Principles and Models of Biological Transport

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Principles and Models of Biological Transport

Morton H. Friedman

Principles and Models of Biological Transport

123

Morton H. Friedman Duke University Department of Biomedical Engineering Durham NC 27708-0292 USA [email protected]

ISBN: 978-0-387-79239-2 DOI: 10.1007/978-0-387-79240-8

e-ISBN: 978-0-387-79240-8

Library of Congress Control Number: 2008931308 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

For Ann, still …

v

PREFACE TO THE SECOND EDITION

Focus, Organization, and Content This book, like the first edition, deals with the mass transport processes that take place in living systems, with a focus on the normal behavior of eukaryotic cells and the organisms they constitute, in their normal physiological environment. As a consequence of this focus, the structure and content of the book differ from those of traditional transport texts. We do not start with the engineering principles of mass transport (which are well presented elsewhere) and then seek biological applications of these principles; rather, we begin with the biological processes themselves, and then develop the models and analytical tools that are needed to describe them. This approach has several consequences. First of all, it drives the content of the text in a direction distinctively different from conventional transport texts. This is because the tools and models needed to describe complex biological processes are often different from those employed to describe more well-characterized inanimate systems. Many biological processes must still be described phenomenologically, using methodologies like nonequilibrium thermodynamics. Simple electrical analogs employing a paucity of parameters can be more useful for characterization and prediction than complex theories based on the behavior of more well-defined systems on a laboratory bench. By allowing the biology to drive the choice of analysis tools and models, the latter are consistently presented in the context of real biological systems, and analysis and biology are interwoven throughout. Owing to its more biological focus, the book includes more biology and physiology than most texts on engineering in the life sciences, and some parts will be easier to follow for readers with some background in biology. To keep the text selfcontained in this respect, an early chapter is devoted to those aspects of cell biology most relevant to biological transport systems. A few words on the use of the term “models” in the title. The explosive growth of molecular biology in the past several decades demands that space be devoted to the molecular-level events that underlie the observables of biological transport. This means that “models” must now be understood to include physical and structural models of transport systems and processes at the molecular level, as well as the mathematical models of transport that continue to be developed to describe biological transport mechanisms at all levels. The text includes chapters that deal primarily with fundamental transport principles, including thermodynamics (Chapters 1, 2, 6); the cell, including intracellular transport (Chapter 3); the biological transport mechanisms, such as channels and carriers, employed by living organisms (Chapters 4, 5, 7), and applications of these mechanisms in control and in the function of tissues and organs such as kidney and vii

viii

PREFACE

lung (Chapters 8–11). A more thorough overview of the text can be gained by reviewing the detailed Table of Contents that follows. The models in latter chapters are derived using the principles presented in earlier ones, so the student appreciates the assumptions that underlie them, and their consequent limitations. Those aspects of transport that are closely associated with specific tissues — for instance, the HodgkinHuxley theory of axonal conduction — are presented in these latter chapters. Certain topics are dealt with less thoroughly in this text, or are outside its scope. Experimental techniques used in transport research are well described in the extensive methods literature and are not discussed here in great detail. Experimental data are presented primarily to illustrate specific transport mechanisms, so the selection of data is representative rather than complete. Transport phenomena that exist only in experimental settings — for instance, isotope interaction effects — are not discussed. On the other hand, the text will provide guidance to the experimenter regarding the appropriate tools to interpret experimental data, and the limits on their applicability. The emphasis of the text is on mass transfer. Both convection and diffusion are included, but purely convective transport, such as the transport of oxygen in large blood vessels or gas flow in the early generations of the respiratory tree, is not. The transfer of momentum, i.e., fluid mechanics, is not dealt with in any depth. Fortunately, there are several recent texts on biological fluid mechanics, particularly in the cardiovascular area. Heat transfer is also outside the scope. Some final comments on style: — the system of units used in the text is centimeter-gram-second (cgs) rather than Système International (SI), since the former units still dominate the literature. Conversions between the two systems are presented in Chapter 1. — the valences of ions are not shown in the text; sodium ion is desig+ nated by Na, not Na . The reader is assumed to know that metals and halogens are ionized in the biological milieu.

Use in the Classroom The text is designed for a first course in biological mass transport and is based on courses I have taught to students primarily, but not exclusively, in biomedical engineering at Johns Hopkins, Ohio State, and Duke. As is appropriate for an engineering course, the student is assumed to have a certain facility with modeling and mathematics. The material is presented at a level easily accessible to upper-class undergraduates or early graduate students. The combination of biological content and engineering approach should be attractive not only to biomedical engineering students, but also to students in chemical, mechanical, and environmental engineering who are interested in the life sciences. The text can also be used to provide a broader perspective for students in physiology and biophysics, or with students in the biological sciences who are interested in a more quantitative approach to transport. Though mathematical modeling is an essential component of the text, the equations are not exceptionally complex, for several reasons. Most biological transport

PREFACE

ix

processes can be described by steady-state equations, and even transients are generally quasisteady with respect to mass transfer. Most membrane transport models assume no variation in the variables of interest in the plane of the barrier. As a consequence, with rare exceptions (most notably Chapter 11 and to a lesser extent Chapter 9), the mathematics is limited to ordinary differential equations. On the life science side, an introductory knowledge of biology and chemistry is assumed, a requisite which my current students meet with a first college course in biology. A third discipline that supports the models developed in the text is thermodynamics, a subject not always included in the crowded curriculum. Accordingly, separate chapters are devoted to equilibrium thermodynamics in solution and nonequilibrium thermodynamics, prior to the treatment of diffusional and coupled transport, respectively. In addition to supporting a core or elective biological transport course, this text could also provide a framework for teaching biology and physiology per se, as well as biological modeling, to students in biomedical, chemical, and mechanical engineering. Transport is relevant to most biological systems and, in contrast to many other subdisciplines within biology, lends itself well to mathematical modeling and engineering analysis. The book includes examples that relate to the cardiovascular system, nerve transmission, kidney function, control systems, secretory activity, and other biological systems and functions. It should be possible to design an undergraduate course in engineering physiology that covers much of the subject in the context of biological transport, perhaps supplemented with a module on cardiovascular fluid mechanics that could be based on any of several recent texts in this area. To support the use of the text in class, exercises are included at the end of each chapter. Some of the exercises are thought problems; some require the student to apply the presented material quantitatively, and others encourage the student to explore beyond the boundaries of the book itself. More exciting and, if it meets its objectives, more useful, a website for the book has been set up on the server at the Pratt School of Engineering at Duke. The URL of the site is http://biotrans.pratt.duke.edu/ The site will be accessible to students, instructors, and other users and is intended to create a “community of the book” that will enhance both teaching and learning. Using the site, community members will be able to post additional exercises; suggest new material for inclusion in courses based on the text, either in general terms or with a link to a specific publication; and share course syllabi. It can also be used to post corrections or amendments to the text itself. Users will be encouraged to register on a listserv so that they can be alerted when something new has been posted on the site. If there is interest, the site can support forums on issues of common concern among the users. The website is intended to complement the text and be a continuing resource in this important field.

x

PREFACE

In Appreciation In a sense, this book has many authors besides myself. It reflects the research efforts of numerous investigators in biology and transport science, and insights from the authors of many former texts. Many of the figures are gifts from colleagues and their publishers, who graciously permitted me to reproduce or adapt their illustrations. Most of the exercises have been vetted by my students at Ohio State and Duke. I would like to single out a few people who either suffered through the writing or without whom it would never have been completed: — the students in my laboratory, to whom I was not always as available as I would have liked during the past few years, particularly Yun Liang, who found and copied all the references that had escaped digitization, and Ji Zhang, who computed and produced all the new numerical plots in this edition; — Tim Oliver, who managed the project and patiently translated my computer files into the pages that follow; the folks at HFS Imaging, who transformed my often ambiguous sketches into the precise figures that appear throughout; and my wife, Ann, who obtained the permissions that allowed us to reproduce the excellent artwork of earlier authors. Durham 2008

Morton H. Friedman

CONTENTS

Preface .....................................................................................................................

vii

1: Equilibrium Thermodynamics Introduction .......................................................................................................................... 1.1. Chemical Potentials and Activities .......................................................................... 1.1.1. Thermodynamic Preliminaries. The Electrochemical Potential ...................................................................................................... 1.1.2. The Interphase Equilibrium Condition ....................................................... 1.1.3. Electrochemical Potentials in Terms of Measurable Solution Variables: Introduction, and the Effect of Electrostatic Potential............... 1.1.4. The Gibbs–Duhem Equation ...................................................................... 1.1.5. Dependence of the Chemical Potential on Pressure.................................... 1.1.6. Dependence of the Chemical Potential on Composition............................. 1.1.7. Units ........................................................................................................... 1.1.8. Activity and Activity Coefficient ............................................................... 1.2. Ion Equilibrium across Membranes......................................................................... 1.2.1. The Nernst Equilibrium .............................................................................. 1.2.2. Origin of the Nernst Potential..................................................................... 1.2.3. Specific Ion Electrodes ............................................................................... 1.2.4. Activity Coefficient Considerations ........................................................... 1.2.5. The Donnan Equilibrium ............................................................................ 1.3. Chemical Equilibrium.............................................................................................. Problems...............................................................................................................................

1 1 2 4 5 5 6 7 11 12 14 14 15 17 17 19 21 24

2: Free Diffusion Introduction .......................................................................................................................... 2.1. Free Diffusion of Nonelectrolytes ........................................................................... 2.1.1. The Teorell Equation .................................................................................. 2.1.2. Integration of the Teorell Equation; Fick's First Law; Solute Permeability..................................................................................... 2.1.3. Unstirred Layers ......................................................................................... 2.1.4. Applications of Solution Theory................................................................. 2.1.5. Fick's Second Law and Convective Diffusion ............................................ 2.1.6. Justification of the Steady-State Assumption: Time Scales in Biological Transport........................................................... 2.2. Free Diffusion of Electrolytes ................................................................................. 2.2.1. Differences between Electrolyte and Nonelectrolyte Diffusion ................. 2.2.2. The Electrodiffusion Equation.................................................................... 2.2.3. Integration of the Electrodiffusion Equation ..............................................

29 30 30 31 36 38 40 41 43 43 44 47 xi

xii

CONTENTS

2.2.4. Some Special Cases .................................................................................... Equilibrium ......................................................................................... Uniform Composition ......................................................................... Diffusion Potential of a Bi-Ionic System............................................ Active and Passive Exchange with a Closed Compartment................ Equal Total Concentrations on the Two Sides of the Membrane: The Constant-Field Equation....................................... 2.2.5. Ionic Permeability and the Resting Potential of the Cell ............................ 2.2.6. Charged Membranes................................................................................... 2.2.7. Limitations of the Electrodiffusion Equation and Its Solutions.................. Problems...............................................................................................................................

49 49 50 51 52 53 57 59 61 62

3: The Cell Introduction .......................................................................................................................... 3.1. Overview ................................................................................................................. 3.2. The Structure of an Animal Cell.............................................................................. 3.2.1. Composition and Structure of the Plasma Membrane................................. Heterogeneity of Cell Membranes ...................................................... The Mosaic Model of the Plasma Membrane ..................................... The Glycocalyx and Surface Charge .................................................. 3.2.2. The Internal Structure of the Cell ............................................................... 3.3. Metabolism: The Production of ATP....................................................................... 3.4. Intracellular Transport ............................................................................................. 3.4.1. Intracellular Diffusion ................................................................................ 3.4.2. Protein Transport ........................................................................................ Transport into the Nucleus.................................................................. Transport into the Mitochondrion....................................................... 3.4.3. Vesicular Transport .................................................................................... Endocytosis......................................................................................... Exocytosis, Secretion and Transcytosis .............................................. 3.5. Cellular Motility and Locomotion ........................................................................... 3.5.1. Actin-Based Movement and Chemotaxis ................................................... 3.5.2. Cilia and Flagella........................................................................................ Problems...............................................................................................................................

67 67 68 68 68 72 74 75 78 84 84 87 87 88 90 91 95 100 100 105 107

4: Facilitated Diffusion: Channels and Carriers Introduction .......................................................................................................................... 4.1. Mechanisms of Channels and Carriers .................................................................... 4.1.1. Hallmarks of Mediated Transport............................................................... 4.1.2. Ion Selectivity of Channels......................................................................... 4.1.3. Energetics of Ion Selectivity, and Steric Effects......................................... 4.1.4. Ion Selectivity of Channels: Summary ....................................................... 4.1.5. The Structure of Ion Channels: Selectivity Filters, Gates, and Energy Profiles..........................................................................

111 112 112 114 116 119 120

CONTENTS

4.1.6. Regulation of the Gating Process................................................................ 4.1.7. Classification of Ion Channels. Aquaporins and Gap Junctions ................. 4.1.8. Carrier Models ............................................................................................ 4.1.9. Carriers and Channels: Convergences and Differences .............................. 4.2. Kinetics of Facilitated Transport ............................................................................. 4.2.1. Models of Ion Channel Transport: Overview ............................................. 4.2.2. Energy Barrier and Binding Models of Channel Transport ........................ Ionic Independence: Absolute Rate Theory........................................ Saturable Channels: Kinetic Analysis................................................. 4.2.3. The Patch Clamp and Two Applications .................................................... Acetylcholine Receptor Channel Kinetics .......................................... Voltage and Current Sensitivity of a Voltage-Activated Sodium Channel ............................................... 4.2.4. Stochastic Properties of Channels: Membrane Noise Analysis .................. 4.2.5. The Simplest Model of Carrier Transport: Assumptions............................ 4.2.6. The Simplest Model of Carrier Transport: Equations................................. 4.2.7. Monosaccharide Transport in the Erythrocyte............................................ 4.2.8. More Complex Carrier Models................................................................... 4.2.9. Exchangers and Cotransporters................................................................... 4.3. Inhibition of Facilitated Transport........................................................................... 4.3.1. Inhibition of Channel Transport: Channel Block........................................ 4.3.2. Inhibition of Carrier Transport ................................................................... Problems...............................................................................................................................

XIII

122 131 137 139 141 141 143 143 146 151 152 154 158 161 162 165 166 170 174 174 175 179

5: Active Transport Introduction .......................................................................................................................... 5.1. Active Transport: General Considerations .............................................................. 5.1.1. Metabolic Coupling and Affinity................................................................ 5.1.2. Classification of Active Transport Processes.............................................. 5.1.3. Identification of Active Transport Processes .............................................. 5.2. Mechanisms of Active Transport............................................................................. 5.2.1. Scalar Active Transport: Overview ............................................................ 5.2.2. Primary Scalar Transport ............................................................................ 5.2.3. Secondary Scalar Transport ........................................................................ Cotransport ......................................................................................... Countertransport ................................................................................. 5.2.4. Vectorial Active Transport, the Curie Theorem and Substrate Activation.................................................................................... 5.2.5. Sodium–Potassium Exchange..................................................................... 5.2.6. Pump Selectivity and Other Properties Shared with Passive Carriers ........ 5.3. Kinetics of Active Transport ................................................................................... 5.3.1. A Simple Secondary Scalar Transport Model: Assumptions...................... 5.3.2. A Simple Secondary Scalar Transport Model: Equations........................... 5.3.3. More Complex Symport Models; the Sodium/Glucose Transporter........... 5.3.4. Primary Scalar Transport ............................................................................ 5.3.5. Flux Equations for Primary Scalar Transport .............................................

185 186 186 188 190 190 191 192 193 195 197 199 202 204 205 206 208 209 213 214

xiv

CONTENTS

5.3.6. Relation between the Coupling Parameter * and the Affinity of the Metabolic Reaction ....................................................... 5.3.7. Vectorial Active Transport and the Nature of Na–K Exchange.................. 5.3.8. Pumps and Leaks ........................................................................................ Problems...............................................................................................................................

215 216 216 220

6: Nonequilibrium Thermodynamics Introduction .......................................................................................................................... 6.1. The Basic Phenomenological Equations.................................................................. 6.1.1. Conjugate Forces and Fluxes...................................................................... 6.1.2. Phenomenological Coefficients and Linear Thermodynamics ................... 6.1.3. Frictional Interpretation of the Phenomenological Equations..................... 6.1.4. A Cautionary Note before Proceeding........................................................ 6.2. Nonequilibrium Thermodynamic Description of Passive Transport ....................... 6.2.1. Setting the Stage ......................................................................................... 6.2.2. The Chemical Potential of the Solvent ....................................................... 6.2.3. A New Set of Forces and Fluxes; Osmotic Pressure................................... 6.2.4. The Kedem–Katchalsky Equations............................................................. 6.2.5. Physical Significance of the Reflection Coefficient: Semipermeable Membranes and the Osmometer, Steric Effects and Sieving........................................................................... 6.2.6. Osmotic Pressure of Solutions; Donnan Osmotic Pressure; Osmotic Effects on Cells ............................................................................ 6.2.7. Passive Transport of Multiple Nonelectrolytes........................................... 6.2.8. Passive Transport of Electrolytes: Electrokinetic Phenomena.................... 6.3. Nonequilibrium Thermodynamic Description of Active Transport......................... 6.3.1. Definition of Active Transport ................................................................... 6.3.2. Coupling between Nonconjugate Forces and Fluxes .................................. 6.3.3. Nonequilibrium Thermodynamics of Motor Proteins and ATP Synthase....................................................................................... 6.4. Limitations of Nonequilibrium Thermodynamics ................................................... 6.4.1. Closeness to Equilibrium: A Limitation Intrinsic to Linearized Nonequilibrium Thermodynamics ............................................ 6.4.2. The Concentration Dependence of the Phenomenological Coefficients................................................................................................. 6.4.3. Closeness to Equilibrium in Biological Systems ........................................ 6.4.4. The Information Content of Nonequilibrium Thermodynamics ................. 6.4.5. Approximations in the Derivation of the Kedem–Katchalsky Equations .................................................................................................... Problems...............................................................................................................................

225 225 225 227 229 230 231 231 232 234 235

237 242 246 247 248 248 250 252 254 254 257 258 259 260 261

7: Models of Transport across Cell Membranes Introduction .......................................................................................................................... 7.1. Transport across the Lipid Bilayer of Cell Membranes........................................... 7.1.1. Evidence for Nonelectrolyte Diffusion across the Lipid Bilayer................ 7.1.2. A Simple Model of Transbilayer Diffusion ................................................

265 265 266 267

CONTENTS

7.1.3. Potential Barriers in the Bilayer.................................................................. Models of Transport through Pores ......................................................................... 7.2.1. Classification of Pore Transport Models .................................................... 7.2.2. Hydraulic Conductivity of a Pore ............................................................... 7.2.3. Hindered Diffusion; Solute Permeability as a Probe of Pore Radius.......... 7.2.4. Other Factors Affecting Estimated Pore Size; The Equivalent Pore........... 7.2.5. Hindered Convection (Sieving); the Reflection Coefficient as a Probe of Pore Radius .................................................................................. 7.2.6. Combined Diffusion and Convection through Pores .................................. 7.2.7. Single-File Transport through Pores........................................................... 7.2.8. The Permeability Ratio of Larger Pores ..................................................... 7.3. Electrical Analogs.................................................................................................... 7.3.1. Equivalent Circuit for the Passive Flux of a Single Ion.............................. 7.3.2. Equivalent Circuit for the Passive Transport of Multiple Ions ................... 7.3.3. The Electrical Analog of a Rheogenic Pump.............................................. 7.3.4. Some Final Remarks................................................................................... Problems............................................................................................................................... 7.2.

XV

269 270 271 272 273 276 279 282 283 287 289 289 290 293 294 295

8: Regulation and Feedback Introduction .......................................................................................................................... 8.1. Regulation of Transport ........................................................................................... 8.1.1. Receptor-Mediated Second Messenger Systems: Cyclic AMP and Antidiuretic Hormone................................................................. 8.1.2. Direct Hormonal Regulation without an Extracellular Receptor: Aldosterone ................................................................................ 8.1.3. Calcium-Based Regulation ......................................................................... Maintenance of a Low Cytosolic Calcium Concentration .................. Calcium Signaling .............................................................................. Calcium Handling in Cells and Compartmental Analysis .................. 8.2. Feedback in Transport Systems: Insulin.................................................................. 8.3. Regulation by Transport .......................................................................................... 8.3.1. Regulation of Cell Volume ......................................................................... 8.3.2. Regulation of Cell pH................................................................................. Problems............................................................................................................................... Appendix 8.1. Nonequilibrium Binding in Compartmental Analysis ................................

299 301 301 306 307 307 307 309 315 317 317 322 325 327

9: Excitable Cells Introduction .......................................................................................................................... 9.1. Nerve ....................................................................................................................... 9.1.1. The Resting Neuron.................................................................................... 9.1.2. The Action Potential: Electrical Aspects .................................................... The Membrane Action Potential: Resting State.................................. Hodgkin and Huxley's Equations for the Dependence of Conductance on Membrane Potential ......................................... Excitation of the Membrane Action Potential.....................................

329 330 331 333 335 337 340

xvi

CONTENTS

The Propagating Action Potential: Cable Theory ............................... The Role of Myelin............................................................................. 9.1.3. The Action Potential: Molecular Aspects ................................................... Hodgkin and Huxley's Channel Model ............................................... More Recent Models of the Potassium and Sodium Channels of Nerve ............................................................. 9.1.4. Synaptic Transmission................................................................................ 9.2. Muscle ..................................................................................................................... 9.2.1. The Resting Muscle Fiber........................................................................... 9.2.2. Excitation and Excitation–Contraction Coupling ....................................... Neuromuscular Transmission and Excitation: The Acetylcholine Receptor ........................................................... Excitation–Contraction Coupling ....................................................... 9.2.3. Molecular Mechanisms of Muscle Contraction .......................................... Problems...............................................................................................................................

343 347 349 349 353 359 367 368 370 370 378 382 387

10: Epithelial Transport Introduction .......................................................................................................................... 10.1. Organization of Epithelial and Endothelial Cell Layers and Some Consequences ................................................................................................ 10.1.1. The Organization of Epithelial and Endothelial Cell Layers ...................... 10.1.2. The Pathways across Leaky and Tight Epithelia ........................................ 10.1.3. Transport in a Parallel Path System ............................................................ 10.1.4. Coupling of Transepithelial Water Flow to Active Ion Transport .............. The Curran Model............................................................................... The Standing Gradient Model............................................................. The Sodium Recirculation Model....................................................... The Cotransporter Hypothesis ............................................................ 10.1.5. The Effect of Unstirred Layers on Transepithelial Diffusion and Osmosis: Concentration Polarization ................................................... 10.1.6. Electrical Analogs of Cell Layers ............................................................... 10.2. Examples of Epithelial and Endothelial Function ................................................... 10.2.1. Absorption .................................................................................................. Absorption of Sugars in the Small Intestine ........................................ Transport of Water and Sodium in the Proximal Tubule of the Kidney ...................................................................... Transport of Water and Sodium in the Distal Tubule of the Kidney .................................................................................. 10.2.2. Secretion ..................................................................................................... 10.2.3. Filtration ..................................................................................................... Problems............................................................................................................................... Appendix 10.1. Convection, Diffusion and Mass Addition in Channel Geometries ...........

391 392 392 395 400 405 405 407 410 411 412 414 418 419 419 422 427 430 434 439 443

11: Gas Transport Introduction .......................................................................................................................... 447 11.1. Partial Pressure and the Equations for Gas Flux...................................................... 448

CONTENTS

XVII

11.2. 11.3. 11.4. 11.5.

449 453 455 456 459 461 462 468 480

Overview of the Gas Transport Process .................................................................. Gas Exchange in the Lung....................................................................................... Oxygen Transport in the Blood ............................................................................... Transport from Red Blood Cells to Tissue .............................................................. 11.5.1. Reaction–Diffusion Processes .................................................................... 11.5.2. The Krogh Tissue Cylinder......................................................................... 11.5.3. Modifications to the Basic Krogh Model.................................................... 11.5.4. Beyond the Krogh Cylinder........................................................................ Problems...............................................................................................................................

References ............................................................................................................... 485 Index ........................................................................................................................ 497

1 EQUILIBRIUM THERMODYNAMICS

INTRODUCTION It is appropriate to begin this text with an examination of the equilibrium state, since departures from this state drive the transport processes that we will discuss in the chapters to follow. The equilibrium state is defined in the first section of this chapter, using the laws of thermodynamics to develop expressions for the activities and chemical potentials of species in the biological milieu. Because of the particular importance of the chemical potential (and the related electrochemical potential) to descriptions of transport, special emphasis is placed on its relation to solute activities and concentrations, and to the other measurable properties of biological solutions. Because of the close relation between equilibrium and transport, it is not surprising that a number of equilibrium thermodynamic concepts have become an integral part of the description of certain transport processes. Two of these concepts are developed in this chapter for later use: ionic equilibria across membranes and at interfaces, and chemical equilibrium and equilibrium constants.

1.1. CHEMICAL POTENTIALS AND ACTIVITIES Among the several equivalent thermodynamic definitions of equilibrium, the one that is based on chemical potential is particularly well suited to the purposes of this text. This is because gradients or differences in chemical potential are the natural driving forces for the fluxes of solute and solvent; they also drive the biochemical processes to which these fluxes may be coupled. By using the chemical potential to define thermodynamic equilibrium, we achieve a certain ease of movement between equilibrium considerations and transport events. In the first part of this section, the condition for thermodynamic equilibrium between two phases is developed in terms of the chemical and electrochemical potentials of the species in each phase. In biological transport, these phases are almost always solutions, and the bulk of the section is devoted to the development of expressions for the chemical and electrochemical potentials in terms of solution composition, presM.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_1, © Springer Science+Business Media, LLC 2008

1

2

CH. 1: EQUILIBRIUM THERMODYNAMICS

sure, and temperature. The section concludes with a brief discussion of solute activity and concentration as they relate to the chemical potential. A limited familiarity with thermodynamics is assumed. Accordingly, a few common thermodynamic terms are used here without definition, and the thermodynamic development includes a small number of equations that are presented without proof and whose derivation can be found in any thermodynamics text. These equations are indicated with an asterisk.

1.1.1. Thermodynamic Preliminaries: The Electrochemical Potential We begin with the First Law of Thermodynamics, in differential form: *dU = dQ – dW.

(1.1)

Here, dU is the change in the internal energy of a system of interest, dQ is a differential quantity of heat transferred into the system, and dW is a differential amount of work performed by the system. The internal energy of the system is increased when heat is added, and decreases when the system performs work. Different systems can perform different kinds of work, including: 1. the movement of volume, or expansion, against an external pressure; 2. the movement of charge, for instance that on an ion, against an electrostatic potential; and 3. the transfer of material. Corresponding to these,

*dW PdV \dq Pi dni ,

(1.2)

i

where the sum includes all species in the system, P is hydrostatic pressure, V is the volume of the system, \ is electrostatic potential, q is electric charge, Pi is the chemical potential of the ith species, and ni is the number of moles of the ith species in the system. Equation (1.2) gives the differential work required to change, by differential amounts, the volume, charge, and composition of the system. We should note that there are other forms of work that systems can do, which we will not include here. Among those having biological relevance, an important work term appropriate to muscle is the work of shortening against a tensile force, F. If l is the length of the muscle segment, the differential work of contraction is –Fdl. The reversible addition or withdrawal of heat changes the entropy of the system, according to the Second Law of Thermodynamics: *dS

dQ T

(1.3)

where S is entropy and T is absolute temperature. By substituting Eqs. (1.2) and (1.3) into (1.1), the Gibbs equation is obtained:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

3

dU TdS PdV \dq Pi dni .

(1.4)

i

The Gibbs free energy of the system is defined by: G = U + PV – TS.

(1.5)

dG = dU + PdV + VdP – TdS – SdT.

(1.6)

The total differential of G is:

Substituting Eq. (1.4) into (1.6), dG \dq Pi dni VdP SdT .

(1.7)

i

Equation (1.7) specifies the change in the Gibbs free energy of a system resulting from reversible changes in its charge, content, pressure and temperature. We shall see shortly that this expression is crucial for defining the conditions under which a system is in a state of equilibrium. In this text, the most common source of charges in solution is ionic solutes. The total charge qi carried by the ith ionic species is proportional to ni, and the first two terms on the right-hand side of Eq. (1.7) can be combined. The coulombs of charge carried by a mol or gram atom of the ith ion is: qi = zi ni,

(1.8)

where zi is the valence of the ith ion and is the Faraday, a conversion factor equal to about 96,500 coul/mol. The total charge in the system, q, is obtained by summing qi over i; its differential is dq zi dni.

(1.9)

dG \ z i dni Pi dni VdP SdT .

(1.10)

i

Substituting Eq. (1.9) into (1.7), i

i

Note that, although both sums are over all species, only charged species (zi z 0) contribute to the first sum. Define the electrochemical potential of the ith species by P i Pi zi\ .

(1.11)

The electrochemical potential becomes identical to the chemical potential if the species is uncharged. Equation (1.10) can now be written as dG P i dni VdP SdT . i

(1.12)

4


From Eq. (1.12), it can be seen that P i measures the contribution of the ith species to the Gibbs free energy of the system: sG ¬ P i sni ®

(1.13)

P, T , nj

where j includes all species except the ith; the electrochemical potential is thus the partial molal free energy of the ith species.

Figure 1.1. A system at equilibrium: P iI P iII .

1.1.2. The Interphase Equilibrium Condition We are now ready to define interphase equilibrium in terms of electrochemical potentials. Consider the system in Figure 1.1, consisting of Phases I and II separated by a barrier that is permeable to species i. The temperature and pressure of the system is uniform. What is the meaning of the statement that the ith species is in equilibrium across the membrane? The thermodynamic condition for equilibrium is defined in terms of the Gibbs free energy: when a system is in a state of equilibrium, the change in Gibbs free energy accompanying an infinitesimal departure from that state is zero. Correspondingly, a species is said to be in equilibrium across a barrier like that in Figure 1.1 if the free energy of the system does not change when a differential amount of the species is transferred from one phase to the other. Consider the transfer of dni moles of species i from Phase I to Phase II. During this process, there is no change in pressure, temperature, or the number of moles of


5

other species in either phase; furthermore, the amount transferred is too small to cause a change in the composition or partial molal properties of either phase. Phase I loses mass and its free energy is reduced proportionately; from Eq. (1.13),

iI dni . dG I P

(1.14)

The mass is received by Phase II and assumes the partial molal properties of that phase; thus, dG II P iII dni .

(1.15) I

II

If species i is in equilibrium, then the total change in system free energy, dG + dG , is zero; thus, the desired equilibrium condition is: P iI P iII .

(1.16)

An unstated key ingredient of this definition of equilibrium is the ability of the species to transfer from one phase to the other. If two phases containing a species in common are on the two sides of a barrier, that species can come to equilibrium (i.e., achieve identical electrochemical potentials in each phase) only if the interface between the phases is permeable to it. As a (rather obvious) corollary, when two phases are separated by a barrier that is impermeable to a particular species, that species cannot equilibrate between the phases. Finally, when phases or systems are described as being in equilibrium with one another, Eq. (1.16) is necessarily satisfied for only those species that can move between them.

1.1.3. Electrochemical Potentials in Terms of Measurable Solution Variables: Introduction, and the Effect of Electrostatic Potential Most biological solutions are characterized not by the chemical or electrochemical potentials of their constituents, but rather by more measurable quantities such as temperature, pressure, electrical potential, and composition. In the next few sections, we derive expressions for the chemical and electrochemical potential in terms of these measurable quantities. This allows us to rewrite Eq. (1.16) in terms of the measurables of the system. The lone contribution of the electrostatic potential to the electrochemical potential is the zi\ term in Eq. (1.11). It remains to express the chemical potential Pi in terms of temperature, pressure, and composition.

1.1.4. The Gibbs–Duhem Equation Before pursuing this task, it is necessary to derive one more key thermodynamic equation: the Gibbs–Duhem equation. We begin with the integrated form of the Gibbs equation: *U TS PV P i ni . i

(1.17)

6


The total differential of U is obtained from Eq. (1.17): dU TdS SdT PdV VdP P i dni ni dP i . i

(1.18)

i

The quantity dU is also given by the Gibbs equation [Eq. (1.4)], which can readily be written in terms of the electrochemical potential: dU TdS PdV P i dni .

(1.19)

i

The Gibbs–Duhem equation is obtained by subtracting Eq. (1.19) from (1.18): SdT VdP ni dP i 0 .

(1.20)

i

The sum in Eq. (1.20) can be broken into two sums: ni dP i ni dPi d\ ni zi . i

i

i

For electrically neutral solutions, ni zi 0 , i

so ni dP i ni dPi , i

i

and the electrochemical potential in the Gibbs–Duhem equation can be replaced by the chemical potential: SdT VdP ni dPi 0 .

