used for partition studies. A black lipid film is usually formed across a small hole in a Teflon sheet. Due to its high solvent content (mostly decane), it is not the best system for partitioning studies, but it offers the advantage of allowing direct measurement of the permeability of the membrane [112]. Principal differences between bulk media–water and membrane–water partition coefficients are listed in Table 2. These differences are essentially based on the several orders of magnitude difference in surface-to-volume ratio. In the liposomal system, charges built up due to sorption of charged species can be electrically ‘neutralised’ by counter-ions from the electrolyte (diffuse double
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Table 2. Comparison of general properties of membrane model systems Bulk media
Membrane
Small equilibrium surface-to-volume ratios Bulk media have to maintain electrical neutrality
Very high surface-to-volume ratios because membranes are only 5–10 nm thick Sorbed charged species build up surface potential shielded by the electrolyte
layer). Hence, the properties of the membrane–water interphase, as opposed to those of the membrane core, become important. There are a variety of experimental methods available to determine liposome–water partition constants [113]. The best established one is equilibrium dialysis [114–116]. Alternatively, ultracentrifugation or ultrafiltration can be used to separate the liposomes from the aqueous phase. The pH-metric method is specifically designed to measure partition constants of ionogenic compounds [117]. Other methods initially developed for neutral compounds, e.g. lipid bilayers immobilised covalently [118] or noncovalently [119] to a solid support material have been evaluated recently for their applicability for HIOCs [120]. Liposomes can also be immobilised in agarose–dextran gel beads and packed in chromatography columns [121]. This method has already been applied to investigate the partitioning behaviour of lipophilic cations [122]. Methods specific for charged molecules include electrophoretic measurements of the x-potential [123] and conductance measurements [7]. 3.4
BIOLOGICAL MEMBRANES
While most model lipid membranes are composed of only one or a limited number of components, biological membranes contain a wide variety of phospholipids and other lipids. In addition, integral membrane proteins can make up the majority of components in a membrane. Biological membranes are extensively reviewed in Chapters 1 and 2 of this volume. At this point, it is only important to mention that for organic compounds and HIOCs, pure phospholipid bilayers appear to be a good surrogate for biological membranes, because organic compounds are mainly localised in the lipid domain of the membranes, although they can still interact and interfere with functional proteins. A comparison of membrane–water distribution ratios of various substituted phenols at pH 7, some of which were neutral, others partially or fully ionised, measured with phosphatidyl choline liposomes and chromatophores (photosynthetically active membranes that contain up to 85% proteins) revealed equal distribution ratios if the values were normalised to lipid content [116]. Hence, sorption to proteins was negligible in this case.
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4
INTERACTIONS OF HIOCS WITH BIOLOGICAL INTERPHASES
4.1 4.1.1
PARTITIONING AND SORPTION MODELS Octanol–Water Partitioning
Kow is the most successful descriptor for hydrophobicity of neutral organic compounds [17,124]. Early attempts to extend the simple concept of Kow from neutral organic compounds to HIOCs have either neglected the partitioning of the charged species [125] or have determined apparent octanol–water distribution ratios Dow (pH,I) at a given pH and with a given ionic strength [126,127]. Dow (pH,I) is defined as the ratio of the sum of all organic species i in the octanol phase, cio , to the sum of all organic species in the aqueous phase, ciw . P cio i P (7) Dow (pH, I) ¼ ciw i
The distribution ratio is strongly dependent on pH and the ionic strength, as is depicted in Figure 2 for the organic acid dinitro-o-cresol (DNOC) and for the organic base 3,4-dimethylaniline (DMA). Despite a large decrease of the Dow at pH values where the charged species predominates, partitioning of the charged species cannot fully be neglected. Depending on the ionic strength, the partitioning of the neutral species is three or more orders of magnitude higher than that of the ionic species. Westall and co-workers [128–130] proposed a detailed model of octanol–water partitioning of HIOCs whose essential equations are depicted in Figure 3a for acids and 3b for bases. For simplicity, all equations for electrolyte partitioning of monovalent and divalent salts [131] have been omitted. Nevertheless, in most cases, in particular in experiments with monovalent salts, pure electrolyte partitioning accounted for less than 10% of the overall partitioning. The lines in Figure 2, which show lipophilicity profiles in the octanol–water system, have been calculated with the full model and the equilibrium constants given in Jafvert et al., and Johnson and Westall [129,130]. Ion pair formation is the dominant process for controlling the partitioning of charged species. Typically, ion pair partitioning is about 10 000 times higher as compared to ion partitioning. Ion pair formation in octanol is also dependent upon the type of counter-ion present. The ion pair of pentachlorophenoxide (PCP ) with Liþ partitions about half a log-unit preferentially into octanol than þ the ion pair with Naþ or K , both of which form about equally strong ion pairs [129]. Moreover, the acidic drug proxicromil exhibited ion pair partitioning that was threefold higher with Liþ than with Naþ [132]. Experiments of ion pair 2þ formation of PCP with divalent counter-ions Mg2þ and Ca suggest that a 2:1 2þ complex of PCP and M is formed, most likely in the form of MPCPþ with PCP as a counter-ion [129]. For bases, so far only chloride has been used as
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3
0.1 mol dm−3 K+
(a)
0.01 mol dm−3 K
2
logD
logDmw
logDow
1
+
0 0.1 mol dm−3 K+ −1 0.01 mol dm−3 K+
−2
0.001 mol dm−3 K+
−3 −4
0.0001 mol dm−3 K 2
4
6
8
+
10
12
pH
3 0.1 mol dm−3 Cl−
(b) logDmw
2
logD
1
0
−1 logDow
−2
2
4
6
8
10
pH
Figure 2. Lipophilicity profiles of (a) dinitro-o-cresol and (b) 3,4-dimethylaniline. Dow values were calculated with the data and model described by Jafvert et al. [129], and Johnson and Westall [130]. Dmw values were calculated with the data and model described by Escher et al. [140]
counter-ion [130]. Again, a difference between the partitioning of the neutral and charged species greater than two orders of magnitude was observed with the ion pair as the dominant complex for all charged species. Only the unsubstituted aniline does not appear to form an ion pair with chloride [130].
222 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES (a)
Ao− /Mo+
HAo KHAow HAw
(b)
K
Ka
+ Hw
KAMow
+
− Aw
BHo+ /Xo−
Bo KBow Bw
AMo
+ A−Mow
K
− BH+X ow
+ Hw+
Kb
BHw+
+
+
Mw
BHXo KBHXow + Xw−
Figure 3. Model for octanol–water partitioning of organic acids (a) and organic bases (b). The figure has been prepared according to the model described by Jafvert et al. [129]; the electrolyte partitioning is omitted for brevity Zwitterion
logDow
10 000
Cation
Anion
1000
Zwitterion
100 0
2
4
6 pH
8
10
12
Figure 4. Scheme of lipophilicity profile of zwitterionic compounds. The line drawn represents the case where the neutral tautomer predominates or the zwitterion is rather hydrophobic, resulting in a bell-shaped profile. The dashed line represents the case where the zwitterion predominates and intramolecular interactions are not possible, resulting in a U-shaped profile. Adapted with permission from [133]: Pagliara, A. et al. (1997). ‘Lipophilicity profiles of ampholytes’, Chem. Rev., 97, 3385–3400; copyright (1997) American Chemical Society
Particularly interesting examples are also the lipophilicity profiles of ampholytes. Depending on the ratio between the neutral tautomer and the zwitterionic tautomer, the log Dow versus pH profile may be bell-shaped or U-shaped [133] (Figure 4). For zwitterions, the shape of the lipophilicity profile depends upon the structure and conformation of the molecule. If the charged groups are situated in proximity and can interact with each other, the zwitterion might be more hydrophobic than the anionic and the cationic species, resulting in a bell-shaped lipophilicity profile. If, however, intramolecular interactions are not possible for steric reasons, the lipophilicity profile is U-shaped [133].
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4.1.2 Membrane–Water Partitioning: General Derivation of Membrane–Water Partition Coefficients of a Charged or Neutral Compound or Species The chemical potential, miw , of a compound i in the aqueous phase, indicated by subscript w, is defined according to: miw ¼ m0iw þ RT ln aiw
(8)
where m0iw is the standard chemical potential and aiw is the activity of compound i in the aqueous phase. If charged lipids are incorporated in the membrane phase (indicated by subscript m) or if charged species are sorbed to the membrane surface, an ~ im will build up according to: electrochemical potential m ~im ¼ m0im þ RT ln aim þ zi F C m
(9)
where zi F C corresponds with the electrostatic contribution to the electrochemical potential in the charged membrane bilayer. At equilibrium, the electrochemical potentials in the two phases are equal (m~im ¼ miw ), and equations (8) and (9) can be combined to give: Dmw G0i ¼ m0im m0iw ¼ RT ln
aim zi F C aiw
(10)
where the standard free-energy change for the phase-transfer reaction, Dmw G0i , is related to the dimensionless partition coefficient, K0i , by a Boltzmann-type expression: Ki0 ¼
aim zRTi F C e aiw
(11)
The activity ai of a given compound or species i is a function of its activity coefficient gi and the mole fraction xi : ai ¼ gi xi
(12)
Activity coefficients in the aqueous phase, giw , of neutral molecules are set equal to one because of the zero charge, and under the assumption that the activity coefficient of the infinitely diluted solution equals the actual activity coefficient. The activity coefficients of the charged species can be approximated with the Davies equation: log giw
pffiffiffi I pffiffiffi BI ¼ zi A 1þ I
(13)
224 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
where zi is the charge, I is the ionic strength (mol dm3 ), A ¼ 0:509, and B ¼ 0:3 at 298 K. The activity coefficients in the membrane phase are set to one. This assumption is justified at low concentrations of ions in the membrane, especially when considering their location at the interface of the hydrophobic and hydrophilic domains [134], but might be inappropriate at concentrations near saturation. The dimensionless partition coefficient Ki0 is based on mole fractions xi or number of moles ni . In the literature, partition coefficients are more often defined as concentration ratios. At low solute concentration and when the adsorbed amounts become very small, the activity coefficients approach zero and the surface potential also becomes insignificant (zi F c ! 0): Ki0
xim nim ¼ xiw niw
(14)
Under these conditions, Ki0 can be converted to the concentration-based dimen sionless partition coefficient, Kimw , with the following equation: ¼ Kimw
cim ¼ fKi0 ciw
(15)
where f is the phase ratio, i.e. the ratio of the molar volume of the aqueous phase, Vw , and the molar volume of the membrane lipids, Vm : f¼
Vm Vw
(16)
Often, the concentration of compound or species i in the membrane is given in molality (mol kg1 ), which yields the membrane–water partition coefficient, Kimw (in units of dm3w kg1 m ): Kimw ¼
mim ¼ frm Ki0 ciw
(17)
where rm is the mass concentration or density of the membrane lipids. The and Kimw are approximately equal because the density of phosvalues of Kimw pholipids, rPL (1:015 kgPL dm3 PL [135]) is close to the density of water. The description of the sorption of charged molecules at a charged interface includes an electrostatic term, which is dependent upon the interfacial potential difference, Dc (V). This term is in turn related to the surface charge density, s (C m2 ), through an electric double layer model. The surface charge density is calculated from the concentrations of charged molecules at the interface under the assumption that the membrane itself has a net zero charge, as is the case, for example, for membranes constructed from the zwitterionic lecithin. Moreover,
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one can assume that the sorption of i does not lead to an expansion of the membrane surface. P zi mim (18) s¼F SPL NA where NA is the Avogadro constant, SPL is the surface area occupied by a single 1 membrane lipid molecule (SPL 0:7 nm2 molecule for phospholipids [136]). The Gouy–Chapman diffuse layer model has been shown to describe adequately the electrostatic potential produced by charges at the surface of the membrane [137]. For a symmetrical background electrolyte, s and c are related by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fc (19) s ¼ 8er e0 RTI sin h 2RT where I refers to the ionic strength in units of mol cm3 , er is the relative permittivity or dielectric constant, and e0 is the permittivity of the free space. Alternatively, in the literature, the constant capacitance model and the Stern model were used to describe the dependence of the surface charge density on the surface potential. In the constant capacitance model, the surface charge is defined as: s ¼ Cc
(20)
where C is the specific capacitance, which varies typically between 0.2 and 1 Fm2 in biological membranes [138]. The Stern model is effectively a serial combination of the constant capacitance and Gouy–Chapman model. All three models yielded similar quality of fits to the experimental liposome–water partitioning data [120]. There are several underlying assumptions for using the combination of the Boltzmann equation and the Gouy–Chapman theory of diffuse double layer or other electrostatic models used to describe the partitioning of charged species into membranes. First, the charged species must not be uniformly distributed throughout the membrane, but rather be sorbed close to the surface of the membrane. This is a realistic assumption as will be demonstrated in detail in Section 4.7. This suggests that in the case of membranes we consider an adsorption mechanism rather than absorption or partitioning. Second, it has to be assumed that the charges are distributed uniformly across the surface, and that neither the standard chemical potential nor the activity coefficient varies with distance from the membrane. This assumption is more difficult to justify even if lateral diffusion of bilayer components is quite fast. The good agreement of experimental data with model predictions, however, demonstrates that this assumption is reasonable.
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Several investigators [7,123] suggested the use of a Langmuir-type saturation model in addition to the electrostatic model to account for saturation effects. The Langmuir model implies that there are a finite number of localised sorption sites [15]: mim ¼
mmax mim im KL ciw or KL ciw ¼ max 1 þ KL ciw mim mim
(21)
where mmax im is the maximum mass concentration of molecules adsorbed and KL is the Langmuir sorption constant, which is equivalent to Kimw . If the adsorption sites are not localised in space (as is the case for sorption to a fluid lipid membrane), then the Langmuir equation could be transformed to the Volmer isotherm [7], mim
KV ciw ¼
mim max ðmim mim Þ max mim mim
(22)
where KV is the Volmer association coefficient. If mim mmax im then the Volmer equation reduces to the Langmuir equation. The combined Langmuir–Stern equation was quite insensitive to changes in the number of molecules maximally adsorbed to the surface [7]. McLaughlin and Harary found saturation at one molecule of negatively charged 2,6-toluidinyl naphthalenesulfonate (TNS) per one to three phosphatidyl choline (PC) molecules [7]. A study on the sorption of pentachlorophenoxide to PC vesicles based on electrophoretic mobility measurements found sorption sites in the size of 4.3 PC molecules [123]. A later study considered both neutral and charged PCP and found sorption sites in the size of one and seven PC molecules, respectively [139]. Such high membrane coverage would lead, however, to a significant membrane expansion (PCP, 50 % saturated, 25 % membrane expansion, PCP , 50 % saturated, 3.6% membrane expansion [140]). Thus, later studies proposed that Langmuir-type sorption cannot be reasonably obtained without membrane expansion, and that experimentally data should be fitted with an electrostatic model alone [120,140]. Neutral compounds show saturation only at membrane burdens, which begin to affect membrane structures. Very hydrophobic and well-shielded anions and cations such as tetraphenylborate or tetraphenylphosphonium already show saturation at a binding density of 1 per 100 lipids [6]. Anyway, these disputes over number of sorption sites are purely academic. Such high concentrations are not likely to be encountered in the environment and are likely to be lethal for all biological organisms. 4.2
THERMODYNAMICS OF MEMBRANE–WATER PARTITIONING
The partition coefficient Kmw is directly related to the free energy of transfer between the aqueous and the membrane phase. The enthalpy and entropy contri-
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butions to the partitioning process can be deduced from a van’t Hoff plot of lnKmw versus the inverse temperature [1]. Although the thermodynamic behaviour depends not only on the nature of the solvent but also of the solute, some general findings can be summarised. The Kow and Kmw usually increase with increasing temperature [141,142]. The thermodynamics of Kow and Kmw are similar in those cases where nonpolar interactions are dominant [143] but the partitioning process is more complex for Kmw than for Kow , because it is influenced by structural changes in the lipid bilayer. Sulfonamides had positive enthalpies, DH, for Kmw and varying sign for Kow , whereas entropies, DS, were positive in both systems [144]. For a series of linear aliphatic alcohols, the enthalpy is positive for shorter alcohols (chain length ¼ 6) but becomes large and more negative for the longer alcohols (chain length ¼ 7) [145]. In addition, if cholesterol is incorporated in the membrane, making the membrane more rigid, then the enthalpy becomes large and positive but decreases linearly with increasing chain length [145]. The strong effect of the lipid type indicates that lipid–lipid interactions play a significant role for the thermodynamics in addition to lipid–solute interactions. The large difference in the hydrophobic binding of the cationic tetraphenylphosphonium and anionic tetraphenylborate is caused by the large changes in the enthalpy of the phase transfer. While the entropic contribution is similar for both ions, the DH is slightly positive ( þ 14:6 kJ mol1 ) for tetraphenylphosphonium (TPPþ ), whereas it is negative (up to 15:1 kJ mol1 ) for tetraphenylborate (TPB ) [6]. It is also interesting to note that variations in the structure of TPB by the introduction of F, Cl, or CF3 on the phenyl rings or replacement of a phenyl group by a cyano-group do not have a large influence on the thermodynamics of sorption, but strongly influence the movement across the membrane. The central energy barrier in the membrane is the lower the better the charge is delocalised over the entire molecule [146]. The thermodynamics of partitioning strongly depend on the physical state of the membrane. Usually more energy is required to insert organic chemicals in bilayers of the gel state as compared with the fluid state. DH and DS of substituted phenols were negative above, and positive below, the transition temperature from gel to liquid crystalline state [143]. For neutral phenols and chlorobenzenes, as well as for sulfonamides at their isoelectric pH, liposome– water partitioning is entropy dominated below, and enthalpy dominated above, the transition temperature [143,144,147,148], but under both conditions large entropy changes were observed, similar to what was observed in fish–water partitioning of chlorobenzenes [149]. 4.3
pH-DEPENDENCE OF MEMBRANE–WATER PARTITIONING
The membrane–water distribution ratio Dmw is defined by the ratio between the sum of the molalities of all species of the considered compound in the
228 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
membrane phase and the sum of concentrations of all species in the aqueous phase. P mim X i Dmw (pH) ¼ P ¼ aiw K imw (23) ciw i i
Dmw can be derived from the partition coefficients of the single species (equation 23) if the fractions of the species, aiw , under given conditions are known (Henderson–Hasselbalch equation). The model for the membrane–water partitioning is depicted in Figure 5a. The pH-profiles of the liposome–water distribution ratios of a representative acid, DNOC (Figure 2a) and a base, DMA (Figure 2b) reveal a much smaller dependence of Dmw on the pH than it is the case for octanol–water partitioning. The Kow of the neutral species and the apparent Dow of the charged species at an ionic strength of 0:010:1 mol dm3 differ by more than three orders of magnitude, while the difference is only one order of magnitude or smaller for the membrane–water system. In addition, there is very little ionic strength dependence on the partitioning of the charged species, due to the lack of ion pair formation in the membrane. Conversely, in octanol, where ion pairing or copartitioning of the counter-ion is a prerequisite for partitioning of a charged species, the concentration of the counter-ion strongly influences the apparent distribution ratio of the charged species. This general trend is important with (a) HAm
HAm
Am−
Am−
HAm
Am−HAm Am−
Am− Am−
K A−
KHA
mw
HAw
(b)
mw
Ka
− w
A
+ H+w
(c)
HBXm
AMm
DAM A−w
DHBX mw
+ Mw+
BH
+ w
mw
+ X−w
Figure 5. (a) Model for membrane–water partitioning of organic acids (equations from [140]). (b) Ion pair formation of the conjugate base of organic acids with alkali ions or metal ions, and (c) of the conjugate acid of organic bases with anionic counterions. Inclusion of ion pair formation in the partitioning model is only necessary under the specific conditions described in the text
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respect to bioaccumulation and toxicity of ionogenic compounds, which will be discussed in Section 4.8. In principle, hydrophobic anions bind more strongly to membranes than structurally similar cations. Tetraphenylphosphonium (TPPþ ) and tetraphenylborate (TPB ) have approximately the same molecular volume and geometry, but TPB binds 5000 times more strongly to lipid bilayers than TPPþ [6]. The difference is caused by the internal dipole moment of a lipid bilayer, which is caused primarily by the ester groups in the phospholipids that link the fatty acids to the glycerol backbone [150]. The resulting dipole is oriented with its positive pole inside the lipid layer, and this favours both absorption and translocation of negatively charged ions [6]. However, this general finding for hydrophobic anions and cations does not necessarily apply for ionogenic pollutants. Many environmentally relevant HIOCs are amphiphilic or form hydrophobic ions that are amphiphilic. Such compounds intercalate into the membrane with their hydrophobic part in the bilayer core and the hydrophilic or ionic domain interacting with the polar and charged head groups of the membrane-building lipids (see Section 3.4). Therefore the better the charge of a compound is accommodated in the polar head groups, the smaller will be the difference between the log Kmw of the neutral and the corresponding charged species, Dmw: Dmw ¼ log Kamw log Kbmw
(24)
where a and b are the acid and conjugate base. Values of Dmw are typically larger for the anion-forming acids than for the cation-forming bases (Table 3) [4,117,120,151,152]. The conjugate base of the phenols can better delocalise the charge over the entire ring system, in particular, when there are electron-withdrawing substituents like nitro-groups. Hence, Dmw is smaller for the phenols than for the carboxy acids, whose charge cannot be delocalised and lacks any direct conjugation with the aromatic ring system. Furthermore, the Dmw values increase in absolute magnitude from primary to tertiary amines (Table 3). This finding can be rationalised in terms of a more pronounced amphiphilicity and better direct interaction of the charged amino group of primary amines with, for example, phosphate groups of the phospholipid molecules of lipid bilayers. Conversely, higher substituted amines have a more shielded charged group that may intercalate deeper into the membrane, but that may have overall less favourable interactions with the membrane. In addition, the steric bulk of several substituents on an amino group appears to disturb the structure of the membrane more than a single substituent. Dmw for a series of (p-methylbenzyl)alkylamines increase with increasing alkyl chain length [153]. This trend was not observed for the corresponding octanol–water partition data, which is additional evidence that the increase in Dmw is caused by an unfavourable steric constraint. The positively charged
230 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES Table 3. Examples of membrane–water partitioning of HIOCs and the difference in partitioning between neutral and charged species Compound
Lipid
Vesicle type
pKaw log Kimw (neutral species)
Phenols 2,4,6-Trichlorophenol 2,3,4,6-Tetrachlorophenol Pentachlorophenol Pentachlorophenol 2,4-Dinitrophenol 2-Methyl-4,6-dinitrophenol 2-s-Butyl-4,6-dinitrophenol
DOPC DOPC DOPC PC DOPC DOPC DOPC
Sonicated SUVa Sonicated SUV Sonicated SUV MLV Sonicated SUV Sonicated SUV Sonicated SUV
6.15 5.4 4.75 4.75 3.94 4.31 4.62
3.99 4.46 5.10
Carboxy acids 5-Phenylvaleric acid Salicylic acid Diclofenac Diclofenac Ibuprofen
DMPC PC Soy-PC DOPC DOPC
Sonicated SUV Extruded LUV Sonicated SUV LUV LUV
Other acids Warfarin
PC
Dmw Ref.
2.64 2.76 3.96
1.49 1.00 0.74 0.70 0.74 0.42 0.61
[120] [120] [120] [123] [120] [120] [120]
4.88 2.98 3.99 3.99 4.45
2.94 2.50 4.50 4.50 3.80
1.44 1.46 1.50 1.90 1.99
[4] [4] [152] [152] [117]
Extruded LUVb
5.00
3.39
1.99 [4]
Amino acids FCCP CCCP
Egg-yolk PC Sonicated SUV Egg-yolk PC Sonicated SUV
6.2 5.95
4.12 3.79
0.00 [154] 0.00 [155]
Anilines 3,4-Dimethylaniline 2,4,6-Trimethylaniline
Egg-yolk PC Extruded LUV Egg-yolk PC Extruded LUV
5.23 4.38
2.11 2.38
0.12 [120] 0.26 [120]
Aliphatic primary amines 4-Phenylbutylamine Amlodipine
DMPC DMPC
Sonicated SUV Sonicated SUV
10.54 9.02
2.41 3.75
0.29 [151] 0.00 [151]
Aliphatic secondary amines Propranolol (p-Methylbenzyl)methylamine (p-Methylbenzyl)propylamine (p-Methylbenzyl)pentylamine (p-Methylbenzyl)heptylamine
PC PC PC PC PC
Extruded LUV Extruded LUV Extruded LUV Extruded LUV Extruded LUV
9.24 9.93 9.98 10.08 10.02
3.24 3.09 3.07 3.50 4.40
0.48 0.55 0.96 1.66 1.69
Aliphatic tertiary amines Lidocaine Tetracaine
PC DOPC
Extruded LUV LUV
7.86 8.49
2.06 3.23
1.15 [4] 1.12 [117]
a
[4] [153] [153] [153] [153]
SUV ¼ small unilamellar liposomes, b LUV ¼ large unilamellar liposomes.
amino groups interact with the phosphate structures of the lipids, and both substituents have to intercalate between the fatty acid chains. This becomes increasingly more difficult the longer or bulkier the two substituents. Dmw of stronger uncouplers, e.g. carbonylcyanide-p-trifluoromethoxyphenylhydrazone (FCCP) [154], carbonylcyanide-m-chlorophenylhydrazone (CCCP)
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[155], and 5-chloro-3-t-butyl-20 -chloro-40 -nitrosalicylanilide (S-13) [156], are not significant, i.e. both neutral and charged species partition equally well into the lipid bilayer because the charge is well delocalised over the entire molecule. 4.4
ION PAIR FORMATION AT THE MEMBRANE INTERPHASE
Ion pairs are outer-sphere association complexes, which have to be clearly distinguished from the organometallic complexes discussed in Section 6. Ion pair formation appears to be much less important in biological membranes as compared with octanol, because the charge of the ions at the membrane interphase can be balanced by counter charge in the electrolyte in the adjacent aqueous phase. The reactions involved in ion pair formation are depicted in Figures 5b for acids and 5c for bases, and the equilibrium constant Kix0 is defined as follows: Kix0 ¼
aixm aiw axw
(25)
The subscripts i and x label the ionic constituents of the ion pair and aixm refers to the activity of the ion pair in the membrane phase. At a constant counter-ion activity, axw , and for the ideal case that giw and gix ¼ 1, equation (25) can be converted to the commonly defined distribution ratio Dixmw : Dixmw ¼
mixm ¼ Kix0 axw fr ciw
(26)
Note that no ion pair is assumed in the aqueous phase. The overall distribution ratio of the charged species is then a combination of the partition coefficient of the charged species and the distribution ratio of the ion pair. Dmw values of the drugs amlodipine and 4-phenylbutylamine were not significantly different in de-ionised water and a 0:02 mol dm3 phosphate/citrate buffer (pH around 7) [151]. The dependence of the overall Dmw of phenols at high pH on the Kþ concentration was rather small [116]. Earlier models of membrane–water partitioning of substituted phenols directly applied the model set-up for octanol–water partitioning [129] but ion pair formation appeared to be not very prominent [116]. A similar set of experimental data was later successfully evaluated with the model described in Section 4.1.2, accounting for the ionic strength effect fully through its effect on the activity coefficients and the membrane electrostatics [120,140]. Austin et al. [132] measured the ionic strength dependence of the liposome– water distribution of several acidic and basic drugs and modelled the data with a combination of electrostatic and ion pair models. They concluded that the increased apparent Dmw values at higher ionic strength were due primarily to the reduction in surface potential and not to ion pairing. Ion pairing was also excluded because the apparent Dmw varied at fixed ionic strength with the
232 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
concentration of drug. The ion pair formation accounted for no more than 5 % of the overall partitioning of the anionic drug proxicromil [132]. In addition, z-potential measurements on PCP showed no significant effect of ionic strength on adsorption characteristics [123]. This finding suggests that no ion pair is formed, but also indicates that screening of the membrane surface by ions has little effect on the energetics of adsorption, i.e. PCP is buried at some depth below the membrane surface. The reason for the insignificance of ion pair formation at the membrane interphase lies in the relative permittivity (dielectric constant) er at the sorption sites of HIOCs in the membrane (see also Section 4.7). Ion pair formation is generally high in solvents of low er (e.g. octanol er ¼ 10), and decreases rapidly until it becomes insignificant when er > 40, e.g. in water [157]. In membranes, er varies from five at the hydrophobic core to about 70 at the membrane surface. Consequently, the insignificance of ion pair formation is a further indication that HIOCs are not deeply intercalated into the membrane but sorb to the region of the polar head groups. Ion pair formation appears to become relevant only for stronger complexing agents. The partitioning of the protonated form of DMA increases significantly in the presence of formic acid/formate buffer at pH 3, and is most likely due to complex formation between the anilinium ion and formate [120]. In addition, hydrophilic substances can be taken up into the membrane when complexes with hydrophobic counter-ions are formed. This property has been exploited for increasing the uptake of peptidic drugs by, for example, salicylate [157]. 4.5 SPECIATION IN THE MEMBRANE: INTERFACIAL ACIDITY CONSTANT The interfacial acidity constant or apparent acidity constant in the membrane phase K am refers to the relative ratio of acidic and corresponding basic species in the membrane at a given (aqueous) pH-value: K am ¼
mAm aHw mHAm
(27)
K am is therefore directly related to the aqueous acidity constant: K am ¼
K Amw Ka K HAmw w
(28)
The pK am is consequently equal to the pK aw shifted by Dmw and can be directly deduced from the lipophilicity profile (Figure 6). Since Dmw is generally smaller in the membrane–water system as compared to the octanol–water
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log Dmw
log Kamw pKa w
log Kbmw pKa
m
Figure 6.
pH
Derivation of the pK am from a lipophilicity profile, a refers to acid, b to a base
system (see Section 4.3), the pK am of an acid is shifted by about one log unit to higher values and that of a base is shifted by about one unit to lower values. The better the charge of a molecule can be delocalised and accommodated in the lipid bilayer, the smaller is the pKa shift. Molecules like CCCP or FCCP do not have a pKa shift at all [154,155]. The pKa shift can be directly measured by the solvatochromic shift of the ultraviolet absorption spectra. For PCP, the pK am is 5.97 in phosphatidyl choline membranes, and increases up to 6.78 in the negatively charged phosphatidyl glycerol membranes [123]. The addition of cholesterol decreases the pK am again slightly in both types of membranes. There exists an inverse relationship between the pKa and the dielectric constant of the medium [123]. This relationship gives an indication that the dielectric constant at the sorption site in the membrane is smaller than in the aqueous phase. 4.6 SORPTION OF HIOCS TO CHARGED MEMBRANES VESICLES AND BIOLOGICAL MEMBRANES Most studies investigating the role of speciation in membrane–water partitioning have been performed with liposomes made up of phosphatidyl choline with varying types and lengths of acyl chains, because PC is a zwitterion over most of the typical pH range of the partitioning experiments. Biological membranes, however, contain a variety of charged or ionisable lipids, e.g. phosphatidyl ethanolamine, phosphatidyl inositol, phosphatidyl serine, phosphatidyl glycerol, or phosphatidic acid (see Table 4 for a list of abbreviations and pKa values). Charged head groups have no influence on the partition behaviour of the neutral species but strongly influence sorption of the charged species. Miyoshi et al. measured the apparent Dmw values at pH 7 of 35 substituted phenols in PC liposomes with 10% cholesterol, 20% negatively charged cardiolipin, and 20% positively charged stearylamine [114]. A selection of their results is listed in Table 5. Cholesterol decreases the Dmw , because it makes the membrane more rigid. The addition of positively charged stearylamine had
234 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES Table 4. Major building blocks of lipid bilayers in biological membranes and their speciation and acidity constants Lipid
Charge at pH 7 pKa (PO4 )
Phosphatidyl choline
Zwitterionic
Phosphatidyl inositol
Negative
Phosphatidylethanolamine Zwitterionic
41 2.5 3.2 2.5–3.1 1.7
Phosphatidylglycerol Phosphatidyl serine
Negative Negative
2.9–3.5 5000 g mol1 had a
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[Image not available in this electronic edition.]