(1.21)

i

1.1.5. Dependence of the Chemical Potential on Pressure The dependence of chemical potential on pressure will be developed first. Consider a solution at constant temperature and composition. Solving Eq. (1.11) for the chemical potential, and differentiating with respect to pressure,

sPi ¬ sP ®

T , ni

sP ¬ s\ . i zi sP ® sP T , ni

(1.22)

The second term on the right-hand side of Eq. (1.22) is zero because the electrostatic potential and the pressure are independent variables. Equation (1.13) is substituted into the right-hand side of Eq. (1.22), which becomes

sPi ¬ sP ®

T , ni

¯ s ¡ sG ¬ ° ¡ °. sP ¡ sni ®P , T , n ° j ¢ ±

(1.23)


7

The free energy is a continuous function of the independent variables, so the order of differentiation may be interchanged:

sPi ¬ sP ®

T , ni

s sni

sG ¬ ¯ ¡ ° ¡ sP ® ° . T , ni °± ¡¢

(1.24)

The partial derivative of free energy with respect to pressure, at constant temperature and composition, is seen from Eq. (1.12) to be V, the volume of the system; thus,

sV ¬ sPi ¬ . sP ®T , n sni ® P, T , nj i

(1.25)

The right-hand side of Eq. (1.25) is the partial molar volume of the ith species, Vi . Thus, at constant temperature and composition, dPi Vi dP . Since the partial molar volume is essentially independent of pressure, we can write the indefinite integral:

Pi Vi P Pic ,

(1.26)

c

where the “constant” of integration Pi is a function of only temperature and composition. In summary, we derived the dependence of Pi on pressure by fixing the remaining independent variables and obtaining a differential relation between chemical potential and pressure. Integrating this relation, we obtained an indefinite integral whose “constant” of integration (constant only with respect to changes in pressure) depends on only the variables that had been fixed, i.e., temperature and composition. The same c technique will now be used to derive the dependence of Pi on composition. Regrettably, the analysis does not proceed as simply — or as directly — as the one we have just performed.

1.1.6. Dependence of the Chemical Potential on Composition We begin by differentiating Eq. (1.26) and substituting the result into Eq. (1.21). c Since the composition dependence of Pi is sought, the remaining independent variable — temperature — is fixed. Consequently, the SdT term in the Gibbs–Duhem equation vanishes, giving

VdP ni (Vi dP dPic ) 0 .

(1.27)

i

The coefficient of dP in Eq. (1.27) is niVi V . This coefficient is zero, since the i

partial molar volumes are defined such that the volume of the system is essentially equal to the sum of the contributions niVi of each species in the system. Thus, the preceding equation simplifies to

ni dPic 0 . i

(1.28)

8


Figure 1.2. A system demonstrating the effect of composition on chemical potential. The water level in Phase I is constant but that in the narrow standpipe can change when solvent crosses the membrane.

With Eq. (1.28) in hand, now consider the system depicted in Figure 1.2. The two phases, one (I) consisting of pure water, and the other (II) consisting of an aqueous solution containing k solutes, are in equilibrium across a membrane that is permeable to only the solvent. We now examine the consequence of a small change in the composition of Phase II. Before the composition change, the solvent, which can transfer between the phases, is in equilibrium across the membrane; that is, its chemical potential is the same in Phase I and Phase II. The chemical potential of the water in Phase I is not affected by the composition change in Phase II. However, the chemical potential of the water in Phase II is altered when the composition of the phase is changed. As a result, water crosses the membrane to restore its chemical potential in Phase II to that in Phase I, that is, to re-establish equilibrium. The transmembrane difference in solvent chemical potential, caused by the change in the composition of Phase II, is the driving force for the flow of water across the membrane. Water does not cross the membrane indefinitely. The chamber containing Phase II has a standpipe that translates any change in the volume of the phase into a change in its hydrostatic pressure. This, in turn, changes the chemical potential of the water in Phase II, according to Eq. (1.26), and equilibrium is eventually reestablished. If we consider a differential process, starting before the composition change and terminating after equilibrium has been reestablished, dP IIw 0 ,

(1.29)


9

since the chemical potential of the water (denoted by the subscript “w”) in Phase II is equal to its chemical potential in Phase I before the process begins, and after it is complete. Equation (1.26) is used to rewrite Eq. (1.29) as Vw dP II dPcw, II 0 .

(1.30)

The second term in Eq. (1.30) is the change in chemical potential caused by the initial composition change, and the first term reflects the compensating pressure change. For this differential process, the effect of the water flow on the composition of Phase II is negligible. The remainder of this analysis will be concerned with the thermodynamics of only Phase II, so the superscript “II” will be omitted in what follows. Our next step is to use the Gibbs–Duhem equation to relate the pressure change in Phase II to the composition change that caused it. The sum in Eq. (1.28) can be split into a sum over the k solutes plus a term for the solvent: k

nw dP cw ni dPic 0 .

(1.31)

i 1

Substituting Eq. (1.30) into (1.31) and rearranging: k

ni dPic nwVw dP

(1.32)

i 1

At this point we must use an empirical relation, the van 't Hoff Law. Refer once again to Figure 1.2. The van 't Hoff Law states that, when Phase II is dilute, the pressure change needed to stop water flow across the membrane is proportional to the initiating change in the composition of Phase II, according to the following equation: k

GP RT Gci .

(1.33)

i 1

Here, R is the gas constant and T is absolute temperature; Gci is the small initial change in the concentration of the ith species in Phase II, and GP is the small compensating pressure change. Though this equation is based on measurements of finite changes in pressure caused by finite changes in concentration, it is reasonable to expect that it holds for differential changes as well: k

dP RT dci .

(1.34)

i 1

The use of the van ‘t Hoff Law in this development means that the final result will be directly applicable to only dilute solutions; thus, additional dilute-solution approximations can now be made without further limiting the usefulness of the result. In particular, since such solutions are almost entirely water, the product nwVw in Eq. (1.32) can be replaced by V, the volume of the solution. The validity of this substitution can be demonstrated with a simple example. For instance, consider a liter of 150 mM NaCl solution, whose density is 1.0046 g/cc. The liter weighs 1004.6 g and con-

10


tains 0.15 mols, or 8.8 g, of NaCl. By difference, it contains 995.8 g water. The molar volume of water is 18 cc/mol, so nwVw = (995.8/18) q 18 = 995.8 cc, close to the actual solution volume, V =1000 cc. Then, with Eq. (1.34), Eq. (1.32) becomes: k

k

i 1

i 1

ni dPic VRT dci .

(1.35)

Furthermore, in dilute solutions, individual solute ions or molecules (“particles,” for short) are far apart, separated from one another by numerous solvent molecules. Thus, each solute particle is oblivious to all other solute particles; solute–solute interactions are absent. In particular, all of the particles of a given solute are oblivious to the presence of other solutes. Under such circumstances, the free energy and chemical potential of a particular solute are not expected to depend on the concentrations of other solutes. In the absence of interactions between different solutes, Eq. (1.35) can hold only if it applies to each species separately:

ni dPic VRTdci .

(1.36)

We now have the desired differential relation between the compositional part of c the chemical potential of the ith solute (Pi ) in dilute solution and a variable (ci) that reflects the composition of the solution. To write the indefinite integral as we did in the previous subsection, we first divide both sides of Eq. (1.36) by ni, and make use of the definition ci = ni /V; then, dPic RT

dci . ci

(1.37) c

We want to integrate Eq. (1.37) to obtain a relationship between Pi and ci. To perform c a definite integration, we need to know the value of Pi that corresponds to a particular c value of ci. We have no means for directly measuring Pi , so we make an arbitrary 0 definition: We define a standard concentration ci (usually 1 M = 1 mol/l for solutes) 0 and define the chemical potential term in this standard state as Pi . We can then integrate Eq. (1.37): Pic

ci

¨

dPic RT ¨

Pi0

ci0

dci . ci

(1.38)

ci . ci0

(1.39)

Integrating and rearranging: Pic Pi0 RT ln 0

Choosing molarity as the unit of concentration, ci = 1, and Eq. (1.39) becomes

Pic RT ln ci Pi0 .

(1.40)


11

Since Pi0 is evaluated at a specified concentration, ci0 , it is a function of only temperature. Substituting Eq. (1.40) into (1.26), we obtain the desired expression for the chemical potential of a solute in a dilute solution:

Pi Vi P RT ln ci Pi0 .

(1.41)

Similarly, from Eq. (1.11), the electrochemical potential is

P i Vi P RT ln ci zi \ Pi0 .

(1.42)

An equation analogous to Eq. (1.41) can be written for the chemical potential of the solvent. It will be derived in Chapter 6, where it is first needed. The requirement that the argument of logarithms be dimensionless appears to be violated in Eq. (1.40) and the equations that follow. Strictly speaking, the argument is 0 ci /ci , which is dimensionless. If the standard state is 1 M, ci must be measured in molarity; i.e., mols/l. When the chemical potential is used to describe the driving force for transport, differences or gradients in this quantity are much more important. It is 0 easy to show that ci is absent from the expression for chemical potential difference, and the gradient is derived directly from Eq. (1.37), which does not contain ci0. As we will see below, more care is needed when using the chemical potential to describe chemical equilibrium.

1.1.7. Units All of the terms that comprise the electrochemical potential have the units of energy per mass. To express all of them in the same units, e.g., cal/mol, some conversions must usually be made: \ has the units of (coul/mol)-volt. 1 coul-volt = 1 joule = 0.2389 cal; thus 1 coul-volt/mol = 0.2389 cal/mol. Biological pressures are often measured in torr (1 torr = 1 mm Hg); thus, VP 2 commonly has the units of (cc/mol)-torr. 1 torr = 1333 dynes/cm and 1 dyne-cm = 1 –7 –5 erg = 0.2389 q 10 cal; thus 1 cc-torr/mol = 1333 ergs/mol = 3.18 x 10 cal/mol. Using R = 1.987 cal/mol-K, RT has the units of cal/mol. SI (Système International, or International System) units are based on the kilogram (kg), meter (m), second (s), ampere (A), and degree kelvin (K). The unit of substance remains the gram-mole, defined as that amount of material that contains as many elementary entities (atoms or molecules, depending on the substance) as there 12 are atoms in 0.012 kg of C. Force and energy are derived quantities; force is meas2 ured in newtons (1 N = 1 kg-m/s ), and energy is measured in joules (1 J = 1 N-m). 2 The SI unit of pressure is the pascal (1 Pa = 1 N/m ); 1 torr = 133.3 Pa. To convert quantities expressed in cal/mol to J/mol, multiply by the mechanical equivalent of heat, 4.187 J/cal. The gas constant in SI units is R = 8.32 J/mol-K.

12


1.1.8. Activity and Activity Coefficient Equation (1.41) is limited in its applicability, since it is based on the van ‘t Hoff Law and a number of approximations that are valid only for solutions more dilute than the physiological solutions with which we will be concerned. This limitation is dealt with by introducing a quantity called the solute activity. The activity is defined by the requirement that:

Pi Vi P RT ln ai Pi0 ,

(1.43)

where ai is the activity of the ith species. The activity of the solute is the value of ai that gives the correct chemical potential when substituted into Eq. (1.43). Thus, by definition, Eq. (1.43) is applicable to concentrated solutions as well as to dilute ones. Theoretical methods are available for predicting solute activity in simple and well-defined systems; however, the biological milieu is not simple and is rarely welldefined. Consequently, activities are generally determined experimentally when they are needed. More often, as will be illustrated in Chapter 2, the distinction between activity and concentration is neglected, Eq. (1.41) is used directly, and the error so introduced is “buried” in the analysis of the experimental data. The difference between activity and concentration is measured by the activity coefficient, Ji: Ji

ai . ci

(1.44)

Comparing Eqs. (1.41) and (1.43), it is seen that the activity coefficient approaches unity (i.e., ai = ci) in dilute solutions. Its deviation from unity is a measure of the extent to which the chemical potential of the solute is affected by interactions among dissolved particles in the solution. The interactions of importance in biological solutions are almost exclusively electrostatic, so the distinction between activity and concentration is generally made only for charged species; the activity coefficients of nonelectrolytes at physiological concentrations are usually assumed to be unity. Two kinds of electrostatic interaction are usually considered. The first of these is the interaction between small ions that are close enough to “see” one other's electric field or the effects of the field on the intervening solvent. The best-known treatment of such interactions is the Debye–Hückel theory and its extensions, which give a theoretical basis for the concentration dependence of the activity coefficient in concentrated electrolyte solutions. These theories predict that the activity coefficient of an electrolyte in solution de2 pends on the ionic strength of the solution, Z = (6izi ci )/2. The activity coefficient initially decreases from unity as the electrolyte concentration is raised (Fig. 1.3). At higher concentrations, the slope of the activity coefficient-concentration curve becomes positive, but this occurs at concentrations beyond the physiological range; thus, in biological systems, ion–ion interactions consistently cause the activity of the solute to be less than its concentration.


13

Figure 1.3. Solid line: concentration dependence of the mean activity coefficient of sodium chloride in aqueous solution at 25ºC. The molality of the solution is the number of mols of solute per 1000 g solvent, and is close to the molarity for dilute solutions. The mean activity coefficient of NaCl is related to the ionic activity coefficients by JNaCl = (JNaJCl)1/2. Dashed line: Mean activity coefficient predicted by the original Debye–Hückel theory. The original theory fails at NaCl concentrations below that corresponding to physiologic ionic strengths. The mean activity coefficient predicted by the extended Debye–Hückel theory, which includes an adjustable parameter that takes ion size into account, is not shown in the figure because it is virtually indistinguishable from the experimental curve.

A second kind of interaction that is more peculiar to biological systems is the binding of a charged solute to a soluble or insoluble particle in the solution. The binding step may or may not be reversible. Often, solute that is bound cannot participate in transport; thus the activity of unbound solute can be underestimated when it is derived from transport-related measurements. When the effect of Debye–Hückel interactions is small relative to the effect of binding, the apparent activity coefficient of the solute is close to the fraction of the solute that is unbound. Indeed, these two quantities — activity coefficient and unbound fraction — are often used interchangeably. A numerical illustration of the effect of binding on the apparent activity coefficient is given in the next section. There is a certain fuzziness in the distinction between the two kinds of interactions described above. The fuzzy middle ground is populated by a large number of biological solutes, such as proteins, charged polysaccharides, and other large, multiply charged molecules. The interactions of these species with small ions are describable by appropriate extensions of the Debye–Hückel theory, yet the presence of multiple charges along the polyion's length can give rise to cooperative effects strong enough to be described as binding, and they usually are. The description of these interactions is one of the more difficult and challenging problems in solution thermodynamics. Irrespective of the nature of the interactions, the activity coefficient in biological solutions is inevitably an experimental quantity. The solutions are too complex to

14


be described by theories appropriate to simple electrolytes, and binding fractions must be measured. Fortunately, our inability to predict the activity coefficient is not as great a liability as one might think. First of all, as noted above, its deviation from unity is small in some cases and buried in the analysis in others. Additionally, as we shall see, the gradient of the chemical potential, and chemical potential differences, are more important determinants of the transport rate than is the chemical potential per se. To replace the activity by concentration in the expression for the chemical potential gradient, it is not necessary that the activity coefficient be unity, but only that it be uniform, a less strict condition that is more closely met in biological systems. This can be seen by taking the differential of the activity term in Eq. (1.43); if the activity coefficient is uniform, the second term in the rightmost side of the equation that follows is zero. d ( RT ln ai ) RT

dc dai dJ ¬ RT i i . ci ai J i ®

(1.45a)

Similarly, the contribution of the activity term in Eq. (1.43) to the difference between the chemical potentials in Phases I and II is RT ln aiI RT ln aiII RT ln(ciI J iI ) RT ln(ciII J iII )

RT ln ciI RT ln ciII RT ln

J iI . J iII

(1.45b)

Here, activity can be replaced by concentration if the activity coefficients are the same in both phases.

1.2. ION EQUILIBRIUM ACROSS MEMBRANES An important application of the concepts developed in the previous section is the description of ionic equilibria. The two such equilibria of greatest biological importance are presented below. The first of these is the Nernst equilibrium, which defines the potential difference across a membrane such that a permeable ion, present on both sides, is at equilibrium. This is followed by a description of the Donnan equilibrium, which arises when two phases are in equilibrium and one or more of the charged species in one phase cannot enter the other phase.

1.2.1. The Nernst Equilibrium The Nernst equilibrium describes the conditions under which an ion is in equilibrium between two phases. For discussion purposes, we will imagine that the two phases are separated by a membrane, and that the membrane is permeable to the ion. The equilibrium condition is given by Eq. (1.16): P iI P iII

(1.16)


15

In most considerations of ionic equilibria in biological systems, the effect of pressure on the equilibrium state can be shown to be very small; in addition, the system is generally isothermal. Under these circumstances, Eq. (1.16) becomes: RT ln aiI zi\ I RT ln aiII zi\ II .

(1.46)

For a nonelectrolyte, zi = 0, and the equilibrium condition given above is satisfied only when the activity of the solute is the same in both phases. This is not the case for an ionic species, since interphase differences in activity can be compensated for by a potential difference across the membrane. The required potential difference for ionic equilibrium is found by rearranging Eq. (1.46): '\

RT aiII , ln zi aiI

(1.47)

where the ' operator denotes the difference between the values of the operand in the I II two phases; e.g., '\ = \ – \ . Equation (1.47) is the Nernst equation, and the potential difference it defines is the Nernst potential of the ith ion, Ei. Each ion in solution has its own Nernst potential, which is given by Eq. (1.47) and depends on the activity of the ion in each phase. When a membrane separates two ionic solutions whose compositions differ, a potential ordinarily develops across the barrier. The factors that determine this potential difference will be discussed in later chapters; for now, it is sufficient to note that the Nernst potential of any given ion in such a system will not in general be equal to the potential across the membrane. Only those ions whose Nernst potential equals the membrane potential are at equilibrium.

1.2.2. Origin of the Nernst Potential To illustrate the use of the Nernst equation, consider two electrolyte solutions separated by a membrane that is permeable to only a single ion, P. The concentrations of the permeant in the two solutions are different. The potential difference between the two solutions is measured by a high impedance device, so no electrical current flows in the external circuit (Fig. 1.4). What potential difference will be measured when the permeant is in equilibrium across the membrane? Since no electrical current passes from one solution to the other through the external circuit, the current through the membrane must also be zero; otherwise, the solutions would accumulate charge, one becoming increasingly positive, and the other becoming increasingly negative. While the current through the external circuit would be carried by electrons, the charges that move across membranes are borne by ions. A general expression for the membrane current is: I zi J i , i

(1.48)

16


Figure 1.4. The Nernst potential. An electrolyte, PA, is dissolved in Phases I and II. There is no current through the external circuit. With the proper electrodes, the voltmeter measures the memI II brane potential \ – \ , which is equal to the Nernst potential of P. The accumulation of charge that gives rise to the Nernst potential when aPII > aPI is shown in the figure.

2

where Ji is the flux of the ith species across the membrane, in mols/cm -s, and I, the 2 current density, is in amps/cm . In the present application, only P can cross the membrane, so I = 0 implies JP = 0. For JP to be zero, the membrane potential '\ must be equal to the Nernst potential of P; then P will be at equilibrium and have no cause to cross the membrane. Thus,

'\ = EP

RT aPII ln . zP aPI

(1.49)

How does the membrane potential arise? For illustrative purposes, assume that P is a cation and that its activity is higher in Phase II than in Phase I. The membrane is sitting in an empty chamber, and suddenly the solutions are added to each side. At that instant, there is no potential difference across the membrane, and P flows from Phase II to Phase I because its activity is higher in Phase II. Since there is no current in the external circuit, Phase II, which has lost some cation, develops a small negative charge, and Phase I correspondingly becomes positively charged. The membrane has become a charge separator — a capacitor. As such, it supports a potential that de-


17

pends on the amount of charge transferred and the membrane capacitance. This potential difference, with Phase I positive with respect to Phase II, opposes the further transport of positively charged P from Phase II to Phase I. The flow of P across the membrane ceases when the membrane potential reaches EP. The amount of ion necessary to establish typical Nernst potentials (up to ca. 100 mV) across biological 2 membranes, whose capacitances are typically of the order of 1-2 Pfarads/cm , is very –7 2 –12 2 small, ca. 10 coul/cm or 10 mols/cm .

1.2.3. Specific Ion Electrodes I

II

To continue with the previous example, observe that, if aP is known, then aP can be found by measuring '\. This is the principle on which the use of specific ion electrodes is based. A solution, which is part of the electrode and contains the ion of interest, is separated from the solution to be analyzed by a barrier whose selectivity for the ion is extremely high. No current is permitted to flow between the solutions, so the electrode potential follows the Nernst potential of the selected ion. The pH electrode, which uses a glass barrier permeable to only hydrogen ions, is the best-known example of this kind of sensor. In most cases, the barrier is not perfectly selective, and ions other than the one of interest can also cross, though with more difficulty. As a consequence, the electrode potential no longer follows Eq. (1.49) exactly. Specific ion electrodes are best calibrated before use against solutions whose known composition is similar to that of the solutions to be analyzed. Equation (1.47) is often written in terms of logarithms to the base ten: Ei

a II 2.303 RT log10 iI . zi ai

(1.50)

At 37ºC, the coefficient RT/ = (l.987 cal/mol-K • 3l0 K / 96500 coul/mol) / 0.2389 cal/coul-V = 0.0267 V = 26.7 mV; multiplying by 2.303, the coefficient of the logarithm in Eq. (1.50) is 61.5/zi mV. Thus, if the activities of a monovalent ion on the two sides of a membrane differ by a factor of ten, then the Nernst potential of the ion is 61.5 mV at 37ºC. The Nernst potential is plotted against the activity ratio in Figure 1.5, for univalent and divalent ions.

1.2.4. Activity Coefficient Considerations For the activity ratio in the preceding equations to be well approximated by the conII I I II centration ratio ci /ci , it is necessary only that Ji | Ji ; the solutions need not be ideal. One characteristic of biological systems favors a uniform activity coefficient, and a second opposes it. First, the good news. As shown in the previous section, theories have been developed that describe quite well the activity coefficient in solutions of physiological ionic strength. Since the vast majority of solute particles in biological solutions are ions, and since the vast majority of dissolved ions are monovalent, the ionic strength closely approximates the

18


total number of moles of dissolved material per unit volume. We shall see in Chapter 6 that powerful and potentially destructive osmotic forces arise when the total concentrations of dissolved particles on the two sides of a membrane are too different. Gradients in ionic strength sufficient to cause substantial activity coefficient gradients would also generate substantial osmotic gradients. Indeed, the ionic strength is quite uniform among the fluid phases in the body (though the composition is not!), suggesting that the activity coefficients should be too.

Figure 1.5. Dependence of the Nernst potential on the activity ratio of monovalent and divalent cations and anions at 37ºC.

The bad news was also alluded to in the previous section. Ion binding by charged macromolecules can significantly affect the apparent activity coefficient of the bound solute. Concentration is usually measured by total assay techniques, such as flame photometry or chromatography, which do not distinguish between free and bound ion. The degree of binding — or the reduction in activity coefficient due to cooperative effects — is not always known and can differ markedly from phase to phase, depending primarily on the number and nature of the macromolecules that are present. A common simplification when binding or cooperative effects are present (as, for instance, within a cell) is to regard the ion in question as residing in two distinct “compartments”: a bound compartment in which it is inactive (Jbound = 0) and a free compartment containing a near-ideal solution (Jfree | 1). Such compartments can be real — for instance, an intracellular organelle that preferentially sequesters an ion — or more diffuse, such as when a compartment represents the electrostatic interactions of an ion with multiple macromolecules distributed throughout the interior of the cell. When the distribution of the ion is represented in this way, the intracellular activity can be replaced by the concentration of free ion. Specific ion electrodes can be very useful for partitioning the total ionic content of a cell between the free and bound compartments, since they measure activity directly. The use of specific ion electrodes to determine the bound fraction in a cell is illustrated in one of the problems at the end of this chapter.


19

As indicated earlier, the Nernst equation can be used to determine whether an ion is at equilibrium across a membrane. This assessment is usually made under the assumption that the activity coefficient of the ion is the same in the two phases that bathe the barrier; it will be flawed if the ion is preferentially bound in one of the phases and a total assay is used to measure concentration. The increasing use of intracellular ion-specific microelectrodes has generated a modest increase in the use of activity (rather than concentration) to characterize biological solutions. This is to be welcomed, since activity is the more correct variable for analyzing transport and interphase equilibrium. Of course, concentration is still used to measure the amount of each species in solution; it is particularly useful when the electroneutrality condition enters the analysis, as in the following subsection.

1.2.5. The Donnan Equilibrium The Donnan equilibrium describes a situation in which electrolyte solutions are in equilibrium across a barrier, and one or both of the phases contain charged species that cannot cross it (Fig. 1.6). For instance, two solutions, one of which contains a charged protein, might be separated by a membrane through which only small ions can pass. A charged (polyelectrolyte) gel in saline solution generates a Donnan equilibrium; the ions from outside can enter the gel and equilibrate between the gel and the solution, while the long-chain charged molecules that make up the gel are covalently linked to one another and cannot enter the saline outside. Another example is an electrolyte solution in which is suspended a biological cell whose cell membrane passes ions but not the charged proteins and other macromolecules in the cytoplasm. Consider the system represented in Figure 1.6a. A membrane separates two phases as in earlier figures. Phase II contains only a 1–1 electrolyte, CA, at a concentration cII = cAII = cCII. Both the cation (C) and anion (A) can cross the membrane, and consequently are present in Phase I as well. Phase I contains charged solutes (e.g., proteins) that cannot cross the membrane; the concentration of charge on these imI permeant solutes is zIcI . The Gibbs–Donnan equilibrium condition defines the equilibrium state of this system. I Before proceeding, it is important to emphasize that cI is the concentration of impermeant charge, not the concentration of impermeant molecules. If the concentration of impermeant molecules in Figure 1.6a is 1 mM and each molecule contains ten I I charged groups, then cI = 10 mM. With this definition of cI , zI is always r1, depending on the sign of the impermeant charge. This treatment of the impermeant charge is essential when Phase I is a gel, and the charge is part of the structure itself. In most biological systems, the impermeant molecules present both positive and negative I charges, and cI is the concentration of net charge. As hinted above, the description of the Donnan equilibrium proceeds more simply when concentrations rather than activities are used. Since both C and A can cross the membrane, both ions must satisfy the equilibrium condition (1.16) for the system as a whole to be at equilibrium. Put another way, their Nernst potentials must each equal the membrane potential, '\; from Eq. (1.47),

20


Figure 1.6. The Donnan equilibrium. In (a), I represents a large, multiply charged macromolecule that cannot cross the membrane. In (b), many charges in the gel are part of its structure and cannot I leave; their total concentration is cI . In (c), a cell is immersed in an external bath or extracellular fluid. In all cases, C and A can freely exchange between the phases.

'\

RT c II RT c II ln I ln I . cC | cA

(1.51)

The two expressions on the right yield c II c AI II . cCI c

(1.52)


21

This ratio, or its reciprocal, is called the Donnan ratio. It is easy to show that when Phase II contains multiple monovalent ions, the Donnan ratio is the same for all ions of like charge. Phases I and II are both electrically neutral; that is, indexing the phases by j, zi cij 0 (j = I, II).

(1.53)

i

Note that the electroneutrality condition is properly written in terms of ionic concentrations, not activities, since the number of charges is sought. The electroneutrality condition applied to Phase II allowed us to equate the concentrations of C and A in that phase; its application to Phase I, which contains the charged impermeants, is cCI c AI zI cII 0 .

(1.54)

Given the concentration of the solution in Phase II and that of the impermeant charge in Phase I, Eqs. (1.52) and (1.54) are solved simultaneously for the concentrations of C and A in Phase I. At this point, the simplicity afforded by using concentrations in Eq. (1.52), rather than activities, becomes apparent. The solutions are: cCI

zI cII (cII )2 4(c II )2

c AI

,

2 zI cII (cII )2 4(c II )2 2

.

(1.55)

The Donnan potential, '\D, is the membrane potential at equilibrium; substituting Eq. (1.55) into (1.51), '\ D

I I 2 II 2 RT zI cI (cI ) 4(c ) , ln 2c II

(1.56)

2

where zI = 1 has been used. Figure 1.7 shows the variation of the Donnan ratio and the Donnan potential with ambient concentration divided by the concentration of imII I permeant charge. As c becomes much larger than cI , the Donnan potential approaches zero and the Donnan ratio approaches unity. This is referred to as suppression of the Donnan effect. The Donnan effect is suppressed in certain analytical and I experimental protocols to minimize uncertainties in the result when cI is unknown or varies widely among preparations.

1.3. CHEMICAL EQUILIBRIUM The concepts that we have just developed to describe the equilibrium of a solute between phases can be used in a very similar fashion to define the conditions under which a chemical reaction is at equilibrium. This is done below.

22


Figure 1.7. Donnan ratio cAI / cII = cII / cCI and Donnan potential vs. cII / cII for zI = +1. When zI = –1, the ordinate gives cII / cAI and –'\D .

Any chemical reaction can be written in the following form: vi Ri l vi Pi ,

reactants

(1.57)

products

where Ri is the ith reactant, Pi is the ith product of reaction, and the {Qi} are the stoichiometric coefficients of the reaction, which specify the relative number of molecules of each species participating in the process. For instance, for the simple oxidation of glucose, where one molecule of the sugar yields six molecules of carbon dioxide and six of water, the stoichiometric coefficient of glucose is one and that of water is six. A chemical reaction is at equilibrium when the change in free energy accompanying the conversion of a differential amount of reactant(s) into product(s) is zero. For an isothermal, isobaric system, this change in free energy can be written as sG ¬ dG dni , i sni ®

(1.58)

where the sum includes the reactants (for which dni is negative) and the products (for which dni is positive). From Eq. (1.13), the partial derivative in the preceding equation is simply the electrochemical potential of the ith species: dG P i dni .

(1.59)

i

Before setting dG = 0, we use the stoichiometric coefficients to relate the dni's of the several participants in the reaction to the dn of an arbitrarily selected product, P*, whose stoichiometric coefficient is Q*. Suppose dn* mols of P* are produced; then the corresponding values of dni for the other participants in the reaction are


23

— for products: v ¬ dni i dn * , v * ®

(1.60a)

v ¬ dni i dn * . v * ®

(1.60b)

— and for reactants:

Substituting Eqs. (1.60) into (1.59), dn * ¬ ¯ dG P i (vi ) P i vi ° , ¡ p °± v * ® ¡¢ r

(1.61)

where r and p denote sums over reactants and products, respectively. At equilibrium, dG = 0. Since the coefficient dn*/v* of the second term on the right-hand side of Eq. (1.61) is nonzero: P i vi P i vi 0 . p

(1.62)

r

Using the equations derived earlier in this chapter [Eqs. (1.11) and (1.43)], the electrochemical potentials of the species involved in the reaction can be written in terms of solution variables: electrostatic potential, pressure, activity and temperature. This procedure splits the left-hand side of Eq. (1.62) into four pairs of sums, which must add up to zero: ¬ ¬ vi zi \ vi zi \ viVi P viVi P ® p ® p r r ¬ vi RT ln ai vi RT ln ai p ® r ¬ vi Pi0 vi Pi0 0 . p ® r

(1.63)

Let us consider each pair separately. The first pair can be rewritten as ¬ \ vi zi vi zi . p ® r Since charge is conserved during the reaction, the parenthesized term above (and hence the first pair of sums) is zero. The second pair of sums can be rewritten as: ¬ P viVi viVi . p ® r

24


Here, the parenthesized term measures the volume change accompanying the reaction. At atmospheric pressure and in solution, this contribution to the free energy change can generally be neglected relative to the remaining pairs of sums. The third and fourth pairs remain. At equilibrium, they must in combination equal zero. The fourth pair depends on only temperature and is termed the standard free energy change of the reaction, 'G0. Equating the third pair of sums to the negative of this quantity, and dividing through by RT, vi ln ai vi ln ai p

r

'G 0 . RT

(1.64)

We now use the properties of logarithms to rewrite the left-hand side of Eq. (1.64) in a somewhat different form:

'G 0 ln aivi ln aivi p r RT ln 3 aivi ln 3 aivi p

ln

3 aivi p

3 aivi

r

w ln K a .

(1.65)

r

The quantity Ka is called the equilibrium constant of the reaction. It is an exponential function of the standard free energy change, so it too depends on only temperature. This constant defines the condition that is satisfied by the activities of the reactants and products when the reaction is at equilibrium: 3 aivi p

3 aivi

Ka ,

(1.66)

r

0

where Ka = exp(–'G /RT). Equation (1.66) is the starting point for incorporating chemical equilibria into biological transport models. The equilibrium constant is dimensionless only if the sum of the stoichiometric coefficients of the products equals that of the reactants. This is generally not the case. The inconsistency of Eq. (1.65) with the requirement that the argument of logarithms be dimensionless has the same explanation as that presented earlier when deriving the 0 0 expression for the chemical potential. If the standard state for the {Pi } (and thus 'G ) is 1 M, then the activities (or concentrations, when ideality is assumed) in Eq. (1.66) must be expressed in moles/l.