Figure 10. Median zebra fish embryo hatching rates as a function of calculated Cu2þ concentrations. Reprinted with permission from [228]: Fraser, J. K. et al. (2000). ‘Formation of copper complexes in landfill leachate and their toxicity to zebrafish embryos’, Environ. Toxic. Chem., 19, 1397–1402. Copyright SETAC, Pensacola, Florida, USA
similar toxic effect to Cu2þ alone, while the toxicity was increased significantly in the raw leachate and the fraction with M < 700 g mol1 , as is depicted in Figure 10. A lipophilic copper complex with neocuproine has been found to increase the toxicity of a trichlorophenol in bacteria, probably due to increased transport of Cu over the membrane [229]. In most of the above-cited studies, it was assumed that the increase of toxicity was due to enhanced uptake of the metal, and that overall toxicity is only due to metal toxicity. However, stable complexes may exhibit specific toxicity by themselves. The Cu2þ complex of 2,9-dimethyl-1,10-phenanthroline has been shown to react with H2 O2 in the cell, thereby producing radicals [221]. Cu(Ox)02 exhibits a specific toxic effect on photosynthesis [230]. Cu-ethylxanthogenate enhances respiration and ATP production [230]. Indirect evidence of hydrophobic complex formation in biological membranes is also given by the modulation of the toxicity of catechol and chlorocatechol by Cu2þ [78]. Whereas toxicity of higher chlorocatechols was decreased by the addition of Cu2þ , it was increased in the case of catechol and monochlorocatechol. Tentative models to explain these findings include complex formation between mono-and di-deprotonated catechols and Cu2þ , both in the aqueous and in the membrane phase.
248 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
6.2 6.2.1
ORGANOMETALLIC COMPOUNDS Mercury
Accumulation of methylmercury in fish is a critical problem in many aquatic systems. A detailed investigation of the octanol–water distribution ratios of neutral mercury complexes has shown that HgCl02 and CH3 HgCl0 exhibit the most hydrophobic character, in comparison in particular with Hg(OH)02 and CH3 HgOH0 [79]. The concentrations of these species depend on pH and on the chloride concentration. The uptake rates of both inorganic Hg and of methylmercury in the marine diatom Thalassiosira weissflogii were dependent on the octanol–water distribution ratios, under various conditions of pH and chloride (Figure 11, [79]). This dependence indicated that Hg was taken up by passive diffusion of the uncharged chloro complexes over the membranes. For methylmercury, bioavailability in fish is not controlled by transfer across gill, skin or intestinal membranes, but rather by digestive processes, among them complexation with amino acids [231]. 6.2.2
Organolead Compounds
Tetraethyllead was used in the past as an antiknock agent in gasoline, but it has been phased out in most countries. Alkyllead compounds have a detergent-like activity on liposomes and black lipid membranes [232]. Tributyllead destroys planar lipid membranes at lower concentrations than tripropyllead, which is again more effective than triethyl- and trimethyllead [232]. Inorganic lead compounds like lead acetate and lead nitrate were effective only at twice as high concentrations [232]. 6.2.3
Organotin Compounds
The octanol–water partitioning of organotin compounds is strongly dependent on the speciation and the counter-ion type and concentration [89]. Consequently, a strong pH dependence of Dow can be observed, as is shown in Figure 12 for TBT and TPT, in the presence of 10 mmol dm3 perchlorate. While for the bromide and chloride complex, partitioning is little more than one order of magnitude smaller than that of the hydroxide species, the nitrite and perchlorate complex differ by more than two orders of magnitude. In contrast, the liposome–water partitioning shows a very weak pH dependence (Figure 12) [233]. The log Dlipw is even slightly larger at low pH than at high pH. While in the octanol–water system, partitioning at low pH is only due to partitioning of the perchlorate complex, the triorganotin cation can directly interact with the lipid bilayer, resulting presumably in a direct complex formation with the phosphate group in the phospholipids [234]. This effect is more pronounced for TPT than for TBT, which is consistent with TPT’s higher
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Uptake rate/ amol cell−1 hr −1 nmol−1 dm−3
20
pCl
8.1 8.1 7.3 6.7 6.7 6.7 6.0 5.6 5.5 5.5 5.3 4.8 4.3 4.2 4.0
0.5 0.5 3.3 3.3 4.3 5.3 3.3 2.2 3.8 3.3 3.7 3.3 2.9 2.4 2.9
(b) Symb. pH
pCl
8.1 5.8 4.5 6.6 6.2 5.8 4.9 4.3 6.6 6.6 5.6 4.2 3.1
0.5 2.3 2.8 3.3 3.3 3.3 3.3 3.3 4.3 5.3 3.9 4.3 4.3
15
10
5
0
0
1
2
20
Uptake rate/ amol cell−1 hr −1 nmol−1 dm−3
(a) Symb. pH
3
15
10
5
0
0
1 log D ow
I 0.7 0.7 1.5⫻10−3 0.08 0.08 0.08 0.01 0.1 0.1 1.5⫻10−3 0.1 0.1 0.1 0.1 0.1
I 0.7 0.1 0.1 0.12 1.2⫻10−3 0.04 1.2⫻10−3 0.2 0.12 0.12 0.1 0.1 0.1
2
Figure 11. Uptake rates of inorganic Hg (a) and of methylmercury (b) by a marine alga as a function of the octanol–water distribution ratio of the Hg-species under various conditions of pH and chloride concentrations. The neutral species HgCl02 and CH3 HgCl0 diffuse through the membranes. Reprinted with permission from [79]: Mason, R. P. et al. (1996). ‘Uptake, toxicity, and trophic transfer in a coastal diatom’, Environ. Sci Technol., 30, 1835–1845; copyright (1996) American Chemical Society
affinity to oxygen ligands, which is also reflected in its lower acidity constant [89]. Partitioning into biomembrane vesicles was even higher at low pH than at high pH [233], indicating additional complex formation of the cationic species with ligands of the intercalated proteins.
250 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
log Dow or log Dlipw (dm3 kglip−1)
5 (a) TBT
4.5 4 3.5
log D lipw
3 2.5 2 log Dow 1.5 1
2
3
4
5
6
7
8
9 pH
pKa
log Dow or log Dlipw (dm3 kglip−1)
5 (b) TPT
4.5 4
log D lipw
3.5 3 2.5 2 1.5 1
log Dow
2
3
4
5 6 pKa
7
8
9 pH
Figure 12. pH-dependence of the octanol–water and liposome–water distribution ratio. (a) TBT, (b) TPT. Reprinted in part from [233], with permission from: Hunziker, R. W., Escher, B. I. and Schwarzenbach, R. P. (1997). ‘pH-dependence of the partitioning of triphenyltin between phosphatidylcholine liposomes and water’, Environ. Sci. Technol., 35, 3899–3904; copyright (2001) American Chemical Society
The number of organic substituents also influences interaction with lipid bilayers. Diphenyltin chloride causes disturbances of the hydrophobic region of the lipid bilayer, triphenyltin chloride adsorbs to the head-group region, and tetraphenyltin does not partition into the lipid bilayer [235–237]. Similar results were found for the butylated tins [238]. In addition, the mono-butyltin was homogeneously distributed within the lipid bilayer [238]. The difference in partitioning behaviour of triorganotin compounds has implications for their toxic effects [239,240]. The surface-active triphenyltin
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Table 6. Bioaccumulation of organotin compounds
TBT TPT a
pKaw
log Kow (R3 SnOH)
6.25a 5.20a
4.10a 3.53a
log BCFss (Apparent) pH 5 2.15b 3.34b
pH 8 2.95b 3.42b
log BCFss (Without metabolism) pH 5 2.32b 3.35b
pH 8 4.40b 3.43b
[89]; b [244]; ss ¼ steady-state
has a stronger hemolytic activity than diphenyltin [241]. The almost equal hydrophobicity of TBT-OH and TBTþ favours membrane permeation of both species and consequently uncoupling of oxidative phosphorylation as mode of toxic action. In contrast, the higher affinity of TPTþ to organic ligands favours binding to sensitive sites in enzymes. Consistent with the physicochemical properties, TBT is a better uncoupler than TPT, and the dominant acute toxic mechanism of TPT is inhibition of the ATP synthetase [239]. Only very few studies have considered the role of speciation of organotin compounds in biological uptake and bioconcentration [242]. Bioconcentration factors of TBT in the midge larvae Chironomus riparius were slightly higher at pH 8 than at pH 5 [243]. The BCF values of TPT are not significantly different at pH 5 and 8, and are twice as high as compared with TBT, whose apparent BCF is decreased due to metabolism [243]. If the BCF-values are corrected for metabolic breakdown of TBT (Table 6), the ratio of the BCF-values of the hydroxides is in agreement with the hydrophobicity. However, the difference between pH 5 and 8 becomes very pronounced for TBT, while still being negligible for TPT [244].
7 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH There is an abundant research on the interactions of HIOCs and metals with biological interphases, in which organic chemicals and metals are treated independently. However, few studies have considered the role of combinations of HIOCs with metals. There is a particular lack of mechanistic approaches. With regard to the metals, the FIAM has been very successful, but it remains to be shown under which conditions additional interactions, such as partitioning of hydrophobic complexes and uptake of specific complexes, are important for metal uptake and toxic effects. In particular, the role of hydrophobic complexes with both natural and pollutant compounds in natural waters has not yet been fully elucidated, since neither their abundance nor their behaviour at biological interphases are known in detail.
252 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
Considering additionally that the risk assessment of mixtures is presently an urgent issue, and that usually mixtures of exclusively organic chemicals or exclusively metals are investigated, in future more emphasis should be placed on the interactions of xenobiotic HIOCs with metals. Major research questions will include how these interactions influence bioavailability of both metals and HIOCs, interactions with biological membranes, uptake, and common toxic effects.
LIST OF SYMBOLS AND ABBREVIATIONS ABBREVIATIONS CCCP BCF DDC DMA DMPC DNOC DOPC EDTA FA FCCP FIAM GSH HA HAw HIOC Lc Li inorg Li org MLi inorg MLi org NA NTA OA PC PCP PE PG PI QSAR
Carbonylcyanide-m-chlorophenylhydrazone Bioconcentration factor Diethyldithiocarbamate 3,4-Dimethylaniline Dimyristoylphosphatidyl choline Dinitro-o-cresol Dioleylphosphatidyl choline Ethylenediaminetetraacetate Fulvic acid Carbonylcyanide-p-trifluoromethoxyphenylhydrazone Free ion activity model Glutathione Humic acid Acid in the aqueous phase Hydrophobic ionogenic organic compounds Biological carrier ligand Inorganic ligand Organic ligand Inorganic complexes Organic complexes Avogadro constant Nitrilotriacetate Oleic acid Phosphatidyl choline Pentachlorophenol Phosphatidyl ethanolamine Phosphatidyl glycerol Phosphatidyl inositol Quantitative structure–activity relationship
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5-chloro-3-t-butyl-20 -chloro-40 -nitrosalicylanilide Tributyltin 2,6-Toluidinyl naphthalenesulfonate Tetraphenylborate Tetraphenylphosphonium Triphenyltin Windermere humic aqueous model [44]
SYMBOLS aiw aim aiw bi cLi inorg cLi org cM, t cM C DH DC Dmw Dmw G0i DS Dixmw Dmw (pH,I) Dow (pH, I) er e0 gi I kf
Fraction of compound or species i in the aqueous phase Activity of compound i in the membrane phase Activity of compound i in the aqueous phase Overall thermodynamic stability constant of the complex MLi Concentration of inorganic ligand Linorg Concentration of organic ligand Lorg Total metal concentration in solution Free metal ion concentration Specific capacitance Enthalpy Potential difference between two phases Difference between the log Kmw of the neutral and the corresponding charged species Standard free-energy change for the phase-transfer reaction between membrane and aqueous phase Entropy Membrane–water distribution ratio of the ion pair ix Apparent membrane–water distribution ratio at a given pH with given ionic strength Apparent octanol–water distribution ratio at a given pH and with a given ionic strength Relative permittivity or dielectric constant The permittivity of the free space Activity coefficient of compound or species i Ionic strength Rate constant for formation of the complex ML
() () () () (mol dm3 ) (mol dm3 ) (mol dm3 ) (mol dm3 ) (F m2 ) (kJ mol1 ) (V) 1
(dm3 kg ) (kJ mol1 ) (kJ mol1 ) (dm3w kg1 m ) (dm3w kg1 m ) (dm3w dm3 o ) () () () () 1
(dm3 mol1 s )
254 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
kd Ka w K am Ki0 Kimw
Kimw
KL KV KLc Kiow mmax im miw m0iw m~im m~iw f rm SPL
s Vm Vw xi zi
Rate constant for dissociation of the complex ML Acidity constant in the aqueous phase Interfacial acidity constant or apparent acidity constant in the membrane phase Dimensionless membrane–water partition coefficient of species or compound i (mole fraction) Concentration-based membrane–water partition coefficient of species or compound i Membrane–water partition coefficient of species or compound i with the concentration given in molality Langmuir sorption constant Volmer sorption constant Equilibrium constant for binding of metal to biological carrier ligand Octanol–water partition coefficient of species or compound i Maximum mass concentration of molecules adsorbed to the membrane bilayer Chemical potential of a compound i in the aqueous phase w Standard chemical potential of a compound i in the aqueous phase Electrochemical potential of a compound i in the membrane phase m Electrochemical potential of a compound i in the aqueous phase w Phase ratio Mass concentration or density of the membrane lipids Surface area occupied by a single lipid molecule in the membrane Surface charge density Molar volume of the membrane lipids Molar volume of the aqueous phase Mole fraction of a compound or species i Charge of a compound or species i
(s1 ) () ()
() (dm3w dm3 m ) (dm3w kg1 m ) (dm3w kg1 m ) (dm3w kg1 m ) () () (mol kg1 m ) () () () () (dm3m dm3 w ) (kgm dm3 m ) (SPL 0:7 nm2 molecule1 for phospholipids) (C m2 ) 1 (dm3m mol ) 1 3 (dmw mol ) () ()
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268 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES 224. Croot, P. L., Karlson, B., van Elteren, J. T. and Kroon, J. J. (1999). Uptake of 64 Cu-oxine by marine phytoplankton, Environ. Sci. Technol., 33, 3615–3621. 225. Block, M. and Pa¨rt, P. (1986). Increased availability of cadmium to perfused rainbow trout (Salmo gairdneri, Rich) gills in the presence of the complexing agents diethyl dithiocarbamate, ethyl xanthate and isopropyl xanthate, Aquat. Toxicol., 8, 295–302. 226. Warshawsky, A., Rogachev, I., Patil, Y., Baszkin, A., Weiner, L. and Gressel, J. (2001). Copper-specific chelators as synergists to herbicides: 1. Amphiphilic dithiocarbamates, synthesis, transport through lipid bilayers, and inhibition of Cu/Zn superoxide dismutase activity, Langmuir, 17, 5621–5635. 227. Palmer, F. B., Butler, C. A., Timperley, M. H. and Evans, C. W. (1998). Toxicity to embryo and adult zebrafish of copper complexes with two malonic acids as models for dissolved organic matter, Environ. Toxicol. Chem., 17, 1538–1545. 228. Fraser, J. K., Butler, C. A., Timperley, M. H. and Evans, C. W. (2000). Formation of copper complexes in landfill leachate and their toxicity to zebrafish embryos, Environ. Toxicol. Chem., 19, 1397–1402. 229. Zhu, B.-Z. and Chevion, M. (2000). Copper-mediated toxicity of 2,4,5-trichlorophenol: biphasic effect of the copper(I)-specific chelator neocuproine, Arch. Biochem. Biophys., 380, 267–273. 230. Stauber, J. L. and Florence, T. M. (1987). Mechanism of toxicity of ionic copper and copper complexes to algae, Marine Biol., 94, 511–519. 231. Leaner, J. J. and Mason, R. P. (2002). Factors controlling the bioavailability of ingested methylmercury to channel catfish and atlantic sturgeon, Environ. Sci. Technol., 36, 5124–5129. 232. Gabrielska, J., Sarapuk, J. and Przestalski, S. (1997). Role of hydrophobic and hydrophilic interactions of organotin and organolead compounds with model lipid membranes, Z. Naturforsch. C., 52, 209–216. 233. Hunziker, R. W., Escher, B. I. and Schwarzenbach, R. P. (2001). pH-dependence of the partitioning of triphenyltin and tributyltin between phosphatidylcholine liposomes and water, Environ. Sci. Technol., 35, 3899–3904. 234. Grigoriev, E. V., Pellerito, L., Yashina, N. S., Pellerito, C. and Petrosyan, V. S. (2000). Organotin(IV) chloride complexes with phosphocholine and dimyristoyll -a-phosphatidylcholine, Appl. Organomet. Chem., 14, 443–448. 235. Langner, M., Gabrielska, J., Kleszcynska, H. and Pruchnik, H. (1998). Effect of phenyltin compounds on lipid bilayer organization, Appl. Organomet. Chem., 12, 99–107. 236. Langner, M., Gabrielska, J. and Przestalski, S. A. (2000). Adsorption of phenyltin compounds onto phosphatidylcholine/cholesterol bilayers, Appl. Organomet. Chem., 14, 25–33. 237. Rozycka-Roszak, B., Pruchnik, H. and Kaminski, E. (2000). The effect of some phenyltin compounds on the thermotropic phase behaviour and the structure of model membranes, Appl. Organomet. Chem., 14, 465–472. 238. Ambrosini, A., Bertoli, E. and Zolese, G. (1996). Effect of organotin compounds on membrane lipids: fluorescence spectroscopy studies, Appl. Organomet. Chem., 10, 53–59. 239. Hunziker, R. W., Escher, B. I. and Schwarzenbach, R. P. (2002). Acute toxicity of triorganotin compounds: different specific effects on the energy metabolism and role of pH, Environ. Toxicol. Chem., 21, 1191–1197. 240. Langner, M., Gabrielska, J. and Przestalski, S. (2000). The effect of the dipalmitolylphosphatidylcholine lipid bilayer state on the adsorption of phenyltins, Appl. Organomet. Chem., 14, 152–159.
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241. Sarapuk, J., Kleszczynska, H. and Przestalski, S. (2000). Stability of model membranes in the presence of organotin compounds, Appl. Organomet. Chem., 14, 40–47. 242. Langston, W. J. (1996). Recent developments in TBT ecotoxicology, TEN, 3, 179–187. 243. Looser, P. W., Bertschi, S. and Fent, K. (1998). Bioconcentration and bioavailability of organotin compounds: influence of pH and humic substances, Appl. Organomet. Chem., 12, 601–611. 244. Looser, P. W., Fent, K., Berg, M., Goudsmit, G.-H. and Schwarzenbach, R. P. (2000). Uptake and elimination of triorganotin compounds by larval midge Chironomus riparius in the absence and presence of Aldrich humic acid, Environ. Sci. Technol., 34, 5165–5171. 245. Tomlin, C. D. S. ed. (1997) Pesticide Manual. Series, Bracknell, Berkshire: British Crop Protection Council. 246. Vazquez, J. L., Merino, S., Domenech, O., Berlanga, M., Vinas, M., Montero, M. T. and Hernandez-Borrell, J. (2001). Determination of the partition coefficients of a homologous series of ciprofloxacin: influence of the N-4 piperazinyl alkylation on the antimicrobial activity, Int. J. Pharmaceut., 220, 53–62. 247. Montero, M. T., Freixas, J. and Hernandez Borrell, J. (1997). Expression of the partition coefficients of a homologous series of 6-fluoroquinolones, Int. J. Pharmaceut., 149, 161–170. 248. March, D. (1990). Handbook of Lipid Bilayers. CRC Press, Baton Rouge, Fl.
6 Transport of Solutes Across Biological Membranes: Prokaryotes ¨ STER WOLFGANG KO Microbiology, Swiss Federal Institute for Environmental Science and Technology ¨ berlandstrasse 133, CH-8600 Du¨bendorf, Switzerland (EAWAG), U
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Membranes of Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Properties and Functions of the Cytoplasmic Membrane . . . 2.2 Intracytoplasmic Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Outer Membrane of Gram-negative Bacteria . . . . . . . . . 2.4 Cell Walls of Gram-positive Bacteria . . . . . . . . . . . . . . . . . . . 2.5 The Envelope of Mycobacteria . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bacteria Devoid of Cell Wall Peptidoglycans. . . . . . . . . . . . . 3 Substrate Translocation Across Membranes: Various Approaches to Solve the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Folding, Membrane Insertion and Assembly of Transport Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Different Driving Forces and Modes of Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Classification of Transport Systems . . . . . . . . . . . . . . . . . . . . 3.4 Various Options for Transporting a Substrate. . . . . . . . . . . . 3.5 Controlling the Number of Active Transporter Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Transport Across the Outer Membrane of Gram-negative Bacteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Porins as Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Porins with Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 TonB Dependent Receptors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Transport Through the Cell Walls of Mycobacteria . . . . . . . . . . . 6 Transport Across the Cytoplasmic Membranes of Bacteria . . . . . 6.1 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 MIP Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Mechanosensitive Channels . . . . . . . . . . . . . . . . . . . . . 6.1.3 Gas Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
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Secondary Active Transporters . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Uniport Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Symport Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Antiport Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Binding Protein-Dependent Secondary Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Primary Active Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 ATPases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 ABC Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.1 Binding Protein-Dependent Uptake Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.2 Systems Without Autonomous Binding Protein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Other Primary Active Transporters (not Diphosphate-Bond-Hydrolysis Driven) . . . . . . . 6.4 Group Translocators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Uptake of Iron: a Combination of Different Strategies . . . . . . . . 7.1 Iron – a ‘Precious Metal’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Iron Transport Across the Outer Membranes of Gram-negative Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Iron Transport Across the Cell Walls of Gram-positive Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Iron Translocation Across the Cytoplasmic Membrane: Various Pathways . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 feo Type Transport Systems for Ferrous Iron. . . . . . 7.4.2 Metal Transport Systems of the Nramp Type. . . . . . 7.4.3 ABC Transporters for Siderophores/Haem/ Vitamin B12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 ABC Transporters of the Ferric Iron Type . . . . . . . . 7.4.5 ABC Transporters for Iron and Other Metals . . . . . 7.5 Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Phylogenetic Aspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Iron Transport in Bacteria: Conclusion and Outlook . . . . . 8 Challenges for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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INTRODUCTION
Many functions and vital processes are linked to biological membranes. To be surrounded by one or more lipid bilayers might be favourable for a micro-
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organism in order to be protected against harsh conditions and environmental stresses. Nonetheless, communication with the environment can be highly important for the survival of a cellular organism in a certain habitat. Information about temperature, osmolarity, pressure, pH, nutrients, antimicrobial agents, etc. will help a microorganism to find the most favourable terms and to adjust its metabolism to the environmental conditions. Moreover, the ability to communicate with members of the same species (e.g. by quorum sensing) is a prerequisite for organisation in populations. Thus, in the processes of sensing and signal transduction, membranes play an important role [1,2]. Membranes also constitute permeability barriers that prevent the passage of many molecules, including essential nutrients. Therefore, it is evident that all organisms have a need for specific transporters. In general, transport of solutes into and out of cells is catalysed by proteins that are embedded in or associated with membranes. This chapter cannot give a comprehensive description and encyclopaedic listing of all existing transport systems in prokaryotes. Mainly, import systems transporting low-molecular-mass substrates will be presented. The uptake of macromolecules like bacteriocins (e.g. colicins) or DNA is an interesting topic in its own right, and will not be discussed in detail. Another major topic, transport out of the cell, is only touched upon. Prokaryotes possess a variety of both more general and highly specific systems that are involved in export of molecules across the cytoplasmic membrane, which can mediate further secretion into the environment. Substrates of these export pathways include proteins (proteases, lipases, various enzymes, cytotoxins, cytolysins, colicins, hemophores) siderophores, amino acids, antibiotics, antimicrobial agents, heavy metals, and many more. Although many of the secretion systems have been studied in detail (e.g. the ‘channel-tunnel’ protein TolC [3,4] the bacterial multidrug efflux transporter AcrB [5], or the arsenate/arsenite export ATPase [6,7]), important questions remain unsolved. For mycobacteria that secrete proteins, which are likely to play an important role in their pathogenicity, there is a lack of knowledge as to how these proteins, and the polysaccharides of the capsule, cross the outer lipid barrier. In summary, the aim of this chapter is to give insights into the nature and composition of membranes serving as biological interphases or interfaces, and to provide an overview on the important types of translocators, thereby demonstrating the diversity of bacterial uptake systems and the different mechanisms of transport and energy coupling. Selected representative examples will be discussed in more detail. In order to illustrate the different strategies of substrate translocation across membranes, and to highlight some unique features of transport, a special focus will be on the iron sequestering systems. In this context, the import of siderophores and haemophores, which are exported, then reshuffled and taken up in a receptor-mediated manner, will be described.
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MEMBRANES OF BACTERIA
Different types of membranes are found in bacteria: . the cytoplasmic membrane (CM) is common for all groups of bacteria. This plasma membrane in prokaryotes performs many of the functions carried out by membranous organelles in eukaryotes. Invagination of the cytoplasmic membrane results in various morphologically different intracytoplasmic membrane structures. . the outer membrane (OM) is characteristic of Gram-negative bacteria. . a special type of membrane forms the envelope of mycobacteria. 2.1 PROPERTIES AND FUNCTIONS OF THE CYTOPLASMIC MEMBRANE The architecture of the CM bilayer is symmetrical, with an equal distribution of the lipids (exclusively phospholipids, mainly phosphatidylethanolamine, phosphatidylglycerol and cardiolipin) among the inner and the outer leaflet. In principle, this holds true for most bacteria, except for those living at extremely high temperatures. For further information, see also Chapter 1 of this volume. A number of vital functions are associated with or linked to the CM: . osmotic and permeability barrier; . coordination of DNA replication and segregation with septum formation and cell division; . energy-generating functions, involving respiratory and photosynthetic electron transport systems, establishment of proton motive force, and transmembrane ATP-synthesising ATPase; . synthesis of membrane lipids (including lipopolysaccharide in Gram-negative cells); . synthesis of the cell wall peptidoglycan murein (see below); . sensing functions (e.g. quorum sensing, chemotaxis, including motility); . assembly and secretion of extracytoplasmic proteins; . location of transport systems (import and export) for specific solutes (nutrients and ions). In Gram-negative bacteria which are characterised by a rather complex cell envelope, the CM is also referred to as ‘inner membrane’ to distinguish it from a second lipid bilayer, termed ‘outer membrane’ (OM). The space between these two layers is called the periplasm (PP). In the periplasmic space, many proteins are found with a variety of functions. Some are involved in biosynthesis and/or export of cell wall components and surface structures (e.g. pili, flagellae,
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fimbriae), some mediate degradation, utilisation and transport of substrates, while others assist in protein folding and targeting. Also in the PP, associated with the CM, one can find the ‘murein sacculus’ (for a review see [8]). This network is formed by the macromolecule peptidoglycan, which confers the characteristic cell shape and provides the cell with mechanical protection. Peptidoglycans are unique to prokaryotic organisms and consist of a glycan backbone of N-acetylated muramic acid and N-acetylated glucosamine and cross-linked peptide chains [9–13]. 2.2
INTRACYTOPLASMIC MEMBRANES
A broad variety of intracellular membrane systems, often organised as distinct organelles, is characteristic for eukaryotic cells (see Chapter 1 of this volume). In prokaryotic organisms, intracellular membranes are restricted to only a few groups of bacteria. In particular, the intracellular membranes of phototrophic bacteria, bearing the photosynthetic apparatus, appear in various morphologies. Vesicles, tubuli and structures resembling the thylakoid stacks of chloroplasts, originate from invagination of the plasma membrane. They have evolved in order to increase the membrane area that harbours the light harvesting complexes. A special type of membrane vesicles, called chromatophores, is found in purple phototrophic bacteria such as Rhodobacter capsulatus. Chromatophores, which can be easily isolated, have been used to study the photosynthetic reaction centre that mediates the conversion of light into chemical energy. Several species belonging to the group of nonphototrophic nitrifying methane utilising bacteria also form extensive intracytoplasmic membrane systems. A similar situation is found in nitrifying and nitrogen-fixing bacteria. In the organisms mentioned above, intracytoplasmic and cytoplasmic membrane are almost identical with respect to composition and respiratory activities. Most recently, a highly unusual membrane composition was reported from anaerobic ammonium-oxidising (anammox) bacteria. In these bacteria, nitrite is reduced, nitrogen gas generated, and carbon dioxide is converted into organic carbon, as the consequence of ammonia reduction. This central energygenerating process can be described as: NHþ 4 þ NO2 ! N2 þ 2H2 O:
The anammox catabolism, an exceptionally slow process generating toxic intermediates (hydroxylamine and hydrazin), takes place in an intracytoplasmic compartment called the anammoxosome. A surrounding impermeable membrane protects the cytoplasm from the toxic molecules produced inside this organelle-like structure. Such a tight barrier against diffusion seems to be realised by four-membered aliphatic cyclobutane rings that have been found
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as dominant molecules in the anammoxosome membrane. This highly unusual feature was never before observed in nature (although three-, five-, six- and even seven-membered aliphatic rings had been reported previously in microbial membrane lipids). The lipids contain up to five linearly fused cyclobutane moieties with cis-ring junctions building a staircase-like formation. These socalled ‘ladderane’ molecules give rise to an exceptionally dense membrane. These results further illustrate that microbial membrane lipid structures can be far more diverse than previously thought [14]. 2.3
THE OUTER MEMBRANE OF GRAM-NEGATIVE BACTERIA
The outer membrane (OM) of Gram-negative bacteria constitutes to a certain extent an osmolarity and a permeability barrier. The OM is highly asymmetrical, with the inner leaflet, oriented to the periplasm, showing a lipid composition that is similar to that of the CM. In contrast, the outer leaflet, facing the external medium, contains a number of additional components, including the lipopolysaccharides (LPSs). LPS molecules consist of three parts: lipid A serving as anchor, the core oligosaccharide functioning as spacer element, and the O-specific polysaccharide consisting of oligosaccharide repeating units. The O-specific polysaccharide moiety is highly specific for the different bacterial (sub)species. LPSs are the major antigenic determinants, preventing the entry of cell-damaging components (like bile salts in the intestine) and they serve as receptors for a number of bacteriophages. The OM serves as an anchor for flagellae, fimbriae, and pili. Such complex extracellular structures are important for locomotion, cell–cell interaction, adhesion to surfaces (binding of pathogens to tissues, and attachment of environmental strains to abiotic surfaces), and formation of biofilms (e.g. dental plaque, Legionella pneumophila in water distribution systems). Proteins can be found as integral components or associated with the OM. Some of them are thought to play an important structural role, and they may contribute to the membrane integrity. They can reach relatively high levels, as with the Escherichia coli major outer membrane protein OmpA, or the major lipoprotein [15]. Last but not least, numerous proteins that are directly or indirectly involved in many transport mechanisms (import, export) are associated with the OM. 2.4
CELL WALLS OF GRAM-POSITIVE BACTERIA
Gram-positive bacteria are devoid of an outer membrane but possess a thick murein layer consisting of up to 40 layers making up to 90% of the cell wall. In some Gram-positive bacteria, teichoic acids are covalently linked to the peptidoglycan. Teichoic acids are polyol phosphate polymers with a strong negative charge. These are strongly antigenic, and are generally absent in Gram-negative
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bacteria. In some species, teichuronic acids are found as well as lipoteichonic acids, which are composed of a glycerol teichoic acid linked to a glycolipid. Additional wall compounds can be polysaccharides, lipids and proteins. Surface components are critical determinants of the interaction of pathogenic Grampositive bacteria with their host [16–18]. 2.5
THE ENVELOPE OF MYCOBACTERIA
Certain species of mycobacteria are the causative agents of tuberculosis and leprosy. The cell walls of mycobacteria are characterised by their unusually low permeability, which contributes to the mentionable resistance of the microbes to therapeutic agents. Two special features seem to be important: an outer lipid barrier based on a monolayer of characteristic mycolic acids, and a capsule-like coat of polysaccharide and protein. The cell walls contain large amounts of C60–C90 fatty acids, mycolic acids, that are covalently linked to arabinogalactan. The unusual structures of arabinogalactan and extractable cell wall lipids, such as trehalose-based lipo-oligosaccharides, phenolic glycolipids, and glycopeptidolipids were described in recent studies [19–21]. An asymmetrical bilayer of exceptional thickness is assembled by incorporating most of the hydrocarbon chains of these lipids. Structural considerations suggest that the fluidity is exceptionally low in the innermost part of the bilayer, gradually increasing toward the outer surface. Differences in mycolic acid structure may affect the fluidity and permeability of the bilayer, and may explain the different sensitivity levels of various mycobacterial species to lipophilic inhibitors. Hydrophilic nutrients and inhibitors are believed to cross the cell wall through channels of recently discovered porins [22]. According to a new concept, the solid and elastic matrix that makes the mycobacterial cell wall a formidably impermeable barrier is the direct consequence of cross-linked glycan strands which all run in a direction perpendicular to the cytoplasmic membrane [23]. The capsule probably impedes access by macromolecules. The structure of the outer lipid barrier seems common to all mycobacteria, fast- and slow-growing, but the capsule is more abundant in slow-growing species, a group which includes all the important mycobacterial pathogens [24]. 2.6
BACTERIA DEVOID OF CELL WALL PEPTIDOGLYCANS
Two groups of eubacteria devoid of cell wall peptidoglycans have been found so far: the Mycoplasma species, which possess a surface membrane structure, and the L-forms that arise from either Gram-positive or Gram-negative bacterial cells that have lost their ability to produce the peptidoglycan structures [25], also absent in the group of Archaea. Some Archaea contain cell walls composed of pseudopeptidoglycan that differs from the ‘normal’ murein, in that one of the backbone components (N-acetylmuraminic acid) is replaced by
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N-acetyltalosaminnuronic acid. Other archaeal species contain cell walls made from thick polysaccharide layers containing acetate, glucuronic acid, galactosamine, and glucose.
3 SUBSTRATE TRANSLOCATION ACROSS MEMBRANES: VARIOUS APPROACHES TO SOLVE THE PROBLEM 3.1 FOLDING, MEMBRANE INSERTION AND ASSEMBLY OF TRANSPORT PROTEINS Evolutionary processes driven by environmental changes and varying conditions have an impact on all components in a living cell. Thus, the primary, secondary and tertiary structure of proteins determines their function and location, giving different properties in different compartments, such as outer membrane, periplasmic space, cytoplasmic membrane or cytoplasm. Proteins can function as monomers or oligomers and can occur in a soluble form, as integral constituents embedded within the membrane, or can be found associated with the lipid bilayer itself or components therein. All proteins that are localised in the periplasm or in the outer membrane, as well as proteins that are secreted into the surrounding medium, have to cross the cytoplasmic membrane at least. A number of more general as well as specific secretion pathways have evolved in all types of bacteria to assist the proteins on their way out. Most of the systems are composed of a number of different components. For all types of export machinery, it is necessary that the polypetides to be transported meet certain criteria. Depending on the type of system, a specific region (signal sequence, export signal) that can be localised at the N- or C-terminal end of the polypeptide, is essential in order to enter a particular secretion pathway. Most polypeptides have to be exported in an unfolded state. Certain proteins, called chaperones, were identified, which help to maintain the correct folded state or at least prevent incorrect premature folding. In addition, it is evident that secreted proteins cannot contain long hydrophobic stretches or domains, because their existence would block the passage through a biological lipid bilayer [26–28]. The OM is a second barrier for proteins to be secreted outside the cell. Specialised integral outer membrane proteins belonging to the usher and secretin families function to allow the secretion of folded proteins in Gram-negative bacteria [29]. Outer membrane proteins face several problems. They have to cross the inner membrane thus implying that they are not allowed to contain hydrophobic ‘membrane anchor’ or ‘stop transfer’ sequences. Moreover, correct targeting to and stable insertion into the outer membrane is important for proper functioning. A number of OM proteins assemble into oligomers even prior to insertion.
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The structural solution for the vast majority of OM proteins is provided in the form of the b-strand, a secondary fold, which allows portions of the polypeptide chain to organise as a b-barrel. In this cylindrical structure, hydrophobic residues point outwards and hydrophilic residues are located inside, which can allow the formation of a water-filled channel [30–33]. The two-dimensional topology of the proteins embedded in the cytoplasmic membrane largely depends on a-helical transmembrane regions with exceptionally high hydrophobicity. Hydrophilic as well as charged amino acids are mainly localised in the connecting loops or at the N- or C-terminus. The orientation of the polypeptide chain in the lipid bilayer is largely dictated by the number and distribution of the positively charged amino acids. The ‘positive inside rule’ of von Heijne [34–36] is based on the observation that the net positive charge of integral membrane proteins (resulting from arginine and lysine residues near the membrane surface) is significantly higher on the cytoplasmic side. Small peptides and simple proteins with only a few membranespanning regions can insert spontaneously into the bilayer. In contrast, the majority of polytopic integral membrane proteins appears to depend on the assistance of components of the general secretion machinery (e.g. SecY protein) [37,38] or specialised chaperones (e.g. YidC protein) [39,40] in order to insert correctly. Interactions with the membrane lipids, as well as intramolecular interactions, determine the three-dimensional arrangement. The first evidence that the composition of phospholipids in membranes may contribute to the topological organisation of polytopic membrane proteins was provided from the group of Dowhan [41]. It was shown that phosphatidylethanolamine (PE) in E. coli membranes assists as a molecular chaperone in the assembly of the lactose permease. This transport protein adopts a partly inverted topology when inserted into membranes devoid of PE. The correct topology and activity of lactose permease could be re-established when PE synthesis was induced after assembly of the polypeptide chain in the membrane [42], demonstrating that alterations in phospholipid composition may have a general influence on membrane proteins, indicating that the topology is not fixed, since it can respond to those changes. Recent results indicate that not only topogenic signals and membrane composition contribute to the proper topology of a membrane protein. The antimicrobial peptide nisin, produced by Lactococcus lactis, kills Gram-positive bacteria via pore formation, thus leading to the permeabilisation of the membrane. Nisin depends on the cell-wall precursor Lipid II, which functions as a docking molecule to support a perpendicular stable transmembrane orientation [43]. To date, very limited information on the atomic structure is available, since crystallisation of hydrophobic membrane proteins remains a challenging problem.
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3.2 DIFFERENT DRIVING FORCES AND MODES OF ENERGY COUPLING With respect to the driving forces and the modes of energy coupling, transport processes can be divided in four major classes: (1) some solutes are able to pass the permeability barrier of a lipid bilayer by passive diffusion (the random movement of molecules from an area of high concentration to an area of lower concentration). This is true for small apolar (lipid-soluble) molecules and small slightly polar, but uncharged molecules like water and dissolved gases (O2 , CO2 , NH2 , H2 S). Other molecules are transported via channels or channel-type proteins (e.g. porins, see Section 4.1) to overcome in a diffusion-controlled movement an otherwise impermeable membrane. The translocation of substrates in this way cannot be against a concentration gradient. (2) in secondary active transport, the translocation step across the membrane is coupled to the electrochemical potential of a given solute. The ion or other solute (electro)chemical potential (e.g. the proton gradient over the cytoplasmic membrane) created by primary active transport systems is the actual driving force, which allows an ‘uphill’ transport of another solute, even against its own concentration gradient. The uptake of a given substrate following this mechanism can be mediated as uniport (also called ‘facilitated diffusion’), as symport (also termed ‘substrate cotransport’), or as antiport in exchange with another solute. (3) primary active transport systems are characterised by coupling translocation of a solute directly to a chemical or photochemical reaction. Primary sources of chemical energy include pyrophosphate bond hydrolysis (e.g. in ATP), methyl transfer and decarboxylation. Other systems are driven by oxidoreduction, light absorption or mechanical mechanisms. (4) a translocation process exclusive to bacterial species involves the phosphoenolpyruvate:sugar phosphotransferase system (PTS), which phosphorylates its carbohydrate substrates during transport.
3.3
CLASSIFICATION OF TRANSPORT SYSTEMS
Various criteria can be applied in order to arrive at a useful classification scheme for the different mechanisms of solute transport through biological membranes. Some authors concentrate mainly on phylogenetic aspects based on sequence data. The amino acid sequences of a considerable number of well-studied transporters from many bacteria are published, and an immense set of primary sequence data will become available within the next few years (primarily from numerous genome projects). Phylogenetic trees of transport proteins are
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compiled on the basis of multiple sequence alignments; consequently, one arrives at different clusters and subclusters. Alternatively, transport systems can be classified according to their mode of energisation (see Section 3.2.) or by focusing on the biochemical characterisation of the translocation process. Looking at the kinetic properties of solute uptake often gives a first clue, since the different modes of substrate import are distinguishable with respect to their transport rates: saturation is typical of carrier-mediated transport, whereas this phenomenon is not observed in simple diffusion (see Figure 1). In many organisms, the situation becomes more complex, since mechanistically different transport may operate simultaneously. In all classification systems, transporters are divided into families and further segregated into subfamilies. Saier and co-workers established a universal classification system called the ‘transport commission’ (TC) system, which is based on both function and phylogeny [44] (see Figure 2). The main types of transporters are presented in Figure 3. 3.4
VARIOUS OPTIONS FOR TRANSPORTING A SUBSTRATE
The expression of a solute transport system depends on the metabolic features and physiological state of an organism, the environmental conditions, the bioavailability of the substrate, and the substrate requirements of the cell. A common observation in bacteria is that a given substrate (or group of similar substrates) can be sequestered by several different uptake routes, including high-affinity, low-capacity systems and at least one low-affinity, high-capacity system. Primary active transporters are generally characterised by their high substrate affinity (low Km ), and low transport capacity (high Vmax ). Many
Transport rate (V )
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Figure 1.
Kinetic properties of transport processes. For details see text
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Transporterindependent diffusion
α−helical protein channels β-barrel proteins Channels
Toxin channels
Porins Gated active channels
Peptide channels
Passage of solutes through membranes via:
Pyrophosphate bond hydrolysis driven Decarboxylation driven
Transporters Primary active Transporters
Oxidoreduction driven Methyl transfer driven Light absorption driven
Carriers Mechanically driven
Uniporters
Secondary active transporters
Cation symporters Cation antiporters Obligatory solute: solute antiporters
Group translocators
Figure 2. Classification of the major types of transport mechanisms across biological membranes based on function and phylogeny (modified after M. H. Saier, 2000; [44])
primary uptake systems operate practically unidirectionally, which is favourable since nutrients can be accumulated several orders of magnitude inside the cytoplasm, even when they are available in extremely small amounts in the external medium. Such systems are typically induced (and/or de-repressed) under low environmental substrate concentrations. In contrast, most of the
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Figure 3. Examples of major types of uptake mechanisms realised in prokaryotic outer membranes (a to c) and cytoplasmic membranes (a, and d to l). The solutes to be transported are shown by filled circles; ‘x’ symbolises another solute which is transported in the same or in the opposite direction. In systems h–k, uptake is driven by the cleavage of ATP to ADP and phosphate. One type of uptake system, l, depends on the energy-rich molecule phosphoenolpyruvate shown as ‘PEP’. (a) simple diffusion; (b) diffusion pores; (c) gated channels; (d) uniporter; (e) antiporter; (f) symporter; (g) binding protein-dependent secondary active symporter; (h) binding protein-dependent ABC transporter; (i) ABC transporter with binding protein fused to integral membrane protein; (k) P-type ATPase; (l) phosphoenolpyruvate-dependent group translocator. For details see the text
secondary systems are known to exhibit low substrate affinity (high Km ), and high transport capacity (low Vmax ). Since transport is coupled to an ion or proton gradient over the membrane, substrate accumulation inside the cytoplasm is normally below 100- to 1000-fold. Under certain conditions, transport may function in the opposite direction. Members of this class of translocators are typically found to be constitutively expressed. Some systems recognise and translocate a broad variety of solutes, whereas others are restricted to a narrow spectrum or exclusively one substrate species. The diversity of prokaryotic nutrient acquisition will be illustrated by a few examples: . glucose, almost ubiquitous in nature and a favourite nutrient for many organisms, can enter living cells by multiple routes. A variety of different uptake systems is also realised in bacteria. The options include the broad
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spectrum of facilitator-type uniport systems, proton- or cation-linked permeases (symport and antiport), ABC-type transporters, as well as phosphoenolpyruvate-dependent phosphotransferase systems. It is not unusual that several parallel systems, all translocating glucose and structurally related carbohydrates, are established in a given microorganism. . in hyperthermophilic Archaea, only transporters of the ABC-type seem to exist so far for the uptake of carbohydrates (e.g. glucose, cellobiose, maltotriose, arabinose, trehalose) [45]. This probably reflects an adaptation to the extreme habitat, enabling the organisms to acquire all available sugars very effectively. . the Gram-negative bacterium E. coli is able to transport proline via two different secondary systems, one of which is Naþ -coupled (putP), while the other is Hþ -coupled (proP). In addition, a high-affinity binding proteindependent uptake system encoded by the proUVW genes exists. Moreover, at least five independent uptake systems exist for the amino acids glutamate and aspartate [46,47]. . many bacteria, ranging from environmental strains to human pathogens, have developed various strategies and specific scavenging systems for iron. This esential element, often being the growth-limiting factor, can be transported as ferrous ion, as ferric ion, or in complexed form (for details, see Section 7). 3.5 CONTROLLING THE NUMBER OF ACTIVE TRANSPORTER MOLECULES It should be envisaged that in an individual bacterial cell more than a hundred different transport systems can be encoded by the chromosome or by suitable plasmids. After sequencing whole genomes, information on genes encoding putative importers and exporters is now available for a growing number of species. It is evident that not all these transporters can be expressed and present at maximum levels at any time. That would be a tremendous waste of energy, and could be a great disadvantage for the physiology of the cell. Moreover, the membranes would be ‘overcrowded’, not leaving enough space for all the different components and functions associated with the cell envelopes (see above). Cells have to express their transport systems in a regulated manner according to their needs, depending on the metabolic state and on environmental conditions. A few major parameters, such as osmolarity of the external medium, extracellular and intracellular pH, energetic status of the cell, internal metabolites, and regulatory proteins affecting the expression of transporters give an idea of the complexity. Only a limited number of transporters are constitutively expressed. Many uptake systems are induced by the substrates that they translocate. The expression of a number of importers is de-repressed when essential nutrients reach a critically low intracellular concentration, whilst
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other systems are repressed when toxic compounds or catabolites exceed critical levels. Despite its importance, regulation cannot be discussed in great detail in this chapter, but regulation is, in principle, possible on different levels: (1) in bacteria, regulation on the transcriptional level seems to be the most important. This often involves proteins that bind to specific DNA regions. Depending on the system, such proteins can act as, for example, repressors, activators, or alternative sigma factors (in combination with the DNAdependent RNA core polymerase) thereby allowing the transcription initiation by de-repression, activation or induction. In some cases, regulatory networks are reported that include cascades of regulatory processes involving sensing and signal transduction from the outer surface to the nucleic acids. Regulation on the transcriptional level can also be achieved by expressing so-called ‘antisense’ RNA molecules that interact with messenger RNA (m-RNA), or by m-RNA stability. (2) regulation on the translational level can take advantage of m-RNA secondary structure, optimal or weak ribosome binding sites, the choice of the start codon, codon usage, and translational coupling (overlapping start and stop codons). (3) regulation can also occur at the level of protein stability, which can be influenced by a number of factors and components inside the cells or in the environment. In addition, the activity of transport proteins can be influenced by modifications such as phosphorylation–dephosphorylation or methylation–demethylation.
4 TRANSPORT ACROSS THE OUTER MEMBRANE OF GRAMNEGATIVE BACTERIA 4.1
GENERAL PORINS AS CHANNELS
Typically, functional porins are homotrimers, which assemble from monomers and then integrate into the outer membrane. The general porins, water-filled diffusion pores, allow the passage of hydrophilic molecules up to a size of approximately 600 Daltons. They do not show particular substrate specificity, but display some selectivity for either anions or cations, and some discrimination with respect to the size of the solutes. The first published crystal structure of a bacterial porin was that of R. capsulatus [48]. Together with the atomic structures of two proteins from E. coli, the phosphate limitation-induced anionselective PhoE porin and the osmotically regulated cation-selective OmpF porin, a common scheme was found [49]. Each monomer consists of 16 b-strands spanning the outer membrane and forming a barrel-like structure.
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The b-strands are connected by loops on the outside, and short turns facing the periplasm. The third loop, L3, has a unique feature, in that it is not exposed at the cell surface but folds back into the barrel. The resulting constriction zone gives the channel an hourglass-like shape. L3 contributes significantly to the permeability properties of the pore (e.g. ion selectivity and exclusion limit) [50]. Experimental data demonstrate that lysine and arginine residues contribute to the selectivity filter in the anion-selective porin PhoE of E. coli [51,52]. The exceptionally high stability of some porins is not only generated by the hydrophobic interface of the monomers. Studies with OmpF showed that loop L2 of each monomer extends into the adjacent monomer [53], leading to an interlocked arrangement of the components. The molecular mechanism of voltage gating, a phenomenon not observed with substrate-specific channels, still has to be solved. 4.2
PORINS WITH SELECTIVITY
The outer membrane of Gram-negative bacteria contains, in addition to the general pores, a number of channels, which facilitate the specific diffusion of certain substrates. Well-studied representatives are the sucrose-specific porin ScrY from Salmonella typhimurium and the maltooligosaccharide-specific maltoporin LamB from E. coli. When purified ScrY porin was reconstituted into vesicles, a high permeation rate for sucrose was observed [54]. Purified LamB reconstituted into lipid bilayers formed ion-conducting channels transporting maltose and maltodextrins (up to maltoheptaose) with a high permeation rate [55,56]. Although similar sugar binding affinities are found for for maltose (10 mmol dm3 ) and sucrose (15 mmol dm3 ) it has been shown previously that sucrose uptake via LamB is negligible [57]. Despite little similarity in the primary structures of LamB and ScrY, superimposition of the threedimensional structures arrives at a similar picture [58,59] (Figure 4). Both sugar-specific porins represent homotrimers with monomers formed out of 18 antiparallel b-strands. The resulting b-barrel, resembling the arrangement of the general porins, contains a constriction formed by loop L3. Characteristic for the sugar-specific porins is a substrate translocation pathway extending from the vestibule (open to the extracellular medium) to the exit (at the periplasmic side). This so-called ‘greasy slide’, built by aromatic residues, is lined by a stretch of polar amino acids termed the ‘ionic track’. The ‘greasy slide’, presumably, interacts with the hydrophobic face of the transported sugars (via van der Waals interactions), whereas the ‘ionic track’ forms hydrogen bonds with their hydroxyl groups [60,61]. It has been suggested that the sugars move down the channel via continuous formation and disruption of these hydrogen bonds [62]. Only a few residues modulating the lumen of the channel define the sugar specificity of LamB and ScrY.
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Figure 4 (Plate 5). Atomic structure of the sucrose specific selective porin ScrY isolated from the outer membrane. Longer loops are directed to the outside, shorter turns are facing the periplasm. Monomer and assembled homotrimer in side view (left and middle); top view of assembled trimer (right). (Reproduced by permission of W. Welte and A. Brosig)
4.3
TONB DEPENDENT RECEPTORS
Most Gram-negative bacteria express outer membrane receptors for the uptake of haem, iron–siderophore complexes, and vitamin B12 into the periplasm [63]. These receptors are larger than the known diffusion-controlled porins, and differ from the latter by their high substrate affinity and specificity. They are characterised by their energy requirement for active ligand transport against a concentration gradient. Energy supply is provided by an inner membraneassociated protein complex composed of the proteins TonB–ExbB–ExbD, which couples substrate translocation across the OM to the membrane potential of CM [63–65]. The molecular mechanism of energy transfer is still unknown. The enterobactin receptor FepA and the ferrichrome receptor FhuA from E. coli were the first TonB-dependent OM proteins to have their threedimensional structures solved [30,66,67]. The crystal structures of both FhuA and FepA revealed an unexpected arrangement with a globular N-terminal portion forming a ‘plug’ or ‘cork’ like domain folding into a 22-stranded b-barrel that spans the entire OM. The structure of FepA is shown in Figure 5. For more details, see Section 7.2. Based on sequence similarity, it appears that transporters of this type may also assist in the uptake and acquisition of solutes unrelated to iron chelating compounds and vitamin B12 [68]. Examples of prospective candidates are outer membrane proteins that are involved in sulfate ester utilisation in Pseudomonas putida [69], or polypeptides playing a role in starch binding at the surface of Bacteriodes thetaiotaomicron [70].