PROBLEMS: CHAPTER 1 1. According to the van ‘t Hoff Law, written as 'P = RT'c, what is 'P in mm Hg when 'c = 1 mM and T = 37ºC?


25

2. Confirm that the coefficient 2.303RT/ in the Nernst equation is 61.5 mV at 37ºC. 3. At constant temperature, the difference between the electrochemical potential of an ion in two phases is 'P i V i 'P RT ' ln ai zi'\ . In cal/mol, what is the contribution to the electrochemical potential difference from the three terms if Vi I II = 60 cc/mol, 'P = 100 mm Hg, ci = 100 mM, ci = 10 mM, Ji = 0.75 in both phases, T = 310 K, zi = +1, and '\ = 100 mV?

4. A membrane separates two solutions, and an ion is in equilibrium across it. In Phase I, the ion concentration is 10 mM, and in Phase II it is 100 mM. The solutions are ideal and the pressure and temperature are the same on both sides. What + – ++ is the potential in Phase II, relative to that in Phase I, if the ion is Na ? Cl ? Ca ? T = 310 K. 5. The electrolyte concentrations inside a nerve axon (the axoplasm) are 14 mM Na, 140 mM K, and 3.5 mM Cl; the external concentrations are 142, 4, and 103 mM, respectively. At rest, the axoplasm is 90 mV negative with respect to the surroundings at 37ºC. Are any of the ions at equilibrium? Assume ideal solutions. 6. I measure the activities of several ions in the solutions on each side of a membrane, and the potential difference across the membrane. I calculate the Nernst potential of each ion and measure several of the ion fluxes across the membrane. For one of the ions, its Nernst potential is not equal to the membrane potential, but its flux across the membrane is zero. How can that be? 2

7. The capacitance of the axonal membrane is 2 Pfarads/cm . How many mols of 2 charge are needed to establish the resting potential across 1 cm of membrane? 8. A membrane permeable to only anions separates two solutions of NaCl at 37ºC. There is no external electrical contact between the solutions, so the current across I the membrane is zero. The concentrations of the solutions are c = 1 mM, dilute II I enough to be considered ideal, and c = 150 mM. The membrane potential \ – II \ is measured and found to be –125 mV. What is the activity coefficient of the chloride ion on Side II? 9. Two solutions of NaCl at 37ºC are separated by a membrane. In Phase I, cNaCl = 10 mM. In Phase II, cNaCl = 100 mM. The system is at open circuit. The membrane I II potential \ – \ = 61.5 mV. Assume all activity coefficients equal unity. (a) Is either ion at equilibrium? (b) The concentration of NaCl in Phase I is doubled. Does the membrane potential change? - yes (to what?), no (why?), I don't have enough information (what do you need to know?) (c) An amount of KCl is added to the original Phase I, sufficient to cause the potassium concentration to be 5 mM. Does the membrane potential

26


change? — yes (to what?), no (why?), I don't have enough information (what do you need to know?) 10. Using literature values, calculate the ionic strength of extracellular fluid and blood plasma. Using Figure 1.3, compare the activity coefficients of NaCl solutions having these values of ionic strength. 11. I stick an electrode in an epithelial cell and find that its potential is –27 mV at 34ºC, relative to the bathing solution which is ideal and contains 5 mM K. I assay for intracellular potassium and find cK = 109 mM. Potassium is in equilibrium across the cell membrane. (a) What is the apparent intracellular activity coefficient of the ion? (b) Assume the intracellular potassium is in two compartments: in one it is bound and unavailable for exchange, and in the other it is free to exchange and has an activity coefficient of 0.9. How much of the 109 mM is in each compartment? 12. A dialysis bag permeable to sodium and chloride contains a charged impermeant. When the bag is allowed to equilibrate with a solution of 100 mM NaCl at 25ºC, the potential in the bag is 15 mV negative with respect to the saline outside. What are the sign and concentration of impermeant charge in the bag? Assume all solutions are ideal. 13. A dialysis bag permeable to sodium and chloride contains a negatively charged impermeant protein capable of binding cations. It is immersed in a solution of 100 mM NaCl at 25ºC, in which the activity coefficient of each ion is 0.78. Sodium and chloride diffuse into the bag and some of the sodium is bound by the protein, neutralizing those negative sites. All of the chloride in the bag is unbound. The solution in the bag can be regarded as ideal. At equilibrium, the total concentration of Cl in the bag is 50 mM and the total concentration of Na is 150 mM. (a) What is the potential in the bag relative to that outside? (b) What fraction of the sodium in the bag is bound? (c) What fraction of the protein sites are bound to sodium? 14. The connective tissue of the normal cornea (the corneal stroma) contains about 15 mM of nondiffusible negative charge. What are the Donnan potential and Donnan ratio between the corneal stroma and its surroundings when the concenII tration of electrolyte in the surroundings, c = 150 mM? 1.5 M? 15 mM? 1.5 mM? Assume T = 37ºC. 15. I drop a rigid dialysis bag containing an unknown concentration of impermeant into a large volume of NaCl solution of unknown concentration. After the bag has equilibrated with the solution, it is found to contain 194 mM Na and 27 mM Cl. T = 37ºC and all solutions are ideal.


27

(a) What are the concentration and the sign of the charge of the impermeant in the bag? (b) What is the concentration of the sodium chloride solution into which the bag was dropped? (c) What was the potential difference between the solution in the bag and that outside? 16. A negatively charged gel (cI = 50 mM) is in equilibrium with a 150 mM NaCl solution at 37ºC. The NaCl solution is separated from a 30 mM KCl solution by a membrane that is permeable to only chloride. The entire system is open circuited. What is the potential difference between the gel and the 30 mM KCl solution? 17. Suppose I drop a negatively charged gel (cI = 50 mM) into a large amount of solution that contains 150 mM Na, 110 mM Cl, and a large macromolecule that produces a net negative charge of 40 mM and cannot enter the gel. What are the equilibrium concentrations of Na and Cl in the gel? 18. Suppose I have three phases in equilibrium: Phase I contains a negatively I charged impermeant, concentration cI ; Phase II contains a 1–1 electrolyte, conII centration c ; and Phase III contains a positively charged impermeant, concentraIII tion cI . Write the fully simplified expression for the potential difference between Phases I and III. All solutions are ideal. 19. Rederive Equations (1.55) and (1.56) for the case in which the cation is divalent. Go as far as you can — at least far enough to set up the equations to be solved — if the math gets hopeless. 20. Consider a chemical reaction at a cell membrane, where a solute A binds to a carrier X in the membrane. The reaction is A + X UAX. I prepare a mixture of solute and carrier that contains 50 mM A and 60 mM X; when the binding reaction has reached equilibrium, cA = 10 mM, cX = 20 mM, and cAX = 40 mM. That is, 80% of the solute is bound. (a) What is the equilibrium constant for the reaction when concentrations are expressed in M? (b) Using Eq. (1.65), show that the same result will be predicted when a standard state of 1 mM is used.

2 FREE DIFFUSION

INTRODUCTION In this chapter, we consider the simplest of transport processes: the passive diffusion of a solute that occurs when its electrochemical potentials on the two sides of a permeable barrier are different. Indeed, this process is so simple that it fails to represent many aspects of transport in living systems. Nonetheless, it does describe some aspects of biological transport quite well, and it also provides a “base case” whose behavior can be compared against that of more complex transport mechanisms. This chapter is divided into two sections; the first deals with free diffusion of nonelectrolytes, and the second with that of ionic species. The principal property variable determining the flux of a nonelectrolyte is its permeability, a quantity that can in principle be related to the diffusion coefficient of the solute. Electrolyte diffusion in free solution is most rigorously described by classical electrodiffusion theory. The flux equations provided by this theory are very complex, and they have not seen nearly as much use as have approximations to them. Accordingly, emphasis will be placed here on the principles of electrodiffusion, and on the approximate solutions and special cases that are most commonly used. The equations of free diffusion can describe a wide variety of transport phenomena, including steady and unsteady transport processes; processes that can be described in one, two or three dimensions and in a variety of geometries; and processes in which chemical reactions and fluid flow take place simultaneously with diffusion. In this chapter, we will discuss a small subset of these, focusing on the tools that are applied to living systems. Comprehensive discussions of diffusional processes can be found in other texts, such as Crank's (1975) classic text, published thirty years ago and still being reprinted! A more limited set of solutions, but with more consistent biological applicability, can be found in Truskey et al. (2004).

M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_2, © Springer Science+Business Media, LLC 2008

29

30

CH. 2: FREE DIFFUSION

2.1. FREE DIFFUSION OF NONELECTROLYTES The first transport process we will consider is the diffusion of a dissolved nonelectrolyte across a membrane or a similar barrier. The diffusive process is driven by the solute's concentration gradient. For now, the nature of the solvent is not particularly important, and it will usually be understood to be water, which is the most common biological solvent. As will be seen in Chapter 7, much of the material developed below is equally applicable to diffusion through a lipid film, such as the hydrophobic region of a cell membrane. In free diffusion through a membrane, the solute particles move about by random Brownian motion, like that in free solution. The solute flux, which is a measurable and reproducible quantity, is essentially the resultant of these separate motions. Even though the path of a single solute particle cannot be predicted, the consequence of an enormous number of these paths is quite reproducible.

2.1.1.

The Teorell Equation

The flux in free diffusion can be written very simply, in a form proposed by Teorell (1953): Flux = Mobility q Concentration q Driving force.

(2.1)

In the most commonly used units, the flux is the number of mols of solute crossing one square centimeter of membrane per second; it is proportional to the product of the solute mobility, which measures its ease of transport and depends jointly on the barrier/solvent and the solute, as well as the temperature; the solute concentration, which measures the amount of material available to participate in the process; and the driving force for the diffusion of the solute. The choice of a proper driving force is dictated by thermodynamic considerations that we will not examine until Chapter 6; for now, we will rationalize that choice by analogy with electrical phenomena. First, we recall that, when the chemical potential of the solute is the same in the two phases bounding the membrane, the solute is in equilibrium, and its flux across the membrane is zero. An analogous situation occurs in electrical circuits; when there is no electrical potential difference, there is no current flow. When the electrical potentials at two points are different, the potential gradient defines a field, and charged particles move in response to it. The force acting on the charges is the negative of the electrical potential gradient. The analogous driving force for solute flux is the negative of the chemical potential gradient: Driving force = –Ps.

(2.2)

Almost every transport process with which we will be concerned can be described in terms of a single spatial coordinate perpendicular to the plane of the barrier. Calling that the x-direction, the driving force becomes:


31

Driving force =

dP s . dx

(2.3)

The implicit assumption in this one-dimensional treatment of gradients and fluxes is that these vectors are oriented perpendicular to the membrane plane and have negligible components parallel to that plane. This is reasonable if the extent of the membrane is much larger than its thickness, as is usually the case. The Teorell equation can now be written:

dP ¬ J s U s cs s , dx ®

(2.4)

where Us and cs are the solute mobility and concentration, respectively. The flux Js is positive in the direction of increasing x. An integral driving force can also be defined, by integrating Eq. (2.3) across the membrane: a

Integral driving force ¨ 0

dP s dx P Is P IIs . dx

(2.5)

In Eq. (2.5), a is the thickness of the membrane. Phase I bathes the face of the membrane at x = 0, and Phase II the face at x = a. From Chapter 1, the integral driving force is zero at equilibrium. The integral driving force would appear to be far more convenient than the differential driving force [given by Eq. (2.3)] for describing transport, because it is based on the chemical potentials in the two phases external to the membrane. Chemical potentials inside the membrane, which must be known to find the local differential driving force, are generally unmeasurable. Fortunately, with a few reasonable assumptions, Eq. (2.4) can be integrated to give an expression that relates the transmembrane flux to the conditions in the ambient solutions. This we now do.

2.1.2. Integration of the Teorell Equation; Fick's First Law; Solute Permeability In integrating Eq. (2.4), the temperature is assumed to be uniform and the effect of ressure on the chemical potential of the solute is neglected; these are quite reasonable assumptions for the systems with which we will be dealing. If, in addition, the solutions are assumed to be ideal, then the chemical potential can be written very simply as μs = constant + RT ln cs.

(2.6)

Differentiating with respect to x,

1 dcs ¬ dP s d ln cs . RT RT cs dx ® dx dx

(2.7)

32


Substituting Eq. (2.7) into (2.4), J s U s RT

dcs . dx

(2.8)

The solute diffusion coefficient Ds is related to the solute mobility through the Nernst– Einstein relation, Ds = UsRT. Ds is often referred to as the binary diffusion coefficient (denoted Dij), as a reminder that its value depends on the identities of both solute and solvent. Introducing the diffusion coefficient into Eq. (2.8), we obtain: J s Ds

dcs . dx

(2.9)

Equation (2.9) is known as Fick's first law of diffusion. Note that the flux is positive if the concentration gradient is negative. In the preceding derivation, it was assumed that Eq. (2.6) holds within the membrane, as though transport proceeded through aqueous pores in which the dependence of chemical potential on solution properties was identical to that in the aqueous solutions at the membrane faces. This is the first of several derivations in which the expressions for chemical or electrochemical potential in free solution will be used to describe the thermodynamic state of solute or solvent inside a transport barrier. The state of solutes and solvent inside a complex, heterogeneous biological barrier is not so neatly defined. Accordingly, it is convenient to think of cs(x) [and \(x) when describing electrolyte transport] as the concentration (and potential) of a free aqueous solution in equilibrium with a thin membrane slice at x. The concentration and potential of this equilibrium solution can be quite different from that of the true solution phase at that point in the membrane; however, since the two phases are defined to be in equilibrium, the chemical potentials of the solute and solvent are the same in each. A notable difference between the concentration of such an equilibrium aqueous solution and the true intramembrane solute concentration arises when the solubility of the solute in the membrane is different from that in the ambient aqueous phases. Such is the case for diffusion through the lipid bilayer of the cell membrane. The relationship between the solute concentration in the lipid and in an equilibrium aqueous solution is expressed in terms of the partition coefficient of the solute between the two phases. Diffusion through lipid layers will be described in Chapter 7.

Fick's first law assumes a somewhat more complicated form when the solutions are nonideal. In that case, the solute chemical potential must be written in terms of activity. The activity, in turn, is the product of the concentration and the activity coefficient. Thus, Eq. (2.7) is replaced by: d ln cs d ln J s ¬ dP s d ln as RT RT . dx dx dx dx ®

(2.10)

For nonelectrolytes, the activity coefficient of the solute can be assumed to depend on only cs, which in turn is a function of x. Thus, the following substitution can be made: d ln J s d ln J s d ln cs ¸ . dx d ln cs dx

(2.11)


33

Substituting Eq. (2.11) into (2.10): 1 dcs ¬ dP s 1 d ln J s ¬ . RT dx d ln cs ® cs dx ® The Teorell equation then becomes d ln J s ¬ dcs J s U s RT 1 . d ln cs ® dx

(2.13)

Define an augmented diffusion coefficient Ds* by d ln J s ¬ . D*s Ds 1 d ln cs ®

(2.14)

For an ideal solution, Ds* = Ds. By substituting Eq. (2.14) into (2.13), a flux equation is obtained that looks almost identical to Eq. (2.9), and can be regarded as a generalization of Fick's first law to nonideal solutions: dc J s Ds* s . dx

(2.15)

Fick's first law, as generalized above, is now integrated across the membrane to yield an expression for flux in terms of the transmembrane concentration difference. To set up the integration, Eq. (2.15) is rewritten as J s dx Ds* dcs .

(2.16)

In the steady state, the solute flux is independent of x. Assume that the same is true of Ds*; Eq. (2.16) can then be integrated across the membrane and solved for Js:

Js

II I D* s (cs cs ) . a

(2.17)

The flux in Eq. (2.17) is based on a unit area of membrane, so it can be continuous at the interfaces x = 0 and x = a only if the entire cross-section of the barrier is available for transport. Furthermore, the assumption that the expressions for chemical potential as a function of concentration are the same in both barrier and bath implies that the solute diffuses through the same solvent as that in the ambient phases. The only barrier for which these assumptions hold would be a thin stagnant water film somehow maintained between two well-stirred aqueous baths. The solute permeability of such a thin film is defined as the solute flux per unit concentration difference: ks0

J s0 J s0 Ds* , a csI csII 'cs

(2.18)

34


where we have used the superscript “0” to indicate that diffusion takes place through a thin aqueous film. The form of Eq. (2.18) has been adopted to describe solute transport in biological systems. For such systems, the barrier is not a thin aqueous film, and the permeability is not given by Ds*/a. The solute permeability of a biological barrier is in general an experimental property, obtained by dividing the measured flux of a solute by its transmembrane concentration difference: ks

J sa , 'cs

(2.19)

where J sa is the measured flux. Radiolabeled tracers are often used to measure permeability; the numerator and denominator of the right-hand side of Eq. (2.19) are replaced by the tracer flux and the transmembrane difference in tracer activity. Even when the solute does not cross the membrane by free diffusion, the experimental permeability is descriptive of the transport behavior of the system. Such empirical permeabilities, though not always easy to interpret in physical terms, are nonetheless useful for comparing solute transport rates and for predicting fluxes under similar conditions. There are some cases in which permeability can be estimated from a diffusion coefficient and membrane thickness. If the barrier is a stabilized thin film of a solvent immiscible with water, the permeability of the solute is determined by the partition coefficient, the film thickness and the binary diffusion coefficient of the solute in the solvent that comprises the membrane [see the second paragraph of the note following Eq. (2.9), and Chap. 7]. If the membrane possesses large interstices or pores such that diffusion through them is the same as that in free solution, the permeability is given by MDs*/a, where M is the void fraction in the membrane. Our inability to predict membrane permeability a priori reflects our ignorance of many factors that influence the transport of a given solute through a given membrane. Some of these factors, particularly applicable to transport through water-filled passages, are itemized below: — The void fraction mentioned above, or the fraction of the presented area of a membrane that is occupied by pores, are often unknown. — If the pores are not highly connected, their resistance to diffusion will depend on their tortuosity; if the solute must diffuse down a tortuous path, it will cross more slowly. — Transport depends critically on the diameter of the passage along the length of the diffusion path. The walls of pores give rise to a viscous drag that retards the diffusional process, and the degree of retardation remains significant for pores as large as ten times the solute diameter. This effect, which will be discussed in detail in Chapter 7, becomes greater when the diameter of the pore is closer to that of the solute. When the pore is only slightly larger than the solute, the latter must cross the membrane by single-file diffusion, and the augmented diffusion coefficient is no longer the


35

appropriate measure of solute mobility. Of course, if the pore size is less than the solute size, then the solute does not cross the membrane at all, irrespective its aqueous diffusion coefficient. — In small pores, there is considerable opportunity for chemical and physical interactions between the solute and the walls. Such interactions include adsorption and binding. Furthermore, if the pore walls are charged, Donnan effects can cause the concentration of charged solutes inside the pore to be markedly different from the concentrations in the adjacent solutions. There can also be interactions between the solvent and the walls of the pore, which can cause the effective solvent viscosity to differ from that in free solution, thereby (see §2.1.4), affecting the solute diffusion coefficient. Once the permeability is specified, the transmembrane flux is predicted by Js = ks'cs.

(2.20)

Equation (2.20) is the expression most commonly used to describe the passive free diffusion of a nonelectrolyte across a barrier. It can also describe the transport of an ion in the absence of an electrical potential gradient. Some illustrative values of solute permeability are given in Table 2.1.

Table 2.1. Nonelectrolyte Permeabilities of Three Cell Membranes a

Ox erythrocyte Urea Glycol Diethylene glycol Glycerol

cm/s 7.8 q 10–5 2.1 q 10–6 7.5 q 10–7 1.7 q 10–8 b

Ehrlich ascites tumor cell (mouse) cm/s Galactose 2.1 q 10–6 3-methyl glucose 1.8 q 10–6 Sorbose 5 q 10–7 Glucosamine 2.5 q 10–7 Human erythrocyte Ethanol Glycerol Thiourea Urea a

c

cm/s 2.1 q 10–3 1.6 q 10–7 1.1 q 10–6 7.7 q 10–7

Davson and Danielli (1952). Crane et al. (1957). c From summary in Lieb and Stein (1986). b

36


2.1.3. Unstirred Layers I

II

Assume that cs > cs ; by our conventions, 'cs > 0, Js > 0, and solute moves from Phase I to Phase II. Before a solute molecule can cross the membrane, it must first find its way from the bulk of Phase I to the membrane interface at x = 0. Two mechanisms are available to accomplish this: diffusion, which is described by equations like those given above, and convection, in which fluid motion carries solute from the main body of the phase to the proximity of the membrane. If one of the phases is the inside of a cell, convection is limited, and exchange between the bulk of the cytoplasm and the membrane interface is largely by diffusion. Similarly, diffusion is usually the dominant transport mechanism in the extracellular space on the other side of the cell membrane. In many experimental situations, however, convection can be introduced by stirring. The influence of stirring does not extend unattenuated to the membrane–solution interface; a thin, effectively unstirred layer adjacent to the membrane remains. Solute crosses this layer only by diffusion, and its flux is properly defined by Eq. (2.18); at x = 0,

Js

DsI I (csb csI ) , I G

(2.21) i

i

where Ds is the solute diffusion coefficient in the ith phase, G is the thickness of the corresponding unstirred layer, and the subscript “sb” denotes the bulk phase conceni tration of the solute; cs is the solute concentration at the interface between the membrane and the ith phase. For nonideal solutions, the augmented diffusion coefficient would be used in place of Ds. Similar considerations apply to the transport of solute from x = a to the bulk of Phase II. The solute concentration profile is shown in Figure 2.1. Three resistances in series separate the two bulk phases. The solute flux is equal to the overall concentration difference divided by the sum of these resistances, each of which is inversely proportional to a permeability: Js

csbI csbII . GI / DsI 1/ ks GII / DsII

(2.22)

Here, ks is the true permeability of the membrane. The apparent permeability of the membrane, Js /'csb, is the reciprocal of the denominator in the preceding equation. The characterization of the unstirred layer (or diffusion layer, as it is also known) as a sharply defined boundary layer containing all of the diffusional resistance outside the membrane is clearly an approximation. When the bounding phase is unstirred, there are concentration gradients throughout. In the presence of stirring, convective effects are absent at the membrane–solution interface and increase with distance from the membrane surface. Notwithstanding the limitations of the unstirred layer concept, it does provide a convenient means for including diffusional resistances outside the membrane in the equation for solute flux, and for characterizing the magnitude of such resistances. The quantity Gi can be regarded as the thickness of a layer of the external phase whose resistance to diffusion is the same as that actually present outside the membrane.


37

Figure 2.1. Concentration profile in the presence of unstirred layers, and in the absence of solvent I II flow. The actual transmembrane concentration difference, cs – cs , is less than the overall concenI II tration difference, csb – csb .

It can be seen from Eq. (2.21) that cs approaches csb as the thickness of the unstirred layer approaches zero; otherwise, the solute flux would become infinite. When these two concentrations are assumed to be identical (an assumption that is often made in practice, and will be made liberally in the chapters to follow), the phase is said to be well-stirred. Although vigorous stirring can reduce the effective thickness of the unstirred layer, it cannot be reduced to zero; the well-stirred assumption is always an approximation. The effects of a variety of stirring motions on solute flux are analyzed in Pedley (1983). In the presence of unstirred layers, the concentration difference driving the transmembrane flux is less than the difference between the bulk phase concentrations (see Fig. 2.1). Solute permeabilities calculated using the latter driving force can be seriously underestimated if the resistance of the unstirred layers is an important fraction of the total interphase resistance. This is more likely to be the case if the membrane permeability is high. The neglect of unstirred layer effects can also lead to errors in the calculated parameters of carrier-based transport systems (Chap. 4). The diffusion coefficients of small solutes in the cytoplasm are not known very well, so it is difficult to make good estimates of the errors in cell membrane permeability caused by intracellular diffusional resistance. In such cases, it is common to

38


assume that there is no diffusional resistance on the cytoplasmic side. The true cell membrane permeability is underestimated when this approach is used; however, permeabilities that are derived in this way can be compared with the permeabilities of other solutes derived similarly, or used to predict flux, as long as the bulk cytoplasmic concentration of the solute is used in the flux equation. We will see in Chapter 6 that, for many solute/membrane combinations, a transmembrane concentration difference induces a solvent flow, termed osmosis, in the direction of the more concentrated solution. The solute concentration profile in the unstirred layer is curved when osmosis (or any transmembrane solvent flow) is present. The effect of unstirred layers on transport in the presence of osmosis is discussed in Chapter 10. Equation (2.22) describes the steady-state transport of solute across a series of resistances, for the case in which two of the resistances are unstirred layers and the third is the membrane itself. The G/Ds ratios in the equation are simply the reciprocals of the permeabilities of the individual unstirred layers. Equation (2.22) can be regarded as a transport equivalent of Ohm's Law for the voltage-driven current through a number of resistors in series; here, the voltage is replaced by the bulk concentration difference, the current by solute flux, and the ohmic resistances by the reciprocals of the permeabilities of each barrier. This description of the flux through series barriers can be applied to many biological transport processes, such as transport though a single layer of cells, where solute enters across one face of the cell, crosses the cytosol, and then exits across the other face; transport through a cell supported by a permeable layer of extracellular matrix; or transport through a series of cell layers, as in epithelia (Chap. 10). In such cases, the general equation for solute flux is Js

'cs , 1 i 1 k si m

(2.23)

where 'cs is the overall concentration difference and ksi is the solute permeability of the ith of m barriers. As above, the reciprocal of the denominator of Eq. (2.23) is the apparent permeability of the composite barrier.

2.1.4. Applications of Solution Theory A considerable body of theory has been developed to describe free diffusion in solution. Most of this theory cannot be directly applied to biological systems, for reasons that have already been presented. One applicable product of solution theory is the Stokes–Einstein equation, which identifies the variables that have the greatest influence on the diffusion coefficient. In general, the diffusion coefficient depends on the solute (naturally), the solvent, the concentration of the solute (or composition, for a multicomponent solution), and temperature. Einstein (1908) used Stokes' Law to derive the following approximate expression for the diffusion coefficient of a spherical solute:


Dx

39

RT , 6 SKsN

(2.24)

where K is solvent viscosity, s is solute radius, and N is Avogadro's number. Stokes' Law describes the drag on a sphere moving through a homogeneous fluid of infinite extent. Implicit in this application of Stokes' Law are the assumptions that solute molecules are much larger than those of the solvent, and that the influence of the solution boundaries (e.g., the walls of a pore) is negligible. Equation (2.24) shows that the most important solute property affecting the diffusion coefficient is its size (and shape; the equation is more complex for nonspherical solutes), and the most important solvent property is its viscosity. Equation (2.24) predicts that the diffusion coefficient is inversely proportional to the solute radius; that is, the sD product is constant. This condition is met by the data in Table 2.2, even though the solute molecules are not much larger than those of the solvent. In biological systems, this simple inverse relation applies only to diffusion through large passages. When the size of the pore is not much greater than that of the solute, the permeability depends on pore radius as well as solute radius. The effect of pore size on solute permeability will be discussed in Chapter 7.

Table 2.2. Test of the Stokes–Einstein Equation

a

Solute

Solute a radius, s nm

Diffusion coefficient in aqueous solution a at 25ºC, D 2 cm /s

sD q 105 nm-cm2/s

Methanol Urea Glucose Glycerol Sucrose Raffinose

0.20 0.24 0.39 0.31 0.45 0.58

1.3 q 10–5 1.16 q 10–5 –6 6.8 q 10 8.3 q 10–6 5.5 q 10–6 4.2 q 10–6

0.26 0.28 0.26 0.27 0.25 0.24

Data from Schafer and Barfuss (1980).

The predicted effect of solvent viscosity on the diffusion coefficient has often been used to interpret and extrapolate experimental permeability data. From the Stokes–Einstein equation, the diffusion coefficient is expected to vary inversely with solvent viscosity. If the temperature dependence of permeability parallels that of the reciprocal of the viscosity of water, this is taken as evidence that the solute crosses the membrane via water-filled pores. The permeabilities of other diffusional transport routes (e.g., across the lipid phase of the cell membrane) are considerably more sensitive to temperature than is the permeability of an aqueous pore. Similarly, if it is known that a solute uses an aqueous pore to cross a membrane, then the temperature

40


dependence of the viscosity of water can be used to predict the solute permeability at one temperature from the measured permeability at a different temperature.

2.1.5. Fick's Second Law and Convective Diffusion Fick's first law is one of the equations most commonly used to describe biological transport by free diffusion. It can readily be generalized to any coordinate system: Js = –Ds cs.

(2.25)

where Js is the flux vector in three-space. In the steady state, the law of mass conservation applied to the species s is

• Js = 0.

(2.26)

• (Ds cs) = 0.

(2.27)

Substituting Eq. (2.25) into (2.26),

Equation (2.27) is the steady-state form of Fick's second law of diffusion, also known as the diffusion equation. When the diffusion coefficient is uniform, the equation simplifies further to 2

cs = 0.

(2.28)

The diffusion equation has been solved in numerous geometries, for a wide variety of boundary conditions. Table 2.3 summarizes some useful forms of the steady-state diffusion equation.

Table 2.3. Some Forms of the Steady-State Diffusion Equation 1. Cartesian coordinates (x,y,z) d 2c a) 1-dimensional: Ds 2s 0 dx s 2 cs s 2 cs s 2 cs ¬ b) 3-dimensional: Ds 2 2 2 0 sx sy sz ® 2.

Cylindrical coordinates (r =radial coordinate, z =longitudinal coordinate, no azimuthal variation) Ds d dcs ¬ a) r-variation only: 0 r r dr dr ® b) r- and z-variation, different diffusion coefficients in r- and z-directions: Dsr s scs ¬ s2c Dsz 2s 0 r r sr sr ® sz

3. Spherical coordinates, r-variation only:

Ds d 2 dcs ¬ 0 r r 2 dr dr ®


41

An additional contribution to the solute flux arises if the solution itself is moving. Let us return to the one-dimensional case described by Eq. (2.9). If the solution is flowing in the x-direction at a velocity v, the solute flux is augmented by a convective term equal to the product of the solution velocity and the local concentration: J s Ds

dcs vcs . dx

(2.29)

The three-dimensional equivalent of Eq. (2.29) is Js = –Ds cs + csv,

(2.30a)

where v is now a vector. Substituting Eq. (2.30a) into Eq. (2.26), the equation for steady-state convective diffusion with a uniform diffusion coefficient becomes Ds 2 cs v c and UC >UA. Then the salt diffuses from Phase II to Phase I under its concentration gradient, and the potential of Phase I becomes positive relative to that of Phase II (i.e., '\ > 0), so as to retard C and increase the driving force for A. When UC = UA, there is no mobility difference to compensate for, and the diffusion potential is zero. Diffusion potentials can cause artifacts in certain electrophysiological experiments, and it is desirable to avoid them. Much use is made of concentrated KCl solutions (salt bridges) in such setups, because the mobilities of potassium and chloride are almost identical. Consider the other extreme, in which the mobility of one ion is much larger than that of the other, say UC >> UA. In this case, the membrane potential given by Eq. II I (2.42) approaches (RT/) ln c /c . Since there is only a single electrolyte in the system, II I the argument of the logarithm is also cC /cC ; thus the membrane potential approaches the Nernst potential of the cation. We shall see that the tendency of the membrane potential to approach the Nernst potential of the more (or most) permeable ion is evident for more complex electrolyte solutions as well. This tendency has been exploited experimentally to “clamp” the membrane potential at a selected value by bathing the two sides of the membrane with solutions containing different concentrations of an ion (often potassium) to which the membrane is particularly permeable. Active and Passive Exchange with a Closed Compartment. In the steady state, the net rate of entry of any species into a closed compartment equals the rate at which it is consumed; otherwise, its concentration in the compartment would change with time. Similarly, when a compound is synthesized in a closed compartment, the synthesis rate (less any consumption of the material inside the compartment) equals the rate at which the substance leaves the compartment. When the solute is neither consumed nor produced within the compartment, its net entry rate must be zero in the steady state. This is the case for most ions. As noted earlier, cell membranes are capable of actively transporting (“pumping”) ions between the interior of a cell and the extracellular fluid. Suppose that two cationic species with the same valence, z, are exchanged across the cell membrane, such that, for each ion of Species 1 that is pumped from Phase II to Phase I, r ions of Species 2 are pumped from Phase I to Phase II. In the steady state, the net rate at which each ion crosses the cell membrane — the active flux plus the passive flux — must be zero. Hence the passive flux of each ion is the negative of its active flux. Thus, for each ion of Species I moving passively from Phase I to Phase II, r ions of Species 2 move passively from Phase II to Phase 1:

J2 = –rJ1,

(2.43).

where r is the coupling ratio or coupling coefficient of the pump. Substituting Eq. (2.43) into Eq. (2.36), and rearranging,

U 2 RT Solving for d\/dx,

dc2 dc d\ d\ U 2 c2 z rU1 RT 1 rU1c1 z . dx dx dx dx

(2.44)


53

¬ rU dc1 U 2 dc2 d\ RT 1 dx dx . dx z rU1c1 U 2 c2 ®

(2.45)

The function in the parentheses on the right-hand side is equal to

d (rU1c1 U 2 c2 ) d ln(rU1c1 U 2 c2 ) dx ; rU1c1 U 2 c2 dx hence,

d\

RT d < ln(rU1c1 U 2 c2 ) > . z

(2.46)

Integrating across the membrane, and letting z = 1,

'\

RT U 2 c2II rU1c1II . ln U 2 c2I rU1c1I

(2.47)

This equation, which is obtained directly from the electrodiffusion equation, was used by Mullins and Noda (1963) to relate the membrane potential of skeletal muscle to the stoichiometry of active Na–K exchange across the muscle fiber membrane. Equal Total Concentrations on the Two Sides of the Membrane: The Constant-Field Equation. Even though the concentrations of individual ions vary widely in the body, the total ionic concentration, N, is quite uniform throughout (Table 2.5). The intracellular ionic content is only 10% less than the extracellular value, so the I II solution of the electrodiffusion equation for N = N is of some interest. This solution is also simpler — and much more frequently used — than the more general solutions given earlier.

Table 2.5. Typical Ionic Content of Intracellular and Interstitial Fluids, and Blood Plasmaa

a b

Ion

Plasma

Na K Ca Mg Cl HCO3 Phosphates SO4 Lactate Total

142 4 1 1 108 24 2 Rb | K > Na > Li K > Rb > Cs > Na Cs > K > Na > Li Na | Li > K K > Rb >> Na | Li

Gramicidin A Alamethicin Acetylcholine receptor, rat Sodium channel, squid axon Potassium channel, squid axon

The inequalities indicate the relative magnitudes of the ion permeabilities in the channel.

Table 4.2. Properties of the Alkali Ions Ion Lithium Sodium Potassium Rubidium Cesium a

Crystal radius, Åa

Hydrated radius, Å

ki/kKb

0.60 0.95 1.33 1.48 1.69

2.31 1.78 1.22 1.18 1.16

0.09 0.07 1 0.74 0.18

1 Å = 1 Ångstrom = 10–10 m = 0.1 nm.

b

Potassium channel in snail neuron (Reuter and Stevens, 1980). Other potassiumselective channels exhibit similar sequences. The sequence here is: K > Rb > Cs > Li > Na.

A review by Diamond and Wright (1969) summarized 15 different permeability sequences that had been observed for the five alkali ions in biological systems. It is hard to imagine, at first, how so many sequences could exist. If the channel is regarded as simply a narrow water-filled pore, one would expect that the smallest ion


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would have the highest permeability, and that the permeability of larger ions would be less. The permeability of ions too large to fit through the pore would naturally be zero. However, we see from Table 4.2 that the relative size of the ions depends on whether one considers the ion in its crystalline form or in aqueous solution. Alkali ions of small atomic number (e.g., Li) have very small crystal radii, but when present in water, they interact more strongly with the water dipoles than do larger members of the family. The range and influence of these dynamic interactions increases as the ionic size decreases, and this is seen in the inverse relationship between atomic number and hydrated radius in the table. The Stokes–Einstein relation [Eq. (2.24)] predicts that the ionic mobility in solution is inversely related to the hydrated radius, and indeed the mobility of the alkali ions increases with atomic number. The interaction represented by the hydrated radius influence the ability of ions to pass through pores, since the cloud of water molecules that surrounds them may need to deform or be stripped away for the ion to get through. In such an event, the permeability sequence would depend on the energy cost of deforming or removing these hydration shells, as the polarized water cloud is known. Furthermore, most channels are lined by charged groups or dipoles (e.g., carboxyl, carbonyl) on the amino acid residues of the channel protein. These give rise to electrostatic forces, which affect, to varying degrees, the permeability of a given ion in the channel. The charges are opposite in sign to that of the ions for which the channel is selective. For instance, cation-selective channels have negatively charged walls, which can electrostatically and reversibly bind cations, or otherwise influence their rate of diffusion through the pore. In addition, the negative charges in the channel wall can, via the Donnan effect, increase the concentration of cations in the channel while excluding anions. The likelihood of an ion shedding its hydration shell, and the influence of charges or dipoles in the walls of the channel, are greatest when the channel is narrow. When it is wide, the ions need not alter their interactions with solvent water, and they pass through in hydrated form, which is the energetically favored state in aqueous solution. In addition, when the pore diameter is large, most of the ions traversing the channel do so at some distance from the wall, shielded from (and accordingly unaffected by) the charges thereon. There are such channels, which function essentially as perforations in the membrane. In these cases, the permeability sequence follows that of the hydrated radii (but in the opposite order, of course): Li < Na < K < }. This description also applies to some instances of transport through the gaps between cells. In narrow channels, ions are more likely to interact with sites at the wall, in some cases hopping from one site to the next as they work their way down the pore. This association–dissociation process is random in the same sense as free diffusion: not every hop is toward the trans side of the membrane, even though the net flux is so directed. Hopping from site to site can increase the mobility of an ion beyond its value in larger pores or in free solution.

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CH. 4: FACILITATED DIFFUSION: CHANNELS AND CARRIERS

4.1.3. Energetics of Ion Selectivity, and Steric Effects The strength of the interaction between binding sites and ions is greater when the interacting charges can approach more closely; that is, when the ions are unhydrated, and when they are small. When the pore is so narrow that ions cannot enter without shedding or significantly attenuating their hydration shells, binding forces become predominant. Two energy differences are involved in this process: — The energy of hydration, 'Eh = Ehydrated ion – Eunhydrated ion. Since the hydrated state is energetically favored, 'Eh is negative. — The energy of binding the unhydrated ion to a site in the channel, 'Eb = Ebound ion – Eunhydrated ion. For an attractive binding force, 'Eb is also negative. Here, we have described the bound state as one in which the ion is “unhydrated,” implying that there are no dipole interactions between the bound ion and the solvent. In fact, ions can experience such interactions at some level throughout their passage through the channel. Thus, Eunhydrated ion is better regarded as the energy of the ion when it has entered the pore and its interaction with solvent is consequently reduced. We will continue to use the term “unhydrated” for simplicity.

Neglect the difference between the energy of the unhydrated ion in the ambient phase and in the pore; then, the energy change associated with the movement of a hydrated ion to the mouth of a pore, its dehydration, and its subsequent binding to a site on the wall, is 'Et = Ebound ion – Ehydrated ion = 'Eb – 'Eh.

(4.1)

This energy difference determines the ease with which an ion can take advantage of the binding site to enter and pass through the channel. The variation of this quantity from ion to ion is closely related to the permeability sequence in narrow channels. Now we can begin to see how the selectivity of a charged narrow channel may not be so predictable. Both ''s on the right-hand side of Eq. (4.1) have the same sign. Since electrostatic forces are responsible for both hydration and binding, both ''s are more negative for the smaller ions. It is not immediately clear how their difference might vary with ion size. The energy of hydration is a property of only the ion, but the energy of binding depends in addition on the field strength of the binding site. Different field strengths can lead to different selectivity sequences, as can be seen from the data in Table 4.3. In Case I, the (magnitude of the) hydration energy of the light ion is three units greater than that of the heavy ion, but the binding energy is also three units larger, so the transition energies ('Et) of the two ions are the same. When the field is stronger (Case II), the binding energies increase proportionately, while the hydration energies remain the same, and the transition energy now favors the binding and permeation of the lighter ion. Case III shows the opposite result of a reduction in field strength.


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Table 4.3. How Field Strength Can Affect Selectivity Case Field strength 'Eb (arbitrary units) Light ion Heavy ion 'Eh Light ion Heavy ion 'E t Light ion Heavy ion Selectivity:

I

II

III

Medium

High

Low

–12 –9

–16 –12

–8 –6

–5 –2

–5 –2

–5 –2

–7 –7 Comparable

–11 –10 Light ion favored

–3 –4 Heavy ion favored

The transition from a hydrated ion in the ambient solution to an unhydrated ion entering the pore to a bound unhydrated ion can be represented by an energy diagram. Figure 4.1 is a graphical representation of the data in Table 4.3. This explanation of the variability in the selectivity sequences of alkali ions was pioneered by Eisenman (1961), who proposed on theoretical grounds the existence of the eleven sequences in Table 4.4. Note that each of the cations can be the most permeable, if the field strength is in the appropriate range. Eisenman's theory has been shown to be capable of explaining most of the 15 alkali ion permeability sequences referred to earlier.

Table 4.4. Eisenman Selectivity Sequences Li> Na > K > Rb > Cs Na> Li > K > Rb > Cs Na > K > Li > Rb > Cs Na > K > Rb > Li > Cs Na > K > Rb > Cs > Li K > Na > Rb > Cs > Li K > Rb > Na > Cs > Li K > Rb > Cs > Na > Li Rb > K > Cs > Na > Li Rb > Cs > K > Na > Li Cs > Rb > K > Na > Li

Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence

XI (at highest field strength) X IX VIII VII VI V IV III II I (at lowest field strength)

Other factors affecting ion selectivity are related to the size of the hydrated or unhydrated ions, and are called steric effects. Two of these are described below: 1. The selectivity of channels between anions and cations is often conferred by a collection of charged sites near the mouth of the channel; negatively

118


charged sites promote cation passage, and positively charged sites favor anion transport. A small ion whose charge is of the "preferred" sign may have such a large hydration shell that it cannot get close to the sites at the mouth of the channel. The associated water molecules get in the way and screen the charge on the site. Such an ion is less likely to enter the pore. 2. If the ion cannot fit through the pore in its unhydrated form, its permeability will be zero. In terms of the energetics described above, it is not correct in this case to assume that the unhydrated ion has the same energy inside and outside the channel.

Figure 4.1. Energy diagram showing how field strength can affect selectivity. For each ion, the energy is relative to that of the hydrated ion in solution. The energy required to dehydrate Ion #1 (–'Eh1) is greater than that required to dehydrate Ion #2 (–'Eh2). The energies of binding for the two ions (–'Ebi, i = 1, 2) depend on field strength; when the field strength is low and –'Ebi = %Ebil , binding favors Ion #2, and when the field strength is high and –'Ebi = %Ebih , binding favors Ion #1.


119

Figure 4.2. Factors affecting ionic selectivity. The curves on the figure represent the variation of ionic diameters with atomic number Z. A point on the Z–d plane corresponds to the transport of an ion whose atomic number is Z through a pore whose diameter is d. The plane has four regions. In Region A, the pore size is less than the diameter of the unhydrated ion, which is therefore too large to pass through the pore. In Region B, the pore size is less than twice the diameter of the unhydrated ion; the ions cannot pass each other in the pore, and cross in single file; this is a saturable transport mode. In Region C, the pore size is greater than that of the hydrated ion and the ions pass through the pore in hydrated form. In Region D, the ion must shed its hydration shell to pass, the permeability depends on field strength, and there can be a steric problem when the ion is small (hydrated diameter is large) and the pore is relatively narrow. The numbers on the horizontal cuts correspond to the numbers in the text.

4.1.4. Ion Selectivity of Channels: Summary The factors affecting the selectivity sequence of a channel for a class of ions (e.g., the alkali ions) are illustrated in Figure 4.2. The ordinate is the size of the pore at its narrowest point (its selectivity filter, which will be discussed subsequently), at which polar groups afford additional selectivity with respect to the sign of the ionic charge. The three curves in the figure represent the variation with atomic number of the ions' unhydrated diameters, twice their unhydrated diameters, and their hydrated diameters. Figure 4.2 shows how the selectivity sequence of a pore can depend on its diameter. Consider, for instance, the four horizontal cuts shown in the figure: 1. When the pore is very narrow, all but the smallest unhydrated ions are too large to cross the pore, which is highly selective for light ions. 2. For a larger pore, the lightest ion is sterically prevented by its hydration shell from nearing the closest binding sites at the pore mouth; the heaviest ions are still too large to fit; and the pore is selective for midsized ions, which are too large to pass each other inside the pore, and therefore cross single-file. 3. The selectivity sequence of the next largest pore depends on the field strength of the binding sites. 4. The permeability sequence of the largest pore is determined by the hydrated diameters of the ions, as described earlier.

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4.1.5. The Structure of Ion Channels: Selectivity Filters, Gates, and Energy Profiles. Figure 4.3 shows several ion channels at different levels of detail. The simplest such channel is formed by a molecule that has a pore down its center, oriented in the membrane thickness direction. The most well-studied channel of this kind, Gramicidin A, forms a pore lined with polar amino acid residues that cause the aqueous channel to be selective for cations. A sketch of the channel, which is comprised of two gramicidin monomers end to end, is shown in Figure 4.3a. The molecule is stabilized in the membrane by hydrophobic interactions of the nonpolar protein residues with the lipid bilayer; hydrogen bonds hold the two monomers in apposition within the membrane. Gramicidin is a member of a class of transport facilitators known as ionophores. Ionophores may be synthetic or natural peptides, and include several antibiotics; some, like gramicidin, function as channels, and others as carriers. Most transmembrane channels are oligomers of several protein molecules, all of which cross the membrane. These can be thought of most simply as assemblies of “rods” rather than tubes, arranged with their axes more-or-less parallel and more-orless perpendicular to the plane of the membrane. The pore lies along the axis of symmetry of the channel complex, occupying the space between and surrounded by the rods. The amino acid sequences of these proteins are such that they are stabilized in the membrane by hydrophobic interactions with the bilayer. The individual protein molecules that form the channel are known as the subunits of the channel. Each subunit consists of several helical domains that cross the membrane. Some sodium and calcium channels are made up of a single protein molecule that folds in such a way that it assumes a subunit architecture, although each “subunit” is actually a distinct domain of the same molecule. A cartoon of an oligomeric channel, voltage activated and selective for potassium, is shown in Figure 4.3b; in this channel, a portion of the pore is lined by a polar segment, called the P-loop, which is not part of a transmembrane domain. The P-loop forms the selectivity filter that preferentially passes potassium ions. A sketch of the acetylcholine receptor (AChR) channel, a cation channel crucial to neuromuscular transmission, is shown in Figure 4.3c. It has five subunits, and each subunit has four transmembrane domains. The receptor is sited in the endplate of the muscle fiber; when acetylcholine is released by the adjacent motor neuron and binds to the receptor sites on the two D subunits of the channel, it opens, allowing an enormous influx of depolarizing sodium ions that ultimately leads to muscle contraction. The other three subunits [the E, G (or H in some species), and J subunits] are each distinct. The AChR channel has been studied extensively and we will return to it later in this chapter and again in Chapter 9. The diameter and charge density of the pore vary along its length. A more detailed model of a potassium-selective channel in a bacterium, based on x-ray crystallography and since shown to be a good model of similar portions of eukaryotic potassium channels, is shown in Figure 4.3d. In this particular channel, the potassium ion relinquishes its hold on solvent water and interacts electrostatically with carbonyl oxygen dipoles that band the wall of the narrow selectivity filter; cation selectivity is provided by the


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acidic residues nearby and at the inner end of the pore. Both the carbonyl groups and the acidic residues are provided by the P-loop. Ions pass through the filter in single file, and more than one ion can be in the filter region at the same time. Repulsive forces between the ions promote swift passage through the filter. The filter is short [only 1.2 nm in length (Doyle et al., 1998)], to minimize its transport resistance. The selectivity filter does not pass Li or Na readily because they cannot pass through in their hydrated forms (see Table 4.2), while the unhydrated forms of the ions are too small to interact fully with the polar groups in the filter. As a consequence, the energy of binding of the lighter ions is less negative, and they are less apt to shed their hydration shells and pass through the pore. The rest of the pore is relatively nonselective. The potassium ion is hydrated in the wider central cavity, which is lined by hydrophobic residues to minimize retarding interactions with the wall. Each pore helix (see Fig. 4.3d) generates a dipole along its length, with the negative end of the dipole at the central cavity; these dipoles, and those of the polarized water in the cavity, help stabilize the ion within the heart of the bilayer. The development of this model earned Roderick MacKinnon, in whose laboratory the work was done, the 2003 Nobel Prize in chemistry. The selectivity filter in ion channels can be charged or lined with polar groups. In general, the wider the selectivity filter, the less selective the channel. The selectivity filter in the highly selective potassium channel is only 0.33 nm in diameter (Dwyer et al., 1980). The AChR channel rejects anions because of a collection of negatively charged regions near its selectivity filter; however, the characteristic dimension of the filter is 0.65 nm (Dwyer et al., 1980), and the channel exhibits only limited selectivity among cations. Though some channels are permanently open, most undergo transitions among conducting and nonconducting states. Transport through these channels is controlled by gates, so called because of their regulatory function; the rest of the pore is permanently open. Most models of gates treat them as physical barriers, blocking or severely narrowing the transport path. The gating process is often controlled by a receptor site, which can be part of the channel or a portion of an associated receptor protein. The gate opens via a conformational change when the site is activated; this is referred to as an allosteric response. For instance, the channel depicted in Figure 4.3b is gated by the S6 domain, which swings closed or open in response to the rotation or tilt of the voltage sensor, S4, whose motion is prompted by changes in transmembrane potential (Cha et al., 1999; Hille, 2001). Owing to the narrowness of the pore, only a small conformational change of the protein is needed, and the process takes only microseconds. The duration of the open state is typically of the order of milliseconds. The AChR channel is thought (Taly et al., 2005) to gate by a twisting motion of the entire pentamer, which narrows the pore. Another gating model, the ball-and-chain model, has been invoked to explain the inactivation of sodium and potassium channels in nerve, and will be described in Chapter 9. Figure 3.2e shows four images of a gated calcium release channel from skeletal muscle, which will be discussed in Chapter 9. The images show that the relative rota-

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tion of the two portions of the channel protein causes the pore to open so that calcium can pass through to initiate contraction. The gates that control the flux through the pore can be located anywhere along the length of the channel, and a channel can contain more than one gate. When ions pass through the pore by hopping from site to site, the energetics of the transport process can be represented by an energy profile such as that in Figure 4.4. The energetically favorable binding sites are represented by local minima in the profile; the depth of these potential wells differs from ion to ion, depending on the selectivity characteristics of the site. As shown in the figure, a gate can be modeled as a barrier whose height depends on whether the gate is open or closed. Rate equations for channel transport by site hopping have been developed on the basis of such energy profiles and will be described later in this chapter.

4.1.6. Regulation of the Gating Process A large number of agents can open or close channel gates. These include ions to which the channel may or may not be permeable, hormones, neurotransmitters (see Chap. 9), intracellular enzymes, changes in membrane potential, and membrane stretch. Some channels respond to these agents for as long as they are present, while others react only transiently. Channels whose gates are opened in this way are said to be activated. Chemical agents activate channels by binding to receptors that are associated with or part of the mediator; as a class, these channels are referred to as ligandactivated or ligand-gated. The ACh receptor channel described above exemplifies this class of mediators. The ligand may bind at the intracellular or extracellular face of the cell membrane. An important class of activators are the intracellular protein kinases that activate proteins by catalyzing their phosphorylation by ATP. Phosphorylation of a channel protein or an associated regulatory protein can increase the probability of the channel opening and the duration of its open state, and it has been proposed (Sperelakis and Schneider, 1976) that phosphorylation is a necessary condition for the opening of certain voltage-dependent channels. Other kinases can depress channel activity. The protein kinases are part of the cell's regulatory system and will be discussed further in Chapter 8. The intracellular mechanisms that regulate transport activity rely on other intracellular ligands as well, including cyclic adenosine and guanosine monophosphates (cAMP and cGMP), ATP itself, G proteins, and calcium ion. An example of a channel that requires phosphorylation to open is the cystic fibrosis transmembrane conductance regulator (CFTR) channel; a mutation in the gene for this protein is responsible for cystic fibrosis. This chloride-selective channel is activated when one of its domains, the R-domain, is phosphorylated by a protein kinase. It has generally been believed that the R-domain ordinarily plugs the channel at its cytoplasmic end, and that phosphorylation unplugs the channel to expose an ATP binding site. Intracellular ATP is then harnessed to provide the energy needed to open the channel. A more recent study (Chappe et al., 2005) suggests that the phosphorylated R-domain activates the channel by interacting with other domains in the


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Figure 4.3. Ion channel models. (a) Gramicidin A, a tubular ionophore. (b) A 1994 model of a voltage-gated potassium channel. [Reprinted with permission from Alberts et al. (1994). Copyright © 1994, Garland Science.] This channel has four subunits, two of which are shown. Each subunit has six membrane-spanning domains, which are numbered. The pore is surrounded by the subunits and partially lined by a loop (the P-loop) of amino acids between Domains 5 (S5) and 6 (S6). One of the six domains (S4) is charged and acts as the principal voltage sensor. (c) Acetylcholine (ACh) receptor channel: (1) an earlier cartoon (J NIH Res 1:152, 1989) and (2) a contemporary reconstruction based on cryoelectron microscopy [Reprinted with permission from Unwin (2000). Copyright © 2000, Royal Society.] It is now known that the ACh receptors are limited to the D subunits, as shown on the more recent image; asterisks mark the ACh binding sites. Each of the five subunits contributes the same transmembrane domain to line the pore. The parenthesized terms are specific to neuromuscular transmission. The recent image shows that sodium ions enter the muscle cell (sarcoplasm) laterally through gaps between the adjacent subunits. (d) Potassium-selective channel (KcsA) in Streptomyces bacteria (Doyle et al., 1998). The ions normally pass from the cytoplasm into the exterior medium. This channel does not possess the voltage activation “hardware” of eukaryotic potassium channels; each of the four subunits has only two transmembrane domains, M1 and M2. The gating region of the pore is near the cell interior. (e) Calcium release channel in skeletal muscle, top and side views [Reprinted with permission from Orlova et al. (1996). Copyright © 1996, Macmillan Publishers.] Rotation within the large cytoplasmic region (CY) of the protein gates the membrane pore that passes through the transmembrane region (TM).

124



125

Figure 4.4. Energy profile for an ion passing through a channel.

protein rather than by dissociating from a blocking site. The need for ATP rather than a simple ligand to open the channel suggests that an unusual amount of energy is needed to do so. Channels can also be activated by the ion to which they are selective; this is known as substrate activation. The solute facilitates its own transport. Inhibition occurs when other solutes compete for the receptor sites and do not cause the gate to open when they are bound to them. The effectiveness of an inhibitor depends on the binding affinity of the receptor site for it, relative to the site's affinity for solutes that open the gate. Inhibitors can also act by binding to a site near the receptor and blocking activator access to it. Often a gate is controlled by multiple binding sites on different subunits, and the dependence of channel permeability on activator concentration (cA) is sigmoidal in shape. At low concentrations, most of the binding sites are unoccupied, and the probability that a given site is occupied is proportional to cA. Assume that n binding sites are associated with each gate and that the gate is much more likely to open if all of them are occupied. If the sites are independent, then the probability that they are all occupied is proportional to c An . When n > 1, this probability, which is also the fraction of channels that are open at any instant, increases rapidly with cA (Fig. 4.5). The fraction of channels that are open cannot exceed unity, and this asymptote is approached as cA becomes large and the fraction of binding sites that are unoccupied becomes negligibly small. We shall see later that the AChR channel exhibits this behavior, which has been termed positive cooperativity. The most widely used techniques for studying channel dynamics all rely on measurements of the electrical properties (e.g., current, voltage) of membrane preparations. Consequently, as illustrated by Figure 4.5, channels are more often characterized by

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their conductance than by their permeability. These properties are different, although the descriptive material that follows makes little distinction between them. The relationships among permeability, conductance, and mobility are presented in Chapter 7, where the distinction is important.

Figure 4.5. Variation of channel-mediated conductance, G, with activating ligand concentration, cA; n = number of gate-controlling sites. The conductance is Gmax when all channels are open.

The response of membrane conductance to membrane potential reflects the action of the transmembrane electric field on a charged segment of the channel or an associated receptor, known as the voltage sensor. The electric fields across biological membranes can be substantial: a cell membrane potential of 100 mV acting across a 57 nm bilayer creates a field of 2 q 10 V/m. A change in the field presumably causes translation or reorientation of the voltage sensor, accompanied by changes in the conformation of the protein and the conductance of the channel. The voltage sensor transduces the potential energy in the electric field into the mechanical work needed to open the gate. The transient motion of the charged sensor in response to a change in the local electric field is thought to be responsible for the brief gating currents that are observed under these conditions (e.g., Armstrong and Bezanilla, 1973). The sensor generally responds to changes in membrane voltage by generating a series of small gating currents, each accompanying a distinct conformational change of a subunit or domain, until sufficient charge is translated to open the channel. Voltage-gated channels are crucial to the function of excitable cells such as nerve and muscle, and we will discuss them further in Chapter 9. Using x-ray diffraction data, Long et al. (2005a,b) computed the structure of a voltage-dependent K channel from rat brain, including the voltage sensor, at 0.29-nm resolution. The structure is shown in Figure 4.6a.


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Figure 4.6. Please see insert for color version. (a) Stereo side view of the Kv1.2 potassium channel. A possibly regulatory E subunit is docked to the cytoplasmic end of the channel. Each subunit is a different color. The TM region spans the cell membrane and is the integral protein component of the channel complex. The T1 domains of the subunits form a tetrameric complex within the cytoplasm that lies on the axis of the pore, so the ions that exit the cell through the pore enter the channel through side portals. Negative charges at the rims of the portals contribute to the cation selectivity of the channel. [Reprinted with permission from Long et al. (2005a). Copyright © 2005, AAAS.] (b) Proposed voltage sensor. The S4–S5 linkers of three of the four subunits are shown in red; the tipping of the linker toward the cytoplasm upon closure can be seen clearly. The S6 segments are shown in blue. Left panel: open state. Right panel: closed state. [Reprinted with permission from Long et al. (2005b). Copyright © 2005, AAAS.]

The pore portion of this channel is quite similar to that of the bacterial channel in Figure 4.3d, including the selectivity filter and the organization of the helical domains. Voltage sensitivity in the eukaryotic potassium channel is provided by several positively charged amino acid residues, called gating charges, on the S4 transmembrane

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segment (Fig. 4.6b). The S4 and S5 segments are connected by a linker region that runs parallel to the bilayer just inside the cell. Long et al. propose (2005b) that electric field-induced motion of the S4 segment causes the nearer end of the linker region between S4 and S5 to move further into the cell, which in turn causes S6 to move in a way that closes the gate. The precise motion of the sensing segment, and the extent to which it moves when gating occurs, are still uncertain. Since gating reflects a conformational change of the protein, it is reasonable to describe the gating process using chemical kinetics. The simplest model would be: E

ZZX C YZZ O, D

(4.2)

where C designates the closed state, O the open state, and D and E are the first-order rate constants for closing and opening, respectively; thus, if Ni is the number of channels in the ith state, dN C DN O E N C . dt

(4.3)

A similar equation can be written for dNO /dt. Let NT equal the total number of channels under consideration, equal to NO + NC. The steady-state value of NO/NT is easily shown to be a function of the two rate constants:

N O ¬ E . N T ® D E s.s.

(4.4)

These rate constants are influenced by membrane potential. Anderson and Stevens (1973) analyzed the current noise at the frog neuromuscular junction and found that the rate constant for channel closing, D, was an exponential function of membrane potential: A'\

D = Be

,

(4.5)

where A and B are (temperature-dependent) constants. This function also describes the voltage sensitivity of other preparations. Through relationships such as Eq. (4.5), the membrane potential can affect the frequency with which channels open and the mean duration of the open state. Naturally, an increase in the fraction of channels that are open at any instant (or, equivalently, the fraction of time that each channel is open) is seen experimentally as an increase in membrane conductance. We shall see in a later section that the membrane potential can also affect the conductance of the open channel by altering the energy profile within the pore. These effects allow membranes containing voltage-activated channels to exhibit a graded response, in which their macroscopic conductance is a smooth function of membrane potential. Gated channels can respond to more than one agent. For instance, potassiumselective channels in rat muscle are activated by calcium and by membrane depolarization. Calcium increases both the frequency of channel opening and the duration of the open state. At a given intracellular calcium level, depolarization further increases


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the frequency of transition to the open state (Fig. 4.7); similarly, the influence of depolarization is increased by raising the calcium level.

Figure 4.7. Effect of intracellular calcium level and membrane potential on the open fraction of K(Ca) (calcium-activated potassium) channels in rat muscle cell membrane. Reprinted with permission from Barrett et al. (1982). Copyright © 1982, Blackwell Publishing.

Mechanically activated channels are a relatively recent discovery. They are found in sensory organs and in nonexcitable cells unrelated to the senses, such as kidney, brain, and blood vessels (Oleson et al., 1988). Some pass only anions, and some only cations, with varying degrees of selectivity. These channels are of considerable interest as potential mediators of the biological and physiological response to mechanical forces in the body, such as hearing, the remodeling of bone under stress, or the response of arteries to shear stress caused by blood flow or tensile stress in hypertension. It appears that mechanosensitive channels use a variety of mechanisms to sense stretch, including a direct response to membrane tension. The cytoskeleton plays an important role as well, modulating the tension experienced by the channel in response to an external force, and transmitting the forces that accompany membrane distortion to the neighborhood of the responsive channel, or directly to the channel itself. A recent review is Hamill and Martinac (2001).

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Figure 4.8. Mechanical activation of a hair cell channel as proposed by Hudspeth and colleagues. (a) Stereo scanning electron micrographs of a sensory hair bundle of a bullfrog. The bundle is ca. 5 Pm in diameter. [Reprinted with permission from Hudspeth (1989). Copyright © 1989, Macmillan Publishers.] (b) Transmission electron micrograph of adjacent cilia and (arrows) attached tip links, which may serve as gating springs. The tip links, thought to be comprised of the protein cadherin (Siemens et al., 2004), connect the top of one cilium to the side of the higher adjacent one. [Reprinted with permission from Jacobs and Hudspeth (1990). Copyright © 1990, Cold Springer Harbor Laboratory Press.] (c) Proposed gating mechanism (Hudspeth, 1989). The tip of the cilium on the left is linked to the gate of a channel in the cilium on the right. An external force causes the two cilia to tip toward the right, so the cilium on the left slides down the cilium on the right, pulling on the gating spring and opening the gate.

Sensory mechanically gated channels, such as those in the hair cells of the inner ear, appear to rely on direct mechanical interaction with the gate. The channels in hair cells are on a bundle of cilia that extend from the cell, and respond directly to soundinduced ciliary motion. A model in which the relative motion of two adjacent cilia


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connected by an elastic gating spring opens a gate in one of them is illustrated in Figure 4.8. A more comprehensive review of the role of mechanically gated channels can be found in Gillespie and Walker (2001). Kinetic analyses of fluctuations in channel conductance suggest that the gating of channels is not a simple open-and-shut case. Channels can have a number of kinetically distinguishable closed states, and multiple open states whose durations or conductances differ. The existence of multiple open states suggests an alternative mechanism for positive cooperativity: when more of the sites controlling a given channel are occupied, the equilibrium among its open states favors those whose durations or conductances are larger. We will see some examples of this behavior later this in this chapter, and in Chapter 9. The kinetics of the transition of channels among closed, open, and inactivated states is a subject of continuing interest. More complex models include inactivated states that can be reached only from the open state, inactivation of channels only some of whose gate-controlling sites are occupied, and transitions among inactivated states.

4.1.7. Classification of Ion Channels. Aquaporins and Gap Junctions To this point, we have focused exclusively on ion channels. The selectivity of these channels varies, depending on the size and charge of the selectivity filter. The least selective channels are water filled and pass all small ions and even small nonelectrolytes. These nonselective channels appear to be very large, 2–3 nm in diameter, and have conductances comparable to those of aqueous solutions. Diffusion through these channels is not very different from free diffusion, and their conductances, in contrast to those of the narrower channels, do not saturate in concentrated solutions. These 8 9 channels can pass 10 –10 ions per second at 100 mV. As noted earlier, the selectivity filter is generally charged or lined with polar groups. As we examine channels with increasingly narrow filters, the filter first excludes ions on the basis of valence (although valence selectivity can also be provided by charges elsewhere in the pore or near its mouth), and then on the basis of size. Valence-selective channels pass only those ions having a particular valence, but are otherwise not very selective. The larger pore size implied by the limited selectivity of these channels is reflected in their conductances, which can be an order of magnitude larger than those of ion-selective channels. Ion-selective channels pass selected ions having a particular valence, and have been subjected to more study than the larger channels. Ion selectivity is governed by the selectivity filter of the channel and the steric factors described earlier. These pathways exhibit a wide range of unitary, or single channel, conductances, mostly in –12 –1 7 the 5–50 pS range (1 pS = 1 picosiemen = 10 ohm ), and pass about 10 ions per second when the driving force is 100 mV. These are discussed in detail in Hille (2001), which is the authoritative text on the subject. These channels can be further classified according to the ion to which they are most permeable, and whether they are ligand or voltage gated. Within the group of ion-selective channels, there is no clear relationship between selectivity and conductance.