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Figure 5 (Plate 6). Crystal structure of FepA, the TonB-dependent receptor for ferricenterochelin. (Reproduced by permission of D. van der Helm and L. Esser)
5
TRANSPORT THROUGH THE CELL WALLS OF MYCOBACTERIA
The very low permeability is one of the most prominent functional features of the mycobacterial cell wall which protects the bacterial cell from noxious substances. Nonetheless, hydrophilic molecules can diffuse through the mycolic acid layer. However, the permeability of the mycobacterial cell wall is 100- to 1000-fold lower than that of most Gram-negative bacteria. It has been shown that special porins localised in the cell wall represent the main hydrophilic pathway [71,72]. Recently, such a porin, MspA of Mycobacterium smegmatis, has been characterised. In vitro studies demonstrate that MspA is an extremely stable oligomeric porin (composed of 20 kDa subunits) that forms waterfilled channels with a conductance of 4.6 nS in 1 mol dm3 potassium chloride. Mycobacteria lacking the MspA porin displayed a nine-fold decreased
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permeability for the zwitterionic b-lactam antibiotic cephaloridine, and the transport of glucose was impaired [73]. Three other porins with properties similar to MspA were identified in the genome of M. smegmatis.
6 TRANSPORT ACROSS THE CYTOPLASMIC MEMBRANES OF BACTERIA 6.1
CHANNELS
Bacteria can survive dramatic osmotic shifts. Osmoregulatory responses mitigate the passive adjustments in cell structure and the growth inhibition that may ensue. The levels of certain cytoplasmic solutes rise and fall in response to extracellular osmolality. Responses to environmental changes necessitate the presence of components that allow sensing and regulated transport of ions and other solutes across membranes. However, the CM of all microorganisms does not tolerate proteins that form pores or pore-like structures that resemble the porins such as those (e.g. OmpF, OmpC, PhoE) found in the OM of Gramnegative bacteria or in the cell wall of mycobacteria. Such permanently open channels would immediately lead to the depolarisation of the membrane, thus resulting in a loss of function of many transporters and energy-generating systems, and so causing cell death. A similar effect can be observed when pore-forming colicins insert into (reviewed in [74–77]) or antimicrobial peptides assemble within the CM [43] in order to form open channels. Nonetheless, in microorganisms, certain types of pore-like proteins with unique features have been reported such as MIP channels, mechanosensitive channels, and gas channels.
6.1.1
MIP Channels
The family of MIP channels is named after the first discovered aquaporin from red blood cells which displays similarity to MIP (major intrinsic protein of mamalian lens fibre) [78]. The discovery of an aquaporin for the first time explained observations which were already made many years ago. Biophysical data provided the hypothesis of pore-mediated water flux. Specialised biological membranes are significantly more permeable to water than artificial lipid bilayers. Movement of water across erythrocyte membranes depends on an activation energy of about 1725 kJ mol1 . This is definitely below the activation energy of 4659 kJ mol1 needed to allow water flow through synthetic bilayers [79]. Members of the MIP family proteins occur in all classes of organism, ranging from bacteria to humans. All MIP channels share highly conserved amino acid residues and are predicted to have six hydrophobic membrane-spanning domains. They are subdivided into three major categories:
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(1) aquaporins sensu stricto, are highly specific for water [80]; (2) glycerol facilitators transport glycerol and possibly other solutes in addition to, or even in preference to, water [81]; and (3) aquaglycerolporins with a mixed function are permeable for glycerol and water. The molecular determinants underlying this specificity are not well understood. Since they are most likely to be involved in osmoregulation and metabolism, MIP channels are thought to affect a wide range of biological processes. Aquaporins are characterised by their extremely high selectivity for water; a simultaneous transport of other molecules or ions does not occur. In particular, protons can cause a special problem, since they are able to move along chains composed of H2 O molecules that are connected via hydrogen bonds (Grotthus mechanism). The mechanism preventing channel passage of protons has been proposed by using the structural model of the aquaporin AQP1 from human erythrocytes, where two asparagine residues are located in the middle of each H2 O channel of the AQP1 homotetramer. The formation of hydrogen bonds between the amido groups of these asparagine residues interrupts the chain composed of H2 O molecules, thus blocking the transfer of protons [82]. Aquaporins seem to be essential for eukaryotes, because of the large size of the multicellular organisms and their need for rapid water movement. In contrast, aquaporins are found only sporadically in bacteria, and it is a current debate as to whether small prokaryotic cells lacking internal organelles require aquaporins, or whether unmediated diffusion of water across their cytoplasmic membranes is sufficient [83]. For example, aquaporin Z exists in strains of E. coli, but homologues were not detected in most other bacteria whose genomes have been sequenced so far. Recent studies indicated that the E. coli AqpZ expression was not affected by up- or downshifts in osmolality, and no evidence was found that AqpZ mediates water permeativity under the conditions tested. Moreover, disruption of the aqpZ gene had no detectable adverse effects on growth and cell viability [83]. The GlpF protein from E. coli is the best-known glycerol facilitator to date. GlpF does not transport water, but is capable of transporting small polyalcohols (e.g. erythritol) in addition to glycerol [81]. A high-resolution structure at the molecular basis of channel selectivity was obtained by Stroud’s laboratory (University of California, Los Angeles) on the glycerol channel GlpF [84]. With ˚ resolution structure available (including three glycerol molecules the 2.2 A captured in the channel) it is possible to address the question as to how the high selectivity of this channel for glycerol can be achieved: the channel is lined with hydrophobic and amphiphatic residues on one side and polar residues on the other. Thus a so-called ‘tripathic’ channel is formed, which is so narrow ˚ ) that the glycerol molecules must pass through in single file. At the (ca. 3.8 A selectivity filter, the second glycerol molecule is tightly packed against a hydrophobic wall. This leaves no space for any substitutions to the glycerol C–H hydrogen position. In addition, both the second and third glycerol molecules
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are pinned by successive H-bonds that have been formed with a pair of donor and acceptor molecules. Due to these constraints only glycerol and water can pass through the pore [85]. L. lactis is the first Gram-positive bacterium in which a MIP channel has been functionally characterised. The lactococcal MIP protein is shown to be permeable to glycerol, like E. coli GlpF, and to water, like E. coli AQPZ. That was the first description of a microbial MIP that has a mixed function. This result provided important insights for reconstructing the evolutionary history of the MIP family and elucidating the molecular pathway of water and other solutes in these channels [86]. 6.1.2
Mechanosensitive Channels
Mechanosensitive ion channels can be looked at as membrane-embedded mechano-electrical switches. They play a critical role in transducing physical stresses at the cell membrane (e.g. lipid bilayer deformations) into an electrochemical response. Two types of stretch-activated channels have been reported: the mechanosensitive channels of large conductance (MscL) and mechanosensitive channels of small conductance (MscS). The MscL family of channels is widely distributed among prokaryotes, including several species of Archaea (e.g. Methanoccoccus jannashii, Thermoplasma acidophilum) and may participate in the regulation of osmotic pressure changes within the cell. MscL from E. coli is the first isolated molecule shown to convert the mechanical stress of the membrane into a simple response, the opening of a large water-filled pore [87]. The crystal structure of MscL from Mycobacterium tuberculosis now allows the analysis of tension-dependent channel-gating mechanisms at the molecular level. The MscL channel is a homopentamer. Each subunit consists of two a-helical transmembrane domains (TM1 and TM2), and a helical region located at the carboxy terminal end that is protruding into the cytoplasm [88]. Sukharev et al. [89] developed structural models in which a cytoplasmic gate is formed by a bundle of five amino-terminal helices (S1), the structure of which was not resolved in the crystal. Cross-linking experiments demonstrated that S1 segments form a bundle when the channel is closed. In contrast, S1 segments interact with another portion of the channel, TM2, when the channel is open. Gating was proposed to be affected by the length of the S1–TM1 linker, in a manner consistent with the model, revealing critical spatial relationships between the domains that transmit force from the lipid bilayer to the channel gate. In this model, helical tilting and expansion of TM1 was also necessary in order to open the channel [89]. Recently, electron paramagnetic resonance spectroscopy and site-directed spin labelling were used to determine the structural rearrangements that underlie the opening of such a large mechanosensitive channel. MscL was trapped in both the open, and in an intermediate closed state, by
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modulating bilayer morphology. Small movements in the first transmembrane helix (TM1) characterise the transition to the intermediate state. Subsequent massive rearrangements in both TM1 and TM2 support the highly dynamic ˚ , lined mostly by TM1, can open state. A water-filled pore of at least 25 A be formed. Members of the MscS family of small-conductance mechanosensitive channels have been identified in Eubacteria, Archaea, and several eukaryotes. McsS from E. coli has been studied in detail by applying the patch-clamp electrical ˚ resolution is availrecording technique [90], and the atomic structure at 3.9 A able [91]. MscS also responds to the tension of the membrane, but it differs from MscL in that it is voltage-modulated. The active channel in the membrane appears as a symmetrical homoheptamer, with each subunit being composed of a membrane-embedded domain and an extramembrane domain extending into the cytoplasm. The membrane domain consists of three transmembrane helices (TM1, TM2, and TM3). In the current model TM3 is lining the pore while TM1 and TM2 display an orientation which is more perpendicular to the membrane, thus playing a role as tension and voltage sensors. Pore opening might be induced by increasing the tension or by depolarisation; pore closing by tension release or hyperpolarisation. The pore extends into the extramembrane region, which forms a large water-filled chamber. This chamber connects to the cytoplasm through eight openings (seven to the side and one central), thereby functioning as a kind of molecular filter [91,92]. 6.1.3
Gas Channels
The ammonium/methylammonium transport (Amt) proteins of enteric bacteria are required for fast growth at very low concentrations of the uncharged NH3 . Homologues exist in all three domains of life. They are essential at low ammonium (NHþ 4 þ NH3 ) concentrations under acidic conditions. The Amt protein of S. typhimurium (AmtB) participates in acquisition of NHþ 4 =NH3 , but cannot concentrate either NH3 or NHþ . In general, Amt proteins appear to be bidirec4 tional channels for NH3 . They are examples of protein facilitators for a gas [93]. The majority of Amt proteins contain 11 transmembrane helices with the C-terminus facing the cytoplasm [94]. 6.2
SECONDARY ACTIVE TRANSPORTERS
The Major Facilitator Superfamily (MFS) [95–97] is the largest secondary transporter family known in the genomes sequenced to date [98]. These polytopic integral membrane proteins enable the transport of a wide range of solutes, including amino acids, sugars, ions, and toxins. Medically relevant members of the family include the bacterial efflux pumps associated with
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antibiotic resistance [99,100]. Although ubiquitous in nature, there are still no high-resolution structures published of secondary transporters (electrochemical-potential-driven porters). The low-resolution structures of a few transporters provide the first evidence that these 12 transmembrane (TM) helix proteins have more than one arrangement of their helices. Different families of 12 TM transporters might well have evolved independently of each other to arrive at the common 12 helical structures that are seen in nature. The existing data already suggest that the 12 helices can be arranged in several ways, thus pointing to the diversity of structures of membrane-transport proteins in nature. 6.2.1
Uniport Systems
Secondary active uniport systems facilitating the permeation of a single solute, dependent on the electrochemical potentials of the solute molecules, are rare in bacteria. Only a glucose uptake system of Zymomonas mobilis has been studied in more detail [101]. 6.2.2
Symport Systems
Lactose permease is a prominent example of the bacterial solute/Hþ cotransport systems. At the same time, the product of the lacY gene from E. coli is one of the best-studied bacterial transporters. LacY is a polytopic membrane protein containing 12 transmembrane helices. By catalysing the coupled stoichiometric translocation of galactosides and Hþ (lactose/Hþ symport) this transporter transduces free energy, which is stored in an electrochemical Hþ gradient, into a sugar concentration gradient. Uptake and accumulation of b-galactosides like lactose works against a concentration gradient. The primary trigger for turnover is binding and dissociation of substrate on opposite sides of the membrane. The permease has been solubilised from the membrane, purified, and reconstituted in membrane vesicles. A large collection of genetic, biochemical and physicochemical methods has been applied. The data lead to a mechanistic model describing the arrangement of the membranespanning elements, unravelling the mode of energy coupling, and defining the passage of molecules through the transporter. Experimental data from the laboratory of Kaback [102] indicate that only six side chains of the 417 residues in lac permease are irreplaceable for active transport. Glutamine 126 (helix 6) and arginine 144 (helix 5) seem to be directly involved in substrate binding and specificity. Glutamine 269 (helix 8), arginine 302 (helix 9), histidine 322 (helix 10), and glutamine 325 (helix 10) are most likely to be involved in Hþ translocation and/or coupling between Hþ and substrate translocation. The permease is protonated in the ground state (histidine 322 and glutamine 269
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share a common Hþ , and glutamine 325 is charge paired with arginine 302). In this conformation, the permease binds the ligand at the interface between helix 4 (glutamine 126) and helix 5 (arginine 144, cysteine 148) at the outer surface of the membrane. A conformational change is induced by substrate binding, thus resulting in a transfer of the Hþ from histidine 322 and glutamine 269 to glutamine 325. Consequently, reorientation of the binding site to the inner surface takes place, accompanied by release of sugar. As the conformation relaxes, glutamine 325 is deprotonated on the inside, due to rejuxtaposition with arginine 302. Then the histidine 322/glutamine 269 complex is reprotonated from the outside surface, to allow reinitiation of the cycle. Recent studies with lactose permease mutant containing a cysteine in place of alanine 122 (helix 4) indicate that alkylation of Cys-122 selectively inhibits binding and transport of disaccharides. By contrast, transport of the monosaccharide galactose remained largely unaffected. The data indicate that Ala-122 is a component of the ligand-binding site and support the idea that the side chain at position 122 abuts on the non-galactosyl moiety of d-galactopyranosides [103]. For further details see [102–105]. A variety of sodium–substrate symport systems are found in bacteria. Sodium cotransport carriers are known to be involved in the acquisition of nutrients like melibiose, proline, glutamate, serine–threonine, branched-chain amino acids and citrate. Some of these also play a role in osmoadaptation. Sodium enters the cell down an electrochemical gradient. There is obligatory coupling between the entry of the ion and the entry of substrate with a stoichiometry of 1:1, leading to the accumulation of substrate within the cell. A combination of spectroscopic, biochemical and genetic methods has been applied to gain new insights into the structure and molecular mode of action of the transport proteins [106]. The melibiose carrier MelB of E. coli is a well-studied sodium symport system. This carrier is of special interest, because it can also use protons or lithium ions for cotransport. The projection structure of MelB has been ˚ resolution [107]. The 12 TM helices are arranged in an asymmetsolved at 8 A rical pattern similar to the previously solved structure of NhaA, which, however, follows an antiport mechanism (Naþ ions out of the cell and Hþ into the cell). Studies with the Naþ /proline transporter (PutP) of E. coli, suggest a 13-helix arrangement in the membrane. In this model, the N-terminus is located in the periplasm and the C-terminus is directed into the cytoplasm. Mutational analysis has identified regions of particular functional importance. For example, amino acids of transmembrane domain 2 of PutP are critical for high-affinity binding of Naþ and proline. It was shown that ligand binding induces widespread conformational changes in the transport protein. In summary, the Naþ /solute symport is the result of a series of ligand-induced structural changes [46].
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Antiport Systems
This section will mainly concentrate on a few subfamilies of the major facilitator family, namely the sugar–phosphate/anion antiporters, the Naþ =Hþ antiporters, and one example of the oxalate/formate antiporters. Structurally dissimilar anions, such as hexose phosphates, hexuronates and glycerol-3-phosphate are the substrates for an anion-exchange mechanism across the membrane [108]. In prokaryotes, the inorganic phosphate (P)-linked systems are the best characterised [109]. Representatives are the E. coli glycerol3-phosphate transporter (GlpT) and the structurally and functionally related E. coli hexose-6-phosphate transport protein, UhpT [110,111]. GlpT mediates glycerol-3-phosphate (G3P) and inorganic phosphate exchange across the cytoplasmic membrane. It possesses 12 transmembrane a-helices. Purified GlpT protein binds substrates in detergent solution, as measured by tryptophan fluorescence quenching, and its dissociation constants for G3P, glycerol-2phosphate, and inorganic phosphate at neutral pH were determined as 3.64, 0.34 and 9:18 mmol dm3 , respectively. GlpT also displayed transport activity upon reconstitution into proteoliposomes. The phosphate efflux rate of the transporter in the presence of G3P was measured to be 29 mmol min1 mg1 at pH 7.0 and 37 8C, corresponding with 24 mol of phosphate s1 (mol of protein)1 [112]. The hexose-6-phosphate transporter UhpT protein also contains 12 transmembrane (TM) regions. Based on experimental data, Hall and Maloney [113] conclude that TM11 spans the membrane as an a-helix with approximately two-thirds of its surface lining a substrate translocation pathway. It is suggested that this feature is a general property of carrier proteins in the Major Facilitator Superfamily, and that, for this reason, residues in TM11 will serve to carry determinants of substrate selectivity [113]. Naþ =Hþ antiporters are ubiquitously found in the cytoplasmic membranes of cells and organelle membranes throughout the prokaryotic and eukaryotic kingdom. They are primarily involved in pH and Naþ homeostasis, since Naþ and Hþ are the most common ions. They play primary roles in cell physiology (e.g. in bioenergetics), and the concentration of protons in the cytoplasm is critical to the functioning of the cell and its proteins. The Naþ =Hþ antiporters cluster in several families, as concluded from the emerging genomic sequence projects (e.g. Helicobacter pylori [114], Vibrio cholerae [115], and Vibrio parahaemolyticus [116]). Two genes encoding Naþ - and Liþ -specific Naþ =Hþ antiporters were detected in E. coli: nhaA [117] and nhaB [118–120]. NhaA is the main antiporter, that has to withstand the upper limit concentration of Naþ for growth (0:9 mol dm3 , pH 7.0) and to tolerate the upper pH limit for growth in the presence of Naþ (0:7 mol dm3 , pH 8.5). NhaB is encoded by a housekeeping gene, which becomes essential only in the absence of nhaA. Structure and function studies have been conducted with purified NhaA.
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NhaA is shown to exist in two-dimensional crystals as a dimer of monomers each composed of 12 transmembrane segments with an asymmetrical helix packing. These studies provide the first insight into the structure of a polytopic membrane protein. A number of Naþ =Hþ antiporters display dramatic sensitivity to pH, a property that confirms their role in pH homeostasis. Amino acid residues involved in the pH response have been identified in the sequence of NhaA, thus pointing to the molecular mechanism underlying this pH sensitivity. Conformational changes transducing the pH change into a change in activity were found in certain loops and at the N-terminus of the protein [121]. The NhaA (Naþ =Hþ antiporter) homologue of V. cholerae seems to contribute to the Naþ =Hþ homeostasis in this pathogenic bacterium, and is therefore presumably involved in the survival and persistence of free-living bacteria in their natural environment [115]. Another subfamily of Naþ =Hþ antiporters appears to predominantly contribute to the alkalinity of many extremophile bacteria. In Bacillus halodurans, a gene was identified which is responsible for electrogenic Naþ =Hþ antiport activity driven by DC (membrane potential, interior negative). The corresponding protein allowed the cells to maintain an intracellular pH lower than that of the external milieu (above pH 9.5). Sequence analyses indicated a significant similarity to the shaA gene product of Bacillus subtilis. Thus the shaA gene most likely encodes a Naþ =Hþ antiporter, which plays an important role in extrusion of cytotoxic Naþ [122]. In addition, results obtained from B. subtilis suggest that shaA plays a significant role at an early stage of sporulation, as well as during vegetative growth. Fine control of cytoplasmic ion levels, including control of the internal Naþ concentration, may be important for the progression of the sporulation process [123]. The three-dimensional structure of OxlT, an oxalate/formate antiporter from Oxalobacter formigenes, most recently solved [124], is explicitly different from that of both antiporter NhaA and symporter MelB (see above). There is an obvious twofold symmetry in the organisation of the 12 transmembrane helices. This supports the previous idea that the MFS proteins evolved from a duplication of a 6-TM protein to form a 12-TM protein. Moreover, it is possible that the 6-TM predecessor was created from a duplication of an ancestral 3-TM protein. 6.2.4
Binding Protein-Dependent Secondary Transporters
As demonstrated above, the typical secondary transporters work as single units, and do not depend on further transport components. One of the first examples of a secondary transport system that requires a periplasmic binding protein is the Naþ -dependent glutamate transporter of Rhodobacter sphaeroides with both transport and binding being highly specific for glutamate. Jacobs et al. [125] reported that growth of a glutamate transport-deficient mutant of
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R. sphaeroides on glutamate as sole carbon and nitrogen source was restored by the addition of millimolar amounts of Naþ . Uptake of glutamate (Km of 0:2 mmol dm3 ) by that mutant was dependent on the proton motive force (pmf) and strictly required Naþ (Km of 25 mmol dm3 ). Transport of glutamate was also observed in membrane vesicles when Naþ , a proton-motive force and purified glutamate binding protein, were present. A similar high-affinity transport system for the C4-dicarboxylates malate, succinate and fumarate was found in R. capsulatus [126]. Again, transport experiments indicated that the proton motive force, rather than ATP hydrolysis, drives uptake. The system is composed of DctP (periplasmic C4-dicarboxylate-binding protein), DctQ, and DctM, the latter two being characterised as integral membrane proteins. DctP, DctQ, and DctM are distinct from known transport proteins in the ABC (ATPbinding cassette) superfamily (see Section 6.3.2). The name TRAP (for tripartite ATP-independent periplasmic) transporters was proposed for this group of uptake systems [126]. Homologous systems were identified in the genomes of a number of Gram-negative bacteria, including Bordetella pertussis, E. coli, S. typhimurium and Haemophilus influenzae. 6.3
PRIMARY ACTIVE TRANSPORTERS
These transport systems use a primary source of energy to drive active transport of a solute against a concentration gradient. Primary energy sources can be chemical, electrical and solar. In this section, systems will be described mainly that hydrolyse the diphosphate bond of inorganic pyrophosphate, ATP, or another nucleoside triphosphate, in order to drive the active uptake of solutes. Transporters using another primary source of energy will be briefly mentioned. 6.3.1
ATPases
F-ATPases (including the Hþ - or Naþ -translocating subfamilies F-type, V-type and A-type ATPase) are found in eukaryotic mitochondria and chloroplasts, in bacteria and in Archaea. As multi-subunit complexes with three to 13 dissimilar subunits, they are embedded in the membrane and involved in primary energy conversion. Although extensively studied at the molecular level, the F-ATPases will not be discussed here in detail, since their main function is not the uptake of nutrients but the synthesis of ATP (‘ATP synthase’) [127–130]. For example, synthesis of ATP is mediated by bacterial F-type ATPases when protons flow through the complex down the proton electrochemical gradient. Operating in the opposite direction, the ATPases pump 3–4 Hþ and/or 3Naþ out of the cell per ATP hydrolysed. P-type ATPases from eukaryotes, bacteria, and Archaea catalyse cation uptake and/or efflux driven by ATP hydrolysis. Bacterial P-type ATPases
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consisting of one, two, or three components are found in the CM. There exists only a single catalytic subunit, the special feature of which is the formation of a phosphorylated intermediate during the reaction cycle [131]. Phosphorylation of the enzyme takes place at an aspartate residue in a highly conserved sequence. The phosphorylation forces the protein into an altered conformation, and the following dephosphorylation allows its return to the original state, so mediating the ion translocation. Distinct systems have been found for uptake of þ 2þ Kþ or Mg , uptake or efflux of Cu2þ or Cu and efflux of Ca2þ , Ag2þ , 2þ 2þ 2þ Zn2þ , Co2þ , Pb , Ni , and/or Cd [132–135]. 6.3.2
ABC Transporters
ABC transporters are multidomain systems that translocate substrates across membranes. A common characteristic is the well-conserved ATP binding cassette (ABC) domain that couples ATP hydrolysis to transport. Members of this group of proteins constitute the largest superfamily of transport components, and they are found in all organisms from Archaea to humans. According to the work of Dassa, who developed a classification based on the ATPase components, the ABC systems can be divided into a number of subfamilies (for details see http://www.pasteur.fr/recherche/unites/pmtg/abc/) [136]. ABC transporters are involved in both uptake and excretion of a variety of substrates from ions to macromolecules. Whereas export systems of this type are present in all kingdoms of life, import systems are exclusively found in prokaryotes. ABC transporters are minimally composed of two hydrophobic membrane embedded components and two ATPase units. 6.3.2.1 Binding Protein-Dependent Uptake Systems Binding protein-dependent uptake systems represent a subfamily of ABC transporters. They are composed of: (1) one or several extracellular (periplasmic) binding proteins; (2) one or two different (homodimer, heterodimer, or pseudo-heterodimer) polytopic integral membrane proteins (IMP); (3) two copies of an ATP-hydrolase (or two different ATPase units) facing the cytoplasm and supplying the system with energy. The best-studied systems to date are those for the uptake of maltose, histidine, siderophores and vitamin B12 . Typical of all the ABC-type importers are the soluble binding proteins, which bind the substrate with high affinity. Early studies revealed that many binding proteins could be extracted from Gramnegative bacteria by an osmotic shock procedure, giving rise to the term ‘osmotic-shock-sensitive’ transport systems. It still holds true that the majority
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of binding proteins of Gram-negative bacteria can more or less freely diffuse in the periplasm. However, in a few cases solute binding proteins are tethered to the cytoplasmic membrane via a hydrophobic a-helical transmembrane domain, or anchored via a lipid tail attached to an N-terminal cysteine residue. Such lipoproteins are also the commonly found primary substrate receptor of ABC transporters from Gram-positive bacteria. An unusual feature was observed in binding proteins (BPs) of Archaea. They can also be tethered to the membrane by means of a hydrophobic transmembrane span that is located at the C-terminus [45]. The structures of a considerable number of BPs have been solved. In general, BPs display very limited sequence homology, since they recognise a wide variety of different ligands. Nonetheless, all these proteins possess bilobate structures. In the majority of cases the two domains are linked by a hinge region formed by two or three flexible b-strands at the bottom of the ligand binding cleft – resembling a ‘Venus flytrap’ mechanism. For this scenario, substrate binding is generally assumed to induce a substantial conformational change from the ‘open’ to the ‘closed’ formation or to stabilise the latter [137]. A few BPs described recently (see Sections 7.3.3 and 7.3.5) differed from the ‘classical’ arrangement, in that the two lobes of the polypeptide chain were combined by a single a-helix. This rather rigid connecting structure does not allow the same degree of flexibility, thus indicating that the mode of BP/ligand interaction might be different in this subclass of BPs. The hydrophobic units are embedded within the CM. Depending on the system they are active as homo dimers, hetero dimers or pseudo-hetero dimers. Based on topological analyses, they contain five to 10 transmembrane regions (for the vast majority, six membrane spanning domains are predicted). As the central part of the translocation complex, the IMPs interact with BP and ATPase. No common motif has been reported concerning the contact sites with the BPs, and the mode of interaction remains unclear. In contrast, a common motif, also known as ‘EAA motif’ was identified in the IMPs of all bacterial permeases belonging to the ABC import systems [138,139]. The EAA motif is part of a structurally conserved region forming a helix-turn helix motif and functioning as the main contact area with the ATPase units. A detailed view suggesting possible mechanisms of ATPase-IMP interaction at the molecular level was obtained in the maltose system [140,141]. For a review see [142,143]. The characteristic Walker A and Walker B motifs that are involved in ATP binding [144] are always found in the ATPase or ABC domains. In addition, a signature motif, also called the LSGGQ motif, is typical of all bacterial ABC domains involved in binding-protein-dependent import. The signature motif is absent in other types of ATPases. A more detailed view also highlighting the special features of ABC transporters involved in iron uptake is provided in Sections 7.3.3. to 7.3.5.
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6.3.2.2 Systems Without Autonomous Binding Protein Bacterial ABC importers possess at least one BP that is essential for the uptake of substrates. In the ‘classical’ arrangement, this BP acts as a separate autonomous entity which is either anchored to the cytoplasmic membrane (typical of Archaea and Gram-positives, rare in the case of Gram-negatives) and/or can diffuse in the periplasmic space of Gram-negative bacteria. Systems with an unusual architecture were found in one family of Gram-negative bacteria and in some families of Gram-positive bacteria (best studied in L. lactis). ABC transporters involved in osmoregulation and osmoprotection, in particular, uptake systems for glycine/betaine and glutamate/glutamine, are characterised by ‘normal’ ATPase subunits but ‘chimeric proteins’ consisting of a hydrophilic portion fused to a hydrophobic integral membrane domain. The hydrophilic portion, localised at either the N- or the C-terminal end or both, displays characteristics typical of the classical BPs. In analogy with systems with known stoichiometry, the functional transport complex contains two or even four substrate binding domains (see Figure 3i) [145,146]. 6.3.3 Other Primary Active Transporters (not Diphosphate-Bond-Hydrolysis Driven) Decarboxylation-driven transporters catalyse decarboxylation of a substrate carboxylic acid, and use the energy released to drive extrusion of one or two ions (e.g. Naþ ) from the cytoplasm. Special enzymes mediate the decarboxylation of oxaloacetate, methylmalonyl-CoA, glutaconyl-CoA, and malonate [147]. Light absorption-driven transporters (e.g. bacterio- and halorhodopsins) pump 1 Hþ and 1 Cl per photon absorbed. Specific transport mechanisms have been proposed [148]. A single methyltransfer-driven transporter representing a multi-subunit protein complex has been isolated from the archaebacterium Methanobacterium thermoautotrophicum [149]. The porter has been characterised as Naþ -transporting methyltetrahydromethanopterin coenzyme M methyltransferase. Oxidoreduction-driven transporters drive transport of a solute (e.g. an ion) energised by the exothermic flow of electrons from a reduced substrate to an oxidised substrate. This subclass of porters includes the NADH:ubiquinone oxidoreductases type I, which couples electron transfer to the electrogenic transport of protons or sodium ions. These multi-subunit complexes consist of 13 or 14 protein subunits [150]. 6.4
GROUP TRANSLOCATORS
The bacterial phosphoenolpyruvate (PEP)-dependent carbohydrate phosphotransferase systems (PTS) are characterised by their unique mechanism of group translocation. The transported solute is chemically modified (i.e. phosphorylated) during the process (for comprehensive reviews see [151,152] and
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Cytoplasm
Periplasm
Mannitol 1-P P
P
Mannitol IIC IIA IIB HPr
EI
Glucose 6-P
P
PEP
P
P
Glucose IIC IIB
IIA Pyruvate P
Mannose 6-P EI
HPr P
P
IIC IIA IIB
Mannose
IID
Figure 6. Uptake of various sugars via the phosphoenolpyruvate group translocator (PTS) mechanism. Please note that all systems shown share the common components E1 and HPr. For details see the text
references therein) (see Figure 6). The first step in a cascade of subsequent phosphorylation and de-phosphorylation events occurs at the expense of PEP. In contrast to ATP, which is the driving force in many biochemical processes, PEP exclusively serves as the primary energy source in carbohydrate uptake. At the same time PEP is an important precursor for the biosynthesis of cell-wall components and aromatic amino acids. In a biochemical cycle linking PTS to glycolysis, two PEP molecules are generated from one sugar. One of these PEP molecules is used for the transport of the next sugar. The transport components at the end of the cascade couple translocation with phosphorylation of the substrates [153]. Uptake systems of the PTS type are widely distributed among bacteria, but they do not occur in archaea, animals and plants. A variety of sugars and sugar derivatives can be transported (e.g. glucose, sucrose, b-glucoside, mannose, mannitol, fructose, lactose and chitobiose). Regarding their composition, PTS systems follow a modular concept (see Figure 6). PTS proteins are phosphoproteins in which the phosphate group is attached to either a histidine residue
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or, in a number of cases, a cysteine residue. The typical PTS is composed of two general cytoplasmic proteins, the protein kinase enzyme I (EI) and the phospho-acceptor histidine protein (HPr). After phosphorylation of EI by PEP, the phosphate group is transferred to HPr. Enzymes II are then required for the transport of the carbohydrates across the membrane and the transfer of the phospho group from phospho-HPr to the carbohydrates. In the subsequent cascade, a variable number (three to four functional units) of the enzymes II (IIA, IIB, IIC, and, in rare cases, IID) are found, which represent sugarspecific enzymes. The autonomous entities can be arranged as one large protein, with membrane-embedded and cytoplasmic domains (e.g. uptake of mannitol in E. coli). Alternatively, two domains can be combined or just one domain can exist as an independent protein. PTS-mediated substrate translocation can be descibed by the reactions: PEP þ EI $ P HisEI þ pyruvate
(1)
P EI þ HPr $ P HisHPr þ EI
(2)
P HPr þ IIA $ P HisIIA þ HPr
(3)
P IIA þ IIB $ P CysIIB þ IIA or P HisIIB þ IIA)
(4)
IIC(IID)
P IIB þ substrateout !
substrate Pin þ IIB
(5)
The free energy of the phosphorylated histidine (P His) or cysteine (P Cys) is comparable with the free energy of PEP (DG80 ¼ 61:5 kJ mol1 ). The reactions (1) to (4) are therefore fully reversible under physiological conditions, whereas reaction (5) is irreversible. The substrate when bound to the domain IIC (or IID) obtains the phosphoryl group from the unit IIB, via unit IIA, which is rephosphorylated by P HPr. Efficient translocation of carbohydrates depends on the phosphorylated IIB domain. The release of the phosphorylated substrate terminates the uptake process.
7 UPTAKE OF IRON: A COMBINATION OF DIFFERENT STRATEGIES 7.1
IRON – A ‘PRECIOUS METAL’
For most living bacteria (lactobacilli being the only notable exception [154]) iron is an essential nutrient. Iron is not readily available under normal conditions, although it is the fourth most abundant metal on earth. In the environment it is mainly found as a component of insoluble hydroxides; in biological systems it is chelated by high-affinity iron binding proteins (e.g. transferrins,
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lactoferrins, ferritins) or exists as a component of erythrocytes (haem, haemoglobin, hemopexin). It has a vital function because it is a component of key molecules such as cytochromes, ribonucleotide reductase and other metabolically linked compounds. Since the redox potential of Fe2þ =Fe3þ spans from 1300 mV to 2500 mV, depending on the ligand and protein environment, iron is well suited to participate in a wide range of electron-transfer reactions. Moreover, it has been shown for a number of bacterial pathogens that sufficient iron is essential [155–159]. It is not therefore surprising that microorganisms have developed a number of different sequestering strategies for this ‘precious’ metal. Under anaerobic conditions, the ferric iron can be transported without any chelators involved. Likewise, at pH 3 the ferric iron is soluble enough to support growth of acid-tolerant bacteria. At higher pH values, iron is mostly found in insoluble componds. Therefore a great variety of low-molecularweight high-affinity iron(III) binding ligands, called siderophores, are produced by many bacterial species and certain fungi. Three major structural types are found: catecholates, hydroxamates, and a-hydroxycarboxylates [160]. The chelators are released in their iron-free forms and then transported as ferric– siderophore complexes. Whereas the internalisation of siderophores is rather well characterised, the mechanism of siderophore secretion to the extracellular environment remains only poorly understood. Zhu et al. [161] identified putative export components in mycobacteria, and, most recently, an important observation was published indicating that E. coli membrane protein P43, encoded by the entS gene (formerly ybdA) a critical component of the enterochelin secretion machinery, is located in the chromosomal region of genes involved in enterobactin synthesis. The EntS protein shows strong homology to the 12-transmembrane segment major facilitator superfamily of export pumps [162]. Certain bacteria (many of them pathogens) are able to use haem-bound iron from haemoglobin, haemopexin, and haptoglobin [156,159,163]. In some bacteria (e.g. Serratia marcescens) a haem-binding protein, called haemophore, is secreted in its apo-form and then taken up in a receptor-mediated fashion [163]. In addition, some species can acquire iron from transferrins or lactoferrins involving specific uptake systems in the cell envelope [24,158,164]. A summary of the most important iron-sequestering systems is given in Figure 7. 7.2 IRON TRANSPORT ACROSS THE OUTER MEMBRANES OF GRAM-NEGATIVE BACTERIA Specific receptors for siderophores and vitamin B12 have been identified in the OM of Gram-negative bacteria. The translocation of these ligands across the outer membrane follows an energy-dependent mechanism and also involves the TonB, ExbB, ExbD proteins anchored in the cytoplasmic membrane. Biochemical and genetic data indicate that these proteins form a functional unit (the Ton complex), which couples the outer membrane receptor-mediated
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Metals (Fe, Zn, Mn, ...)
Ferric siderophore
Haempexin Haemoglobin Heme Haemophore
Siderophore
Transferrin Lactoferrin
Haemophore
Cytoplasmic membrane Periplasm Outer membrane Metals (Mn, Zn, Fe ...)
Ferrous iron
Figure 7. The summary of systems involved in (or related to) the uptake and assimilation of iron represents a schematic view of a typical Gram-negative bacterium. The OM receptors, as well as the proteins of the Ton complex, are not present in Gram-positive bacteria, and were not found in mycobacteria and members of the mycoplasma group
transport to the electrochemical potential across the inner membrane (reviewed in [63–65,165,166]). Certain pathogenic Gram-negative bacteria living in their hosts under conditions where the availability of free iron is strongly limited, and the iron acquisition by siderophores is not appropriate, acquire haem and iron from host haem-carrier proteins. The main mechanisms involve either direct binding (like siderophores or vitamin B12 ) to specific OM receptors, or the release of bacterial haemophores that take up haem from host haem carriers and shuttle it back to specific TonB-dependent OM receptors [159,163,167]. The S. marcescens hemophore is a monomer which binds haem with a stoichiometry of 1, and an affinity lower than 109 mol dm3 . The crystal structure of the holoprotein has been solved, and found to consist of a single module with two residues interacting with haem [168]. The secretion of haemophores depends on specific export systems of the ABC-type [169,170]. TonB-dependent OM receptors also play an essential role in the utilisation of iron that is bound to transferrins and lactoferrins. At present, the mechanism by which the iron is released from these polypeptides has not been elucidated [158,164].
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In E. coli the passage across the outer membrane is the rate-limiting step for the siderophores on their way from the environment into the cytoplasm [171]. Furthermore, the siderophore receptors display higher substrate specificity than the proteins of the ABC systems mediating further transfer of siderophores into the cytoplasm. Often there exist more different TonB-dependent OM receptors in a bacterial cell than corresponding permeases. This difference can be very striking, as, for instance, in Pseudomonas aeruginosa, where 34 putative TonB-dependent receptors were identified in the genome [172], whereas only four iron-related ABC transporters seem to exist in this organism. Interestingly, Caulobacter crescentus, a Gram-negative bacterium that grows in dilute aquatic environments has no OmpF-type porins that would allow hydrophilic substrates to diffuse passively through the outer membrane. Instead, there is evidence that as many as 65 OM receptors of the high-affinity TonB-dependent type may catalyse energy-dependent transport of a number of solutes [173]. The structures of several TonB dependent outer membrane transport proteins have been investigated, thus allowing a more detailed insight into energy-coupled uptake mechanisms. The proteins FhuA, FepA, FecA and BtuB from E. coli, whose crystal structures are available, display a striking similarity with respect to their overall organisation in the lipid bilayer. A barrel-like structure anchored in the membrane forms a channel which is (partially) closed by a globular domain referred to as a ‘cork’, ‘plug’, or ‘hatch’ domain (see Figure 8). The FhuA receptor of E. coli transports the hydroxamate-type siderophore ferrichrome (see Figure 9), the structural similar antibiotic albomycin and the antibiotic rifamycin CGP 4832. Likewise, FepA is the receptor for the catecholtype siderophore enterobactin. As monomeric proteins, both receptors consist of a hollow, elliptical-shaped, channel-like 22-stranded, antiparallel b-barrel, which is formed by the large C-terminal domain. A number of strands extend far beyond the lipid bilayer into the extracellular space. The strands are connected sequentially using short turns on the periplasmic side, and long loops on the extracellular side of the barrel. The N-terminal domain of FhuA folds into the barrel, thereby forming a ‘plug’ or ‘cork’ which obstructs the free passage of solutes through the otherwise open channel. The cork domain consists of a mixed four-stranded b-sheet and a series of short a-helices, thus delineating a pair of pockets within FhuA. The extracellular pocket is larger and open to the external medium, while the periplasmic pocket is smaller and in contact with the periplasmic space. An aromatic pocket near the cell surface representing the initial binding site of ferrichrome undergoes minor changes upon association with the ligand, revealing two distinct conformations in the presence and absence of ferrichrome [66,67]. As in FhuA, the special feature of the FepA structure is also the N-terminal domain, which folds into the barrel pore (Figure 5). The core of that ‘plug’ is a four-stranded mixed b-sheet, formed by a central b-hairpin, which is flanked on one side by an antiparallel b-strand and on the other side by a parallel b-strand.
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Figure 8 (Plate 7). Structure of the Escherichia coli FhuA protein serving as receptor for ferrichrome and the antibiotic albomycin. (a) side view; (b) side aspect with partly removed barrel to allow the view on the ‘cork’ domain; (c) top view. A single lipopolysaccharide molecule is tightly associated with the transmembrane region of FhuA (reproduced by permission of W. Welte and A. Brosig)
FecA, the transporter of ferric citrate, is composed of three functional domains. In addition to the barrel and the plug, the receptor contains an extra portion at its N-terminus [174]. This domain, comprising 80 residues,
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Figure 9. Structure of the siderophore ferrichrome (and derivatives) produced by certain fungal species
is typical of a subclass of TonB-dependent receptors. It resides in the periplasm and is involved in a special signal transduction process [175]. The three-dimensional structure FecA displays a number of special features. Probably due to its flexibility, the N-terminal extension could not be located in the crystal structure. However, based on the crystallographic data obtained with and without a bound ligand, a bipartite gating mechanism was described [174]. The subsequent formation of two gates allows for a rational distinction between the binding event and the transport process. A conformational change of the extracellular loops takes place upon binding of the ligand diferricdicitrate that closes the external pocket of FecA. Ligand-induced allosteric transitions are then propagated through the outer membrane by the plug domain, signalling the occupancy of the receptor in the periplasm. These data establish the structural basis of gating for receptors dependent on the cytoplasmic membrane protein TonB [174,176]. Since the proteins in the OM have no direct access to energy-producing pathways, active transport steps depend on activation of TonB by the proton electrochemical potential of the inner membrane. Activated TonB, which together with the ExbB and ExbD proteins is part of a membrane-anchored complex, can then bind to the outer membrane iron transporters, transducing energy to them. In the absence of the proton gradient or TonB, ligands are unable to cross the outer membrane, but still bind with high affinity to their ˚ resolution of the C-terminal transporters [177]. The crystal structure at 1.55 A domain of TonB from E. coli has been reported recently. The structure displays a novel architecture with no structural homology to any known polypeptides, and there is evidence that this region of TonB (residues 164–239) dimerises. The dimer of the C-terminal domain of TonB appears cylinder-shaped, and each monomer contains a single a-helix and three b-strands. The two monomers are
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intertwined with each other, thus leading to the formation of a large antiparallel b-sheet composed of all six b-strands of the dimer [178]. The stoichiometry of the TonB-ExbB-ExbD complex is not yet solved, and it remains open if TonB functions as a dimer in vivo. 7.3 IRON TRANSPORT ACROSS THE CELL WALLS OF GRAMPOSITIVE BACTERIA Whereas many aspects of receptor-mediated transport of siderophore-chelated, haem-bound and transferrin/lactoferrin-associated iron across the outer membrane of Gram-negative bacteria are fairly well understood, comparatively little information is available on suitable systems for iron acquisition in the envelope of Gram-positive bacteria. Siderophores may diffuse through the multilayered murein (¼peptidoglycan) network without major problems, but many questions remain open as to how pathogens belonging to this group of bacteria access iron from host iron sources. One group of surface proteins, associated with the cell walls of Grampositives, is characterised by a signature sequence located near the C-terminus, reading ‘LPXTG’ or ‘NPQTN’. Proteins containing one of these known cellwall-anchoring motives will be attached to the murein. This sorting step involves specific enzymes designated sortase A and sortase B, respectively, displaying a transpeptidase function [179,180]. As illustrated below, a subclass of the cell-wall-anchored proteins may act as receptors for iron-containing compounds such as haem, haemoglobin, transferrin and lactoferrin. The Gram-positive bacterium Streptococcus pneumoniae is an important cause of respiratory tract infections, bacteremia, and meningitis. In this strain, the cell wall anchored pneumococcal surface protein A (PspA) has been demonstrated to bind lactoferrin [181]. PspA and closely related proteins in a variety of pneumococcal isolates are most likely involved in the sequestration of iron from lactoferrins, and finally contribute to the virulence of these bacteria. However, the means by which the pneumococcus acquires iron at the mucosal surface during invasive infection is not well understood at the molecular level [182]. Staphylococci have evolved sophisticated iron-scavenging systems, including cell-surface receptors for transferrin. In Staphylococcus epidermidis and certain S. aureus strains, an iron-regulated transferrin receptor (Tpn) has been described as cell-surface-associated glyceraldehyde-3-phosphate dehydrogenase, which not only retains its glycolytic enzyme activity, but also possesses NADribosylating activity and binds diverse human serum proteins. Tpn was introduced as a member of a newly emerging family of multifunctional OM proteins that are putatively involved in iron acquisition and contribute to staphylococci virulence [183,184]. Conflicting results were published recently by another laboratory, providing evidence that mutants lacking Tpn were still capable of
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binding transferrin. Instead, the stbA gene was identified to encode a putative cell-wall-anchored transferrin receptor [185]. Interestingly, in a different S. aureus strain, IsdA, a protein almost identical to StbA, was characterised as a haem-binding surface protein. Moreover, the isdA gene is located in a genomic region which also encodes a haemoglobin receptor protein and other haem-binding proteins, some of which display similarity to known transport proteins related to iron uptake [186]. A novel type of haem-associated cell-surface protein was identified in Streptococcus pyogenes [187]. The Shp protein shows most of the characteristics (signal sequence at the N-terminus and a hydrophobic putative transmembrane region followed by a positively charged C-terminus) of the cell-wall-anchored proteins described above, but it is missing an obvious sorting signal. The involvement of Shp in haem acquisition is likely, since genes encoding components of ABC transporters clustering with iron uptake systems were identified in the same region of the genome. 7.4 IRON TRANSLOCATION ACROSS THE CYTOPLASMIC MEMBRANE: VARIOUS PATHWAYS Based on experimental data and analysis of sequences available from the databases, we can conclude that different routes for the translocation of iron across the cytoplasmic membrane are possible in bacteria. They can mediate the importation of ferrous iron, and of ferric iron, both in its ionic form and coupled to siderophores or haem. Three of the transport systems represent members of the binding protein-dependent type (a subfamily of ABC transporters or traffic ATPases) (see Section 6.3.2). 7.4.1
feo Type Transport Systems for Ferrous Iron
E. coli has an iron(II) transport system, feo, which may make an important contribution to the iron supply of the cell under anaerobic conditions. Kammler et al. [188] identified the iron(II) transport genes feoA and feoB. The upstream region of feoAB contained a binding site for the regulatory protein Fur, which acts with iron(II) as a corepressor in all known iron transport systems of E. coli. In addition, a Fnr binding site was identified in the promoter region. The FeoB protein (70 kDa) was localised in the cytoplasmic membrane. The sequence revealed regions of homology to ATPases, which indicates that ferrous iron uptake may be ATP driven. Genes with significant similarity to these have been found in the genomes of a great number of bacteria. Biphasic kinetics of Fe2þ transport in a wild-type strain of H. pylori suggested the presence of high- and low-affinity uptake systems. The high-affinity system (apparent Ks ¼ 0:54 mmol dm3 ) is absent in a mutant lacking the feoB gene. Transport via FeoB is highly specific for Fe2þ , and was inhibited by FCCP,
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DCCD and vanadate. This indicates an active process energised by ATP. Ferrozine inhibition of Fe2þ and Fe3þ uptake implied the concerted involvement of both a Fe3þ reductase and FeoB in the uptake of iron supplied as Fe3þ . It is concluded that FeoB-mediated Fe2þ represents a major pathway for H. pylori iron sequestration [189]. In addition, growth experiments on the human pathogen L. pneumophila using artificial media, as well as replication studies within iron-depleted Hartmannella vermiformis amoebae and human U937 cell macrophages, provided evidence that the FeoB transporter is important for extracellular growth and intracellular infectivity [190]. 7.4.2
Metal Transport Systems of the Nramp Type
Another example of a metal transporter family is the Nramp-family. The Nramp transporters (natural resistance associated macrophage proteins) are transmembrane proteins found in many eukaryotic and prokaryotic organisms, including Archaea, bacteria, yeast, insects, mammals and higher plants. Nramp1 was the first identified gene of this family, characterised in mice as a protein involved in host resistance to certain pathogens [191–193]. Nramp2 was shown to be the major transferrin-independent iron-uptake system of the intestine in mammals. It is capable of mediating influx of transition-metal 2þ divalent cations, including Fe2þ , Mn , and probably, Cd2þ , Co2þ , Ni2þ , 2þ 2þ Cu and Zn [194]. Smfp1, Smfp2, and Smfp3, the homologous protein in yeast, are rather selective for Mn2þ but have been recently linked to the uptake of other heavy metals, including copper, cobalt and cadmium [195]. Hence, like mammalian Nramp transporters, yeast Smf proteins exhibit a broad specificity for both essential and non-essential toxic metals. The three different Smf proteins are distinguishable by their distinct cellular localisations: Smfp1 was found in the cytoplasmic membrane, Smfp2 in intracellular vesicles, and Smfp3 at the vacuolar membrane, indicating that the transporters play different roles in metal metabolism [196]. The only bacterial Nramp proteins characterised so far are highly selective for Mn2þ . This is found for the E. coli MntH, which is a proton-dependent divalent cation co-transporter with a preference for Mn2þ (Km approximately 0:1 mmol dm3 [197,198]). In Salmonella enterica serovar Typhimurium and E. coli, the mntH gene is regulated at the transcriptional level by both substrate cation and H2 O2 . In the presence of Mn2þ , MntH expression is prevented mainly by the manganese transport repressor, MntR. MntR strongly interacts with an inverted-repeat motif on the DNA located between the 10 polymerase binding site and the ribosome binding site. In the presence of Fe2þ , the Fur repressor blocks expression of mntH, acting through a Fur-binding motif overlapping the 35 region [199,200]. In the presence of hydrogen peroxide, mntH is activated by the OxyR protein, which binds to a consensus motif just upstream of the putative promoter [200]. Biochemical and genetic studies
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demonstrate that the MntH transporter in the Gram-positive bacterium B. subtilis is selectively repressed by Mn(II). This regulation requires the MntR protein acting under high Mn(II) conditions as repressor of mntH transcription [201]. 7.4.3
ABC Transporters for Siderophores/Haem/Vitamin B12
ABC transporters involved in the uptake of siderophores, haem, and vitamin B12 are widely conserved in bacteria and Archaea (see Figure 10). Very few species lack representatives of the siderophore family transporters. These species are mainly intracellular parasites whose metabolism is closely coupled to the metabolism of their hosts (e.g. mycoplasma), or bacteria with no need for iron (e.g. lactobacilli). In many cases, several systems of this transporter family can be detected in a single species, thus allowing the use of structurally different chelators. Most systems were exclusively identified by sequence data analysis, some were biochemically characterised, and their substrate specificity was determined. However, only very few systems have been studied in detail. At present, the best-characterised ABC transporters of this type are the fhuBCD and the btuCDF systems of E. coli, which might serve as model systems of the siderophore family. Therefore, in the following sections, this report will mainly focus on the components that mediate ferric hydroxamate uptake (fhu) and vitamin B12 uptake (btu). The fhu genes of E. coli constitute an iron-regulated operon starting with the fhuA gene, which encodes the outer membrane receptor for ferrichrome and albomycin mentioned in Section 7.2. The fhuC, fhuD, and fhuB genes – organised downstream from fhuA in this order – are essential for further translocation across the inner membrane [202]. Genes contributing to iron acquisition can also be located on mobile genetic elements (e.g. pathogenic islands) inserted into the chromosome or on episomal DNA, like the fat genes of the fish pathogen Vibrio anguillarum [203,204]. As mentioned above, transport of siderophores across the cytoplasmic membrane is less specific than the translocation through the outer membrane. In E. coli three different outer membrane proteins (among them FepA the receptor for enterobactin produced by most E. coli strains) recognise siderophores of the catechol type (enterobactin and structurally related compounds), while only one ABC system is needed for the passage into the cytosol. Likewise, OM receptors FhuA, FhuE, and Iut are needed to transport a number of different ferric hydroxamates, whereas the FhuBCD proteins accept a variety of hydroxamate type ligands such as albomycin, ferrichrome, coprogen, aerobactin, shizokinen, rhodotorulic acid, and ferrioxamine B [165,171]. For the vast majority of systems, the substrate specificity has not been elucidated, but it can be assumed that many siderophore ABC permeases might be able to transport several different but structurally related substrates.