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While Hille (2001) provides a comprehensive summary of the various families of ion channels and their members in excitable membranes, we present here a less detailed classification that includes non-excitable cells. The channels are sorted by the ion they chiefly transport. Several of these have been described already and more will be discussed in context of their function later in this text. — voltage-gated sodium channels. These channels are critical to action potential propagation in nerve. — epithelial sodium channels (ENaCs), responsible for absorption and secretion of the ion. — voltage-gated delayed rectifier potassium channels (e.g., the Kv1.2 channel, Fig. 4.6). These channels are responsible for most of the potassium current seen during action potential propagation in nerve. — voltage-gated fast transient potassium channels. These channels, also called A-type channels, open transiently and facilitate the translation by the nervous system of stimulus level into action potential frequency. — voltage-gated slow M-type potassium channels, that close during synaptic transmission to increase the responsiveness of the postsynaptic neuron (Chap. 9). — inward rectifier potassium channels. These channels exhibit their greatest conductance when the cell potential is more negative than the resting potential (the cell is hyperpolarized), and the potassium current is directed into the cell. — ATP-sensitive potassium channels. These inward rectifiers are found in smooth muscle and cardiac tissue, and play a role in the regulation of insulin secretion; they are gated according to the intracellular ADP/ATP ratio. — calcium-activated potassium channels. These ubiquitous channels are important mediators of the regulatory role of calcium in all cells, and are responsible for spontaneous pacemaker activity by individual excitable cells. — voltage-gated calcium channels (e.g., the calcium release channel in skeletal muscle, Fig. 4.3e). Extending beyond their role in muscle function, these channels mediate the entry of this important regulatory ion into all cells. — ligand-gated calcium channels, such as the IP3-activated calcium release channel in endoplasmic reticulum, which plays an important role in calcium signaling, as do calcium-dependent calcium channels in the plasma membrane of nonexcitable cells. — mechanically gated cation and chloride channels, discussed in the previous subsection. These channels are involved in the regulation of blood vessel diameter and cell volume. — various neurotransmitter-gated channels that pass sodium, potassium and chloride (e.g., the cation-selective AChR channel, Fig. 4.3c). Neuro-


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transmitter-gated chloride channels play a role in inhibitory chemical synapses (Chap. 9). — other ligand-gated cation channels involved in sensory transduction, such as photoreception and olfaction. — voltage-gated chloride channels. Some of these channels appear to play a role in kidney function; others contribute to cell volume regulation. — ligand-gated chloride channels. Some of these channels, which are important in secretion, are gated by calcium; others are regulated by cyclicAMP through protein kinases (Chap. 8). Kinase activation of the CFTR chloride channel was described earlier. A sampling of monovalent cation channels, including their unitary conductances, ionic selectivities, and number density on the cell surface, is presented in Table 4.5. The channels classified above facilitate the passive flux of solutes, primarily ions, between the cytosol and the extracellular domain. Not all channels share these characteristics. We have already met one such exception, in the previous chapter: the nuclear pore that allows solutes of all sizes to pass between the cytosol and the nucleoplasm. We now briefly describe two more classes of channels: aquaporins, which facilitate the flux of solvent, and gap junctions, which bridge the cytosols of adjacent cells. Aquaporins. Water crosses the cell membrane via several kinds of facilitators. These include the large nonselective channels mentioned above that also pass ions and small electrolytes; certain regulatory channels such as the cAMP-regulated Cl channel; and possibly some carrier systems. However, the primary transmembrane route for water is via specialized water channels called aquaporins, which are specific for H2O. Transport through large pores has been studied for many years, and solvent flux through them can be described using the formalism of nonequilibrium thermodynamics, to be discussed in Chapter 6. Aquaporins, which we describe below, are a more recent discovery. Before describing aquaporin channels, we should note the evidence that such channels, through which water flows rather than diffuses, exist. First, the notion that water crosses the membrane through an aqueous path rather than across the lipid bilayer is strongly supported by the observation that the temperature dependence of water transport follows the reciprocal of aqueous viscosity; the temperature dependence of transport through a lipid film is much stronger than this. Although this observation excludes transport through the bilayer, an expected result given the low solubility of water in lipid, it does not tell us whether solvent traverses the alternative route, an aqueous channel, via flow or by diffusion. This is because an inverse relation between solvent flux and viscosity is consistent with transport by either diffusion (recall the Stokes–Einstein equation in Chap. 2) or flow (see the expression for hydraulic conductivity in Chap. 7).

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Table 4.5. A Selection of Channels that Pass Monovalent Cations

Channel Na channel, squid axon Na channel, frog node of Ranvier Delayed rectifier K channel, squid axon Gramicidin A cationselective channel Ca-activated K channel, human erythrocyte Na channel, mouse skeletal muscle Inward rectifier K channel, frog skeletal muscle Acetylcholine receptor cation channel, rat muscle Ca-activated cation channel, rat heart Ca-activated high conductance (BK or maxi-K) K channel, rat muscle Maxi-K channel, rabbit sarcoplasmic reticulum

Unit conductance, a pS

Number density, channels/Pm2

Selectivity

Reference

4 8

330 1900

Na ~ Li >> K > Rb > Cs Li ~ Na >> K > Rb,Cs

(a, b) (c, d)

12

30

K > Rb >> Cs, Na, Li

(a)

15

Cs > Rb > K > Na > Li

(e)

18, 17

K >> Na

(e, f)

23.5

65

(g)

26

1.3

(h)

35, 25b

Cs > K > Na > Li

(e)

35

K ~ Na

(e)

187, 220

K > Rb >> Cs,Li,Na

(f, i)

130

K > Rb > Na > Li

(e)

Number density is obtained by dividing the maximum conductance per unit area of membrane by the unitary conductance of the channel. The number density is underestimated if not all of the channels are open when the maximum conductance is measured. (a) Conti et al. (1975); (b) Chandler and Meves (1965); (c) Conti et al. (1976); (d) Hille (1972); (e) taken from Latorre and Miller (1983); (f) taken from Petersen and Maruyama (1984); (g) Patlak (1988); (h) Matsuda and Stanfield (1989); (i) Blatz and Magleby (1984). a 1 pS = 1 picosiemen = 10–12 ohm–1.

The evidence for flow being the dominant mode of water transport across the cell membrane comes from experiments in which a water chemical potential difference was imposed across the membrane by adding a solute to one side, thereby generating a flow of solvent into the phase containing the added solute. This flow, termed osmosis, will be explained in much more detail in Chapter 6. The osmotic flow was five times greater than the diffusive rate measured using tritiated water as a tracer. It is estimated that up to 90% of the water flux across the erythrocyte membrane 9 crosses through aqueous channels; transport rates can exceed 10 water molecules per second (Agre and Kozono, 2003). The first identified aquaporin was isolated from erythrocyte membranes in 1988 (Denker et al., 1988). For this discovery, Peter Agre, in whose laboratory it was made, won the Nobel Prize in chemistry in 2003. Since 1988, aquaporins have been found in many other tissues, including the kidney, eye, brain, and gut. The original aquaporin protein, aquaporin-1, has a molecular weight of 28 kDa and consists of four subunits, each comprised of six membrane-spanning domains that surround a pore. Loops between two pairs of domains in each subunit extend part of the way into the bilayer —


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one loop from the intracellular side and one from the extracellular side — to line the pore (Fig. 4.9a). The entire oligomer consists of a cluster of four identical water channels, one from each subunit. This model of the aquaporin-1 channel is known as the hourglass model (Jung et al., 1994) because the loops that form the pore converge within the membrane. Most aquaporins are selective for water, but some do pass small nonelectrolytes. They are essentially impermeable to ions.

Figure 4.9. Aquaporin-1 (AQP-1). (a) Oblique view of a single subunit (Jung et al., 1994). The six domains are numbered in sequence. The loops between Domains 2 and 3 and between Domains 5 and 6 (denoted by a finer line for clarity) line the pore. The link between Domains 4 and 5 is partially obscured. (b) Stylized bipolar orientation of water molecules within the pore (Jensen et al., 2005). The water “wires” are aligned by hydrogen bonds (dashed lines), and the orientation of the molecules is altered by a strong positive dipole caused by amine groups lining the pore.

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Molecular dynamics simulations (de Groot and Grubmuller, 2001; Jensen et al., 2005) indicate that the water molecules pass through the channel pores in a single file, interacting through dipole forces with one another and with polar carbonyl and amine groups that line the pore. The water dipoles in the file are oriented in what has been termed a bipolar structure, with oxygen toward the center of the pore and the hydrogen atoms toward the ends of the pore (Fig. 4.9b). Selectivity to water relies on a complex electric field in the pore that results from the particular structure of the protein; only a small molecule with a large dipole moment, like water, can follow the + field and pass through. Protons, the most likely ion to pass the channel (as H3O ), are excluded by a positively charged region at the center of the pore. The dipole moment of a polar molecule measures the energy of its interaction with an electric field. When a molecule encounters a complex electric field, the force on it that tends to align it with the field is larger if its dipole moment is large. If, in addition, the molecule is small, less force is needed to align it. Consequently, a small molecule with a relatively large dipole moment is particularly nimble when trying to follow a complex electric field.

Gap Junctions. Gap junctions provide an aqueous path connecting the cytoplasms of adjacent cells, and are found in most tissues. The junctional complex in each cell consists of six subunits of a protein called connexin, which are arranged like the staves of a barrel to surround a pore that is about 1.5 nm in diameter (Fig. 4.10); each subunit consists of four transmembrane domains. The pore extends from the cell interior through the cell membrane and into the gap of about 2–4 nm between the outer leaflets of the cell membranes of the adjacent cells. The hemichannels in each cell membrane line up to produce a continuous aqueous path, consisting of 12 connexin subunits, between the interiors of the two cells.

Figure 4.10. Organization of gap junctions. 1 Å = 0.1 nm. Reprinted with permission from Purves et al. (2001). Copyright © 2001, Sinauer Associates.


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Gap junctions pass ions and small molecules up to 1 kD in size; since it is a conducting path (typical unitary conductances are in the 30–300 pS range), it also dissipates differences between the intracellular voltages of neighboring cells. Like ion channels that communicate with the extracellular space, gap junctions are gated and can fluctuate between open and closed states. Some are voltage gated, with conductances that depend on the difference between the intracellular potentials of the communicating cells. Their permeability can also be affected by hormones like dopamine, and by intracellular composition, decreasing when the cytosolic pH becomes too acidic or when intracellular calcium levels rise. Decreases in cytosolic pH and increases in intracellular calcium are signs of cell injury. It appears that the closure of the junctions acts as a quarantine at the cellular level, initiated in this case by the affected cell itself. Gap junctions allow cells to “communicate” with one another, and provide a pathway for electrical potential to spread, and solutes to diffuse, between cells. This serves several functions: — transmission of signals between excitable cells via gap junctions is more rapid than the chemical synapse that will be described in Chapter 9. This is particularly valuable when cellular activity must be synchronized, as in cardiac contraction and peristalsis. — many organs, like stomach and liver, are designed to respond to neurotransmitter signals from the autonomic nervous system, but not every cell in the organ is innervated; gap junctions transmit the signal from cells at which nerves terminate to cells where they do not. — there are natural cell-to-cell variations in metabolism that lead to differences in intracellular composition; these differences are dissipated when there is a path between cells for solute transport. — in embryonic development, cells destined to form specific organs communicate through gap junctions so that they can develop in a coordinated fashion.

4.1.8. Carrier Models A stylized model of the most common form of transmembrane carrier is depicted in Figure 4.11a. Like channels, these transport mediators span the membrane and consist of multiple subunits and/or domains. The operation of the dimeric, alternating access, carrier in the figure is straightforward. Suppose the concentration of the carrier substrate is higher in Phase I than Phase II. The carrier flips randomly between the two states in the figure. When the carrier is open to Phase I, it can bind a substrate molecule from that phase. Binding to one of the two sites on the carrier sterically inhibits binding to the other one, so carriers usually transport one molecule of substrate at a time. When the carrier flips to face Phase II (a process that substrate binding can accelerate; see §4.2.9), the substrate dissociates from the carrier and enters that phase. More than 1000 different carriers are believed to operate in this way.

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Figure 4.11. Stylized and molecular models of transmembrane carriers. (a) Alternating access model. An aqueous path extends from an external phase (I or II) to the receptor sites (solid circles) when the carrier is open in that direction. The “gate” that is formed at the abutment of the two protein molecules flips from one end of the pore to the other when the carrier undergoes a conformational change. Generally, only one substrate molecule fits in the channel between the two oligomers. The notation in the figure will be described in the following section of this chapter. (b) Two-gate model of the Cl–HCO3 exchanger (Knauf and Pal, 2003). See description in text. (c) Molecular model of the lactose permease of E. coli. The aqueous internal cavity of the molecule is open to the cytoplasmic side. A bound substrate molecule can be seen as a black shape between the domains within the membrane. The kissing arrows indicate the motion of the two domains that flips the gate from the external face to the internal one. Reprinted with permission from Abrahamson et al. (2003). Copyright © 2003, Federation of the European Biochemical Societies.


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An alternative carrier model (Knauf and Pal, 2003), which has been proposed for the Cl–HCO3 exchanger in red blood cells, in shown in Figure 4.11b. This exchanger will be described later in the chapter and is essential to CO2 transport in blood. In this model, the carrier is a channel with gates at both ends; the gates open and close randomly, but in a coordinated fashion such that they are not both open at the same time. When the internal binding site is unoccupied, an anion can enter the channel through an open gate, and bind. If the gate through which the anion entered closes, the anion is trapped in an occluded state, and it can cross the membrane if the first gate that opens subsequently is not the one through which it entered. Naturally, solute can cross either of these carriers in either direction. When only one substrate species participates in the transport process, the net flux of the substrate will be from the phase in which its concentration is higher to that in which it is lower, just as in free diffusion. The kinetics of single-substrate carrier transport will be modeled mathematically in the next major section of this chapter. The molecular structure of an alternating access carrier in the E. coli bacterium has recently been elucidated by Abramson et al. (2003) at 0.35-nm resolution, using xray diffraction techniques, and is shown in Figure 4.11c. This particular carrier actually functions as a countertransport system that accumulates lactose in the cell through secondary active transport, which will be discussed briefly in this chapter and in more detail in the next. The protein contains a single sugar binding site and is made up of two domains, each of which consists of six membrane-spanning helices connected by a flexible loop. The authors suggest that the translocating flip of the carrier can consist of a 30º clockwise rotation of the domain on the left, and a complementary 30º counterclockwise rotation of the one on the right, around an axis perpendicular to the plane of the figure and passing near the binding site. We mentioned earlier that carriers used to be thought of as ferryboats, binding substrate and crossing the bilayer from one side of the membrane to the other. Although this model is not the best description of native carriers, it is the mechanism used by carrier ionophores such as valinomycin, which contains a polar pocket for its ionic cargo and presents a hydrophobic exterior that allows it to pass through the bilayer with relative ease. Carrier ionophores are also ion selective; valinomycin, for instance, is specific for potassium.

4.1.9. Carriers and Channels: Convergences and Differences The convergence of carrier and channel models referred to at the beginning of this section is illustrated by the carrier models of Figure 4.11. Consider the alternating access model. Some of the characteristics of this model are unique to carrier transport: the solute is bound to a specific site and remains bound while transiting from one side of the membrane to the other (i.e., while the carrier changes conformation), and at no time is there an unobstructed path across the entire membrane. On the other hand, there are real similarities between the two: when the protein domains in the alternating access model block access to the binding sites from one phase or the other, the effect is much like that of a gate at the end of a channel; transport is enabled by a small con-

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formational change that opens a “gate”; and the binding sites on a carrier protein barely move, like the sites on the wall of a pore. Indeed, the two-gate model (Fig. 4.11b) stipulates that the carrier is effectively a channel with a gate on each side of the binding site. Some of the differences between the behavior of carrier and channel transport systems are summarized below. These are generalizations, and there are exceptions to almost all of them. 1. Carrier transport rates are generally more sensitive to temperature (i.e., have a higher activation energy) than channel-mediated fluxes. Solute passing through an aqueous channel experiences a viscous drag as it moves relative to the solvent in the pore. This is only one of the resistances the solute encounters but, as in free diffusion, its activation energy is much less than that of the conformational change of a carrier protein. The latter figure is typically 15–20 kcal/mol. 2. Carrier transport rates are less than channel-mediated rates. A single car2 5 rier might move 10 –10 substrate molecules across the membrane each second; we have seen that ions can pass through a single channel at rates 6 9 of 10 –10 per second. 3. When the carrier binding sites are not highly selective for a single solute, the flux of the primary substrate can be influenced more strongly by other solutes and their transmembrane concentration differences. Competition for binding sites on such a carrier leads to competitive inhibition, which reduces the transport rate of the primary substrate. Inhibition will be discussed in a later section in this chapter. The effect of one solute on the transport rate of another also underlies several mechanisms of active transport, which will be presented briefly in this chapter and discussed in greater detail in the following one. Two of these mechanisms are cotransport and countertransport. In cotransport, two solutes, the primary substrate and the cosolute, are both bound to the carrier and cross from Phase I (the cis side) to Phase II (the trans side) together. The flux of the primary substrate from Phase I to Phase II is accelerated by increasing the concentration of cosolute in Phase I. In countertransport, the carriermediated flux of the primary substrate is stimulated by adding cosolute to the trans side of the membrane. Carriers that perform countertransport are known as exchangers. Although solute interactions have a greater effect on carrier transport, channels are not immune to them. No channel is absolutely selective for a single species. For instance, the permeability of potassium in sodium channels is perhaps 5% of that of sodium. When more than one ionic species can compete for the transport sites in a pore, inhibitory effects are seen that are not very different from the inhibition of carrier transport. As a consequence, the selectivity of a channel can depend on the relative concentrations of the competing ions. Channels can also be blocked by ions that can enter the pore but cannot pass through the selectivity filter.


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Furthermore, as mentioned earlier, the rates of transport through gated channels are influenced by a variety of solutes that affect the dynamics of the gate. 4. Carriers transport ions and nonelectrolytes across membranes, but most channels pass only ions or water. Nonelectrolytes can traverse large and nonselective channels, but not the narrower, more selective ones; thus, if a facilitated transport system is found to be selective for a particular nonelectrolyte, especially a large nonelectrolyte, then the mediator is almost certainly a carrier. As noted earlier, carriers, which rely on specific binding interactions, are generally more selective regarding their substrate than are channels.

4.2. KINETICS OF FACILITATED TRANSPORT The transport of solutes via carriers and channels has been the subject of extensive kinetic analysis. Almost all of the carrier models that have been proposed obey very similar kinetics. Mathematical models of channel transport are more diverse, reflecting a broader range of assumptions regarding the biophysics of the transport process. Channel models tend to be specific to particular mediators, which can differ with respect to gating kinetics, the role of charges in the pore and electrostatic effects in general, and the relative sizes of the pore and solute. Several models of channel gating and transport will be presented below, as well as two less specific aspects of channel kinetics: energy barrier models and current noise analysis. Related topics reserved for later chapters include: (1) the use of models of solute diffusion and convection through narrow channels to estimate the size of pores in cell membranes (Chap. 7); (2) the Hodgkin–Huxley model of the kinetics of gated channels in nerve (Chap. 9); and (3) a continuum model of flow through large channels, developed to describe glomerular filtration (Chap. 10). With respect to carrier transport, we begin with the simplest model of an alternating access carrier and demonstrate its application to sugar transport across the red cell membrane. After that, we discuss more complex models, ending with models in which the carrier can bind multiple substrates. This will lead to a discussion of inhibition in the final section of this chapter, but in this section it is used to introduce the cotransporters and exchangers whose role in active transport will be an important part of the chapter to follow.

4.2.1. Models of Ion Channel Transport: Overview Whereas all models of carrier transport are essentially straightforward extensions of the alternating access model, models of channel transport fall into two distinct classes, which can be thought of as macroscopic and microscopic. The macroscopic approach uses the continuum description of ion transport embodied in the electrodiffusion equation and its solutions. The details and statistical aspects of channel function are ig-

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nored and the channel-containing membrane is regarded as a simple barrier with an experimentally determined permeability or conductance. Since these latter properties are obtained experimentally, the continuum approach can work well as long as the conditions to which it is applied do not differ too much from the conditions under which the membrane properties were measured. Permeability and conductance obtained in this way can of course be interpreted in terms of more microscopic channel properties, but these properties do not enter explicitly into the macroscopic description of the behavior of the membrane in which the channels are embedded. The most commonly used solutions of the electrodiffusion equation are the constant-field equation [Eq. (2.55)], and equivalent circuits or electrical analogs, which will be developed in Chapter 7. This was the approach taken by Hodgkin and Huxley (1952) in their groundbreaking study of conduction in nerve, which we will cover in Chapter 9. The microscopic approach, which will be the focus of the subsections to follow, builds the transport behavior of a channel-carrying membrane from the statistical properties of a large number of individual channels, each operating independently. All of these methods take gating into account and the consequent statistical aspect of transport through a single channel or a collection of channels. These models employ rate theory, noise analysis, and formal kinetics. An important tool for testing these models is the voltage clamp: measuring transmembrane current and its fluctuations while the potential difference across a portion of a channel-containing membrane is fixed. These data can be used to parameterize kinetic models of the transitions that accompany ion binding within channels, channel opening and closing, and the conformational changes that accompany activation. Although all of these models recognize the importance of gating and are developed in the context of solute movement through a single channel, they do fall into two distinct classes. The energy barrier and kinetic models of channel transport that we discuss first regard the pore as a sequence of energy barriers and wells that represent gates, filters, and binding sites. This profile is static, and the problem is either (1) to find the fluxes across an array of such pores when the boundary conditions are specified or, alternatively, (2) to determine the shape of the energy profile or the rate constants for barrier transit from experimental flux data. The effect of opening or closing a gate is represented by a change in the height of the energy barrier it presents to the approaching ion. A second class of models focuses on the statistics of the gating process itself. The channel is generally assumed to have a fixed conductance (which can be treated by the models just described) when in any given open state, and the challenge is to develop or parameterize a model of gating kinetics that describes experimental results. The seminal advance in this area was the voltage clamp, which allowed time-dependent measurements of the conductance of single channels in a small patch of membrane, with submillisecond resolution. When a larger patch of membrane containing voltageactivated channels is clamped at a fixed potential, the transmembrane current is characterized by a mean value on which are superimposed fluctuations as the individual channels in the membrane open and close. The use of both the mean current and current fluctuations to illuminate the gating process will also be described below.


143

Finally, it should be noted that, beyond these latter models, high-performance computations can be made of molecular transport through discrete water-filled channels whose shape, charge distribution, and dynamics are specified in detail. These molecular model calculations may or may not retain a continuum description of the solution phase. With continuing increases in computing power, they are an exciting area of contemporary biophysical research, but they are outside the scope of this text.

4.2.2. Energy Barrier and Binding Models of Channel Transport We saw earlier that the binding sites along the wall of a narrow channel can be modeled as a series of energy barriers and wells. This is illustrated in Figure 4.12a. The barriers at the membrane–solution interfaces must be crossed when the ion enters or leaves the pore, while the barriers within the membrane are crossed when the ion hops from site to site. For analysis purposes, it is generally assumed that, apart from the gates, the energy profile in the membrane does not change with time and is unaffected by the binding of ions to sites in the pore. We begin this section by using absolute rate theory to develop expressions for the ion transport rate across a barrier system when the concentrations in the bounding phases are dilute enough that individual ions do not interfere with one another in the pore. Such a system exhibits independence: the transport of any given ion is unaffected by the presence or flux of any other ion in the system, and the flux equation assumes a form similar to that in free diffusion. Next, we develop a kinetic model that recognizes that there is a limited number of pores in any real system, and only a limited number of ions can be bound at any time. Since ions compete for binding sites, the independence principle no longer applies. This system is seen to demonstrate saturation, which has been observed in real channel systems. Another process in which ionic independence fails is single-file diffusion. In the simplest model of single-file diffusion, the ions all move down the pore by binding to the same sequence of sites. They cannot pass one another, and an ion cannot hop to an adjacent site if it is occupied. In dilute solutions, the rate equations for single-file diffusion approach the free diffusion form, because each ion is not likely to find another ion bound to an adjacent site. We will touch on this briefly in a description of occupancy states. Independence also fails if the energy barrier between two sites is altered by the binding of an ion elsewhere in the pore. There are certainly local effects of this kind. One model proposes a “repulsive” interaction between bound ions: the binding of one ion makes it more likely that a nearby ion will hop further away — or out of the channel.

Ionic Independence: Absolute Rate Theory. The kinetics of transport through a series of energy barriers and wells can be described by the theory of absolute reaction rates (e.g., Johnson et al., 1954). According to this theory, the rate at which ions of the ith species move from the jth well (or binding site) to the (j + 1)st well is

ki , j l j 1 ci , j ki , j l j 1 ,

(4.6)

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Figure 4.12. (a) Energy barrier model of an ion channel. There is no electrostatic potential gradient across the membrane, so PiI PiII ( Pi is defined in the text). The energies of hydration and binding associated with ion selectivity at the entrance to the pore are indicated. (b) Effect of an applied potential on the energy profile. Curve 1: energy profile at the jth and j + 1st wells in the absence of an applied field. Curve 2: increment in local electrochemical potential, relative to that at x = a, caused by an applied transmembrane potential, '\. The potential profile is drawn as linear, though it need not be. Curve 3: energy profile at the jth and j + 1st wells in the presence of the applied field. Curve 3 is obtained by adding to Curve 1 the electrostatic contribution to Pi from Curve 2.


145

where ci,j has been termed the internal concentration of the ith species in the jth well. The rate constant for the transition is

ki , j l j 1 Ae

( Pi , j / j 1 Pi , j ) / RT

,

(4.7)

where A is a constant and Pi w Pi0 ziᑠ\ is the concentration-independent part of the electrochemical potential (again neglecting pressure effects). As shown in Figure 4.12a, the first electrochemical potential in the exponent is that at the top of the barrier between the two wells, and the second is that in the well. The potential difference is the height of the barrier as seen from the jth well, and is the activation energy of the transition. The exponential term is the fraction of the ions in the jth well that have sufficient energy to cross the barrier. The preexponential term A can be regarded as the rate at which ions attempt the transition. The rate of transition across the barrier in the opposite direction is given by a similar expression, except that the activation energy is the height of the barrier as seen from the (j + 1)st well: J i , j 1l j ci , j 1 Ae

( Pi , j / j 1 Pi , j 1 ) / RT

.

(4.8)

The internal concentrations {ci,j} are not the same as concentrations in free solution. For a membrane that contains a large number of identical pores, ci.j can be regarded as the number of these pores, per unit membrane area, whose jth sites are occupied by an ion of the ith species, divided by Avogadro's number. To relate the internal concentration to a free solution concentration such as ciI , consider the exchange of solute between Phase I and the first well inside the membrane. From Eq. (4.6), the net flux into the membrane at x = 0 is J i J i ,Il1 J i ,1l I ciI ki ,Il1 ci ,I ki ,1l I .

(4.9)

When Phase I and the first well are in equilibrium, Ji = 0. Substituting Ji = 0 into the preceding equation, ci ,1

ciI ki ,Il1 ki ,1l I

(equilibrium) .

(4.10)

The rate constants in energy barrier models are sensitive to changes in membrane potential. Assume that a fraction, Dj, of the potential drop across the membrane occurs between the jth and (j + 1)st wells; Dj is referred to as the electrical distance between the wells. As can be seen from Figure 4.12b, the imposition of a potential difference across the membrane causes the activation energy on one side of the barrier to increase by ziᑠDj'\/2 (assuming the barrier is midway between the wells and the local field is constant); the activation energy on the other side falls by the same amount. For a cation, the transition from the jth well to the (j + 1)st well is favored by an increase in '\. The effect of membrane potential on ki , j l j 1 is obtained using Eq. (4.7): zi ᑠD j '\ ¬ . ki , j l j 1 (ki , j l j 1 )'\0 exp 2 RT ®

(4.11)

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The rate constants in the opposite direction are given by an equation identical to Eq. (4.11), except that the sign of the exponent is changed. The energy barrier model has been used to interpret the current–voltage relationship of potassium channels in excitable cells. Like the constant-field equation, absolute rate theory predicts that the conductance of these channels increases when the extracellular potassium level is high and the intracellular potential is made more negative. This is known as inward rectification. Unless the membrane is assumed to contain very few barriers, an excessive number of rate constants (or, equivalently, barrier heights) must be evaluated to describe the transport process in quantitative terms. For instance, seven independent rate constants are needed to model a three-site channel that passes a single species. Because the number of experimental conditions that can be varied is small, the number of distinct rate constants that can be evaluated experimentally is also limited. Thus it has often been assumed that many of the barriers in a multibarrier membrane are identical; this reduces the number of distinct rate constants, and hence the number of transport experiments that are needed to characterize the energy profile. Another alternative is to model the pore as containing a limited number of internal binding sites, and correspondingly few barriers; we will see some examples of this in the next subsection. In analyzing the effect of potential differences across the membrane, assumptions must also be made regarding the partitioning of the potential drop across the several barriers; i.e., the {Dj}. The series of barriers that determines the permeability of a given channel is not the same for all ions. The factors affecting channel selectivity come into play here; indeed, the barrier heights that determine the rate constants at the membrane–solution interfaces can be related to the energies of hydration and binding discussed in the previous section. Saturable Channels: Kinetic Analysis. In the previous treatment of transport through a sequence of binding sites, no account was taken of the possibility that an ion might find an adjacent site occupied by another ion; all that was necessary for an ion to proceed over a barrier was that it have sufficient energy to do so. As pointed out earlier, such models implicitly assume that the solutions adjacent to the membrane are dilute enough — or the channels are numerous enough — that such interference is so rare that it can be ignored. In these systems, each ion proceeds independently of the others and is unaffected by the presence of other species. One characteristic exhibited by real biological channels is saturation; that is, for a given trans concentration of an ion, the ionic flux increases asymptotically to a limit as the cis concentration is raised. Systems in which ions cross independently are not expected to exhibit saturation; thus, when it is seen, it is taken as evidence that the transitions of individual ions across the membrane are no longer independent of one another. In the context of energy barrier models, independence fails when increasing numbers of ions occupy binding sites in the pores, thereby impeding the free passage of ions that characterizes independence. This is increasingly likely to happen as the number of ions seeking entry into the pores (i.e., the ambient ionic concentration) increases relative to the number of channel sites per unit area of membrane.


147

To deal with occupancy issues, it is useful to model transport as a kinetic process, in which there is a finite concentration, CBT, of binding sites. Denoting the binding sites by B and the ion by S, and assuming that the ion binds to only a single site in transiting the membrane, the kinetic scheme is: k

k

1

2

II 1 2 ZZZ X ZZZ X S I B YZZ Z SB YZZ ZS B. k k

(4.12)

In the steady state, the concentration of occupied binding sites is constant; thus, the net rate at which S enters from Phase I and is bound equals the net rate at which S is released from the pore into Phase II. These rates, per unit area, are equal to the net flux across the pore array, JS.: J S k1cSI cB k1cSB k2 cSB k2 cSII cB .

(4.13)

Furthermore, the total concentration of sites is CBT: cB + cSB = CBT.