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At present, FhuD is the best-characterised siderophore binding protein. Binding of different iron(III) hydroxamates to the mature FhuD protein has been demonstrated [205,206]. Changes in the intrinsic fluorescence of purified FhuD allowed the estimation of the dissociation constants (KD ) for ferric aerobactin (0:4 mmol dm3 ), ferrichrome (1:0 mmol dm3 ), ferric coprogen (0:3 mmol dm3 ), ferrioxamine A (79 mmol dm3 ), ferrioxamine B (36 mmol dm3 ), ferrioxamine E (42 mmol dm3 ), and albomycin (5:4 mmol dm3 ). FhuD contributes to a large extent to the substrate specificity of transport through the cytoplasmic membrane. Recently, the characterisation of FepB from E. coli by intrinsic fluorescent measurements revealed a significantly lower KD (30 nmol dm3 ) for ferric enterobactin [207]. The crystal structure of FhuD complexed with gallichrome has been solved at ˚ [208] (see Figure 11). The binding of the siderophore to FhuD is mediated 1.9 A by hydrophilic and hydrophobic interactions. The ligand binding site represents a shallow groove between the N- and C-terminal domains of the kidney-beanshaped bilobate protein. Remarkably, the polypeptide chain crosses between the N-terminal and C-terminal domains only once. The linker connecting the two domains is a kinked a-helix, which spans the entire length of the protein. The N-terminal domain consists mainly of a twisted five-stranded parallel b-sheet and the C-terminal domain is composed of a five-stranded mixed b-sheet. Both sheets are sandwiched between layers of a-helices. From the extensive, predominantly hydrophobic, domain interface in FhuD it is
Ferric siderophore Ferric siderophore
OM-Receptor
OM BP
BP
Ton complex
PP
IMP
CM ADP ATP
ADP
ATPase
ATP
Gram-positive bacteria
IMP
ADP ATP
ATPase
CM
ADP ATP
Gram-negative bacteria
Figure 10. Schematic view of the uptake of ferric siderophores by Gram-positive and Gram-negative bacteria. Please note that the murein (peptidoglycan) network associated with the cytoplasmic membrane is not shown. For details see text
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concluded that binding and release of the ligands probably do not cause large scale opening and closing of the siderophore binding site. Thus, FhuD differs from the majority of BPs in that it does not adopt the ‘classic’ fold that has been observed in almost all BPs that are structurally characterised to date (see Section 6.3.2). FhuD adopts a novel fold (missing the flexible hinge region) and represents a new class of BPs [208]. FhuD forms a distinctive family, together with all BPs that transport siderophores, haem, and vitamin B12 . Members of this siderophore family are clearly distinguishable from any other component involved in the uptake of metals. Despite considerable differences in size (28 to over 40 kDa) and very limited similarity in their amino acid sequences, the BPs of the siderophore family possess characteristic signatures, also pointing to an internal homology. Moreover, the recently solved crystal structure of the vitamin B12 binding protein BtuF [209,210] demonstrated a significant structural similarity when the FhuD and BtuF structures were
Figure 11. Crystal structure of the hydroxamate binding protein FhuD from Escherichia coli complexed with gallichrome. (Reproduced by permission of H. J. Vogel and T. E. Clarke)
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superimposed. Vitamin B12 is bound in the ‘base-on’ conformation in a deep cleft formed at the interface between the two lobes of BtuF. Comparison of the ligandbound and the apo structures lead to the conclusion that the unwinding of a surface located a-helix in the C-terminal domain of BtuF may take place upon binding to BtuC. This conformational change could be important for triggering the release of B12 into the transport cavity and further passage through the BtuC2 D2 complex in the CM [210]. At present, only two structurally characterised BPs share some topological similarities with FhuD and BtuF: Mn2þ (and possibly Zn2þ ) binding PsaA from S. pneumoniae, and the zinc-binding protein TroA from Treponema pallidum [211,212] (see Section 7.3.5). It is assumed that FhuD binds and delivers ferric hydroxamates to the FhuB transport protein in the cytoplasmic membrane. Genetic and biochemical experimental approaches (e.g. protease protection and cross-linking experiments) indicate a physical interaction of FhuD with FhuB [202,206]. Likewise, the vitamin B12 binding protein is thought to transport the ligand to the transport components in the membrane. This picture is in good agreement with the observed formation in vitro of a stable complex between BtuF and BtuCD (with the stoichiometry BtuC2 D2 F). After Locher et al. [209,213] had determined a high-resolution atomic structure of BtuC2 D2 (the first structure of an ABC importer complex composed of integral membrane proteins and ATPases), modelling of the individual crystal structures suggested that two surface exposed glutamates from BtuF may interact with arginine residues on the periplasmic surface of the BtuC dimer in the CM [209]. These glutamate and arginine residues had already been reported to be conserved among BPs and IMPs related iron and B12 uptake [202]. It therefore can be assumed that they may play a more general role in protein–protein interaction and the triggering of conformational changes. FhuB is an extremely hydrophobic polytopic integral membrane protein. During substrate translocation, FhuB plays a central role in the system, interacting not only with FhuD and FhuC but also with the different ferric hydroxamates. For many years, FhuB was unique among the integral membrane proteins, in that it is about double the size (70 kDa) of comparable components from other ABC transporters and it consists of two major domains displaying significant homology to each other. Both halves (FhuB[N] and FhuB[C]) of the polypeptide are essential for transport; deletion of either domain results in loss of activity. FhuB[N] and FhuB[C] are still functional when produced as two distinct polypeptides [171]. Recently, FhuB-like IMPs have been identified in V. cholerae, Rhizobium leguminosarum, and R. capsulatus. Complementation studies with the dissected FhuB strongly suggest that each of the two FhuB halves has the potential to insert independently into the lipid bilayer, where it diffuses and associates stochastically with the complementary subunit present. The mode of recognition and interaction of the transport components is probably very similar in most ABC transporters. Little is known, however, about
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the interaction of the integral membrane proteins at the molecular level. The interaction of the other siderophore family IMPs may in principle follow the same rule as the FhuB[N]–FhuB[C] interaction. Most IMPs involved in siderophore uptake might form hetero dimers. In haeme and vitamin B12 transport systems the formation of homo dimers is most likely, since only one IMP was detected in the relevant genomic regions of e.g. E. coli, V. cholerae, P. aeruginosa, Shigella dysenteriae, Yersinia enterocolitica and Y. pestis. Several areas of striking homology are present in the primary structures of the siderophore family IMPs. One of these regions includes a glycine residue at a distance of about 100 amino acids from the C-terminus. It corresponds with the conserved Gly [139], which is part of the ‘E A A - - - G - - - - - - - - - I - L P’ motif defined by Dassa and Hofnung [138] (see Section 6.3.2). This conserved region (CR), present twice in FhuB, plays a general role in the translocation process [214]. The homologous regions, especially the conserved glycine residues, are believed to be structurally and/or functionally important for the other siderophore family uptake systems as well. The topology of FhuB differs from the equivalent components of other ABC transporters, in that each half consists of 10 membrane-spanning regions. The location of N- and C-termini is cytosolic. The CR is also oriented to the cytoplasm. However, in contrast to the ‘classical’ arrangement, this putative ATPase interaction loop is followed by four instead of two transmembrane spans [215]. A schematic topology model is presented in Figure 12. It is assumed that the hydrophilic regions may – entirely
Figure 12 (Plate 8). Transmembrane arrangement of the polytopic FhuB protein in the cytoplasmic membrane as determined by the analysis of ß-lactamase proteins C-terminally fused to various portions of FhuB. The FhuB protein is composed of two times 10 membrane-spanning regions connected by loops contacting the periplasm or the cytoplasm. These loops were predicted to entirely or partly fold back into the overall structure. The conserved regions (CR) typical of all prokaryotic importers belonging to the ABC transporter family are shown in dark gray. They are important for the interaction with the FhuC protein – the ATPase supplying energy for the siderophore translocation process
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or partially – fold back into a channel-like structure built by the transmembrane spans. At present, it cannot be decided to what extent the ‘loops’ are accessible from the periplasm or cytoplasm. Sequence analysis data suggest a similar arrangement for all IMPs of the siderophore family. Support for this idea came again from the structural data of the BtuC2 D2 complex [213]. Remarkably, each of the hydrophobic BtuC units contains 10 transmembrane regions. This arrangement is in perfect agreement with the predicted FhuB topology model, containing altogether 20 membrane-spanning segments. See also Figure 13. The analysis of the primary structure of FhuC had suggested a function as an ATP-binding component. FhuC was one of the first ATPases (of bindingprotein-dependent import systems) in which highly conserved residues in the ‘Walker A’ and ‘Walker B’ consensus motifs were altered. A total loss of function in all these FhuC derivatives indicated that FhuC indeed acts as an ATP-hydrolase, thereby energising the transport process, most likely via inducing conformational changes in the components of the permease complex [165,171,216]. Since the ATPases are the components which are the most conserved among all ABC transporters, it is highly likely that the structural features and the mechanism of energisation are very similar in all these systems. It was concluded from previous studies, in particular in the histidine and in the
Figure 13 (Plate 9). Crystal structure of the BtuC2 D2 complex involved in the uptake of vitamin B12 . Two copies of the polytopic integral membrane protein BtuC and of the ATPase subunit BtuD are shown, together with bound ATP (reproduced by permission of K. Locher). For more details see the text
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maltose uptake systems, that the active ATPase subunits function as a dimer. The crystal structures of some ATPases have been solved in absence of the integral membrane proteins (for further information see [142,217–221]. They all adopt a similar L-shaped structure with two arms, one containing the signature motif, and the other the Walker A and B motifs. Different arrangements of the ATPase proteins in a putative dimer had been proposed, but it remained unclear which orientation is realised in the active permease complex: ‘back to back’, ‘head to tail’ or ‘head to head’. In the reported structure of the BtuC2 D2 complex the ‘head to tail’ orientation is realised, with a surprisingly small interface between the two ATPase units and with the Walker A motif of one monomer facing the LSGG motif of the other. Each nucleotide-binding site contains residues from both monomers. This architecture now supports previous biochemical data obtained with the functional ABC transporters, in that it provides a sound basis for the cooperativity observed in the nucleotide-binding domains [222,223]. In addition, the participation of the highly conserved family signature motif (LSGGQ) in ATP binding and hydrolysis becomes more understandable. Interaction of ATPase FhuC with IMP FhuB was first demonstrated by dominant negative effects on transport of FhuC derivatives with single amino acid replacements in the putative ATP-binding domains. Furthermore, immunoelectron microscopy with anti-FhuC antibodies showed FhuB-mediated association of FhuC with the cytoplasmic membrane [216]. Moderate overexpression of the FhuB derivatives (point mutation in the CRs) in a fhu wild-type strain displayed a negative complementing phenotype to various extents, as shown by growth tests, and transport rates. These experimental data already indicated that the CR is mainly involved in the interaction with FhuC. Again, a detailed molecular view identifying amino acid residues as potential candidates for protein–protein interaction and suggesting possible mechanisms of ATPase-IMP interaction was feasible by analysing the structure of the BtuC2 D2 for vitamin B12 uptake (Figure 13). In this arrangement, the BtuC homo dimer resembles the two halves of FhuB, whereas the BtuD units are equivalent to the FhuC components. Since proteins trapped in a crystal structure represent only a snap shot of a dynamic process, further studies will be necessary in order to unravel the details of the actual translocation process. 7.4.4
ABC Transporters of the Ferric Iron Type
The first transporter of this type characterised as an iron-supply system that functions in the absence of any siderophore was the Sfu system of S. marcescens [224]. Later, similar systems were reported from Neisseria gonorrhoea and Neisseria meningitidis, and have been detected by analysing the genomes of a variety of bacteria, e.g. Actinobacillus pleuropneumoniae, B. halodurans, Campylobacter jejuni, Ehrlichia chaffeensis, Halobacterium sp., H. influenzae,
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Brachyspira hyodysenteriae, Pasteurella haemolytica, P. aeruginosa, V. cholerae and Y. enterocolitica [156–158,164]. The Sfu/Fbp-like systems contribute to the virulence of pathogenic bacteria. They are thought to mediate the further transport into the cytoplasm of ferric iron that is acquired from lactoferrin or transferrin and delivered into the periplasm in a receptor-mediated Ton complex-dependent fashion [157–159]. A rather uniform organisation of ferric iron transport genes seem to be the rule for most bacterial species studied so far. A putative iron-regulated operon contains genes encoding the substrate-binding protein, the IMP, and the ATPase – in this order. The ferric-binding protein (FbpA) is one of the major iron-regulated proteins, and is highly conserved in all species of pathogenic Neisseria [157]. The first crystal structure was solved for the FbpA homologue from H. influenzae (HitA), and was found to be of the ‘classical’ arrangement with a flexible hinge region. Interestingly, iron binding in HitA and transferrin appears to have developed independently by convergent evolution. From structural comparison of HitA with other prokaryotic BPs and the eukaryotic transferrins, it is concluded that these proteins are related by divergent evolution from an anion-binding common ancestor rather than from an iron-binding ancestor. The iron-binding site of HitA incorporates a water molecule and an exogenous phosphate ion as iron ligands [157]. The IMPs of the ferric iron type display an internal homology, in which each half, smaller than IMPs of most other ABC transporters, harbours a CR as putative interaction site with the ATPase. Interestingly, in the B. hyodysenteriae Bit system the hydrophobic membrane domains are expressed as two separate proteins. The ATPases from ferric iron transport systems show typical characteristics, and are supposed to follow the same mechanism of energising the translocation step of substrates into the cytosol. No detailed studies have been reported at the molecular level. 7.4.5
ABC Transporters for Iron and Other Metals
The third group of ABC type importers related to iron uptake in bacteria was discovered a few years ago. Transport systems of the metal type are present in many bacterial species. Only a small number of uptake systems are primarily involved in the acquisition of iron. Many have a higher specificity for metals like zinc or manganese. For some systems it has been clearly shown that they are essential for iron acquisition (e.g. Yfe of Y. pestis and Sit of S. typhimurium). As in the ABC transport systems mentioned above, the genes encoding components of metal-type ABC transporters are often organised in operons. The expression of the vast majority seems to be regulated by the degree of metals present in the environment, often depending on the metals to be transported. A number of repressors acting at the transcriptional level with
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different metal binding specificity and different recognition sequences on the DNA have been identified. The crystal structures of binding proteins PsaA from S. pneumoniae and TroA from T. pallidum have been solved at 2.0 ˚ solution, respectively [211,212]. Both proteins consist of an N- and and 1.8 A C-lobe, each composed of b-strand bundles surrounded by a-helices. The two domains are linked together by a single helix. Also found for the structurally similar siderophore binding protein FhuD the structural topology was fundamentally different from that of other ‘classical’ ABC-type binding proteins, in that PsaA and TroA were lacking the characteristic ‘hinge peptides’ involved in conformational change upon solute uptake and release. Experimental evidence suggests that the BPs of the metal-type systems do not completely share metal specificity, as S. typhimurium SitA binds primarily iron and Yersinia pestis YfeA iron and manganese. PsaA from S. pneumoniae is presumed to bind primarily Mn2þ , and possibly Zn2þ , T. pallidum TroA and S. pneumoniae AdcA bind primarily Zn2þ , and Synechocystis MntC binds Mn2þ . The variation in metal specificity amongst the metal-type BPs is reflected by the variation in those residues (His, Asp, Glu) that are sequence related to the metal-coordinating residues, allowing the BPs to be grouped into several subclusters. The hydrophobic components of the metal-type system display characteristics typical of IMPs from most ABC transporters. The same holds true for the corresponding ATPases.
7.5
OTHER SYSTEMS
L. pneumophila, a facultative intracellular parasite of human alveolar macrophages and protozoa, causes legionnaires’ disease. Two genes related to virulence were detected, iraA encoding a 272-amino-acid protein that shows sequence similarity to methyltransferases, and iraB coding for a 501-amino-acid protein that is highly similar to di- and tripeptide transporters from both prokaryotes and eukaryotes. Experimental data suggest that IraA is critical for the virulence of L. pneumophila, while IraB is involved in a novel mode of iron acquisition, which may utilise iron-loaded peptides [225]. Treponema denticola, which is strongly associated with the pathogenesis of human periodontal disease, does not appear to produce siderophores, so it must acquire iron from other sources. Recently, this Gram-negative bacterium has been shown to express two homologous iron-regulated outer membrane proteins with haemin binding ability. These proteins, HbpA and HbpB, both of the size of 44 kDa, do not show any similarity to TonB-dependent OM receptors, and thus may be part of a previously unrecognised iron-acquisition pathway [226].
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The Gram-negative anaerobic bacterium Porphyromonas gingivalis has been implicated as a major pathogen in the development and progression of chronic peridontitis. The iht gene locus of this organism is involved in iron haem transport. In addition to a TonB-dependent OM receptor (IhtA) and a typical ABC transport system (IhtCDE), IhtB was characterised as an OM haem-binding lipoprotein. Experimental data and sequence analysis suggest that IhtB is a peripheral outer membrane chelatase involved in iron uptake [227]. 7.6
PHYLOGENETIC ASPECTS
The almost identical design suggests a common origin of all ABC systems. However, the members of the three iron-transport families, the siderophore/ haem type, the ferric iron (Fbp/Sfu) type, and the metal (Fe, Zn, Mn) type, are clearly distinguishable with respect to the primary structure of the different components. The integral membrane proteins and the substrate binding proteins display significant similarity only within their families. In particular, BPs and IMPs of the siderophore family seem to be totally unrelated to any other known ABC transporter. By contrast, the ferric iron type proteins display a low but significant homology to the equivalent components that are involved in the utilisation of, for example, sulfate, spermidine and putrescine. The ATPases of different families show a higher degree of conservation, but still cluster in distinctive groups. All three major families can be divided in subfamilies. The formation of subfamilies is not species specific, and components of a given cluster can be found in Gram-positives, Gram-negatives and Archaeae. Some bacteria possess uptake systems of all the ABC types mentioned in this chapter. For example, the pathogenic microbe H. influenzae is able to sequester iron via siderophore-type systems, ferric iron systems, and metal-type systems. Similarly, strains of Yersinia use multiple routes to take up iron bound to siderophores (e.g. yersiniabactin) and haem, as well as unliganded iron by the ferric-iron-type Yfu system and the metal-type Yfe system. No iron-uptake systems of the ABC transporter type were identified in the genomes of Mycoplasma genitalium and Mycoplasma pneumoniae. In contrast, among the 19 ABC transporters of the related species Ureaplasma urealyticum six presumed different Fe3þ and/or haem transporters were identified [228]. 7.7 IRON TRANSPORT IN BACTERIA: CONCLUSION AND OUTLOOK Various strategies of iron acquisition are realised in a great number of microbes. Of the different uptake systems, the three different groups of ABC
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transporters (siderophore/haem/vitamin B12 type, ferric iron type, metal type) are of particular interest. They can serve as model systems to study general aspects of: . . . . . .
evolution and biodiversity; gene expression and regulation; protein structure and folding of polypeptides; topology and membrane insertion of proteins; intra- and inter-molecular interactions; and substrate binding and translocation mechanisms at the molecular level.
In addition, the unique features of iron-uptake systems make the components involved ideal candidates: . to examine their potential as targets for antimicrobial agents; . to investigate their role in virulence mechanisms of pathogens; and . to deliver siderophore–drug conjugates to microbes causing infections in humans and animals.
8
CHALLENGES FOR FUTURE RESEARCH
The most challenging aspects related to transport phenomena can be summarised as follows: . solve the three-dimensional structures of representatives from all major groups of transport proteins, particularly those of the hydrophobic components embedded in the cytoplasmic membrane; . unravel the dynamics of transfer processes; . understand transport mechanisms at the molecular level; and . create data sets in order to predict protein structure and function based on sequence information, and to identify potential interaction partners.
ACKNOWLEDGEMENTS The author apologises to those whose papers and important studies were not cited because of space limitations. The author is grateful to W. Welte and A. Brosig (Konstanz, Germany), D. van der Helm and L. Esser (Norman, USA) H. J. Vogel and T. E. Clarke (Calgary, Canada), and K. Locher (Pasadena, USA) who provided excellent material for some of the figures presented in the chapter.
322 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
GLOSSARY ACRONYMS ABC Amt AqpZ ATP BP btu CM CR DCCD FCCP feo fhu Hpr IMP LPS MFS MIP MscL MscS Nramp OM PE PEP pmf PP PTS TC TM TRAP
ATP binding cassette Ammonium/methylammonium transport Aquaporin Z Adenosine triphosphate Binding protein Vitamin B12 uptake Cytoplasmic membrane Conserved region Dicyclohexylcarbodiimide Carbonyl cyanide para-trifluoromethoxyphenylhydrazone Ferrous iron uptake Ferric hydroxamate uptake Histidine protein (from PTS system) Integral membrane protein Lipopolysaccharides Major facilitator superfamily Major intrinsic protein Mechanosensitive channels of large conductance Mechanosensitive channels of small conductance Natural resistance associated macrophage proteins Outer membrane Phosphatidylethanolamine Phosphoenolpyruvate Proton motive force Periplasm Phosphotransferase system Transport commission Transmembrane domain Tripartite ATP-independent periplasmic
SYMBOLS DGo Dc KD
0
Km Vmax Ks
Standard-free-energy difference at pH 7 Membrane potential Concentration of ligand to reach half maximum binding Concentration at which the transport rate reaches half its maximum (Vmax ) Maximum transport rate Saturation concentration
(kJ mol1 ) (mV) (mol dm3 ) (mol dm3 ) 1 (mol min1 mg ) 3 (mol dm )
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7 Transport of Solutes Across Biological Membranes in Eukaryotes: an Environmental Perspective RICHARD D. HANDY School of Biological Sciences, The University of Plymouth, Drake Circus, Plymouth, PL4 8AA, UK
F. BRIAN EDDY Environmental and Applied Biology, School of Life Sciences, The University of Dundee, Nethergate, Dundee, DD1 4HN, Scotland UK
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamental Processes in Solute Transport . . . . . . . . . . . . . 2 Solute Adsorption: Example of Naþ Binding to the Gill Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solute Import into Epithelial Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ion Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Co-transport on Symporters. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Counter-transport on Antiporters . . . . . . . . . . . . . . . . . . . . . . 4 Intracellular Trafficking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Solute Export from Epithelial Cells . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Export from the Cell to the Blood via Ion Channels and Antiporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Export from the Cell to the Blood by Primary Transport: ATPases . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion and Environmental Perspectives . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physicochemical Kinetics and Transport at Biointerfaces Edited by H. P. van Leeuwen and W. Ko¨ster. ß 2004 John Wiley & Sons, Ltd
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1
INTRODUCTION
This chapter attempts to illustrate the principles of solute transport in eukaryote cells, and then explains the intimate relationship between cellular ion transport and environmental chemistry. The information presented herein is thus complementary to that presented in Chapter 6 of this volume, for prokaryotes. Since this book is about the interface between the environment and biological surfaces, we will draw on examples from epithelia where this is a particular concern, such as fish gills and intestine. However, the principles of solute transport that we discuss are universal, and will apply to a wide variety of epithelia, including those that do not have direct contact with the external environment (e.g. renal epithelium). The gill is also a particularly useful model to consider, because, under certain environmental conditions, the direction of solute transfer may be reversed (e.g. in freshwater versus seawateradapted fish), and, of course, the gill epithelial cells have intimate contact with the aqueous environment. The fundamental principles of solute transport are demonstrated with reference to sodium (Naþ ) transport initially, which is arguably the most characterised solute transport process of all eukaryote cells [1–8]. Sodium transport also ultimately depends on at least one solute transporting protein that is ubiquitous in eukaryotes (the Naþ pump [9–14]). We then illustrate how these principles apply to other solutes of environmental concern, particularly in relation to divalent ions and trace metals. Solute transfer across membranes has been identified in many organisms, but the details of the transport mechanisms that are involved have often been neglected, although they may be of fundamental importance in understanding environmental toxicology and chemistry. In this chapter, we use the chemical symbol to describe metals where we do not wish to imply a particular charge or oxidation state (e.g. Cu for copper generally), and only give valency where a particular chemical species is relevant (e.g. Cuþ , Cu2þ , Al3þ , Fe2þ , etc.). 1.1
FUNDAMENTAL PROCESSES IN SOLUTE TRANSPORT
All eukaryote cells are faced with differences in intracellular solute composition when compared with the external environment. Many eukaryotes live in seawater, and have cells which are either bathed in seawater directly, or have an extracellular body fluid which is broadly similar to seawater [3]. Osmoregulation and body fluid composition in animals has been extensively reviewed (e.g. [3,15–21]), and reveals that many marine invertebrates have body fluids that are iso-osmotic with seawater, but may regulate some electrolytes (e.g. SO2 4 ) at lower levels than seawater. Most vertebrates have a body fluid osmotic pressure (about 320 mOsm kg1 ), which is about one-third of that in seawater (1000 mOsm kg1 ), and also regulate some electrolytes in body fluids at
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much lower concentrations than seawater (e.g. Naþ , Cl , Ca2þ , Mg2þ , SO4 ). Amongst the vertebrates, the elasmobranchs (sharks, skates and rays) are notably different, with body fluids that are iso-osmotic with seawater; although only one-third of it is attributed to salts, with the majority of it being due to urea. Importantly, most eukaryote cells ‘see’ an extracellular medium which is of relatively fixed composition, either by virtue of the huge buffering in the medium (e.g. oceanic seawater), or because specialised osmoregulatory organs (e.g. gills, gut, kidney) regulate body fluid composition. Many multicellular eukaryotes have regulatory mechanisms which maintain relatively fixed body fluid compositions that protect the intracellular compartments from change (Claude Bernard’s ‘Constancy of the internal milieu’). However, there are a number of key differences in solute concentrations between the intracellular and extracellular environment, which arise mainly from: (1) the effects of large fixed anions (e.g. cellular proteins) which tend to repel diffusible anions (e.g. Cl ), creating anion differences across the cell membrane (Donnan equilibria); (2) relative differences in the permeability of biological membranes to solutes (e.g. higher Kþ permeability compared with Naþ ). In general, solutes with a small hydrated radius and absent/low charge density will diffuse more easily [22]; (3) the source of the solute. Cells may generate intracellular solutes during metabolism, e.g. glucose from glycogen stores, or urea or ammonia from protein catabolism. The combination of these events may create both chemical and electrical gradients across the cell membrane, which must be overcome by energy expenditure if the solutes are to be moved against these electrochemical gradients. The absolute rate of flux of a solute will also depend on the surface area of the cell membrane and the particular types of lipids and proteins that constitute the cell membrane in a particular cell type. The process of solute transport through a cell may involve several steps, as shown in Figure 1: (1) (2) (3) (4)
adsorption of the solute on to the surface of the cell membrane; import of the solute across the cell membrane into the cell; intracellular trafficking and/or storage in membrane-bound compartments; export of the solute from the cell.
An idealised eukaryotic epithelium is represented in Figure 1. This might, for example, be the gut mucosa, the reabsorbing portion of a renal tubule system, or a gill epithelium. The solute must move from the bulk solution (e.g. the external environment, or a body fluid such as urine) into an unstirred layer
340 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES Unstirred layer Bulk solution
Unstirred solution Mucus A− M+
A−
M
+
M+ A−
M+
Me A+ A−
+
K1
M+ A−
X M+ K4 A− M+
1. Adsorption
M+
A−
A−
A
K2
K3
Blood
paracellular
K6
A−
M+
−
M+ M+
M+ +
M+ A−
M+
M+
A−
Epithelia
SP
4. Export ATP
K5
3. Cytosolic carriers?
AP
Kn
X+
2. Import
Intracellular stores
M+
M+
Tight junction
M+
Figure 1. Solute transfer across an idealised eukaryote epithelium. The solute must move from the bulk solution (e.g. the external environment, or a body fluid) into an unstirred layer comprising water/mucus secretions, prior to binding to membranespanning carrier proteins (and the glycocalyx) which enable solute import. Solutes may then move across the cell by diffusion, or via specific cytosolic carriers, prior to export from the cell. Thus the overall process involves: 1. Adsorption; 2. Import; 3. Solute transfer; 4. Export. Some electrolytes may move between the cells (paracellular) by diffusion. The driving force for transport is often an energy-requiring pump (primary transport) located on the basolateral or serosal membrane (blood side), such as an ATPase. Outward electrochemical gradients for other solutes (Xþ ) may drive import of the required solute (Mþ , metal ion) at the mucosal membrane by an antiporter (AP). Alternatively, the movement of Xþ down its electrochemical gradient could enable Mþ transport in the same direction across the membrane on a symporter (SP). A , diffusive anion such as chloride. K1–6 refers to the equilibrium constants for each step in the metal transfer process, Kn indicates that there may be more than one intracellular compartment involved in storage. See the text for details
comprising water/mucous secretions (see Section 2), prior to binding to membrane-spanning carrier proteins which enable solute import. The solute may also interact with other ligands in the cell glycocalyx, not just the membranebound proteins, but these are not shown for clarity. Solutes may then move across the cell by diffusion or via specific cytosolic carriers. In the case of ‘nonreactive’ solutes like Naþ or glucose, these probably move across the cell by diffusion. However, some solutes, such as transition metals, are highly reactive with structural components in the cell (e.g. with haem centres in proteins) and must be moved around the cell by specific carriers (usually peptides). These carriers or molecular chaperones enable controlled delivery of the solute to the relevant part of the cell. The driving force for solute transfer
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is often an energy-requiring pump (primary transport) located on the basolateral or serosal membrane (blood side), such as an ATPase. Outward electrochemical gradients for other solutes (Xþ , Figure 1) may drive import of the required solute (secondary transport). These primary and/or secondary transport systems may be electrogenic (moving an unequal number of charges across the cell membrane), thus creating voltage differences that may contribute to paracellular absorption (diffusive flux between the cells of the epithelium) of solutes down the electrochemical gradient. This latter movement of solutes between the cells depends critically on the permeability of the tight junctions that hold the cells together, and this is greatly influenced by the Ca2þ content of the medium as well as the driving force for diffusion by this route [23]. Exclusion of diffusive anions (A ), such as chloride, may also contribute to voltage differences via Donnan effects in the mucus layer and within the cell (see below). Figure 1 also illustrates some of the thermodynamic considerations in solute transfer across the cell. K1–6 represent the equilibrium constants (log K is inversely related to affinity) for all the steps in solute transfer. The steps K1–4 represent those for movement of free solute into the unstirred water and mucous solution (K1 and K2), binding to the mucoproteins (K3), and from mucoprotein to importer (K4), while K5 depicts binding to cytosolic carriers, and K6 binding to an exporter. It is sometimes difficult to differentiate the thermodynamic steps in adsorption experimentally, and K1–3 may be given one overall binding constant (as in the gill models, see Section 2). It would, however, be a gross oversimplification to assume that transport is achieved simply by each step having a higher binding affinity than the previous one. This is clearly not the case, as transporters on both sides of the cell may have similar binding affinities for a given solute. After considering the reversibility of each ligandbinding event, it is also necessary to consider the local solute concentration and ligand availability at that particular position in the cell. It is the overall effect of solute and ligand availability, and binding affinity, that enables solute movement to the next step in the overall process. This of course requires the accurate measurement of free solute and ligand concentrations in different parts of the cell. We are at least some way towards measuring these with spectrofluorometric techniques that can measure free-ion movements across cells (e.g. Ca2þ sparks in excitable cells [24]). An in-depth critical evaluation of the various parameters and processes that must be considered in modelling of biouptake is given in Chapter 10 of this volume. 2 SOLUTE ADSORPTION: EXAMPLE OF Naþ BINDING TO THE GILL SURFACE The first step in the movement of any solute across a cell membrane is the provision of a readily available supply of solutes to the membrane surface. The
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overall process of adsorption on to the cell membrane surface is influenced by several factors [22, 25–28] including: (1) (2) (3) (4)
the free solute concentration in the environment; the number and type of solute binding ligands on the epithelial surface; the rate of solute uptake and any associated replacement of surface ligands; unstirred layer formation on the extracellular surface of the cell membrane.