(4.14)

Equations (4.13) and (4.14) constitute three equations in cB, cSB, and JS. Solving for the flux, JS

C BT (k1cSI k1D) , 1 D

(4.15a)

where

cSB k1cSI k2 cSII D , cB k1 k2 or

J S C BT

k1k2 cSI k1k2 cSII . k1 k2 k1cSI k2 cSII

(4.15b)

I

For a given cis concentration cS , the flux is maximized when the trans concentration II cS = 0. Under these conditions, the flux simplifies to

J S C BT I

II

k1k2 cSI . k1 k2 k1cSI

(4.16)

When cS is small and cS = 0, D is also small, indicating that most binding sites are unoccupied. Equation (4.16) shows that, under these conditions, the flux approaches I proportionality to cS , as in free diffusion (one of the exercises at the end of the chapter demonstrates that this is also the case for energy barrier models when the ions cross the membrane independently). Equation (4.16) also shows that, even when the ion is absent from Phase II, there is a maximum transit rate across the pore irrespective of

148

CH. 4: FACILITATED DIFFUSION: CHANNELS AND CARRIERS I

the upstream concentration; the pore saturates. At very high values of cS , virtually all the pores are occupied (D is large; cB 20 ms) exposure to acetylcholine, but this does not happen during normal physiological function]. However, it usually moves along a series of only four of the states in sequence: R, AR, A2R, and A2R*. The other states and additional reaction paths are largely excluded because the rate constants that describe the transitions among the six states do not favor these transitions; this is why open channels with unoccupied receptors are rare. This is a common characteristic of many complex biological systems: many reaction pathways are possible, but the rate constants define the fewer paths that the system almost always follows. The parameters of the reaction pathway given by Eq. (4.20) are given in the problem set at the end of this chapter. We will have more to say about the role of the AChR channel in neuromuscular transmission in Chapter 9. Voltage and Current Sensitivity of a Voltage-Activated Sodium Channel. We noted earlier that voltage-activated channels open when a voltage sensor in the protein moves in response to a change in transmembrane potential. This motion must therefore cause the open state to become energetically more favorable — that is, of lower free energy — than it was before. We can estimate this change in free energy and predict how the fraction of channels in the open state might change with membrane potential. To relate the fraction of channels that are open to the free energy of the opening process, consider the opening process as an equilibrium reaction: C U O . Let Ni be the number of channels in the ith state (i = C, O) and NT be the total number of


155

channels, equal to NO + NC. From Chapter 1, the equilibrium constant for channel opening is K

'G 0 ¬ NO , exp NC RT ®

(4.21)

0

where 'G is the standard free energy change of the transition to the open state. Substituting NC = NT – NO, and solving for the fraction of channels that are open, 1

'G 0 ¬¯ N O ¡ ° . ¡1 exp N T ¢¡ RT ®°±°

(4.22)

Assume that, when the voltage sensor moves to open the channel, the accompanying movement of charge is equivalent to n electronic charges crossing the membrane, of thickness a, in which the channel is embedded. The electric field across the membrane is '\/a, where '\is the membrane potential. Thus, the potential energy given up by the sensor when it moves is nᑠ ¸ a ¸

'\ nᑠ'\ a

(using the Faraday instead of the electronic charge to be consistent with the use of 0 0 mols as the measure of quantity). Replacing 'G in Eq. (4.22) by 'G – nᑠ'\, 1

'G 0 ¬ nᑠ'\ ¬¯ NO exp ('\) ¡¡1 exp °° . ® NT RT RT ® ¢¡ ±°

(4.23)

Taking the reciprocal of both sides, rearranging, and taking the logarithm, N ¬ 'G 0 nᑠ'\ ln T 1 N O RT ® RT

(4.24)

Let f be the fraction of channels that are open under a particular membrane potential 1 ¬ '\. Equation (4.24) indicates that ln 1 is linear in '\, with a slope from which f ® the number of gating charges can be calculated. Hartshorne et al. (1985) used a voltage clamp to record the opening kinetics of sodium channels from rat brain, from which they obtained f as a function of '\(Fig. 4.18). Fitting this data to a form of Eq. (4.24) employed by French et al. (1984), they determined that the apparent gating charge, n, was approximately 4. From Eq. (4.17), the unitary current through an open sodium channel is iNa J Na ('\ ENa ) .

(4.25)

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Figure 4.18. Fraction of voltage-gated sodium channels from rat brain that are open, as a function of the potential across the bilayer into which they are incorporated (Hartshorne et al., 1985). The curve through the data is fitted by eye.

For a region of membrane containing NT sodium channels, the current INa is equal to the product of the unitary current and the number of channels that are open. Using Eq. (4.23), 1

'G 0 ¬ nᑠ'\ ¬¯ ° exp I Na ('\ ) J Na ('\ ENa ) N T ¡¡1 exp ° . ® RT RT ® ¢¡ ±°

(4.26)

Equation (4.26) shows that the membrane potential affects the current carried by voltage-gated sodium channels through its influence on both the driving force for transport and the fraction of channels that are open. To predict this dependence, it is necessary to know the unitary conductance of the channel, the sodium activities on both sides of the membrane (needed to calculate ENa), the total number of channels, the standard free energy change of channel opening (or the equilibrium constant, K), and the apparent gating charge. For now, we just consider the shape of the current–voltage curve of a membrane patch containing these channels. The easiest way to get a sense of the shape of the current–voltage curve is to plot the driving force and open fraction separately, as is done in Figure 4.19. The driving force is a linear function of voltage; the number of open channels is a sigmoidal function of '\as it varies from very large negative values (NO/NT o 0) to very large positive values (NO/NT o 1). The slope of the driving force line is J and its '\-intercept is ENa; the asymptote of the number of open channels is NT, and K and n determine the steepness of the latter curve and its location along the '\axis. For very large negative values of '\, more negative than ENa, the current is zero because essentially all of the channels are closed. As the potential increases, channels start to open, but '\ < ENa, so the current is negatively directed, and becomes more so


157

as more channels open with increasing membrane potential. However, as the membrane potential approaches the sodium Nernst potential from below, the driving force for ion flux falls sufficiently to cause the magnitude of the negative current to decrease, and it becomes zero when '\= ENa, since the ion is then at equilibrium. As '\continues to increase, both the driving force and the number of open channels increase; as the open fraction approaches unity, the slope of the current–voltage curve approaches the constant value of JNT.

Figure 4.19. The effect of membrane potential on membrane current through voltage-sensitive channels. (a) The separate effect of potential on the driving force for transport. (b) The separate effect of potential on the number of channels that are open. (c) The combined effect of potential on membrane current.

158


In this analysis, the only transition undergone by the channel is from the closed state C to the open state O. We have noted several times that ion channels generally pass through several closed states before opening. In this simple model, C can be thought of as the closed state that immediately precedes channel opening. A more detailed model would apply equations analogous to those derived above to each transition of the channel, each with its own standard free energy change and equivalent gating charge. Also, the use of constant values of the {Ni} ignores the temporal variations in these quantities as individual channels open and close. Thus, an assumption of the analysis is that there are enough channels in the patch being modeled that such fluctuations perturb the mean only slightly. We will not make this assumption in the next subsection, where information regarding channel kinetics is derived from the current fluctuations under voltage clamp.

4.2.4. Stochastic Properties of Channels: Membrane Noise Analysis When the potential difference across a membrane containing a large number of channels is fixed, the membrane current fluctuates around a mean value as individual channels open and close. The statistics of these fluctuations contains information about the unitary conductance of the channel and the kinetics of the gating process. Assume that a patch of membrane contains N independent channels and that the unitary conductance of a single open channel is J. Let p be the probability of a given channel being open at any instant. On average, Np channels are open at any time, and the mean conductance is G NpJ .

(4.27)

Since the channels operate independently, the number of channels, n, that are open at any instant follows the binomial distribution. Therefore, the variance of n is V2n Np(1 p) .

(4.28)

Since the instantaneous conductance of the patch, G, is nJ, its variance is VG2 Np(1 p)J 2 .

(4.29)

Divide Eq. (4.27) into (4.29), to eliminate N:

VG2 (1 p)J . G

(4.30)

If the channels are closed most of the time, then p cA , and JA I II = 0 when cA = cA . The dependence of the transport rate on substrate concentrations in the ambient phases is sketched in Figure 4.22. In free diffusion, these graphs would be straight lines. The maximum flux attainable by the system is Vm = DcXT/2, achieved when the substrate concentration in Phase II is zero and the concentration in Phase I is raised to a value high enough to saturate the carrier system.

164


Figure 4.22. Dependence of carrier transport rate on substrate concentrations. The cAI -intercept of the lower curve shows that the flux is zero when c AI and c AII are equal (to C). When c AII = 0 and cAI = K, JA is half of the maximum flux, Vm.

When the concentrations in both phases are low, such that c iA > K), divided by the specific activity. It can be seen from Eq. (4.46) that this “rate” is equal to the true maximum flux of which the carrier is capable — DcXT/2 — multiplied by the area-to-volume ratio of the cell. More comprehensive measurements of sugar transport across the erythrocyte membrane, including measurements of efflux as well as influx, and their dependence on sugar concentration, indicate that the solute that enters the cell does not reside in a single intracellular compartment, but is partitioned among three compartments: a submembranous compartment, a binding complex associated with clusters of carrier protein, and the cytosol (Cloherty et al., 1995). The identification of these distinct compartments was made using compartmental analysis, a powerful tool for describing the interaction of fluxes and concentrations in complex biological systems. Compartmental analysis will be introduced in Chapter 8, and the compartmental parameters determined by Cloherty et al. will be used in one of the exercises. Note that the organization of the compartments in the red cell does not affect the determination of the maximum transport rate, which depends on only the amount of tracer entering the cell, irrespective of the compartment within which each tracer molecule subsequently resides.

4.2.8. More Complex Carrier Models The usual practice in characterizing a transport process thought to operate by a carrier mechanism is to see if its behavior fits the predictions of the simplest model (provided, of course, that the assumptions of the model make sense for the case of interest). The most common way to do this is to measure the unidirectional flux through the carrier as a function of the concentration of substrate on the side from which the flux originates. This can be done most easily using tracers and Eqs. (4.42) and (4.43). Substituting Eq. (4.43) into (4.42) and taking the reciprocal of both sides,


1

J

I l II A

¬ D 1 K I 1 . * J A Vm c A ®

167

(4.47)

Thus, a plot of the reciprocal of the unidirectional flux against the reciprocal of the cis solute concentration will be a straight line if the carrier follows the simple model. The I slope of the line is K/Vm, and the 1/cA -intercept is –1/K. If the carrier data do not fit the simple model, some of the assumptions of the model can be relaxed. Relaxation of these assumptions is inevitably accompanied by an increase in the number of unknown parameters of the system, and one must be careful when fitting a limited amount of experimental data to models having many adjustable parameters. The modifications to the model can relate to the boundary conditions at the membrane surface or the mechanism of the carrier itself. Regarding the former category, Thomson and Dietschy (1977) showed that when unstirred layers are in series with a simple carrier, the dependence of the substrate flux on the bulk concentrations is very different from Eq. (4.39). Some of the complexities that have been added to the model of the carrier itself are described below. 1. The Rate Constants Are Substrate Dependent. As indicated earlier, the rate constants for X and AX may differ. In this event, the rate constant Dc is replaced by two constants — DcX and DcAX — which govern the flux of free and loaded carrier, respectively. The flux equations are given by Eq. (4.37), with Dc replaced by D aj . Equations (4.38b) and (4.38c) are modified accordingly. Solving for JA,

JA

a DXa XT K (c AI c AII ) DAX . a c AII c AI K (c AI c AII )( DXa DAX a ) 2 DXa K 2 2 DAX

(4.48)

The transport system remains symmetric. Equation (4.48) reduces to (4.39) when DcX = DcAX. See also #5 below. 2. The Equilibrium Constants Are Asymmetric. The equilibrium constants i at the two sides of the membrane may differ, in which case K is replaced by K in Eq. (4.36). 3. The Rate Constants Are Asymmetric. The rate constant for transitions from Side I to Side II may be different from that for transitions from Side II to Side I. Such asymmetry is plausible: these mediators have different conformations when facing each side of the membrane, and it is not unreasonable that different conformations might exhibit different transition rates, as well as different substrate affinities. The solution of the rate equations when the rate constants depend on the direction of the transition but are the same for X and AX is left as an exercise. When the rate constants for X and AX are also assumed to differ, the single rate constant of the simplest model is replaced by four rate constants, and Eq. (4.37) becomes

J j D aj c Ij Da j c IIj ( j X , AX ) .

(4.49)

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Equations (4.38b) and (4.38c) change accordingly. In Eq. (4.49), the unidirectional rate constants are distinguished by their subscripts: Dcj is the rate constant for the transition of the jth form of the carrier from Side I to Side II, and Dc–j is the rate constant in the opposite direction. Asymmetric carrier systems exhibit rectification: when the concentrations of the solutions bounding an asymmetric carrier are interchanged, the magnitude of the substrate flux changes. The “efficiency” of the rectifier can be expressed in terms of the ratio of the fluxes in the two directions (Krupka and Deves, 1983). The equilibrium constants and transition rate constants of asymmetric systems are not independent. Facilitated diffusion is a passive process, and the substrate flux must be zero when c AI c AII . It can be shown that, for this requirement to be met, the following must hold: II

I

DcAXDc–XK = Dc–AXDcXK .

(4.49a)

This condition is met trivially for symmetric systems. Asymmetry can arise if the substrate A is charged and is being transported through an electrostatic potential difference (see #5 below), or if a significant standard free energy change accompanies the transition of the bound or free carrier from one side to the other. In the latter case, the partition of free carrier between the two sides at equilibrium is

'G 0 ¬ c XI ¬ II K X exp RT ® cX ®

,

(4.50)

equilibrium

0

where 'G is the difference between the standard state free energy of the carrier when it is facing Side I, and that when it is facing Side II, and KX is the equilibrium constant II I for the transition between X and X . At equilibrium, the rates of transition in each direction are equal: I

II

DcXcX = Dc–XcX (equilibrium), whence Dc–X = KXDcX (equilibrium). 0

If 'G is small, KX | 1, and the two rate constants are approximately equal. Otherwise, the rate constants are asymmetric. 4. The Uptake and Release of Substrate Are Not at Equilibrium. In this case, Eq. (4.36) is replaced by rate equations for the binding and unbinding processes. Setting the substrate flux equal to the net rate of binding at Side I, and that of release at Side II, I II J A kbI c AI c XI krI c AX krII c AX kbII c AII c XII ,

where kbi (kri ) is the rate constant for binding (release) at Side i.

(4.51)


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Weiss (1996a) presents the most general case of single substrate binding, in which binding and release are regarded as rate processes as in Eq. (4.51), and the transition rate constants are asymmetric and substrate dependent. The requirement that the flux II I I II be zero when 'cA = 0 implies DcAXDc–X kr kb = Dc–AXDcXkr kb . An interesting result is that the relation between flux and the bounding concentrations can be expressed in terms of only four parameters that in turn are functions of the four interfacial rate constants, the four transitional rates, and XT. Thus, it is impossible to obtain all the rate constants of the model from experimental flux measurements. 5. The Carrier–Substrate Complex Bears a Charge. If, for instance, X is neutral and A is an ion, then AX is charged. In that event, the kinetics of the transition of the complex across the membrane are influenced by the membrane potential, since the potential difference adds to the carrier concentration difference, in a fashion similar to the energy barrier models of channel transport described earlier. If X is neutral and AX is charged, DcX z DcAX and Dc–X z D'–AX because the charge status of the two species differs, and D'AX z D'–AX because the electrostatic potential gradient favors one transition and opposes the other. The flux equations for ion transport via a charged complex, when the uptake and release reactions are not at equilibrium, are summarized in Lauger (1980). 6. The Carrier Mechanism Follows the Two-Gate Model. If the carrier follows the two-gate model (Fig. 4.11b), the kinetic model of transport includes two additional states: the occluded state, with or without a substrate molecule in the space between the two gates. Referring to Figure 4.21, the vacant occluded state is interI II I II posed between X and X , and the occupied occluded state lies between AX and AX . The kinetic model is straightforward and is included among the exercises at the end of this chapter. 7. The Carrier Can Bind Multiple Substrates. When a carrier can bind and transport more than one substrate (call the second one B), its repertoire of functions is greatly increased. The versatility afforded carriers by the ability to bind multiple substrates is even greater if the binding affinities (i.e., the equilibrium constants for binding) or the transition rates of the carrier depend on what is bound to it.

When a carrier can transport more than one solute, it may function in several ways: if it carries two substrates in the same direction, it is called a cotransporter; if it carries them in opposite directions, it is called a countertransporter or exchanger. The rates of transport through such mediators depend on the transmembrane differences in the concentrations or electrochemical potentials of both A and B. Under some conditions, the carrier may employ the electrochemical potential gradient of one substrate to move a second substrate against its electrochemical potential gradient; this process is called secondary active transport and will be discussed in the next chapter. On the other hand, if both A and B bind to the same site, the transport of one substrate can be inhibited by the other as the two compete for available sites on the carrier; inhibition will be discussed in the next section of this chapter.

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Exchangers and cotransporters are found in great variety and in every cell. They deserve a subsection of their own.

4.2.9. Exchangers and Cotransporters In discussing exchangers and cotransporters, we will focus again on the red blood cell. The major anions in the erythrocyte — chloride, sulfate, bicarbonate, phosphate — are in equilibrium across the red cell membrane. The anions enter and leave the cell by a process known as exchange diffusion. The exchanger rarely crosses the membrane without an accompanying substrate molecule. Two examples of the operation of an exchange diffusion carrier are represented in Figure 4.23. Figure 4.23a shows the normal function of the carrier. The mediator can transport two substrates, A and B. It can bind either substrate at either side of the membrane, but is very unlikely to undergo a conformational transition to the other side unless its single binding site is occupied. This restriction on the behavior of the carrier can arise if the energy barrier for the conformational change of the unbound exchanger is greater than those for the exchanger–substrate complexes. The rate constants for the transition are sensitive functions of the barrier height, so D'X would be much less than D'AX or D'BX.

Figure 4.23. Exchange diffusion processes. (a) Heteroexchange. In this example, cBI and c AII are negligibly small, so the carrier rarely transits from Side I to Side II as BX, or from Side II to Side I as AX. (b) Tracer measurement of the exchange rate. The tracer is initially absent from Side II, and the flux of tracer from Side II to Side I remains negligible. The tracer flux is proportional to the unidirectional flux of A from Side I to Side II. In all cases, the unidirectional fluxes of unloaded carrier are minor and are not represented in the cycles.

Consider the behavior of such a carrier when the concentration of B in Phase I, and the concentration of A in Phase II, are negligible. Then the carrier binds A at Side I, transits to Side II, releases A into Phase II and binds B, and then returns to Side I, where it releases B into Phase I and once again binds A. This process is termed heteroexchange and can involve more than two substrates. If a substrate is present on both sides of the membrane, the carrier can carry it in either direction, just as in the


171

single substrate model described earlier. As noted above, inhibition can take place if either bounding solution contains both substrates. Returning to Figure 4.23a, the transition rate constants of the exchanger are generally asymmetric and substrate dependent. For an exchanger whose purpose is to transfer a molecule of A from Side I to Side II, in exchange for a molecule of B transported from Side II to Side I, the rate constants DcAX and Dc–BX will be greater than Dc–AX and DcBX. When these inequalities are very strong, X will carry only A from Side I to Side II, and only B from Side II to Side I. This is an example of obligatory, or pingpong exchange. A similar result is obtained if the relative binding affinities of the carrier for the two solutes depend on whether the binding sites are facing Side I or Side II; we will see examples of this in the next chapter.

Figure 4.24. Kinetic representation of a two-substrate ping-pong system. The substrates are A and B. Only the carrier states are shown.

The most well-studied exchanger in erythrocytes exchanges one chloride ion for one bicarbonate ion; bicarbonate leaves the cell and chloride enters it. This exchanger plays an important role in maintaining the volume and internal pH of most cells (Chap. 8), in anion transport across epithelial cells, and in carbon dioxide transport in the blood (Chap. 11). The transport is obligatory and can be represented by the kinetic I cycle in Figure 4.24, with AX representing carrier bound to extracellular chloride and II BX representing carrier bound to intracellular bicarbonate. Evidence suggests that chloride and bicarbonate bind to different sites on the protein; at any time, both sites are exposed to the same side of the erythrocyte membrane, and they alternate in tan-

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dem between the two sides as the carrier undergoes conformational change. The ex4 changer normally transports about 5 q 10 ions per second. A cotransporter in the erythrocyte membrane that we will discuss in Chapter 5 transports one potassium ion, one sodium ion, and two chloride ions into the cell. This carrier, like the anion exchanger, is electrically neutral, and no net charge is transferred across the cell membrane. Other molecules can “piggyback” on exchangers that are carrying ions across the membrane. An example of such a carrier is the serotonin transporter, which takes up this neurotransmitter after synaptic transmission has occurred. This exchanger actually accumulates serotonin in the cell, pumping it in against its concentration gradient via secondary active transport, a mechanism that will be discussed in detail in the next chapter.

Figure 4.25. Kinetic representation of the serotonin (S) transporter, modified from, and following the rules stipulated by, Rudnick (1996). Normal cyclic behavior is counterclockwise. The unidirectional arrows are written to reproduce the stipulated rules. In fact, these transitions are reversible, but with equilibrium constants that strongly favor the indicated directions. In Rudnick's model, the carrier binds chloride, transmitter and sodium in sequence from the extracellular synaptic cleft (Chapter 9) and discharges transmitter, sodium, and chloride in sequence on the cytosolic side. Potassium is then bound and the carrier undergoes a conformational change to expose its binding sites to the extracellular region once more.


173

The serotonin transporter cycle is shown in Figure 4.25. The exchanger brings one sodium ion and one chloride ion into the cell and carries one potassium ion out. Serotonin enters the cell in its cationic form, so the transporter is electrically neutral. Each transporter molecule possesses four distinct binding sites, one for each substrate. The transition rate constants of the exchanger depend on which of its binding sites are occupied, and this dependence governs its function. One proposed set of “rules” for the serotonin transporter (Rudnick, 1996) leads to the behavior in the figure; these stipulate that: — the transporter can flip from the extracellular side to the intracellular side only when sodium, chloride, and serotonin are all bound — the transporter can flip from the intracellular side to the extracellular side only when its binding site for potassium is occupied, or when none of the sites are occupied As shown in the figure, Rudnick's model also stipulates a specific sequence of binding and unbinding steps on the two sides of the membrane. Figure 4.23b illustrates the usual technique for measuring the exchange rate of the carrier. The concentration of a selected substrate, A, is the same on both sides of the membrane. A small amount of radiolabeled substrate is added to the solution on one side of the membrane (Side I in Fig. 4.23b). The exchange rate is computed from the rate of accumulation of tracer in Phase II. When the concentration of A is the same on both sides of the membrane, and no other substrate concentration difference is imposed, the unidirectional flux of A from Phase I to Phase II, measured by the tracer, is the same as that from Phase II to Phase I, and is termed the exchange flux; the net flux of A, equal to the difference between the two unidirectional fluxes, is zero under these conditions. Exchange diffusion systems are subject to trans stimulation, sometimes called accelerative exchange diffusion. Consider the cycle depicted in Figure 4.23a. Here, the flux of A from Side I to Side II is enhanced by raising the concentration of B in Phase II. Increases in cBII promote the binding of B to the carrier, facilitating its transition to Side I. As a result, the concentration of binding sites available to the molecules of A in Phase I is raised. Viewed another way, raising the concentration of B in Phase II increases the overall carrier transition rate from Side II to Side I. In the steady state, this rate must be the same as that from Side I to Side II. Thus, the carrier cycles more frequently, increasing the flux of A as well. In a ping-pong system, every carrier transition carries a substrate molecule. The equation for the exchange rate of a single-substrate ping-pong system is derived in Dalmark (1975); its form is similar to that of the flux equation for the simplest carrier: J Aex

Mc AI . K1/ 2 c AI

(4.52)

In ping-pong transport, and in contrast to the simple carrier model described earlier, both M and K1/2 depend on the transmembrane concentration ratio, c AI / c AII . Equation

174


(4.52) predicts a relatively simple dependence of exchange rate on substrate concentration, provided that the concentration ratio is fixed. The obligatory nature of ping-pong transport was used in an interesting way by Jennings (1982) to estimate the number of chloride exchangers in the red cell membrane. Cells containing radiolabeled chloride were placed in a medium, none of whose anions were readily carried by the exchange system. Under these conditions, few carriers that are facing outside flip to the inside; carriers initially facing inside carry one ionic load into the medium, and stop. The number of transport sites per cell was estimated from measurements of the initial loss of labeled chloride from the cells.

4.3. INHIBITION OF FACILITATED TRANSPORT Both channels and carriers are subject to inhibition: the decrease, limit, or block of their action or function as transport mediators. Here we focus on the inhibition of facilitated transport by a solute other than an “intended” transport substrate. Transport may also be inhibited by physical forces, such as by voltage clamping of gated channels, as described earlier. The mechanism of inhibition depends on the identity of the mediator. Inhibitors can interfere with channels by blocking the path of the desired substrate, either by virtue of their size or by attaching to binding sites intended for the substrate; this is called channel block. Inhibitors can enter from either end of the channel. They can also affect gating kinetics. Inhibitors of carrier transport operate primarily by interfering with substrate binding; ligand-activated channels can also be inhibited in this way.

4.3.1. Inhibition of Channel Transport: Channel Block It should be clear by now that much of the research on channel behavior has been motivated by the important role of ion channels in nerve and muscle function. Correspondingly, a large number of studies of channel block have dealt with the effect of toxins, anesthetics, and other drugs on native channels. These studies and others involving substances that are not natural to the organism, though they have lent considerable insight into channel structure and function, are outside the scope of this text, and the reader is once more referred to Hille (2001) for a thorough discussion of the subject. An example of channel inhibition by competition between naturally occurring ions is the voltage-dependent block of inward-rectifying potassium channels by intracellular ions. As noted earlier, these channels are called inward rectifiers because they exhibit a greater potassium conductance when the electrochemical potential gradient for potassium is such that the ion flux is directed inward. Block is seen as a further reduction in outward current. We saw earlier that potassium ions pass through these channels in a queue. Other intracellular ions, such as magnesium and larger molecular ions, will also migrate into the channel, but will not easily pass through the selectivity filter. The interfering ion will be trapped at the filter, with potassium ions backed up behind, and the transport channel will remain blocked until the blocking ion either passes through the filter or, by Brownian motion, works its way out of the pore and back into the cytosol.


175

Hille and Schwarz (1978) modeled this kind of inhibition using an energy profile model consisting of three wells representing a series of binding sites accessible to both potassium and the inhibitor, terminating in an energy barrier at the extracellular end of the pore that the inhibitor could not cross. The model hypothesized that more than one binding site could be occupied at a time, and that the presence of an ion in one well affected the rate constants for the binding or unbinding of a second ion at an adjacent well; the latter was included to reflect electrostatic repulsion between ions of like charge at closely spaced binding sites. A family of current–voltage curves predicted by the model is shown in Figure 4.26. It is notable that the outward potassium current exhibits a maximum as the intracellular potential becomes increasingly positive; at high membrane potentials, it is virtually impossible to extract a plugging cation from the pore once it has reached the impermeable energy barrier.

4.3.2. Inhibition of Carrier Transport In the models of carrier transport analyzed in the previous section, the transport rate depended on the ambient concentrations of only the designated substrate, A. Often, other solutes can compete with A for the carrier's binding sites, or block A's access to these sites. When this happens, JA decreases. Competitive Inhibition. In competitive inhibition, other solutes, in addition to A, can bind to X. To examine mathematically the implications of substrate competition, consider the simplest case, in which a carrier has a single binding site that reversibly binds B as well as A. The equations of the previous section are modified as follows: I II 1. Conservation of Carrier. Two additional concentrations, cBX and cBX , are added to the left-hand side of Eq. (4.38a). 2. Steady State. The governing equation becomes JX + JAX + JBX = 0, where I II J BX D a (cBX cBX ) . The transition rate of BX is assumed to be the same as that of X and AX. 3. Equilibrium. Two additional equations describe the BX U B + X equilibrium at the two sides of the membrane: cBi cXi K B (i I, II) . i cBX

(4.53)

These last two additional equations are needed, since there are two new unknowns, I II cBX and cBX . Solving for JA, JA

D aXT 2

¯ c AI c AII ¡ °. ¡ c I K (1 c I / K ) c II K (1 c II / K ) ° B B A B B ± ¢ A

(4.54)

Weiss (1996a) presents the flux equation for competitive inhibition with rate constant asymmetry, while retaining the assumption that the transition rates of the carrier remain unaffected by substrate binding. The dependence of the flux on the terms in the bracket is the same as in Eq. (4.54), but the maximum flux is equal to DcjDc–jXT /(Dcj+Dc–j), which reduces to DcXT/2 when the rate constants in each direction are equal.

176


Figure 4.26. Energy barrier model of voltage-dependent block. (a) Energy barrier model. The outside phase contains potassium; the inside phase contains potassium and an cationic inhibitor, B. Potassium can cross all barriers, and B can cross all but the leftmost one. (b) Model predictions of potassium flux as a function of membrane potential (inside relative to outside). Curves are shown for fixed internal concentrations of B and K and an external concentration of K that varies over a factor of 100, with Curve 1 corresponding to the lowest value and Curve 5 corresponding to the highest value, equal to the fixed internal concentration. The units of the ordinate, which measures outwardly directed potassium flux, are arbitrary. Reprinted with permission from Hille and Schwartz (1978). Copyright © 1978, The Rockefeller University Press.


177

Comparing Eq. (4.54) to (4.39), we see that the binding of B effectively increases II the value of K in both terms in the brackets. When cA = 0, the second term vanishes I and K in the first term is increased by KcB /KB. Referring to Figure 4.22, an increase in I K raises the value of cA needed to achieve a flux equal to Vm/2; thus, the flux curve is I shifted to the right, reducing the transport rate of A for any value of cA . The reduction I in the rate of transport of A increases as cB is raised. The effectiveness of B as an inhibitor depends on the ratio K/KB, which is a measure of the selectivity of the carrier. For instance, if the carrier's affinity for B is stronger than it is for A, then KB is small compared to K, and the influence of a given concentration of B on the flux of A is relatively large. When B binds irreversibly to the carrier, the inhibitor eventually occupies all the carrier binding sites and, in the only attainable steady state, JA = 0. This result is obtained from Eq. (4.54) by setting KB = 0. Carriers subject to competitive inhibition exhibit trans stimulation even when the transition rate constants are symmetric and independent of the occupancy of the carrier binding sites. The flux given by Eq. (4.54) is proportional to the difference between two ratios, each of which is reduced by the addition of B. By our convention, the first ratio — evaluated at Side I — is the larger, and adding B to Phase I lowers JA. However, adding B to Phase II reduces the second ratio, increasing the difference between the two, and raising JA. This instance of trans stimulation can be understood in physical terms as follows: II . This increases the driving force for B competes with A for X at Side II, reducing c AX the transition of AX from Side I to Side II, causing the transition rate JAX to increase. Equation (4.54) is the basis of a simple experimental test for the presence of competitive inhibition. The flux of radiolabeled A (call it A*) from Phase I to Phase II is measured at several inhibitor concentrations, cBI . The concentration of tracer in Phase II is negligible, and Eq. (4.54) simplifies to J A*

ac AI * , c AI * b

(4.55)

where a = DcXT/2, a constant. The second term in the denominator, b = K(1 + cBI /KB), is a linearly increasing function of inhibitor concentration. Rearranging Eq. (4.55), J A* a

bJ A* . c AI *

(4.56)

Thus, in the presence of competitive inhibition, plots of JA* vs. JA* / c AI * are linear, with an intercept (a) that is independent of inhibitor level. The slopes of these plots (–b) are negative, and become more so as the concentration of inhibitor in Phase I is raised. Substrate B can competitively inhibit the transport of A without crossing the membrane. Suppose DcBX = 0. Then, the carrier is effectively immobilized when B is bound to it. Modifying the earlier equations accordingly (DcBX = 0, JBX = 0), and assuming that cBII = 0, the flux of A is

178


JA

D aKXT K B (c AI c AII ) . ( K c AII )(2 K B c AI 2 KK B KcBI )

(4.57)

It can be seen from Eqs. (4.54) and (4.57) that the effect of the inhibitor can be overcome to some extent by raising the concentration of A in Phase I; indeed, when A is absent from Phase II, the maximum substrate transport rate (which is obtained by letting c AI become large) is equal to the maximum rate in the absence of inhibitor.

Figure 4.27. Kinetic representations of two inhibited carrier systems. For clarity, only the carrier states are shown. (a) Two substrates (A1, A2) and an immobilizing inhibitor, B. (b) Irreversible inhibition by an inhibitor that can bind to all forms of the carrier. An expression for the carrier inactivation rate in terms of the inhibition rate constants and the equilibrium constants for substrate binding is given in Krupka and Deves (1983).


179

The cases shown above are among the simplest models of reversible competitive inhibition. Many more situations have been treated by Krupka and Deves (1983). These include: the transport of two competing substrates in the presence of an immobilizing inhibitor; the formation of a ternary complex of substrate, inhibitor, and carrier; and a combination of these, whose kinetic scheme is shown in Figure 4.27a. Krupka and Deves also treated the effects of asymmetry in the equilibrium constants for inhibitor binding, KB, and competition between immobilizing inhibitors on the same side of the membrane. As remarked earlier, when the inhibitor binds irreversibly to the carrier, there is no substrate flux in the steady state. The experimental measurement of interest in this case is the rate at which the carrier is inactivated. The kinetic representation of an irreversible inhibition process is shown in Figure 4.27b. Noncompetitive Inhibition. Carrier-mediated fluxes can also be inhibited by solutes that do not bind to transport sites. For instance, the flux of A can be inhibited by added B if the latter binds to a site close to A's transport site, thereby blocking A's access to the carrier. The monosaccharide carriers in the erythrocyte can be blocked by large molecules, such as glycosides and corticosteroids, and noncompetitive inhibition is thought to be the mechanism by which this takes place.

PROBLEMS: CHAPTER 4 1.

The selectivity of a membrane for three of the alkali metals is Na > Li > K. Recognizing that smaller ions have more negative energies of hydration and binding, provide a set of values of these energies, in arbitrary units like those in Table 4.3, that explains this selectivity sequence.

2.

Derive Eq. (4.4) from the preceding equations. Show that the fraction of channels that are open depends only on E/D, and sketch the dependence. Why does doubling both rate constants not affect NO/NT? What does doubling both rate constants do to the kinetics of channel opening?

3.

The membrane of the squid axon is called the axolemma. Suppose a piece of axolemma is clamped between two identical physiological solutions of sodium chloride and a 90-mV potential difference is applied across it. What is the current density through the sodium channels in the membrane? The needed channel properties are in Table 4.5.

4.

Represent transport through a pore that contains a single binding site by an energy profile in which the barriers on each side of the site are of different height. In general, transport through such a pore is defined by four rate constants. (a) Solve for the steady-state flux across the membrane in terms of the rate I II constants and the bounding concentrations ci and ci ; i.e., derive Eq. (4.15b).

180


(b) Using the relationships between the rate constants and the energy levels of the barriers and wells with respect to the external solution, show that the flux across the membrane is proportional to 'ci. (c) Repeat (a) for the case in which there is a transmembrane potential difI II ference '\ = \ – \ and the binding site is electrically midway between the faces of the membrane. 5.

In Eq. (4.16), the ion flux depends on the total concentration of channels, the concentration of ion on the cis side, and three rate constants. Describe how the ion flux varies as each of the rate constants is separately increased, and explain why, in physical terms.

6.

Start with the data for the conductance dependence of the glycine receptor channel in Figure 4.13. (a) Relate Km and Jmax for a single channel to the rate constants in Eq. (4.16). Km is the Cl activity at which the flux through the channel is half of its maximum value. (b) Show how the conductance–activity curve changes as each of the rate constants is separately doubled, and explain the change.

7.

Construct the state diagram for a channel containing three sites in series, where the sites are so close that two adjacent sites cannot be occupied at the same time.

8.

The parameters of the mouse ACh receptor reaction pathway given by Eq. (4.20) have been derived by Auerbach and Akk (1998) from patch clamp measurements 8 on transfected kidney cells at 22–24ºC; they are k1 = k2 = 1.1 q 10 /M-s, k–1 = k–2 4 4 3 = 1.8 q 10 /s, E = 5 q 10 /s and D = 1.2 q 10 /s. Using these data, (a) Calculate the fraction of doubly bound receptors that are open in the steady state; i.e., N A2 R* /( N A2 R* N A2 R ) . (b) Calculate the mean lifetime of a single opening. (c) What fraction of the transient closures to the A2R state result in reopening? Why is this state seen as a brief flicker when the channel reopens?

9.

A useful parameter of voltage-dependent gating is '\0.5, the membrane potential at which half of the channels are open. Use Eq. (4.24) to relate '\0.5 to the standard free energy change of channel opening, and rewrite Eq. (4.23) in terms of 0 '\0.5 instead of 'G .

10. Pick the data points off Figure 4.18 and fit them to Eq. (4.24) to find the effective gating charge of the sodium channel in brain, and the equilibrium constant for channel opening. Then compute the predicted open fraction at each potential from the best fit and plot the results on the original figure to see how well the model fits the original data.


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11. Assume that the sigmoidal curve describing the open fraction of a voltageactivated potassium channel is very steep, so that the fraction of open channels increases discontinuously from zero to one at a threshold voltage Et. Sketch the current–voltage curve for NT of these channels, when (a) Et < EK (b) Et > EK EK is the Nernst potential of potassium across the membrane. 12. Sketch a current–voltage curve, like that in Figure 4.19c, for a membrane that contains voltage-gated cation channels that are increasingly likely to be open as the membrane potential becomes more negative. Explain any differences from the curve in the figure. 13. From the variance of sodium current fluctuations at the frog node, Sig4 worth (1980) determined that the node possessed N = 4.2 q 10 channels, each carrying a unitary current of i = 0.34 pA (picoamperes) under the voltage clamp employed. (a) Sketch the variation of the current variance with mean current, as given by Eq. (4.32). How would you find N and i from this curve? (b) The driving force for the current, including the Nernst potential of Na, was 59 mV; what is the conductance of a single channel? (c) The maximum current seen during these experiments was 8.2 nA; what fraction of the channels was open at that time? 14. From the spectral density curve in Figure 4.20, derive the rate constant for gate closing, and the unitary conductance, of the endplate channel studied by Anderson and Stevens in 1973. 15. Using the values of J and D obtained by Anderson and Stevens (1973) (Fig. 4.20), calculate the quantity of charge that passes through an AChR channel during a open state of average duration when the potential across the channel is 100 mV. 16. Derive the expression for average state duration given by Eq. (4.35). 17. According to Eq. (4.39), the maximum flux attainable by a simple carrier is D'XT/2. Provide a physical explanation of why this is so, for an alternating access carrier. 18. For the simplest carrier model, what fraction of the carriers are in each state (i.e., I I II what are cAX /XT, ...) when cA = cA ? 19. A membrane containing a simple carrier specific for A separates a solution of the substrate (Phase I) from a solution in which there is no A. When the concen-

182


tration of A in Phase I is 5 mM, the flux across the membrane is 1 q 10 I 2 mols/cm -s; when cA = 20 mM, the flux is doubled. (a) What is the equilibrium constant at the interfaces? (b) What is the maximum attainable flux? I (c) What is the flux of A when cA = 0 and the concentration on the other II side of the membrane, cA , is 30 mM? –9

–10

2

20. A simple carrier system transports A at a rate of 6 q 10 mols/cm -s when the concentration of A on the cis side of the membrane is 6 mM, and there is no A on the trans side. When the trans concentration of A is raised to 3 mM, the flux falls to one-third of what it was before. What are K and Vm for this carrier system? 21. Transport across a monolayer of cells can often be treated as transport across two membranes in series: one membrane faces one bath, and the second membrane faces the other bath, with the cytoplasm between the two. Suppose that the cells in such a monolayer produce a carrier for a particular solute, A, and that the concentration of the carrier in both membranes is the same. The concentration of A I in one bath is cA , and the concentration in the other bath is zero. Write an expresI sion for the intracellular concentration of A in terms of cA , and the Vm and K of the carrier. Assume that Vm and K are the same in each membrane. 22. A carrier system whose parameters are Vm and K is placed between two solutions I II and c Ab . There are unstirred layers of thickwhose bulk concentrations are c Ab ness d at each membrane surface; the solute diffusion coefficient in the unstirred layers is DA. (a) Write an expression that can be solved for the solute flux JA as a function of the bulk concentrations in the bathing solutions, the unstirred layer thickness, the diffusion coefficient in the unstirred layer, and the transport parameters of the carrier. You do not need to solve for JA. (b) Show that JA approaches the solution given by Eq. (4.39) as d approaches zero. 23. Derive Equation (4.41). 24. You run an experiment on the uptake of sugars by red blood cells, a system you suspect is describable by simple carrier kinetics. You use radioactive tracers to make your flux measurements, which give you the flux Js as a function of the I II sugar concentration on Side I (cs ), when cs = 0. How would you plot these data to get a straight line, and how are K and DcXT related to its slope and intercept? 25. Demonstrate mathematically that Eq. (4.42) for unidirectional flux follows from the unidirectional rate of transition of the bound carrier from Phase I to Phase II, I Dcc AX.


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26. The values of K and Vm for glucose transport across the cell membrane of human 2 erythrocytes are 5.6 mM and 0.0403 μmol/cm -min. What is the initial rate of decrease of cytoplasmic glucose concentration when the plasma glucose level is reduced to half of the 4.3 mM concentration inside the cell? The cell can be modeled as a disk 7.2 microns in diameter and 2.2 microns thick. 27. The glucose concentration in the cytoplasm of the erythrocyte described above is 10 mg/100 cc and that in the blood is 85 mg/100 cc. In the steady state, the rate of entry of glucose into the cell equals the rate of glucose consumption by metabolism. What is the metabolic rate of the red cell, in kcal/hr-cell? One mol of glucose yields 456 kcal of usable energy. Both the bath and the cytoplasm may be regarded as well mixed. 28. (a) Derive Eq. (4.48) for the carrier flux when the rate constants are substrate dependent. (b) Show that JA is symmetric for this case. (c) Show that Eq. (4.48) reduces to Eq. (4.39) when DcX = DcAX = Dc. 29. Derive an equation for the flux through the simplest carrier when the equilibrium constants at the two sides of the membrane are different, and show that the carrier flux remains zero when the concentrations of substrate are the same on both sides of the membrane. 30. Derive an equation for carrier flux when the rate constants for the transition of the carrier from Side I to Side II differ from those for the transition from Side II to Side I, assuming that the rate constants for AX and X are the same. 31. (a) Construct the state diagram for the two-gate channel model. Denote each state by abc, where a is the state of the inside facing gate (Open or Closed), b is the state of the substrate binding site (occupied by A or Empty), and c is the state of the outside facing gate (O or C). Thus, the state of the channel when both gates are closed and the binding site is occupied is CAC. Recall that both gates cannot be open at the same time. (b) Assume that the transitions in the state diagram in which substrate binds to or dissociates from the channel binding site (E U A) are at equilibrium with dissociation equilibrium constant K, and that all gating transitions (O U C) have the same rate constant Dc. Write the equations from which the flux of A through the system, JA, can be found, usout in ing XT, cA and cA as for the simple carrier. (c) Solve the equations for JA. 32. Explain why the Cl–HCO3 exchanger in the normal red cell is expected to be asymmetric, and give one reason that the Na–K–2Cl transporter might be.

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33. Explain the shape of the curves in Figure 4.26, and their dependence on external potassium concentration. 34. Using Eq. (4.54) for carrier transport in the presence of competitive inhibition, derive an equation for the flux of A as a function of 'cB when the concentration of A is the same on both sides of the membrane. Sketch the variation of JA with 'cB for positive and negative values of the latter quantity. 35. Using Eq. (4.54) for carrier transport in the presence of competitive inhibition, II I derive an equation for the flux of A as a function of cB when cB = 0. Sketch the II variation of JA with cB . 36. A membrane contains two sets of carriers for the same solute, A. They are similar in all respects, including their equilibrium constant for binding A, except that one has a binding site that allows it to be irreversibly inhibited by B. In the absence of –9 2 B, the flux through the membrane is 6 q 10 mols/cm -s when the concentration of A at Side I is very large and the concentration of A at Side II is zero. The flux I –10 2 is halved when cA is reduced to 40 mM. The flux falls to 8 q 10 mols/cm -s II I when cA is raised to 20 mM, maintaining cA at 40 mM, and enough B is added to inactivate the inhibitable fraction. What fraction of the carrier is inhibitable?

5 ACTIVE TRANSPORT

INTRODUCTION In free diffusion, and in facilitated transport, the driving force for the flux of a particular solute depends on only the concentrations or thermodynamic potentials of that species at the two faces of the transport barrier. Such transport processes are passive, and can have either of two outcomes: 1. If at least one of the phases is not practically infinite in extent and the solute is neither consumed nor synthesized, then the process will “run down” to equilibrium as the passive flux acts to equalize the chemical potentials on the two sides of the barrier. 2. If the solute is consumed or synthesized on one side of the membrane, then a nonequilibrium steady state can be achieved, such that the rate of solute transport across the membrane equals the rate of consumption or synthesis. The most cursory inspection of biological systems quickly reveals situations very different from either of those described above. In some cases, it is found that species move “uphill” from a phase in which their chemical (or electrochemical) potential is low to one in which it is higher. In other cases, a nonmetabolized solute is present on both sides of a membrane across which it can pass, but no net flux of the species can be demonstrated even though its chemical (or electrochemical) potential on one side of the membrane is different from that on the other side. These situations, illustrated in Figure 5.1, are most commonly the result of an interaction between metabolism and transport called active transport. Active transport is the subject of this chapter and is unique to biological systems. This chapter is organized rather like the preceding one. The first section is a general overview of active transport and includes a few words about its driving force. The distinction between primary and secondary active transport is presented, and means for identifying the presence of active transport are briefly reviewed. The second section describes the transport mechanisms in some detail. Finally, in the last section, kinetic analyses are presented for some of the more tractable cases, and the effects of internal slippage and external leaks on the net flux produced by active systems are discussed. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_5, © Springer Science+Business Media, LLC 2008

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186

CH. 5: ACTIVE TRANSPORT

Figure 5.1. Two transport situations that cannot be explained by the passive transport concepts in the previous chapters. (a) Uphill transport. (b) A nonequilibrium state without net flux.

5.1. ACTIVE TRANSPORT: GENERAL CONSIDERATIONS The two situations in Figure 5.1 can easily be distinguished from passive transport effects by locating them on a plot of flux versus electrochemical potential difference. This is done in Figure 5.2. In all the passive processes we have considered so far, the flux of a solute is zero when it is in equilibrium across the membrane; that is, Ji = 0 i = 0. The flux increases as the electrochemical potential difference inwhen 'P creases, and is always directed toward the phase in which the electrochemical potential is less. For thermodynamic reasons to be described with greater rigor in the next i is termed the conjugate driving force for the passive species flux, Jip. chapter, 'P

5.1.1. Metabolic Coupling and Affinity In Figure 5.2, the curve for passive transport lies in the first and third quadrants and passes through the origin. In contrast, uphill transport processes occupy the second (or i -axis that appear to fourth) quadrant, and give rise to nonzero intercepts on the 'P violate the equilibrium condition. As shown in Figure 5.2, both situations in Figure 5.1 reflect a displacement of the flux curve from that for a purely passive process:

J i J ip J ia .

(5.1)

In thermodynamic terms, the quantity J ia reflects a coupling of the flux of the ith species to a driving force that is not its conjugate driving force. When the nonconjugate driving force that gives rise to J ia derives from a metabolic reaction, we refer to this component of the total flux as the rate of active transport of the ith species.


187

Active transport systems can be regarded as transducers that convert chemical energy, obtained from metabolic reactions, into transport work. This is generally accomplished by using chemical energy from ATP to induce a change in the conformation of a carrier protein, which is also the ATPase that catalyzes the reaction. The energy is transferred to the protein by phosphorylation, in which the high-energy terminal phosphate on the ATP is relocated to the carrier. When the energy from ATP drives active transport directly (as in primary active transport; see below), the conjugate driving force for the active flux is the affinity, A, of the phosphorylation reaction:

A = 'Preaction vi Pi vi Pi , reactants

(5.2)

products

where {vi} are stoichiometric coefficients, as in Chapter 1. Since these reactions necessarily provide energy, A > 0.

Figure 5.2. Solute flux vs. electrochemical potential difference. (a) Passive transport. (b) Flux in the presence of an active transport contribution. The static head condition corresponds to that in Figure 5.1b. In the present figure, the active transport rate J ia is much less sensitive to %P i than is the passive flux J ip .

Many active transport systems also convert chemical energy into electrostatic potential energy by contributing to the potential difference across the membrane. The membrane potential can influence a variety of cellular and transport events (e.g., the modulation of voltage-dependent channels) not directly related to the active process itself.

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5.1.2. Classification of Active Transport Processes When phosphorylation by ATP is directly coupled to the flux of the actively transported solute, the process is referred to as primary active transport. This mechanism is exemplified by a number of ion transport systems in cell and organelle membranes, and is shown schematically in Figure 5.3a. Generally, this process causes the chemical potential of the transported solute to be higher on the side of the membrane toward which active transport is directed; hence, such systems are commonly described as “pumps,” pumping “uphill.” Primary active transport creates a driving force for passive, downhill backflux of the substrate. This passive flux can in turn be coupled to the flux of a second species, driving it uphill against its chemical potential gradient. Such a coupled process is called secondary active transport and is illustrated in Figure 5.3b. The solute whose pump-generated gradient drives the secondary flux is called the cosolute of the secondary transport process. Secondary transport is used by cells to take up many solutes, such as amino acids and sugars, for which there are no specific ATPases. The most common cosolute used by secondary transport systems is sodium ion.

Figure 5.3. Two classes of active transport. (a) A primary transport system that moves sodium ions from the inside (i) of the cell to the outside (o), and potassium ions in the opposite direction. (b) A secondary transport system that uses the passive influx of sodium to accumulate a sugar, S, in the cell. Na is the cosolute.

Primary active transport carriers that enzymatically phosphorylate themselves are known as P-type ATPases. In animal cells, only three primary transport systems of this kind are known. These are: 1. The sodium–potassium exchange pump, which exchanges two potassium ions for three sodium ions per ATP molecule hydrolyzed. Potassium is transported into the cell and sodium is carried out. This ubiquitous pump is responsible for the nonequilibrium state of these ions across the cell


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membrane; the pump creates a large transmembrane difference in the electrochemical potential of sodium, which drives most of the secondary transport systems that bring nutrients and metabolites into the cell. We will review the mechanism of this pump later in the chapter. 2. The plasma membrane calcium pump, which transports calcium out of the cell, reducing its intracellular concentration to submicromolar levels, while the extracellular calcium level is ca. 1.2 mM. The transporter has a much higher affinity for calcium when it is facing the interior of the cell, allowing it to bind the ion from cytosolic levels of 0.1–0.2 μM [note that this usage of “affinity” refers to the strength with which the carrier binds substrate, rather than the chemical energy available for transport defined in Eq. (5.2)] The sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump, which is found in endoplasmic reticulum and muscle and will be described in Chapter 9, differs structurally from the plasma membrane pump but probably operates by a similar mechanism. The plasma membrane calcium pump transports one calcium ion out of the cell per ATP molecule hydrolyzed. The pump can be activated by other phosphorylating sources, such as the kinases that are part of the cell's regulatory system (Chap. 8). 3. The hydrogen–potassium exchange pumps that have been found in stomach, colon and kidney. The gastric H,K–ATPase acidifies the stomach with hydrochloric acid. The steps in H–K exchange are similar to those of the Na–K exchanger described below, including the existence of two conformational states of the protein and the presence of occluded states (Sachs et al., 1995). The pump transfers one proton into the secretory canaliculus, which empties into the stomach, and one potassium ion into the cell, per molecule of ATP hydrolyzed. Since primary active transport systems are often referred to as pumps, it is understandable that two key features of the active process are commonly described using centrifugal pump nomenclature. These are: level flow, which describes the transport rate when the transmembrane difference in the electrochemical potential of the substrate is zero; and static head, which is the opposing electrochemical potential difference that reduces the flux to zero. These two operating points of the pump are shown in Figure 5.2. At level flow, the flux is generated solely by nonconjugate driving forces. In primary transport, the nonconjugate force is the affinity of the ATP reaction; in secondary transport, it is the electrochemical potential difference of the cosolute. At static head, the absence of net flux can be due to backflux (“leak”) through a passive path in parallel with the pump, “slippage” of the pump itself, or both. It has been argued that secondary active transport is not strictly active, since the nonconjugate force to which the substrate flux is coupled is the electrochemical potential difference of the cosolute and not the affinity of a metabolic reaction. We will examine this argument from a thermodynamic perspective in the next chapter.

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5.1.3. Identification of Active Transport Processes i curve As noted earlier, active transport processes cause a deviation from the J i 'P for passive transport. Therefore, to demonstrate the presence of active transport, it is necessary to measure the flux and the electrochemical potential difference of the transported species. Since these pumps reside in cellular and intracellular membranes, at least one of the required chemical potentials is intracellular. The measurement of the appropriate intracellular activity poses problems that have not been fully solved, even with the use of intracellular microelectrodes. As discussed in Chapter 1, when some of the solute is bound or otherwise reduced in activity, and a total assay is used, the chemical potential of the solute can be overestimated. An added complication is the likelihood that the distribution of active solute is not uniform throughout the cell. For instance, suppose that only a portion of the cell is accessible to solute entering from (or leaving for) the extracellular phase, so that the solute is entirely or disproportionately contained in this “transport” compartment. The concentration of solute in this compartment cannot be assayed separately. The errors in measured activity due to compartmentalization cannot be eliminated by using ionselective electrodes, and may be compounded by compartment-to-compartment variations in the activity coefficient of the solute. An uncertain fraction of the water in the cytosol is associated with the large number of polar macromolecules that inhabit the cell. This structured water is a poorer solvent than the free fraction. The presence of structured water acts to lower the size of the transport compartment. In many cases, the uncertainties in the operative chemical potential difference are so large that one cannot be certain that apparent deviations from passive transport behavior are real. As a result, the identification of active transport is usually made by observing the effects of various agents on the transport rate. For instance, the supposed pump should be slowed or stopped by low temperatures, metabolic inhibitors such as cyanide, or transport inhibitors such as ouabain. However, even these approaches can be ambiguous, particularly when metabolic inhibitors, which reduce the availability of ATP and thereby affect a multitude of cellular processes, are used. The kinetic mechanisms of active transport proteins are usually examined in isolated preparations or as expressed in other cells, such as oocytes, in which flux and activity measurements can be more readily be made

5.2. MECHANISMS OF ACTIVE TRANSPORT The mechanisms by which solutes are actively transported across membranes can be regarded as extensions of the mechanisms of carrier-mediated diffusion discussed in the previous chapter. The main difference is that, in active transport, ATP hydrolysis or the transport of a cosolute is coupled into one or more of the steps in the carriermediated process, driving it to maintain a nonequilibrium state. Any of these steps may be involved: the binding of the substrate to the carrier at the cis side of the membrane, or its release at the trans side, or the conformational change of the carrier itself. This section will be devoted to these mechanisms.


191

The transport models we will discuss in the first part of this section and in much of the following section will be developed as extensions of the simple carrier model of the previous chapter. These will demonstrate the basic mechanisms on which biological active transport relies. The real systems are often more complex than the models we present below, and will be described along the way. The term scalar active transport has been used to describe transport driven by alterations in the binding and unbinding rate constants at the membrane interfaces; when modifications in the translocation kinetics are involved, the term vectorial active transport has been used. Real active transport mechanisms can have both scalar and vectorial aspects. For clarity, we will begin by examining a model that is unambiguously scalar, in which ATP or a cosolute act on only the substrate affinity of the carrier, and on only one side of the membrane. Throughout, we will assume that the pump is oriented to drive substrate from Phase I (the cis side) to Phase II (the trans side).

5.2.1. Scalar Active Transport: Overview In the simplest model of passive carrier transport that was analyzed in Chapter 4, the flux of carrier-bound solute across the membrane was proportional to the concentraI II tion difference c AX c AX . If c AI c AII , the concentration difference is positive, and solute flows passively from Phase I to Phase II. In scalar active transport, ATP or the cosolute modify the interfacial carrier concentrations so that 'cAX is positive even when the concentration of A, the substrate, is less in Phase I than in Phase II. This cannot happen in facilitated transport, even if the equilibrium constant for binding is different at the two faces of the membrane. How does it happen in active transport? Suppose the carrier exists in two forms, X and Y, normally in equilibrium with one another. The forms can be chemically different, or differ only in conformation. Form X is the transport state, which has a high affinity for the substrate. Form Y has a much lower affinity for the substrate, and binds A weakly, if at all. The transport cycle for such a carrier is shown in Figure 5.4. It is not very different from those in Chapter 4. The transformation of the carrier from Y to X is termed activation, occurs at Side I, and promotes carrier binding of substrate. Substrate crosses the membrane passively as AX. After the substrate has been released at Side II, X deactivates to Y, which returns to Side I, again passively. In this model, X does not cross the membrane unaccompanied by A. We will assume for now that Y does not bind A. The principles discussed below continue to hold (though the analysis becomes more difficult) if X can cross the membrane, or if Y can also transport A. We will discuss these alternatives later in this chapter. I The cyclic process shown in the figure can be driven in either of two ways: c AX can be raised by facilitating the conversion of Y to X at Side I, thereby increasing the II availability of active carrier; or c AX can be lowered by accelerating the conversion of X to Y at Side II. Either of these effects will increase the driving force for the translocation of AX.

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Figure 5.4. Cyclic representation of a scalar active transport carrier. The activation step takes place at Side I, and deactivation takes place at Side II. Dashed lines indicate the processes by which the pump can be driven: (1) primary activation [Eq. (5.3a)] or secondary cotransport [Eq. (5.5)]; (2) primary deactivation [Eq. (5.3b)] or secondary countertransport [Eq. (5.6)]. Either the activation step or the deactivation step is driven, but not both.

In principle, both primary and secondary scalar transport mechanisms are possible. However, while secondary systems are well described by the models that follow and their natural extensions, the sequence of reactions and conformational changes that P-type ATPases undergo are far more complex than the one-step scalar activation process contemplated here. For the sake of completeness and to increase our understanding of metabolic coupling, we will describe how a primary transport process can follow this simple model, but will emphasize secondary transport as the dominant example of the scalar mechanism.

5.2.2. Primary Scalar Transport In this simple model of primary scalar transport, the cycle in Figure 5.4 is driven by phosphorylation of the carrier. Activation is described by the following reaction: I

I

Y + ATP UX + ADP + Pi ,

(5.3a)

where X is the high-affinity form of the carrier. The superscript on the carrier symbol denotes the side of the membrane at which the reaction takes place. A corresponding equation for deactivation at Side II can also be written: II

II

X + ATP UY + ADP + Pi .

(5.3b)


193

The notion that a protein can be “deactivated” by phosphorylation seems contradictory, but it's only a matter of terminology. The two forms of the carrier, X and Y, are distinguished only by their different transition rate constants (only Y crosses the membrane without A) and substrate affinities (only X binds A). Thus “deactivation” by ATP signifies only that phosphorylation alters one or both of these properties, which is quite plausible.

How can coupling promote (for instance) the activation of Y at Side I? To answer this question, we compare the reaction given by Eq. (5.3a) to the activation process in the absence of coupling, which is simply I

I

Y UX .

(5.4)

The free energy change accompanying Eq. (5.3a) is the sum of the free energy changes of (1) the activation step in the absence of coupling [Eq. (5.4)] and (2) hydrolysis of the ATP. Since the hydrolysis of ATP liberates energy, the standard free energy change of the latter reaction is negative; thus, its effect is to make the free energy change of the activation step more negative. As was shown in Chapter 1, this causes the equilibrium constant of the activation reaction to be larger, raising the ratio of activated to inactivated carrier at Side I. More colloquially, the hydrolysis of ATP drives the activation reaction to the right.

5.2.3. Secondary Scalar Transport Secondary scalar transport is the principal means by which nonelectrolytes such as sugars and amino acids, and certain inorganic solutes, are actively taken up by cells. The immediate driving force for secondary scalar transport is a transmembrane gradient of the electrochemical potential of a cosolute, B. The cosolute is usually an ion. Often, it is sodium that is driven by the Na–K exchange pump to a nonequilibrium distribution across the cell membrane. The passive backflux of this ion carries the substrates of many secondary transport systems into the cell. In one form of secondary transport, the carrier is activated by binding the cosolute: I

I

I

Y + B UX .

(5.5)

X subsequently binds the substrate, A, at a different binding site, and then transits to Side II. The complex that undergoes the transition consists of carrier, cosolute, and substrate. This process is called cotransport, or symport, or positive coupling, because B and A cross to Side II together. A kinetic representation of a cotransport system is shown in Figure 5.5a. As far as transport rates are concerned, there is no difference between the substrate A and the cosolute B in this simple model. Here, the two solutes are cosubstrates, both of which must be bound to Y at Side I before it can transit to Side II. A distinction remains in terms of mechanism if the association of one species with the inactive carrier must precede that of the other species; for instance, if binding the first species in-

194


creases the affinity of the carrier for the second. Finally, there is an energetic difference between A and B if B is giving up energy by traveling down its electrochemical potential gradient and A is gaining energy by virtue of its transport into a region where its (electro)chemical potential is higher.

Figure 5.5. Kinetic representation of secondary scalar transport systems. The rate processes in these figures are reversible; their net directions when the pump is transporting A from Phase I to Phase II are shown with heavier arrows. (a) Cotransport. In this representation, B is bound first at Side I, and A is released first at Side II; alternatively, the binding and dissociation sequences can be random. (b) Countertransport. B binds to X at Side II to form Y. The systems as shown here are obligatory: the cotransporter transits between Sides I and II only as Y, in which form neither A nor B are bound, or as AX, in which form both A and B are bound; the countertransport carrier transits between Sides I and II only when either A or B (but not both) are bound.

The fact that B passes through the pump from Phase I to Phase II is crucial to the coupling process. In secondary active transport, the energy that drives the pump is effectively stored in the electrochemical potential gradient of the cosolute. The cosolute releases this energy by crossing from the phase in which its electrochemical potential is high to the phase in which it is low. In countertransport, the cycle is similar to that in cotransport, except that the cosolute binds to X at Side II. II

II

II

X + B UY .

(5.6)


195

Thus, while X = BY in cotransport, Y = BX in countertransport. The cosolute crosses the membrane from Side II to Side I and is released into Phase I. By increasing cYII , the electrochemical potential gradient of the cosolute drives the transition of Y to Side I, increasing the availability of the carrier on that side. When B is released at Side I, the carrier, once more in the active form X, can now bind A. For this pump to function, the electrochemical potential of B must be larger in Phase II than in Phase I. The flux of the cosolute is opposite to that of the substrate. Accordingly, this process is termed countertransport, antiport, or negative coupling. It is illustrated in Figure 5.5b. Cotransport. Cotransport exhibits a number of identifying features: 1. Raising the concentration of B in Phase I increases the active flux of A. This is called cis stimulation of the pump, and occurs because an increase in the concentration of B drives the reaction given by Eq. (5.5) to the right. Thus more activated carrier is available to bind A. (Similarly, countertransport is trans stimulated by raising cBII .) 2. Raising the concentration of A in Phase I, while holding that in Phase II constant, increases the rate at which B crosses the membrane via the pump. As with any carrier system, the effect of raising c AI is to increase the rate at which the substrate crosses the membrane from Phase I to Phase II. If Y cannot transport A, then substrate crosses only in the company of B, so the flux of cosolute by this pathway also increases. This is called reciprocal stimulation. 3. When the chemical potential of the substrate is the same on both sides of B changes the sign of JA. This is so the membrane, changing the sign of 'P because the only driving force for the flux of A at level flow is the electrochemical potential gradient of the cosolute. 4. When A and B cross from Phase I to Phase II only by cotransport, the flux of A is stoichiometrically related to that of B; that is,

J B vB J A .

(5.7)

where Ji is the flux of the ith solute through the pump. The coupling coefficient vB is the number of molecules of B that accompany each molecule of A through the pump. 5. The rate of secondary active transport does not respond as rapidly as that of primary transport when inhibitors are added to the solutions bathing the tissue. Rather, the transport rate decays slowly as cosolute flows through the pump while the driving gradient, no longer maintained by the inhibited primary transport process, dissipates. If the cosolute gradient is maintained artificially by addition of cosolute to the appropriate phase, the rate of active transport of A and its dependence on the concentrations of B are not affected by inhibitors of primary transport. Similarly, reciprocal stimulation can be demonstrated in the presence of inhibitors, provided that cosolute is available at Side I.

196


6. The overall free energy change associated with the secondary active transport process must be negative. For instance, in cotransport, more free energy must be lost by the cosolute, as it passively returns to Phase II, than is gained by the substrate. This is a necessary condition for any irreversible process. For cotransport of a nonelectrolyte by an ion, the following inequality must therefore be satisfied: IB P IIB ) J A (P IIA P IA ) . J B (P

(5.8)

This condition was once a source of controversy, because it appeared to fail for some systems that were thought to operate by cotransport. It turned out that this seeming inconsistency was due to an overestimate of the intracellular electrochemical poten IIB in Eq. (5.8). Early estimates of this quantity were based tial of cosolute sodium, P on total assay concentrations; however, it is now known from ion-selective microelectrode measurements that the intracellular sodium activity can be less than half of the total sodium concentration. When the intracellular electrochemical potential of sodium is computed correctly, the inequality is satisfied.