In the context of environmental biology, these processes have been mostly investigated at the surface of fish gills with respect to Naþ transport [29–33] and the uptake of toxic metals [34–38]. The general anatomy of the gills is beyond the scope of this text (see reviews [39–44]), but, as with any model system for adsorption processes, it is important to define the components that make up the surface interface. These are the bulk water (solution that freely exchanges with the external environment), unstirred layers of waters (relatively nonmobile solvent layers adjacent to the membrane surface), mucus (secreted by the epithelia), the glycocalyx on the cell surface and associated external binding sites on membrane-bound ion transporters (Figures 1 and 2). These are common components of biological interfaces, but authors often use varied terminology. For example, in fish gill research these surface layers (often including the bulk water in the opercular cavity) are collectively called the ‘gill microenvironment’ [45,46]. Authors generally may not differentiate the secreted mucus layer from the water (or other layers of body fluids) component of ‘unstirred layers’. This situation might arise from the difficulties in experimentally measuring the solute composition of the adjacent solute and mucus layers in anatomically complex epithelia such as the gills [32, 47,48]. However, differentiation of the water/mucus unstirred layer may be of functional significance, given the vast differences in the composition, rheology and ion-exchange properties of mucus solutions as compared with simple salines. Importantly, the relative permeabilities of chloride to sodium are about 10% less in mucus solutions when compared with simple salines, and the mobility of Naþ is about 50% less in mucus than in salines [49,50]. Mucus is therefore much more than a simple unstirred barrier to diffusion. Distinguishing between adsorption on to the cell surface and the actual transfer across the cell membrane into the cell may be difficult, since both processes are very fast (a few seconds or less). For fish gills, this is further complicated by the need to confirm transcellular solute transport (or its absence) by measuring the appearance of solutes in the blood over seconds or a few minutes. At such short time intervals, apparent blood solute concentrations are not at equilibrium with those in the entire extracellular space, and will need correcting for plasma volume and circulation time in relation to the time taken to collect the blood sample [30]. Nonetheless, Handy and Eddy [30] developed a series of ‘rapid solution dipping’ experiments to estimate radiolabelled Naþ
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Unstirred layer
Bulk water
Unstirred water Mucus Cl−
Cl− Na+
Cl−
Cl−
Cl− Na+ − Na+ Cl
Na+
+
Na
ATP
2K+
Cl-
Na+
H+
+
Cl Na Na+
HCO3−
HCO3−
CO2 −
3Na+
Cl−
CA
CO2
Blood
Na+
Na+
Na+
Epithelia
Cl−
Cl−
ATP
H+
CA
CO2
CO2
+
Na
Cl−
Tight junction
6−18 mV Mucous layer Donnan potential 1−3 mV Trans-epithelial potential
Figure 2. Sodium and chloride uptake across an idealised freshwater-adapted gill epithelium (chloride cell), which has the typical characteristics of ion-transporting epithelia in eukaryotes. In the example, the abundance of fixed negative charges (mucoproteins) in the unstirred layer may generate a Donnan potential (mucus positive with respect to the water) which is a major part of the net transepithelial potential (serosal positive with respect to water). Mucus also contains carbonic anhydrase (CA) which facilitates dissipation of the [Hþ ] and [HCO 3 ] to CO2 , thus maintaining the concentration gradients for these counter ions which partly contribute to Naþ import (secondary transport), whilst the main driving force is derived from the electrogenic sodium pump (see the text for details). Large arrow indicates water flow
adsorption to the gills. These experiments showed that the combined steps of Naþ adsorption to the gill surface and uptake to the blood (absorption) took low exopolymer coating. Conditional stability constants have been determined for cadmium binding to humic acid in freshwater, log K 6.5 [27], which may be comparable to binding to humic acid coated particles. The experiments demonstrated the importance of cadmium uptake from particles rather than from the dissolved phase. The authors recognised that the overall conclusion was similar to previous studies [28], but there remain inconsistencies in the uptake levels which may be related to the heterogeneity of the systems. Uptake from the intestine into the mucosal cells was not investigated. It was presumed that the material was digested extracellularly by hydrolytic enzymes and the released metal was taken up by facilitated diffusion. 3.2
SEPARATION OF DISSOLVED AND COLLOIDAL FRACTIONS
In a study involving several contaminated freshwater streams in New Jersey Pinelands, Ross and Sherrell [8] have used CFF, with a 10 kDa (ca. 3 nm) cutoff, to separate the filtrate (0:45 mm. A detailed discussion of the association of lead with particles came to no firm conclusions, but there appeared to be seasonal factors in the magnitude of the distribution coefficients, perhaps related to the total suspended matter. 3.3 BIOACCUMULATION OF METALS AS COLLOID COMPLEXES AND FREE IONS – A KINETIC MODEL In a study of the bioaccumulation of metals as colloid complexes and free ions by the marine brown shrimp, Penaeus aztecus [29] the colloids were isolated and concentrated from water obtained from Dickinson Bayou, an inlet of Galveston Bay, Texas, using various filtration and ultrafiltration systems equipped with a spiral-wound 1 kDa cutoff cartridge. The total colloidal organic carbon in the concentrate was found to be 78 1 mg dm3 . The shrimps were exposed to metals (Mn, Fe, Co, Zn, Cd, Ag, Sn, Ba and Hg) as radiolabelled colloid complexes, and free-ionic radiotracers using ultrafiltered seawater without radiotracers as controls. The experiments were designed so that the animals were exposed to environmentally realistic metal and colloid concentrations. The data were analysed using a kinetic model, proposed by Farringdon and Westall [30]. The equation for metal uptake in shrimp is as follows: dcs ¼ k1 cw k2 cs dt
(2)
where cw is the activity (Bq ml1 ) of the radiotracer in the treatment solution, and cs is the activity in shrimp tissues, and k1 and k2 are the rate constants for the uptake and clearance of the radiotracers in the animal. Then, at equilibrium: dcs ¼ k1 (cw (o) cs ) k2 cs dt
(3)
¼ k1 cw (o) (k1 þ k2 )cs
(4)
The rate equation is solved to give the uptake in the shrimp at time t: cs (t) ¼ cw (1) Ksw (1 e(k1 þk2 )t )
(5)
where Ksw is a partition coefficient k1 =k2 at equilibrium (dc=dt ¼ 0) and cw (1) is the concentration in the water phase at equilibrium. The concentration factor (CF) at equilibrium can then be expressed as: CF ¼
Ksw cs (1) ¼ cw (1) cp cp
(6)
368 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
where cp indicates the concentration of shrimp mass in the water (g ml1 ), and cw (1) and cs (1) are the activities associated with water and shrimp at equilibrium respectively. During the uptake phase, CF can be expressed using: CF ¼
Ksw cw (1)(1 exp [ (k1 þ k2 )t]) cw (o) cw (1)(1 exp [ (k1 þ k2 )t])
(7)
where cw (o) is the initial concentration in the water and cw (1) ¼ cw (o) cs . The results showed that the shrimp accumulated only low levels of Ba2þ , Zn2þ and Co2þ from colloid–metal complexes, and Sn2þ accumulated less from the colloid than from the free-ion experiment. However, most free-ion metals were not accumulated at significantly higher levels than the comparable colloidal forms (in one-tailed t-test, p > 0:05 using equal or unequal variances as appropriate). Tissue analyses showed that most of the metal which had been associated with the colloid was accumulated in the hepatopancreas, whereas most free-ion metals, except Mn and Ag, were found in the stomach. It appears that the shrimp is able to discriminate between colloidal and free-ion species, but the route of uptake for the colloidal metals is uncertain. It was considered that primary uptake was at the gills, but it was not clear whether the complex dissociated or was endocytosed in the hepatopancreas. Two other examples of the behaviour of chemicals in sediments and the effects on benthic organisms will be reviewed here to illustrate the complexity of these problems. The first deals with silver and the second with tributyltin (TBT). 3.4
BIOACCUMULATION OF SILVER
Although the abundance of silver in the Earth’s crust is comparatively low (0:07 mg g1 ), it is considered an environmental contaminant and is toxic at the nanomolar level. As an environmental pollutant it is derived from mining and smelting wastes and, because of its use in the electrical and photographic industries, there are considerable discharges into the aquatic environment. Consequently, there have been studies on the geochemistry and structure of silver–sulfur compounds [31]. Silver, either bound to large molecules or adsorbed on to particles, is found in the colloidal phase in freshwater. In anoxic sediments Ag(I) can bind to amorphous FeS, but dissolved silver compounds are not uncommon. A more detailed study of silver speciation in wastewater effluent, surface and pore waters concluded that 33–35% was colloidal and ca. 15–20% was in the dissolved phases [32]. Although much of the discharged silver will remain in the soil or waste water sludge at the discharge site, some will be transported in the aquatic phase as the free ion, or as colloidal suspended material. In freshwater it is found largely
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bound to sediment, but as the water enters estuarine sites and the marine environment, the speciation will change as the chloride ion levels increase, and silver will form chloro-complexes [33]. The toxicity of silver is largely related to the free-ion activity model. Although there appears to be no evidence for biomagnification, i.e. from food chains, in aquatic organisms, Luoma [24] believes that all studies of metal uptake should always be considered in terms of the appropriate food chain. An example of such a study showed that although algae accumulate considerable concentrations of metals, mainly by adsorption on to the cell surfaces, in the case of silver particles of colloidal size, the silver was not released even at pH 2.0 in the laboratory, and trophic transfer from the algae was considered unlikely in estuarine invertebrates [34]. Accumulation from the sediment by benthic organisms appears to depend on local speciation of the silver. It has been shown that in laboratory experiments with sediments amended with silver sulfide, Ag2 S, there was practically no silver accumulation in the oligochaete, Lumbriculus variegatus, and this was attributed to the low solubility of the silver sulfide [35]. In marine organisms, the bioconcentration factors for silver ranged from 8–65 for the deposit feeder, Macoma baltica, 40–180 for the filter feeder, Mytilus edulis, to 5000 for zooplankton, 24 000 in the tissue of the crustacean, Crangon crangon, and values up to 200 000 in marine algae. Although adsorption on to the cell surface is most important, especially with algae, many of the other organisms do ingest particles. Silver assimilation efficiency from food was lower in the mussel Mytilus edulis and oysters, Crassostrea virginica and Crassostrea gigas, where uptake was considered to be largely from the dissolved phase than in the clam, Macoma baltica. However, the mechanism of absorption, i.e. whether it is by extracellular digestion followed by absorption of the digested pollutant or by endocytosis of particulate matter, can only be deduced from the general basic zoology of the feeding and digestion of these organisms [36,37]. It is clear that in natural waters, ionic silver and some silver complexes were readily adsorbed on to particles with less than 25% as dissolved ions, complexes or colloids. It was also considered that the exposure route for particulate silver had not been fully explored [33]. An earlier study [38] of the adsorption of silver on to particles from intertidal sediment collected from San Francisco Bay, showed that removing bacteria, organic matter, iron and manganese oxides did affect the rate at which the silver was removed from solution, but not the total amount of silver adsorbed over 24 hours. Similar results were found with oxic sediments collected from 17 English estuaries [39]. The bioaccumulation of silver in three sediment-dwelling organisms consisting of the deposit-feeding clams, Macoma balthica, Scrobicularia plana and the polychaete, Nereis diversicolor was greater than other elements such as copper and mercury that were also present in the system. However, in
370 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES
considering the bioavailability of silver in marine environments, it is important to consider its speciation [33]. In the marine environment, silver can be present as Agþ , but also forms neutral and anionic chloro-complexes that are present in both the aqueous phase and adsorbed on to particles in the sediment. In these studies it was shown that Macoma balthica accumulated silver preferentially, but the only conclusion that could be drawn was that the bioavailability correlated with an easily extracted fraction of silver from the particles [39]. 3.5
BIOACCUMULATION OF TRIBUTYLTIN
Tributyltin (TBT), which was used in timber treatment, was introduced as an antifouling agent in the 1960s and 1970s, but, by the mid-1970s, there were already reports of its toxicity to invertebrates. One such form of this occurred in the female dogwhelk, Nucella lapillus, which grew a penis, a phenomenon known as imposex. The occurrence of this phenomenon led to population declines. There was also a near collapse of oyster mariculture on the French Atlantic coast. In early 1982, the link with tributyltin-containing antifouling paints led France to restrict their use to vessels greater than 25 metres long. Most countries had adopted these restrictions by 1989. This had the effect of reducing contamination, but the concentrations in seawater and sediment were still high enough to cause acute and chronic toxicity to aquatic benthic organisms. There is some evidence for the transfer from the sediment back into surface waters. A total ban of TBT use in 2003 and its presence on boats by 2008 has raised doubts concerning the toxicity of a proposed copper-containing triazine replacement [40], even suggesting that any ban on TBT is postponed until the safety of new compounds is assessed for their environmental impact. Much of the work on TBTs in the 1980s was reviewed in Volume 3 of this series [41], when the relationship between TBT concentrations in the clam Scrobicularia plana and TBT levels in estuarine sediments was already well established [42]. The nature of the adsorption and desorption of TBT in estuarine sediments was investigated by Langston and Pope [43], since reported partition coefficients, Kd , (the ratio of TBT between sediments and water) varied by several orders of magnitude. Partition coefficients were determined using various concentrations of TBT, different levels of suspended solids, a range of salinities and pHs, as well as sediments with a range of organic carbon contents. The sorption on to sediment was rapid with most of the TBT adsorbed on to sediment within 10 min and equilibrated in about 2 h. Plots of Kd versus concentration of TBT in water were not linear, as Kd values decreased with increasing concentration of contaminant as the proportion of TBT bound to particulates declined from 85% at low TBT values to 63.5% at high levels. The equilibrium concentrations of TBT ranged from 6.9 and 6377 ng Sn dm3 . However, a plot of Kd versus the log of the TBT concentration in water showed a linear relationship (r ¼ 0:962):
M. G. TAYLOR AND K. SIMKISS
Kd ¼ 4:6 104 (10:2 103 ( log [TBT]))
371
(8)
Interestingly, the data for all the sediments examined from estuarine and coastal waters fitted this equation. Although the concentration of TBT was always greater in sediments rather than the overlying water, there was less desorption at pH 6–7 and at low salinities, demonstrating that freshwater sediments retained higher levels of TBT. The organic component providing attachment sites for the hydrophobic regions of TBT was also an important factor. The results reflected the properties of TBT which has characteristics of both a hydrophobic organic species as well as that of a metal. It has been suggested that the plasma membrane may be the site of toxicity of TBT as might be expected from an organometallic compound [44], but there is no doubt that many benthic invertebrates accumulate TBT from the sediment, although the precise route of uptake remains unclear.
4
MEMBRANE TRANSPORT PROCESSES
The most important and universal characteristic of the cell is its ability to manipulate the location of ions and molecules. It achieves this by exploiting their hydrophilic and hydrophobic properties so as to compartmentalise them for different functions within a variety of cell membranes. This phenomenon is clearly seen with the plasma membrane – a lipid bilayer that is virtually impermeable to hydrated molecules. The influx and efflux of ions and small metabolites across this membrane occurs through aqueous pores in protein molecules that act as selective channels, pumps or transporters. The chemical specificity of these transport systems appears to depend upon the dimensions of the narrowest region (the selectivity filter), the possibility of interactions between ions during multi-occupancy of the channel, the size of the hydrated/dehydrated ions and the relative attraction of the transported molecules to the channel walls [45]. Some pollutant ions may enter the cell through an inappropriate channel (e.g. Cd through a Ca channel), but these are infrequent and typically dealt with inside the cell by binding to specific proteins (such as metallothioneins) and subsequent detoxification. Lipophilic molecules are more difficult for the cell to regulate as they cross the plasma membrane by the relatively indiscriminate process of partitioning from the environment into the lipid bilayer. There are, however, a number of metabolic pathways that introduce hydrophilic groups into such molecules and then conjugate them to polar compounds that facilitate their excretion. There is also considerable interest in multidrug resistance transporters or P-glycoproteins that reside in the membrane itself and which actively transport lipophilic molecules out of the cell or out of the membrane itself into the extracellular fluids [46,47]. By these means, the cell is able to regulate the influx and efflux of materials from the
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extracellular environment. The mechanisms involved in membrane transport processes have been extensively studied over the past 50 years, and the theoretical basis of transport by channel proteins and lipid partitioning is well established [48,49]. Physiologists have variously attempted to distinguish between the different types of carriers and pore systems, according to the accessibility of their binding sites, their saturation kinetics or their stoichiometric coupling properties, but none of these criteria can be used to satisfy a clear distinction. There is, in addition, a third mechanism by which materials may enter the cell. This is called endocytosis.
5 5.1
ENDOCYTOSIS INTRODUCTION
Endocytosis is a process whereby portions of the external surface membrane of a cell can invaginate and pinch off to form membrane-bound vesicles that pass into the interior of the cell. Dynamin, a large GTPase, is believed to have a role in the scission of the vesicle in clathrin-mediated endocytosis, but the exact process remains unclear. However, the effects of point mutations in the GTPase and GTPase effector domains (GED) have recently been analysed [50]. The mutants were used in an in vivo assay involving COS fibroblast cells, and it was found that none of the GED mutants had a significant inhibitory effect on the endocytosis of transferrin, although it was still considered that GED may have a role in the oligomerisation of dynamin. GTPase activity was also measured, and a kinetic analysis showed that the GED mutants had similar kcat (turnover rate) and Km (binding affinity) values to the wild type 3.1 0.5 s and 7:8 2:5 mmol dm3 respectively [50]. Most of the GTPase mutants had similar or higher Km values, but these were not supported by similar activities in the hydrolysis of GTP. It was concluded that dynamin oligomerisation and GTP binding alone did not facilitate endocytosis, but GTPase hydrolysis in combination with an associated conformational change are also part of the process involved in vesicle scission. Dynamin also appears to have a role in phagocytosis in macrophages, in the formation of phagosomes at the stage of membrane extension around the particle [51]. These vesicles may enclose substances present in the external medium or molecules previously adsorbed on to the cell’s surface. The significance of this route was generally underestimated, until it became clear that it is part of an extensive system of intracellular vesicles that are involved in an elaborate signalling network associated with the recycling of membrane components. Its involvement in the uptake of xenobiotic molecules is still poorly appreciated, largely because in the human its activities are restricted to only a few cell types. In many organisms, however, it is the dominant route for the
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uptake of molecules. As with other transport systems it is often extremely difficult to distinguish experimentally between the endocytotic pathway and other cellular routes. Thus, the enhanced uptake of contaminants in the presence of particulate sources may be due to endocytosis, but it might equally well be due to the ingestion of such particles with extracellular digestive processes releasing adsorbed contaminants. There are two types of endocytosis largely distinguished by the size of the endocytosed vesicle. Phagocytosis (cellular eating), which is an actin-mediated process, involves the ingestion of large particles ranging from insoluble particles and cell debris to microorganisms. These particles are usually >250 nm diameter, and they become enclosed in vesicles termed phagosomes, which often occur in only a small number of specialised cells. In contrast, pinocytosis (cellular drinking) occurs in most eukaryotic cells, and involves the ingestion of fluids and solutes and colloids via small vesicles >150 nm in diameter. A specialised form of pinocytosis is receptor-mediated endocytosis (RME). A specific receptor on the cell surface binds to a recognised ligand, and the receptor ligand complex is endocytosed into the cell. Ligands include molecules such as low-density lipoproteins, hormones and the iron-binding protein transferrin, which is a particularly interesting example, in that it is believed to be able to bind and transport other metals and is discussed in Section 5.6. 5.2
ENDOCYTOTIC PATHWAYS
Materials that have been endocytosed are delivered to lysosomes by multiple pathways for further processing and the release of nutrients and trace metals for cellular biosyntheses or exocytosis. These pathways are illustrated schematically in Figure 4. The pathway taken by the endocytosed matter appears to be regulated in the endosomes: the organelles referred to as sorting stations [52]. There are four classes of these organelles: early endosomes, late endosomes, recycling vesicles and lysosomes. The pH of early endosomes is 6.3–5.8, only slightly acid but adequate for the initial dissociation of receptor–ligand complexes and their recycling back to the cell surface. It has been proposed that endocytosed calcium has a role in the acidification process [53]. As the pH decreases, there is a concomitant release of calcium from the endosome. It is suggested that acidification can only occur when there is an initially high Ca2þ concentration in the endosome. One suggested role for the calcium is to maintain charge balance following the influx of protons via a Hþ the influx of protons Hþ pump by the efflux of Ca2þ ions. Another possibility is that the Ca2þ enables other channels such as Kþ and Cl to remain open to allow for charge compensation. The more acidic late endosomes (pH ca. 5.5 or less) are sometimes referred to as pre-lysosomes. Digestion of the food material or other macromolecules
374 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES R C
E
P
CP
CV E
H+
L
Ex
Figure 4. A schematic of endocytotic pathways in a cell. P ¼ pinocytosis; Ex ¼ Exocytosis; R ¼ receptor; C ¼ clathrin; CP, CV ¼ coated pit and coated vesicle; E ¼ endosome; L ¼ lysosome. Open arrow indicates recycling of clathrin and receptors. Solid arrows indicate pathways. See the text for discussion
occurs in the lysosome with hydrolytic enzymes, in an acidic environment which may have a pH as low as 4.6 in many cells. The exact sequence of transport and hydrolytic activity is still the subject of research, but involves a series of progressively more hostile environments. Many of the investigations into endosomal pathways have concentrated on receptor-mediated endocytosis, as in the iron–transferrin–receptor complex, and it is not clear how the systems vary depending on whether or not the pathway is clathrin-dependent or clathrin-independent [54]. Most of the work on endosomes has involved mammalian systems, but studies on the ciliate, Paramecium multimicronucleatum [55] have shown the presence of parasomal sacs with a cytosolic coat resembling clathrin, as in the coated pits of mammalian cells. It is not clear if these structures are functionally similar, but a typical Paramecium will have up to 8500 coated pits on its surface, with a coating which was morphologically identical to the triskelions seen in the clathrin cages of higher organisms. These appeared to be lost quite quickly. As in mammalian cells the early endosomes appeared to have a sorting role, but differed from mammalian cells in some respects. A dual labelling experiment revealed two populations. One, a pre-endosomal vesicle (188 41 nm diameter) contained both the marker, horseradish peroxidase (HRP) and the goldlabelled antibody to a component of the plasma membrane, whilst the second appeared to be an early endosome-derived vesicle (90 17 nm diameter). These vesicles were formed from coated evaginations on the early endosome, and retained only the HRP marker. These vesicles were also located deeper in the
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cytoplasm. This system was quite different from the feeding phagosome/lysosome pathway. 5.3
PHAGOCYTOSIS
Phagocytosis is common in eukaryotes. It is the basic means by which many organisms obtain their food especially for single-celled organisms such as protozoa and slime moulds. In higher eukaryotes, phagocytosis is an important process in the response to foreign matter in specialised cells such as circulating monocytes and neutrophils, as well as tissue-associated macrophages and some epithelial cells. Phagocytosis is an important mechanism for the organism to rid itself of bacteria and pathogenic material, as well as cell debris and remnants of apoptosis. However, it can also provide a route for the uptake of pollutant particulate material. It is seen to be especially important in the incorporation of airborne particulate material, which often has serious health consequences (see Section 6.4). In terrestrial invertebrates, food is obtained either from particulate matter in the soil or from molecules dissolved in interstitial water. Most of these organisms have extracellular digestion, with nutrients and foreign material being absorbed by one or more of the routes available for transport across membranes, such as diffusion, channels or pinocytosis. There have been few studies to establish which route is taken. The phagocytic process is initiated by the interaction of specialised plasma membrane receptors with specific ligands, localised on the surface of the particles. Contact of a phagocyte with a suitable particle causes a local accumulation of actin-rich cortical material at that site. The ligand–receptor complex triggers the local reorganisation of the submembranous actin-based cytoskeleton that mediates the engulfment of particles. The actin filaments form part of the pseudopodia, which can then surround and engulf the particle, which is then digested in a lysosome. The scission of the particle is mediated by the dynamin family of GTPases [51]. Physical contact with the particle surface is necessary but not sufficient on its own to trigger phagocytosis, because some form of recognition is necessary, as, for example, in antibodies binding to the Fc receptors on a macrophage. Some form of chemical identification is, therefore, important as a particle approaches a cell, is recognised, adheres to the cell, is engulfed and then digested in a phagosome (see also Chapter 2 in this volume). Many of the receptors that are involved recognise the surface components of microorganisms or identify the opsonin molecules that coat foreign particles. Nonspecific coatings include polysaccharides such as lectins, while the more specific opsonins include IgA and IgG antibodies and complement fragments C3b and C3a, and, in addition, the complement receptor C3 (CR3) of the integrin superfamily. Phagocytotic-competent receptors for fibronectin and vitronectin also belong to the integrin superfamily.
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The first event following the binding of a particle to the cell surface is the clustering of the receptors, which are the trigger for the activation of several cellular proteins, including the receptors and the submembranous cytoskeleton. Actin-driven engulfment requires the progressive recruitment of receptors that work in a zipper-like manner linking the ligands on the surface of the particles. The outcome is that the actin microfilaments are polymerised, filling the pseudopods of the phagocytic cups that embrace the particle. Subsequently, the actin depolymerises and allows fusion of the phagosome with an endosome [56]. The pathways of phagocytic signal transduction remain at present unknown, due to their complexity. There have been several studies to determine which surface properties on unopsonised particles are important in their uptake by phagocytic cells. The driving force has often been related to the use of liposomes as carriers of drugs and other macromolecules, but may also be important for the removal of particles in body fluids. The two steps of the phagocytosis, binding to the surface of the phagocyte and subsequent internalisation of the particle, have been subject of binding and kinetic studies. In order to study which properties of the membrane were important in the interaction with endocytic cells, liposomes are prepared to ensure that they are of colloidal size, diameter range 80–110 nm. In particular, the lipid composition of the surrounding membrane has been investigated [57]. Studies appear to indicate that negatively charged groups, such as phosphatidyl serine (PS) and phosphatidyl glycerol, are more effective than neutral lipids, such as phosphatidyl choline (PC), in increasing the binding of the colloidal particle to the cell membrane and the subsequent endocytosis. However, the net negative charge was not the sole determinant in its binding to the cell surface, as the uptake also depended on the nature of the head group. Nevertheless, a higher overall charge density appears to promote uptake in some cells [57]. In a murine macrophagelike cell line J774, preliminary studies indicated that the rate of endocytosis after binding is faster than the actual rate of binding. It is suggested that the number of bound liposomes may be controlling the overall uptake. In a more detailed study [58] it was found that there were 6.9 times more negatively charged than neutral liposomes, associated with 106 J774 cells at 37 8C. Binding studies at 37 8C were conducted using inhibitors of endocytosis. Affinity constants defined as, K ¼ kN, where K is the product of k representing the binding affinity for a liposome and a binding site, and N the number of binding sites in a cell, were determined. Scatchard plots were linear over a 1 limited range of lipids. The affinity constant, K 1012 dm3 mol and number of binding sites, 3000 for the negatively charged PS/PC/Chol liposomes, were an order of magnitude greater than for the neutral PC/Chol liposomes. It was found that at steady state the number of liposomes on the cell surface at 37 8C was similar to the values at 4 8C or 37 8C without endocytosis. This implies that, following endocytosis, the binding sites are rapidly recycled back the cell surface, so leaving the number of binding sites relatively constant.
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A more recent study of the binding of multilamellar colloids to macrophages (J774 cells) has shown that cationic colloids bind to the cell surface more efficiently than neutral ones [59] but are not endocytosed. Biocompatible particles were synthesised with a mean size of 200–300 nm, to allow endocytosis. Particles were labelled with 2 103 mol dm3 calcein for analysis by fluorescence microscopy. After exposure of the cells to the particles, the fluorescence images indicated that the cationic colloids were strongly adsorbed on to the surface of the macrophage, and analysis of the adsorption curve suggested a Langmuir isotherm, which assumes one class of adsorption sites. Kinetic studies were attempted at 4, 15, 25 and 37 8C, but the colloids tended to aggregate at all temperatures above 4 8C. Four models were used to determine the binding mechanisms from the kinetic data. A detailed analysis of binding at 4 8C, was made. Models were set up involving one or two surface sites which also satisfied the overall kinetics but the analyses were not definitive. Although it was demonstrated that the cells were capable of endocytosis of fluorescence-conjugated transferrin, there was no evidence for the endocytosis of the cationic colloids. 5.4
PINOCYTOSIS
Pinocytosis is a process whereby external fluids can be taken into a cell. This process could, therefore, facilitate the uptake of contaminated material that had been digested extracellularly to give soluble pollutants. Pinocytosis is a constitutive process that occurs continuously. However, it is generally not a saturable process that limits the intake of material, since any membrane material that is removed by endocytosis is matched by an exocytotic process returning material to the membrane surface. There are two proposed pathways for pinocytosis. The first is a clathrin-dependent pathway producing micropinosomes, while the second is an actin-dependent process producing macropinosomes. In a review, [54] devoted largely to the clathrin-independent pathway, it was considered that pinocytosis in many mammalian cells occurred by both clathrin-dependent and clathrin-independent mechanisms. Examples of clathrin-independent pinocytosis were given for kidney intercalated cells, cultured fibroblasts and cultured hepatocytes. In many amoebae, nonselective pinocytosis appeared to be the dominant mode of endocytosis [54]. Although the process has been studied most in single-celled organisms such as Dictyostelium, there is some evidence that, for example, fibroblasts have similar mechanisms for macropinocytosis [60]. However, in a study of the uptake of 10 nm gold-labelled asialorosomucoid and 5 nm gold-labelled bovine serum albumin into rat hepatocytes, electron microscopy revealed that the uptake of asialorosomucoid was clathrin dependent (i.e. into small clathrin-coated vesicles), whilst the uptake of bovine serum albumin was a clathrin-independent process [61].
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There have been many studies designed to establish whether the endosomal pathway to lysosomes, and the recycling of receptors is the same with clathrindependent and clathrin-independent pinocytosis. The pathway appears to be determined in the endosome, with perhaps the lysosome being the default pathway [54]. During this sorting process, some endosomes are transported to the Golgi apparatus and become associated with secretory vesicles. 5.5
CAVEOLAE
An interesting feature of many cells is the permanent presence on the plasma membrane of flask-shaped regions termed caveolae. They are abundant in certain capillary endothelial cells, and appear to have a role in cholesterol binding, although many other functions have been suggested [62]. 5.6
RECEPTOR-MEDIATED ENDOCYTOSIS
Receptor-mediated endocytosis (RME) is a specialised form of pinocytosis related to the particular needs of cells. The receptor–ligand complexes form at clathrin-coated pits at the plasma membrane. These are pinched off to form clathrin-coated vesicles. The clathrin coat is lost quite rapidly and allows fusion with early endosomes. Clathrin-coated pits are on the cytosolic face of the plasma membrane and are in specialised regions occupying up to about 2% of the total plasma membrane area of cells such as hepatocytes and fibroblasts. The clathrin structure consists of triskelions which facilitate the curvature of the membrane to form the pits. Many internalised ligands have been observed in clathrin-coated pits and vesicles, and it is believed by many that these structures function as intermediates in the endocytosis of many ligands bound to cellsurface receptors. Some receptors are clustered over clathrin-coated pits even in the absence of ligands. Other receptors diffuse freely in the plane of the plasma membrane, but undergo a conformational change when binding to the ligand, so that when the receptor–ligand complex diffuses into the pit it is retained there. More than one complex can be seen in the same coated pit or vesicle. The removal of clathrin can facilitate the recycling of the membrane receptors. Some receptors can make the round trip in and out of the cell every 20 min, making several hundred journeys without being denatured. Others make only two or three before the receptor and ligand are degraded in the lysosome, thus restricting the number of receptors on the cell surface. Thus, a crucial factor in endocytosis involves the recycling of cell-surface receptors and membranes, as well as the maintenance of cell polarity. This receptor-mediated endocytotic pathway has been especially well studied in the uptake of iron from blood plasma. Iron, because of its very low-solubility product (< 1017 at pH 7.4), is transported in plasma bound to the iron-binding protein transferrin. Two Fe3þ ions bind to each transferrin molecule. Entry into
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mammalian cells occurs by receptor-mediated endocytosis, whereby the iron transferrin binds to a receptor, forming an iron–transferrin–receptor complex in a clathrin-coated pit. Fusion with a primary endosome allows the recycling of the transferrin–membrane receptor and the clathrin associated with the coated pit. This is accomplished by a reduction of pH in the endosome. The transfer of iron to the cytosol also requires the activity of a ferrireductase (not yet identified in mammalian cells) as well as a Fe2þ transporter, since the iron in the cytosol exists primarily as Fe2þ . However, the receptor-mediated endocytosis of iron–transferrin studies [63] does not explain the initial uptake of iron from nutrients in the intestinal (villus) cells, since apotransferrin is generally not available in the lumen, except in a limited amount from biliary excretion. Work on other iron transport mechanisms has mainly been reported in the last five years. One development has been the cloning and characterisation of a protoncoupled metal ion transporter in the rat DCT1 [64]. The divalent cation/metal ion transporter (DCT1) was found to transport not only Fe2þ , but also Zn2þ , Mn2þ , Cu2þ , Co2þ , Cd2þ and Pb2þ . DCT1 operates in the acidic environment found in the proximal duodenum and is similar to that found in transferrin receptor mediated endosomes. DCT1 was found to have considerable homology with the Nramp proteins. Nramp1 is a macrophage protein involved with resistance to infectious diseases. However, Nramp2 has a broader spectrum of activity, and has a key role in the metabolism of transferrin-bound iron by transporting free Fe2þ from the endosome to the cytoplasm. Nramp2 has been observed in recycling endosomes and also in the plasma membrane by immunofluorescence and confocal microscopy [65]. It appears that DCT1 is the rat homologue of Nramp2 that was determined in the mouse and human. The conclusion is that although the initial uptake of iron and other metals takes place by a transferrin-independent process (as described above), on crossing the basolateral membrane, the metals (Zn2þ , Mn2þ , Cu2þ , Cd2þ and Pb2þ ) have a similar affinity for transferrin as Fe2þ and are circulated in the serum as metal transferrin complexes. This suggests that the receptor-mediated endocytotic pathway of iron for cellular uptake may be available for other metals. For example, it has been shown that 80–90% of aluminium in blood plasma is present as the aluminium transferrin complex, which can cross the blood brain by receptor-mediated endocytosis [66]. 6 EVIDENCE FOR THE ENDOCYTOTIC UPTAKE OF CONTAMINANTS 6.1
INTRODUCTION
Most amoeboid protozoa assimilate particles by invagination of the plasma membrane in an actin-based phagocytotic process and digest them in food
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vacuoles. Freshwater ciliates are suspension feeders ingesting organic matter and bacteria, but, because of the presence of a pellicle, the plasma membrane has a fixed area. It is, therefore, a good example where the membrane materials have to be continually recycled from used food vacuoles to be incorporated into the cytopharyngeal membrane and form a new food vacuole [67]. In sponges, digestion is exclusively intracellular, and this is also the dominant system in the cnidaria (jellyfish and corals) and triclad turbellaria (flatworms). It is well developed in molluscs, especially in lamellibranchs (bivalve clams), cephalopoda and gastropods (snails), as well as echinoderms (starfish and sea urchins). Digestion is predominantly extracellular in annelids (worms), crustacea (crabs and shrimps) and chordates (vertebrates). Intracellular digestion clearly depends upon the food material being broken down into particles that are smaller than the enveloping cell. This implies that such organisms feed on small particles or possess some way of disrupting larger particles by physical attrition or extracellular digestion. In a review of the feeding and digestion in the Bivalvia, Owen [37] has described the feeding mechanisms of each lineage of bivalves. There is a formal distinction between deposit and suspension feeders, but many deposit feeders, such as some species of Macoma and Scrobicularia plana, supplement their food intake by filtering suspended matter. The contents of the guts of these animals include algal material, organic detritus and particulate mineral material. Although digestion may be largely extracellular, there is a considerable body of evidence for the role of the midgut or digestive diverticula in intracellular digestion (Figure 5). Much of the evidence for endocytosis and lysosomal digestion in these bivalves derives from Owen’s ultrastructural work [69,70]. From these studies, he was able to identify the digestive cells of the digestive tubules from experiments involving feeding iron oxide and carbon particles. Phagosomes have not been observed in cells from all species, but it is likely that ingested particulate matter in bivalves is at least partially endocytosed and digested intracellularly. Engulfing and ingesting foreign particles is clearly an important form of nutrition, but the process has the advantage of also being able to remove particulates and destroy potential pathogens. It is in this context that the process persists in the vertebrates, where macrophages in the lungs remove particulates and a variety of cells endocytose invading bacteria. Given this diversity of examples, what evidence is required to assess the significance of endocytosis in the uptake of contaminants via colloids and particles? One consequence of the uptake of some contaminants is a reduction in the ability of lysosomes to retain the dye Neutral Red. As a consequence, the Neutral Red retention time has been developed as an index of lysosomal membrane fragility and thus of toxicity. The test has been used on the digestive cells involved with intracellular digestion of endocytosed food following administration of organic contaminants, such as polycyclic aromatic hydrocarbons [71]. The phenomenon has been reported many times after exposure to
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[Image not available in this electronic edition.]
Figure 5. Digestion strategies in molluscs. Modified from [68]: Decho, A. W. and Luoma, S. N. (1996) ‘Flexible digestion strategies and trace metal assimilation in marine bivalves’ in Limnol. Oceanogr., 41, 568–572. Reproduced with permission
xenobiotics and other contaminants, and has recently been suggested as a biomarker for exposure of earthworms to zinc and copper in the soil [72,73]. The cause of the increase in lysosomal fragility is not well understood, and the link with endocytosis is not rigorously established, as the contaminants often appear to inhibit phagocytosis in haemocytes in Mytilus edulis [74]. It has been suggested that the leakage of Neutral Red may be as a result of some impairment of the membrane proton pump in haemolymph cells of the freshwater snail Viviparus contectus after exposure to copper [75]. There have been many studies on the role of haemocytes in invertebrates, but their role in ingested material is not clear. Haemocytes are the molluscan analogue of the vertebrate macrophage. They are present in the haemolymph and appear to be able to migrate through epithelia, and they are found in the mantle fluid of bivalves and appear to be involved in the uptake of particulate matter. In a study of the clam Tridacna maxima that had been injected intramuscularly with a suspension of carbon particles, it was found that within 24 h the extracted haemocytes were laden with the particles. They had cleared the haemolymph of the particles within 48 h [76]. In the tridacnidae some zooxanthellae are
382 PHYSICOCHEMICAL KINETICS AND TRANSPORT AT BIOINTERFACES Table 2.
Experimental approaches demonstrating endocytosis of contaminants
Protozoa (1) Endocytosis of particulate material by Tetrahymena increases growth rate from dissolved nutrients by seven to eight fold [78]. (2) Lead precipitates ingested by Tetrahymena [79]. (3) Lead material tracks through digestive vacuoles and enters cytoplasm of Tetrahymena [79,80]. (4) Tetrahymena mutants lacking functional phagocytosis require high levels of iron and copper supplements [81]. Table 3.
Experimental approaches demonstrating endocytosis of contaminants
Metazoa (1) Physiological studies of digestion in marine bivalves show that it progresses via two separate routes: a rapid intestinal system using extracellular processes, and a slower system involving the digestive gland and endocytosis [82]. (2) The use of pulse-chase methods identifies differences in the timing and efficiency of absorption between the two pathways, enabling the study of uptake in the endocytosis system [83]. (3) Ultrastructural and radioisotope studies on the phagocytosis of particles by gills [84] or lungs [85] demonstrate the endocytosis of particles by mobile macrophages, and their subsequent lysosomal attack.
endocytosed and digested within the lysosomes of amoebocytes that occur in the circulation or intertubular spaces of the diverticula [77]. Table 2 gives some of the experimental approaches demonstrating endocytosis of contaminants in Protozoa, and Table 3 summarises the experimental approaches for Metazoa. 6.2
ENDOCYTOSIS BY AQUATIC INVERTEBRATES
There is considerable evidence that organisms ingest colloidal and particulate material, notably in suspension and deposit feeders. However, there have been few definitive studies demonstrating that these materials might be taken into the cell by endocytotic processes, especially for benthic organisms in sediments. Frequently, the evidence presented is circumstantial, except perhaps in the case of single-celled organisms. An attempt has been made to produce a mathematical model based on the rate constants of the processes involved in the fate of endocytosed material from uptake to the fusion with the endosome and the subsequent degradation in lysosomes [86], but there are few examples of its use. In a review of feeding and digestion in the Bivalvia [37], it was proposed that the accumulation of metal-bound particulates in the digestive gland was a twophase process reflecting extracellular and intracellular digestion, and Viarengo [87] has reached similar conclusions. In a pulse chase study of the uptake of radiolabelled metals (Ag, Cd, Cr, Hg, Se) by the zebra mussel Dreissena
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polymorpha, the egestion rate was determined by counting the radioactivity in faecal pellets. After an initial egestion of Cd and Cr, a second egestion pulse was found for Ag, Se and Hg, and it was concluded that this was indicative of intracellular digestion [88]. Earlier it was shown that a two-phase digestion system operated in both the deposit-feeding Macoma balthica and the suspension-feeding clam Potamocorbula amurensis. In an experiment investigating the uptake of Cr(III), it was found that no Cr(III) was taken up during the rapid intestinal pathway involving extracellular digestion, but it was taken up in the slower digestive gland pathway requiring intracellular digestion [68]. The time courses for the retention of 51 Cr-labelled beads in bivalves are shown in Figure 6. Following Owen’s work [37] it is presumed that the uptake of the particulate material was by endocytosis. In their early work on the uptake of insoluble iron particles by Mytilus edulis, George et al. [84] suggested that the gill and mantle cells may be important sites of phagocytosis, since the material could easily be seen on electron micrographs. The role of endocytosis in the accumulation of particle-bound metals is considered to be especially important in lamellibranch molluscs, and metal deposits are frequently observed in residual bodies of these cells. X-ray micro analyses of electron micrographs of the amoebocytes, harvested from the mantle fluid of the oyster Ostrea edulis, frequently identify copper associated with sulfur, and zinc associated with phosphate, in distinct inclusion bodies [90]. This suggests that these metals may have been endocytosed, since amoebocytes are phagocytic cells. 6.2.1
Assimilation Efficiency Models
Various attempts have been made to rationalise the terminology used to define the response of aquatic invertebrates to chemical contaminants. Thus, terms Intestinal faeces
Glandular faeces
Lag period
0
Ingestion
25
50
Time/h
Figure 6. Time courses in the retention of 51 Cr beads in the bivalves, Potamocorbula amurensis and Macoma balthica. During the lag period there was no significant release. Modified from [89]
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such as concentration factors, bioaccumulation and assimilation efficiency (AE) have been used in an attempt to overcome some of the difficulties associated with the term ‘bioavailability’. These concepts have been critically reviewed by Fisher and Reinfelder [91]. AE is essentially a ratio of the element of interest retained in the organism compared with the original uptake. In demisponges, where digestion is largely by amoebocytes, it is sometimes referred to as retention efficiency when applied to the food particles, such as bacteria, diatoms or detritus: retention efficiency ¼ (ambient exhalant) 100=ambient
(9)
where ambient represents the concentration of particles near the inhalant surfaces and exhalant refers to the concentration of particles from the oscular stream of the sponge [92]. More commonly, AE is defined as the ratio: AE ¼ (ingestion excretion egestion)=ingestion
(10)
Initially, it was calculated from mass-balance data measuring ingestion, excretion and egestion. Excretion data were determined by analysis of the depuration water. The inherent experimental difficulties in collecting faecal pellets for egestion measurements from aquatic organisms has led to a pulse-chase radiotracer feeding technique. Animals are fed for a time shorter than their gut passage time with radiolabelled isotopes, and so the amount ingested can be quantified. This is followed by a depuration stage, when the animal is given unlabelled food until the egestion period is past the gut evacuation time. In a study of the uptake of selenium by Macoma balthica, faecal pellets and water from the depuration stages were counted, and the data from the two methods confirmed to within 10% [93]. The bioavailability of selenium to a benthic deposit-feeding bivalve, Macoma balthica from particulate and dissolved phases was determined from AE data. The selenium concentration in the animals collected from San Francisco Bay was very close to that predicted by a model based on the laboratory AE studies of radiolabelled selenium from both particulate and solute sources. Uptake was found to be largely derived from particulate material [93]. The selenium occurs as selenite in the dissolved phase, and is taken up linearly with concentration. However, the particle-associated selenium as organoselenium and even elemental selenium is accumulated at much higher levels. The efficiency of uptake from the sediment of particulate radiolabelled selenium was 22%. This contrasts with an absorption efficiency of ca. 86% of organoselenium when this was fed as diatoms – the major food source of the clam. The experiments demonstrated the importance of particles in the uptake of pollutants and their transfer through the food web to molluscs, but the mode of assimilation was not discussed.
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The AE methods have been used to determine the effects of different algae as food sources in the bioaccumulation of radiolabelled essential (Co, Se, Zn) and nonessential trace metals (Ag, Am, Cd, Cr) in the mussel Mytilus edulis [94]. Assimilation of essential metals was correlated with carbon assimilation, but not nonessential metals. The distribution of the metal in the alga and the gut passage time in the mussel was found to be important. An interesting outcome of the observed depuration patterns was that there could be a two- or three-phase process. This were interpreted as indicative of extracellular followed by intracellular digestion [95]. A possible third phase might be due to metabolic loss of metals. Several models have been developed for a kinetic approach to bioaccumulation that would model the trophic transfer of contaminant in animals from ingested food. A first-order kinetic model has been proposed, which considers uptake from both dissolved and particulate phases [95]. A particular application of that model is to separate the pathways for metal uptake in marine suspension and deposit feeders since: dcA ¼ ðku cw Þ þ ðAE IR cf Þ ðke þ gÞcA dt
(11)
where cA is the chemical concentration in the animals (mg g1 ), t is the time 3 of exposure (d), ku is the uptake rate constant from the dissolved phase dm 1 1 3 g d Þ, cw is the chemical composition in the dissolved phase mg dm , AE is the chemical assimilation efficiency from ingested particles, IR is the ingestion rate of the animal mg g1 d1 , cf is the chemical concentration in ingested particles mg mg1 , ke is the efflux rate constant (per day) and g is the growth rate constant (per day). As the body size of the organism increases, the new tissue mass dilutes the toxicant. Without the growth-rate constant, the derived elimination rate overestimates the actual elimination, as it incorporates both ke and g in the growing organism [96]. By using steady-state conditions, it is possible to calculate and separate the concentration in the animal of the contaminant derived from the dissolved and particulate phase, and so estimate the fraction coming from each source: ðku cw Þ ðkew þ gÞ
(12)
ðAE IR cf Þ ðkef þ gÞ
(13)
cw, ss ¼ cf , ss ¼
cw, ss and cf , ss (in mg g1 ) are the chemical concentrations in organisms from the water and from food, and kew and kef are the corresponding efflux rates. A further step is to determine the assimilation efficiency from metals in particles digested extracellularly and intracellularly. This is derived from the
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metal depuration rate. For example, in studies with bivalves it was found that there were three different losses with respect to time. Firstly, a rapid loss representing passage through the gut and including extracellular digestion; secondly, a slower loss attributed to intracellular digestion following phagocytosis of fine particles; and thirdly, a loss that could be attributed to the physiological turnover of assimilated metals. The assimilation efficiencies of cadmium, chromium and zinc have been investigated in the green mussel Perna viridis and the clam Ruditapes philippinarum when fed with one of five species of phytoplankton or seston [97]. The five different phytoplankton and seston (particle size 363 mm) were kept in 0:22 mm filtered seawater enriched with nutrients, to which had been added radiolabelled 109 Cd, 51 Cr and 65 Zn for four to seven generations before being harvested, washed in filtered seawater and fed to the bivalves for 30 min in pulse-chase experiments. The bivalves were depurated for three days, during which time individual specimens were counted for radiolabel. Faeces were collected every 1 to 5 h during depuration, and the metal gut passage time calculated. An estimate of the cytoplasmic metal content of the phytoplankton was determined after removal of surface-adsorbed metal by EDTA. The AEs for each metal for each feeding strategy were calculated. They were consistently lower for the seston source. The AE was higher for the clam than the mussel for Cd and Zn. Chromium was the least assimilated metal, but the AEs were comparable for both species and seemed to be related to the gut passage time. Egestion of each metal was determined in relation to the production of faeces, and appeared to be complete within 60 h. See also Figure 6. A biphasic pattern of egestion was found, corresponding firstly with the rate of extracellular (faster) digestion, and secondly, to intracellular (slower) digestion. It was suggested that fine particles resulting from extracellular digestion were then phagocytosed by the digestive cells. Both phases were present for cadmium, but no second phase was detected for chromium and zinc. This did not necessarily imply that there was no intracellular digestion in the latter case, but that the absorption of the metal was very efficient, with little loss to the faeces. These models suggest new ways of assessing environmental impact, and they may assist our understanding of how endocytosis of particles can influence the uptake of metals. 6.2.2
Further Bioaccumulation Studies
Other examples of work that have attempted to address the role of endocytosis of particulate matter in the uptake of contaminants from the environment include the assimilation of 210 polonium. This is the final a-emitting daughter nuclide in the natural decay of 238 uranium, and marine organisms are believed to be the major source of 210 Po in 70% of the Portuguese human population [98]. In this work, Mytilus edulis were fed the alga, Isochrysis galbana, labelled
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with 210 Po, for 18 h, and were then fed unlabelled algae for up to 20 days. The mussels were sampled at intervals, and the tissues, digestive gland, mantle/gill, foot and remainder of the soft tissue counted for 210 Po. There was a triphasic loss of labelled polonium from the mussel soft tissue, with the first occurring on day 1, a second, slower loss occurring between days 5 and 7 days, and an even slower loss ultimately attaining control levels at 20 days. The experiment was interpreted as showing rapid extracellular digestion involving little absorption, followed by slower intracellular digestion with a higher rate of absorption. In a study of the importance of the sediment in the uptake of TBT, Langston and Burt [42] exposed the bivalve Scrobicularia plana for a month to 14 Clabelled TBT in water or water plus sediment. The distribution coefficient for the partitioning of the TBT between sediment and water was determined giving values of KD between 104 and 2 104 depending on the acidity of the water – the lower values occurring at pH < 6 [43]. Uptake into the bivalve in the presence of sediment accounted for more than 90% of body burden compared with water uptake of less than 10% [99]. These results demonstrate how sediment materials can enhance the bioaccumulation of pollutants into organisms. It would appear that the TBT is carried into the bivalve as particles that release the contaminant after being endocytosed. There is, however, no direct evidence for this interpretation, and an alternative explanation would be that digestion in the alimentary tract released the TBT and that a low intestinal pH facilitated its uptake. In an attempt to establish unequivocally that bivalves could endocytose particles and release adsorbed contaminants, ion-exchange resins and the mineral hydroxyapatite were used as particles that could be dosed with radioisotopes and fed to the suspension feeder, Mytilus edulis. The adsorption of cadmium and zinc ions on to the particles were fitted to a Langmuir adsorption isotherm, and parameters indicating the coverage of the particles and the adsorption constants were calculated [100]. The particles were then suspended in aerated artificial seawater and introduced to the mussel, Mytilus edulis for two hours and washed for 30 min in clean seawater [101]. In an extended experiment, some mussels were kept in the seawater for up to seven days, being sampled at intervals for subcellular fractionation and histological studies of the digestive glands. The histological studies of the digestive gland sampled after 18 h showed a large number of particles in the digestive cells, indicating that the particles had indeed been endocytosed. After 24 h, similar sections showed a total absence of particles, showing that they had then been exocytosed back into the alimentary tract. The subcellular fractionation studies showed that with time the distribution of metals shifted from the particulate fraction to the cytosol fraction, where they were presumably bound to a metallothionein type protein. A linear plot showed that the levels of the metals in the cytosol could be related to the logarithm of the binding constant to the particle [100].
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An ultrastructural study of the digestive gland of the cephalopod Sepia officinalis using ferritin as a tracer, also provided evidence for endocytosis and intracellular digestion of large proteins (and perhaps particles) inside the digestive cell [102]. The animals were fed on shrimps that had been injected with ferritin in seawater, and the tissues were examined after 4 h and 18 h. It was shown that the ferritin was captured by electron-dense endocytotic vesicles that fused to form heterophagosomes that were then digested in heterolysosomes. The presence of iron in the digestive cells was confirmed by positive staining by the Perl’s Prussian Blue method in light microscopy. 6.3
ENDOCYTOSIS BY TERRESTRIAL INVERTEBRATES
Most terrestrial invertebrates have limited access to water and feed on solid matter. As a consequence, they take up most of their nutrients by ingestion of foodstuffs that are also the vehicle for ingestion of contaminants. Many of the class ‘a’, metals that are taken up are found in membrane-bound granules in the cells of the hepatopancreas, although uncertainties remain as to the initiation of granule formation. Other metals, such as the class ‘b’ metal cadmium, may be in the granule or may be bound to a metallothionein type protein. There are studies on the uptake of contaminant metals into terrestrial molluscs from contaminated sites [103] and from food to which metal salts (Ca, Co, Fe, Mn, Zn) had been added [104]. The metals accumulate in the animal’s digestive gland where histological and ultrastructural studies indicate that a major route of uptake is by phagocytosis [105,106]. 6.4 6.4.1
ENDOCYTOSIS OF AIRBORNE PARTICLES Introduction
The most convincing evidence for the endocytosis of pollutant particulate matter into cells comes from the inhalation of airborne particles by humans. A full account of the composition, properties and distribution of particles in the atmosphere is given in Volume 5 of this series [3]. The particles can arise from natural sources such as volcanoes, or from anthropogenic industrial processes. The particles may be localised around mines, or can be distributed widely, depending on prevailing winds and precipitation, as occurred following the Chernobyl nuclear reactor accident. Among the more extensively studied particulates is asbestos, and it is now more than 40 years since it was implicated in mesothelioma [85]. Recognition that this was an environmental problem came when many victims who had not worked directly with asbestos but had lived near the asbestos fields in South Africa, or had other close contact with workers, developed asbestos-related conditions. The asbestos fibres are rapidly phagocytosed into lung epithelial cells. Although there is general agreement
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that asbestos fibres have the potential for DNA damage, the actual mechanisms are more elusive. Many mechanisms have been proposed, including the ability of the fibres to disrupt the phagolysosomal membrane, releasing hydrolytic enzymes which may poison the cell. A role has also been proposed for the iron content associated with the asbestos fibres in its ability to generate freeradical reactions. However, asbestos fibres themselves are considered to be physical carcinogens, as their size and shape appear to be the most important factor in inducing tumours [107]. Long (>4 mm), thin (