It is easy to show that the electrochemical potential difference of the cosolute can also be written in terms of the potential difference '\ across the B = z Bᑠ('\ E B ) . membrane and the cosolute Nernst potential, EB: 'P

In cotransport, at least two molecules or ions bind to the carrier on one side of the membrane (this can also be the case in countertransport if the stoichiometry of the exchange is not 1:1). The sequence of binding can be random or ordered. If the affinity of the inactive carrier for A is lower than that for B, and is enhanced by the prior binding of B, then most carriers will bind B first and then A. This characteristic is exemplified by the Na–K–2Cl (NKCC) cotransporter, a secondary symporter that is found in nearly all cell types and pumps chloride into the cell against a modest electrochemical potential difference. The stoichiometry of the electrically neutral pump is given by its title; the carrier transits from one side to the other only when no ions, or all four substrates, are bound to it. The ions bind and unbind in a specific sequence, mediated by the effect of the binding of one ion on the affinity of the carrier for the next one in the sequence. Figure 5.6 illustrates a model of this sequential process proposed by Lytle et al. (1998). When the fluxes of A and B from Phase I to Phase II are coupled stoichiometrically, Eqs. (5.7) and (5.8) can be combined to provide an upper bound on the B : accumulation of A that can accomplished by a pump whose driving force is 'P II I B . PA – PA < vB 'P

(5.9)

B , then either vB > 1, or If the chemical potential difference of A is found to exceed 'P A is actively transported across the membrane by other mechanisms. This test was used by Bichara et al. (1983) to infer the presence of multiple active transport systems for hydrogen ion in the proximal tubule of the rabbit kidney.


197

Figure 5.6. Model of Na–K–2Cl cotransport. In normal function, the net sequence of states proceeds clockwise. The empty carrier E0 opens to the extracellular space, revealing an Na binding site near the intracellular gate. Sodium binding prompts a conformational change revealing a binding site for Cl. Chloride binding similarly prompts K binding, which in turn reveals a binding site for the second Cl. After the carrier passes through an occluded state E4 closed to both the extracellular and intracellular fluids, the ions leave the carrier for the cytosol in the order in which they were bound from the external phase. After the second chloride has entered the cell, the carrier is once more in the E0 state. Reprinted with permission from Lytle et al. (1998). Copyright © 1998, The American Physiological Society.

When the chemical potentials are expressed in terms of experimental quantities, pressure effects are neglected, and the cosolute is assumed to be a monovalent cation, Eq. (5.9) becomes ln

aBI ¬ a AII ᑠ ln . v '\ B a II RT a AI ® B

(5.10)

The maximum accumulation ratio a AII / a AI is never achieved in real systems, because of inefficiencies in the pump and passive leaks of substrate from Phase II, which in symport is usually the inside of a cell. The accumulation ratio is also reduced if the substrate is consumed in the cell. Therefore, the nonconjugate driving force needed to obtain a specified accumulation ratio is always greater than the theoretical minimum. The losses that degrade the performance of active transport systems will be discussed at greater length in the next section. Countertransport. Similar considerations apply to antiport systems, including those, like the Cl–HCO3 exchanger described in the previous chapter, that exchange

198


ions across the cell membrane. Here we consider the Na–Ca exchanger, in which the influx of three cosolute sodium ions (from Phase II to Phase I) drives the removal of one calcium ion from the cell (i.e., QNa = 3). Using the notation presented earlier, AX is Ca-bound carrier and Y = BX is Na-bound carrier. It is believed that, for this exchanger, the same set of binding sites carries each ion. Since unbound carrier can bind either ion on either side of the membrane, not all transitions carry the “intended” ion. Tracer studies have demonstrated the so-called Na–Na and Ca–Ca exchanges that are a consequence of this imperfect selectivity. The net loss in free energy accompanying a single cycle of the exchanger is

a II ¬ a II ¬ ᑠ 2ᑠ ln Ca Q Na ln Na '\ '\ p 0 , I I aNa RT ® a Ca RT ® I

(5.11)

II

where '\ \ – \ is the cell membrane potential as usually defined. When the left-hand side of Eq. (5.11) equals zero, no change in free energy accompanies Na–Ca exchange; accordingly, the system is at equilibrium, there is no net flux of either ion, and static head has been achieved. The membrane potential such that the exchanger is at equilibrium for a specified intracellular and extracellular composition is called the reversal potential of the exchanger. Solving Eq. (5.11) for the minimum intracellular calcium level sustainable by the exchanger, which occurs at static head, I II ln aCa,min ln aCa Q Na ln

II aNa ᑠ'\ (Q Na 2) . I RT aNa

(5.12)

Equation (5.12) shows the leverage afforded by the coupling coefficient and membrane potential in reducing the minimum intracellular calcium level. Exponentiating both sides of Eq. (5.12), and setting QNa = 3, 3

I ¬ ᑠ'\ ¬ I II aNa aCa,min aCa II exp . aNa ® RT ®

(5.13)

Because of the larger number of cosolute molecules carried by each cycle of the pump, a tenfold transmembrane ratio of cosolute activities translates into a thousandfold ratio of calcium activities at static head. This effect derives from the activity dependence of the chemical potential, is based on only pump stoichiometry, and does not require that either solute be charged. The third factor in the right-hand side of Eq. (5.13) arises because each cycle of the pump brings one net positive charge into the cell, which is favored by a negative intracellular potential; if QNa were equal to two, the pump would be electrically neutral and this term would vanish. Examples of symporters include the carriers that allow absorptive epithelia such as intestine and kidney to accumulate amino acids and sugars; these systems, for which sodium ion is the cosolute, will be discussed in Chapter 10. Typical antiporters include the Cl–HCO3 and Na–Ca exchangers that have already been discussed. The effect of membrane potential on transporter performance seen for the Na–Ca ex-


199

changer is also seen for sodium-driven cotransporters. Even when the substrate is negatively charged (e.g., certain amino acids, carboxylic acids, sulfate and phosphate ion), the coupling coefficient is such that each cycle of the pump carries net positive charge into the cell. It has been proposed that the concentration gradient of an amino acid brought into the cell via a secondary transport system can drive by countertransport the accumulation of a second amino acid by the cell. The uptake of the second amino acid by this mechanism has been termed tertiary active transport. Figure 5.7 shows the coupled cycles for this tertiary transport process and the primary and secondary processes that accompany it. Aronson (1983) has suggested that a similar mechanism may be partially responsible for hydrogen secretion by the kidney proximal tubule; the tertiary process in this case is driven by the efflux of organic anions that are accumulated in the tubule cell by a sodium gradient-driven cotransport system.

5.2.4. Vectorial Active Transport, the Curie Theorem and Substrate Activation In vectorial coupling, a nonequilibrium state is maintained by driving the translocation of the substrate from one side of a membrane to the other. Only primary active transport systems exhibit this kind of coupling: the energy acquired by the ATPase as a result of its phosphorylation is used to drive the conformational change that transports the substrate. Before describing the mechanisms by which vectorial coupling can take place, we must first consider the limitations on these mechanisms that follow from the Curie theorem. The Curie theorem states that forces of one tensorial order cannot couple with fluxes of a different order (tensors are outside the mathematical scope of this text, but the following reasoning does not depend on any specialized knowledge). The affinity of a chemical reaction is a scalar, having no physical direction. A scalar is a zerothorder tensor. Thus the energy released by ATP hydrolysis can couple into other chemical reactions, such as the activation or deactivation of a carrier, which are also scalars. The transition of a carrier from one side of a membrane to the other is clearly vectorial, having a direction, and is accordingly a first-order tensor. The Curie theorem says that the latter process cannot be driven by a chemical reaction. The Curie theorem notwithstanding, mechanisms of vectorial coupling have been proposed that are biochemically plausible. A universal feature of these mechanisms is anisotropy: the driving reaction is still a scalar, but it functions differently at each face of the membrane. Indeed, the phosphorylation reaction that drives primary active transport is generally believed to take place only when the ATPase is facing a preferred side of the membrane. Such an enzymatic reaction can easily be anisotropic, since the conformation of the enzyme when it faces Side I of the membrane differs from its conformation when it is facing Side II. This anisotropy gives a directional aspect to the chemical process, evidently sufficient to satisfy the Curie theorem.

200


Figure 5.7. Tertiary amino acid transport. The potassium exchange process is omitted from the primary transport cycle. Carrier #1 is activated by Na; Carrier #2 is deactivated by AA1.


201

In vectorial transport, the energy transferred to the ATPase by phosphorylation is used to drive loaded carrier from Side I to Side II (or, in principle, unloaded carrier in the opposite direction); there is no need to postulate the existence of active and inactive forms (though they can be added to the picture). In the first instance, I

II

AX + ATP UAX + ADP + Pi .

(5.14)

Here, the ATPase, X, exists in only one state (there is no inactive state) and binds A via an association reaction identical to that of the simple carrier of Chapter 4. It catalyzes its phosphorylation by ATP only when it is bound to A and only when it is facing Side I; this provides sufficient anisotropy to satisfy the Curie theorem. The loaded and phosphorylated carrier then undergoes a conformational change, exposing the bound substrate to Phase II. Similarly to primary scalar transport [Eq. (5.3a)], the energy in the terminal phosphate bond of the ATP drives the reaction to the right. The phosphate is subsequently released. The transport cycle corresponding to Eq. (5.14) is shown in Figure 5.8. The carrier transition that is not driven by ATP proceeds as a passive process in the opposite direction; for the cycle shown here, cXII c XI , and X returns to Side I passively.

Figure 5.8. Cyclic representation of vectorial active transport, as described by Eq. (5.14).

The enzymatic activity of the ATPase that carries A from Side I to Side II must be greater when A is bound to it; otherwise, it would be equally likely to transition to Side II when its binding site is unoccupied. Like other enzymes, its activity could increase as a result of a conformational change, in this case induced by binding A. Such an enzyme is said to be activated by the substrate.

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The kinetics of real primary transport systems are more complicated than either the scalar or vectorial examples given above, and indeed include features of both. This will be apparent from the description of the ubiquitous sodium–potassium exchange pump that follows.

5.2.5. Sodium–Potassium Exchange The sodium–potassium exchange pump is a primary active transport system driven by a cell membrane ATPase that carries sodium ions out of the cell and potassium ions in. The Na–K ATPase was discovered by Jens Skou in 1957, for which he won the Nobel Prize in Chemistry in 1997 (transport science has produced a large number of Nobel laureates, but sometimes the wait is lengthy). The stoichiometry of the pump is such that, for each molecule of ATP that is hydrolyzed, three sodium ions and two potassium ions are usually carried across the cell membrane. Although this is the most common stoichiometry, it can vary from 3:2, depending on the concentrations of the ions on the two sides of the cell membrane (Lauger, 1991). Both species are transported against their electrochemical potential gradients. The pump is present in almost all animal cells and is of immense physiologic importance, being responsible for functions as diverse as intestinal absorption and signal transmission in nerve. It produces the sodium electrochemical potential gradient that drives most secondary scalar transport processes, and plays an important role in cellular pH and volume regulation. As another indication of its importance, it consumes perhaps one-third of the ATP produced by the cell, and more in excitable tissue, where its maintenance of transmembrane differences in ion electrochemical potential is essential to nerve and muscle function. The commonly accepted mechanism of the pump, known as the Post–Albers scheme (Albers et al., 1963; Post et al., 1965), has both scalar and vectorial aspects. Na–K ATPase is the carrier and exists in two conformations. In Form E1, the enzyme's binding sites are at Side I (the cytoplasmic side) and have a higher affinity for sodium than potassium. In Form E2, the sites face Side II (the extracellular side) and prefer potassium. The sodium and potassium movements across the membrane take place consecutively in a sequence of elementary steps: I

E1•ATP + Na U Na • E1 • ATP,

(5.15a)

Na • E1 • ATP UNa • E1 ~ P + ADP,

(5.15b)

Na • E1 ~ P UNa • E2 – P,

(5.15c)

II

(5.15d)

E2 – P + K UK • E2 – P,

(5.15e)

K • E2 – P UK • E2 + Pi ,

(5.15f)

Na • E2 – P UE2 – P + Na , II


203

K • E2 + ATP UK • E2 • ATP,

(5.15g)

K • E2 • ATP U K • E1 • ATP,

(5.15h)

I

K • E1 • ATP U K + E1 • ATP.

(5.15i)

The symbols “Na” and “K” in these equations represent three sodium ions and two potassium ions, respectively. Here, Eqs. (5.15a) and (5.15b) describe substrate binding to, and phosphorylation of, the ATPase. Magnesium ion is necessary for the phosphorylation step to take place. Equation (5.15c) represents the consequent conformational change of the protein from Form E1 to Form E2. During the course of this change, which requires that Na be bound to the enzyme, some energy is extracted from the phosphate bond that was formed when the enzyme was phosphorylated. Thus the E2–P bond is less energetic than the E1~P bond. Facing Side II and in Form E2, the enzyme no longer has a high affinity for sodium, and releases it (5.15d). Since it does have a high affinity for potassium when it is in Form E2, the ATPase binds this ion from Phase II (5.15e). Thereupon, the enzyme releases its phosphate (5.15f) and binds a fresh molecule of ATP (5.15g) prior to returning to its original conformation. When the conformational change is complete (5.15h), the carrier loses its affinity for potassium, which it releases into Phase I (5.15i). The ATPase is now ready to bind sodium once more. The overall stoichiometry of the normally functioning pump is: 3NaI + 2KII + ATP U 3NaII + 2KI + ADP + Pi .

(5.16)

The overall reaction given by Eq. (5.16) ordinarily proceeds to the right, hydrolyzing ATP, accumulating potassium in the cell, and pumping sodium out. However, by the law of mass action, the reaction can be driven to the left if the opposing ion electrochemical potentials get too high (e.g., large cNaII/cNaI or cKI/cKII ). In this event, the pump operates in reverse and phosphorylates ADP to produce ATP. The primary Ca pump does this too, and it is the normal mode of operation of the ATP synthase in the mitochondrion.

There are several variants and elaborations of the Post–Albers scheme described above, some of which involve additional intermediate states of the enzyme. Robinson (1997) presents a cyclic model of the pump that includes 21 steps, but generally follows the sequence in Eq. (5.15). In another version (Sachs, 2003), K•E2 can either follow the bind–transit–release sequence in Eqs. (5.15g–i), or transit to the E1 form first, and then release its potassium before binding ATP. Included in these more recent models are occluded states, similar to those seen in the two-gate carrier model (Fig. 4.11b) and the Na–K–2Cl cotransporter model (Fig. 5.6), in which bound ions can access neither ambient phase. Furthermore, the binding of the three sodium ions, and that of the two potassium ions, to the transport protein is clearly more likely to proceed one ion at a time, rather than all at once, and the same is true for their release.

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A modification of the Post–Albers scheme that includes occluded states and sequential binding and unbinding has been offered by Balshaw et al. (2001), based on recent work by others in the field. Referring to the sequence of steps in Eq. (5.15), — sodium binds sequentially, first two ions, and then, more slowly, the third. Some transitions take place when only the first two sodium ions have bound to the enzyme. — phosphorylation of E1 is accompanied by occlusion of the (2 or 3) sodium ions. — after the enzyme has assumed the E2 form, the extracellular gate opens and the sodium ions are released sequentially: last in–first out. — potassium also binds sequentially, with positive cooperativity. Some carrier transitions take place with only one potassium ion bound to the enzyme. — dephosphorylation of E2 is accompanied by occlusion of the potassium ion(s). — the intracellular gate opens when the enzyme assumes the E1 form, and the potassium ions are released sequentially into the cytosol. Models similar to Eqs. (5.15) have been used to describe the function of other Ptype ATPases. As noted earlier, the reaction cycle of the SERCA pump will be described in Chapter 9. Eisenberg and Hill (1985) have pointed out an important relationship between the kinetic parameters of the transport process and the efficiency of the pump. In particular, the rate constants of competing steps that would decrease this efficiency are small. This ensures that the binding and release of substrates occur at the proper point in the cycle. For example, the rate constant for the reaction E1 • ATP o E1 ~ P + ADP is much less than that for Eq. (5.15a); as a consequence, E1 has time to bind sodium before autophosphorylating. Similar relationships among rate constants ensure that ions are released from the enzyme before it transits back to the side at which they were bound. In the authors' words, the rate constants “channel the enzyme” along a reaction path in which the hydrolysis of ATP is tightly coupled to the transport process.

5.2.6. Pump Selectivity and Other Properties Shared with Passive Carriers Since the proteins responsible for active transport function similarly to carriers, they share many of the characteristic carrier properties described in the previous chapter. Selectivity. Similar to passive carriers, the ATPases that participate in active transport are selective for their substrates; accordingly, each primary transport pump relies on a different transport ATPase. ATPases that exchange different ions across the cell membrane possess two sets of binding sites, one for each ion. Each site would be expected to have a higher affinity for its substrate when it is facing the side of the


205

membrane that is the cis side for that ion (for sodium, the inside of the cell; for potassium, the outside). Indeed, in the Na–K pump, there is considerable asymmetry in the binding constants for both ions. The overall selectivity of the Na–K ATPase reflects the binding constants of both sets of sites and is not perfect (although, for simplicity, the schemes described above are written as though it is); internal K competes for the Na sites in the E1 state, and external Na competes for the K sites in the E2 state. Indeed, 20% of the ions bound from the cytoplasm by the Na–K ATPase in nerve are potassium (Trachtenberg et al., 1981). Similarly, in red blood cells, the Na–K ATPase supports sizeable Na–Na and K–K exchange fluxes, which can occur only if these ions pass in both directions through the pump. Saturability and Inhibition. Other carrier properties discussed in the previous chapter, such as saturability and inhibition, are also exhibited by pump-associated carriers. As with passive carriers, the maximum pump rate is ultimately limited by the highest frequency with which the protein can undergo conformational change. Transport by the Na–K pump is inhibited by the cardiac glycoside ouabain at a concentration of 10–5 M. This is in the same range (10–6–10–5 M) as that required to inhibit the dephosphorylation of ATP by the membrane ATPase. Other agents also have corresponding effects on the transport rate and the activity of the ATPase. Other enzymes responsible for specific transport processes have been identified in this way. Ouabain binds to the ATPase at only the outside of the cell. Red blood cells can be reconstituted with their cell membranes inside out, i.e., with the cytoplasmic face of the membrane exposed to the bathing medium. The pump is also turned around, and these cells actively accumulate sodium. This uptake is not inhibited by ouabain in the bathing medium, presumably because the ouabain binding sites now face the interior of the cell. Structure. The ATPases also share some of the structural features of the carriers discussed in the previous chapter. ATPases are oligomers and, as we have already seen, have preferred orientations in the membranes in which they are embedded. Among the P-type ATPases, the structure of the SERCA pump has been characterized in greatest detail, and will be presented in Chapter 9. In addition to sharing structural features with passive carriers, the physical processes by which ATPases are thought to translocate species from one side of the membrane to the other are among those described earlier for carrier transport. The pump conformations include occluded states like those in the two-gate carrier model, and other models of active transport are similar to the alternating access model, modified to include distinct binding sites for multiple substrates.

5.3. KINETICS OF ACTIVE TRANSPORT As in the previous section, we begin with scalar active transport. Secondary scalar transport is dealt with first because it is the most natural extension of the carrier transport models discussed in Chapter 4. We then move on to primary scalar transport and

206


vectorial transport. This section closes with a discussion of the leakage processes internal and external to the pumps that limit the degree of substrate accumulation that they can achieve.

Figure 5.9. Kinetic model of a simple symport system.

5.3.1. A Simple Secondary Scalar Transport Model: Assumptions Figure 5.9 is a kinetic model of the simplest symport system, in which activation precedes substrate binding, and A binds to only X. The kinetic equations that describe this system are extensions of the equations that govern the simplest carrier, and are based on similarly extended assumptions: 1. The carrier, in either the active or inactive form, is restricted to the membrane; Y reversibly binds only B, and X (= BY) reversibly binds only A. For simplicity, it is assumed that only one molecule of A or B binds to the carrier in each cycle. 2. The only species that can cross the membrane are Y and AX; X and unbound A cannot cross the membrane. 3. The binding and unbinding reactions of cosolute and substrate are at equilibrium at both sides of the membrane. The equilibria at the two membrane interfaces are defined by two dissociation equilibrium constants, yielding four equations: Activation: cBi cYi K a (i = I, II). c iX

(5.17a)

c iA cXi K (i = I, II). c iAX

(5.17b)

Substrate binding:


207

As in Chapter 4, activities have been replaced by concentrations, and the assumption that the interface reactions are at equilibrium implies that the translocation of the carrier is rate limiting. 4. A steady or quasisteady state exists. 5. The rate at which each form of the carrier passes from one side of the membrane to the other is given by Eq. (4.37):

J j D a(c Ij c IIj ) (j = Y, AX).

(4.37)

A weakness of this assumption that is particularly relevant to active transport is its failure to include the effect of the transmembrane potential difference on the transition rate constants. The membrane potential is rarely small under physiological conditions, though it can be brought to zero in experimental preparations by short circuiting. Almost all active transport processes move charges across the membrane. Symport is no exception, since the cosolute is usually sodium. Consequently, the active transport rate is generally affected by the membrane potential. For example, the sodium-coupled solute flux into vesicles made from epithelial brush border membranes can be raised by lowering the potential inside the vesicle (Freel and Goldner, 1981). Parent et al. (1992a) measured the substratecoupled sodium flux through sodium/glucose cotransporter molecules expressed in frog oocytes, under voltage clamp, and fit the results to a saturating function analogous to Eq. (4.42); the maximum sodium current and the apparent dissociation equilibrium constant both exhibited a strong dependence on the membrane potential. Since the maximum sodium current is proportional to Dc, Parent et al. attributed the dependence of maximum current on membrane potential to a voltage-sensitive transition rate. The voltage dependence of the apparent equilibrium constant was taken at face value, i.e., as evidence of the effect of membrane potential on the activation equilibria at the membrane interfaces. They also found that negative membrane potentials enhanced sugar uptake, a result to be expected from the increased sodium driving force. The omission of membrane potential from the analysis limits the applicability of the flux equations that follow to strongly depolarized or short-circuited membranes (unless, of course, the pump is not voltagesensitive). Fortunately, many experimental investigations of active transport are carried out under just such conditions. One cannot uncritically assume that pump kinetics measured at short circuit apply when the normal transmembrane potential is present. To model rheogenic (i.e., currentgenerating) active transport across a membrane that is not short-circuited, it is necessary to relate the fluxes to driving forces that are expressed in terms of the electrochemical potential. The sensitivity of pump rates to membrane potential also arises from the conformational changes of the carrier that effect transport. These

208


changes generally require the movement of charged residues and dipoles that are part of the pump protein but are not directly involved in substrate binding or translocation. The movement of these groups, like that of the binding sites and substrate, is influenced by the electric field in which the transitions occur. As one consequence, the potential difference across the membrane can affect the transition rate constants and hence the equilibrium distribution of carrier between Side I and Side II. The influence of the electric field on the transition rate constants was examined mathematically by Lauger (1984), who represented the pump molecule as an assembly of point charges. This effect is of somewhat less concern for most contemporary carrier models (e.g., Fig. 4.11a), in which the binding sites move only a short distance, and in a direction that is more or less in the plane of the membrane. The analysis of antiport is not substantially different from that of symport and is included as an exercise. There is one important difference between the two: since in antiport A and B are never bound to X at the same time, the carrier need have only a single binding site for both substrate and cosolute. The assumptions of the symport model are modified as follows: 1. The activated carrier X binds either A or B, and neither solute binds to any other form of the carrier; when A binds to X, it forms the carrier–substrate complex, AX; when B binds to X, it forms the “inactivated” form of the carrier, BX = Y. 2. As in the symport model, the only forms of the carrier that can cross the membrane are Y and AX. 3. The activation equilibrium constant Ka is replaced by the deactivation i i i equilibrium constant Kd = cX cB /cY .

5.3.2. A Simple Secondary Scalar Transport Model: Equations 1. Conservation of Carrier. The carrier has six possible states: inactive (Y), activated (X = YB), and bound to substrate (AX = AYB), at Side I or Side II: I II cYI cYII c XI cXII c AX c AX XT .

(5.18a)

2. Steady State. For the simplest case, we assume, as we did for the simplest passive carrier model, that all rate constants have the same value, Dc. This assumption can be relaxed as in Chapter 4; see also the next subsection: I II D a(cYI cYII c AX c AX )0 .

(5.18b)

Recall that the activated carrier does not cross the membrane without substrate. 3. The Transport Rate is equal to the flux of AX: I II J A D a(c AX c AX ).

(5.18c)


209

Equations (5.17) and (5.18) constitute seven equations in seven unknowns: the concentrations of the six carrier states, and JA. The substrate flux depends on the concentrations of A and B in Phases I and II, and the properties of the carrier system (XT, Ka, and K). Solving for JA , JA

D aX T K a K (cBI c AI cBII c AII ) (cBI c AI K a K Kc BI )(cBII c AII K a K ) (cBII c AII K a K KcBII )(c BI c AI K a K )

.

(5.19)

The effect of the cosolute gradient on the flux of A is seen more clearly at level flow; when the substrate concentration in both phases is cA,

JA

D aXT K a Kc A (cBI cBII ) , den

(5.20)

where den is the denominator in Eq. (5.19). Since the cosolute concentrations appear in the denominator, the substrate flux at level flow is not as simple a function of these concentrations as it might first appear. The level flow solution is, however, symmetric with respect to cosolute concentration; when cBI and cBII are interchanged, the sign of JA is reversed, but the magnitude of the substrate flux remains the same.

5.3.3. More Complex Symport Models; the Sodium/Glucose Transporter Many of the assumptions on which the preceding equations were based can be relaxed, usually at the price of increased complexity. Since symport is a carrier-based process, the simple transport model developed above can be subjected to all of the complicating modifications applied in Chapter 4, such as the introduction of ligandsensitive transition rate constants, asymmetries, or nonequilibrium at the interfaces. Some modifications that are more specific to active transport are described below: 1. The Activated Carrier Can Cross the Membrane without Substrate. The I II quantity Dc(cX – cX ) is added to the left-hand side of Eq. (5.18b), and the expression for substrate flux becomes

JA

c AI cBI ( K a cBII ) c AII cBII ( K a cBI ) D aXT K ¸ . 2 [ K ( K a cBII ) c AII cBII ][c AI cBI K ( K a cBI )]

(5.21)

In an analysis of the Na/glucose cotransporter, Centelles et al. (1991) point out that the ability of an activated carrier to cross the membrane without substrate constitutes a “short circuit” of the cyclic behavior of the system (see Fig. 5.10, below), which in the case of this particular symport system seriously compromises the coupling of substrate glucose transport to the energy contained in the transmembrane difference in the electrochemical potential of cosolute sodium. They conclude that this transition is prevented in the cotransporter either by minimizing the transition rate constant for X (as in the simple model given above), or through kinetic parameters that allow activation to occur only after the substrate has bound to the carrier.

210


Figure 5.10. Kinetic representation of a general cotransport system. X is BY. The dissociation constants for the three interfacial binding equilibria are shown. The kinetics of the simplest system are within the shaded region demarcated by heavy dashed lines.

2. The Inactive Form of the Carrier Can Also Bind and Transport A (as AY). I The presence of this side reaction introduces two more unknown concentrations, c AY II and c AY . Minor modifications must be made to Eqs. (5.18a–c) (one of the exercises at the end of the chapter asks the student to make them). Two more equations are needed and are provided by the equilibrium constant for the side reaction:

ciA cYi K s (i = I, II). ciAY

(5.22)

Competitive inhibition does not ensue when both A and B can bind to Y, because the binding sites for the substrate and the cosolute are distinct. Activation of the carrier by the cosolute is reflected in a greater affinity of X for A compared to that of Y for A; i.e., K > Ka.

For the cosolute gradient to be well coupled to the active transport of A, the activated carrier must have either a greater affinity for substrate than Y, or a faster transition rate. Heinz and coworkers (Heinz et al., 1972, Geck and Heinz, 1976) classified symport processes as affinity-type or velocity-type, depending on which effect of activation dominates. In a pure affinity-type process, the transition rates of all forms of the carrier are the same, but K is much less than Ks. The simplest model given above is of this kind; Ks is so large that AY is absent. In the simplest velocity-type process, K = Ks, but DcAX >> DcAY. Truskey et al. (2004) modeled a limiting case of a velocity-type scalar transport process in which a side reaction is present but only Y and AX can cross the membrane; nine independent parameters were needed to define this transport system. 3. When The Reactions at the Interfaces Are not at Equilibrium, the equilibrium constants are replaced by rate constants for binding and dissociation of the substrate and cosolute, as was done in the previous chapter for a passive carrier. The rate constants must be consistent with the equilibrium constant. At equilibrium, the rates of binding and dissociation are equal; for instance, for substrate binding at Side I, I kbI c AI cXI krI c AX (equilibrium),

(5.23)

where kbi (kri ) is the rate constant for binding (release) at Side i. Referring to Eq. (5.17b), I

I

I

K = kr /kb .

(5.24)

212


Similar equations apply at Side II, and for activation. 4. The Cosolute and/or Carrier are Charged. The implications of this added complexity have already been discussed. 5. The Coupling Coefficient QB is not Unity.

Figure 5.11. Kinetic model of the sodium/glucose cotransporter in rabbit intestine (Parent et al., 1992b). C is the unbound carrier, equivalent to Y; CNa2 is the carrier activated by two sodium ions, equivalent to X; and SCNa is the activated carrier bound to glucose, equivalent to AX. The superscripts denote the external and internal phases between which transport takes place. All transitions and binding reactions are defined by rate constants as given in the figure; interfacial equilibrium is –1 not assumed. The units of the transition rates are s ; activation is regarded as a third-order reaction –2 –1 and the units of the rate constants are mols s ; sugar binding to the activated carrier is a secondorder reaction and the units of the rate constants are mols–1s–1. E ᑠ'\ / RT , where '\ is cell potential.

The sodium/glucose cotransporter, an important contributor to intestinal absorption, exemplifies most of the complexities described above. Two sodium ions accompany each transition of the activated carrier, which carries one glucose molecule (S) at a time; thus, the activated carrier X = CNa2, where C = Y is the inactive carrier, and the loaded carrier AX = SCNa2. The inactive carrier possesses two negative charges, so the activated carrier is electrically neutral. Parent et al. (1992b) have thoroughly characterized the kinetics and potential dependence of this transport system, using the intestinal transporter of rabbit, and their results are summarized in Figure 5.11. All three forms of the carrier — C, CNa2, and SCNa2 — can cross the membrane. The transition rate constants for CNa2 and SCNa2 are symmetric, but the rate constants for the transition of the charged form of the carrier, C, are asymmetric and potential dependent. The rate constants for the activation/deactivation and substrate binding/unbinding reactions in the two carrier conformations are all different, and the activa-


213

tion/deactivation rate constants when the carrier sites are exposed to the external medium are potential dependent. Centelles et al. (1991) examined the fluxes through vesicles containing porcine renal sodium/glucose transporters and favor a different model. In contrast to the sixstate model of Parent et al., they found that all eight states in Figure 5.10 were important. In the model preferred by Centelles et al., the inactivated carrier C can bind either the cosolute or substrate first. If it binds the substrate first, it can cross to some extent as SC (this is absent from the Parent model); however, if it binds sodium first, it rarely crosses as CNa2 (recall that the transition rate for the sodium-bound carrier was thought to be small, to reduce short-circuiting of the glucose flux). Therefore, it must either lose the sodium, or bind glucose, before transitioning to the other side of the membrane. Binding cooperativity promotes the formation of the ternary complex SCNa2 from either CNa2 or SC.

5.3.4. Primary Scalar Transport The equations that describe primary transport can be derived from a model and assumptions that are very similar to those used to treat secondary transport. The only important difference between the equations for primary and secondary scalar transport is in the activation/deactivation equilibria at the two sides of the membrane. Whereas in secondary transport, these equilibria are given by Eq. (5.17a), the corresponding equations in primary transport naturally do not involve B, and they are different for each side of the membrane. Consider, for instance, the simple illustration given earlier in which the primary transport process is driven by activation at Side I [Eq. (5.3a)]. For simplicity, write the metabolic reaction to which the activation process is coupled as R (reactants) o P (products). Then, at Side I, the activation step is I

I

I

I

Y +R UX +P.

(5.25a)

In Eq. (5.3a), R is ATP and P is ADP + Pi . The carrier deactivates passively at Side II: II

II

X UY .

(5.25b)

The equilibrium constants for these reactions are designated KA and KD, respectively: KA

cXI cPI , cYI cRI

(5.26a)

cYII . cXII

(5.26b)

KD

We assume that the equilibrium constant between X and Y, in the absence of activation, is symmetric. Thus, if there were no coupling to ATP hydrolysis at Side I, then:

214


c I ¬ X c I ® Y

no coupling

1 . KD

(5.27)

The ratio cXI / cYI is a measure of the fraction of A-free carrier sites at Side I that are activated; call it the activation ratio. Its value in the presence of ATP is obtained by rearranging Eq. (5.26a), assuming Eq. (5.25a) is at equilibrium: c XI K A cRI . cYI cPI

(5.28)

A parameter, *, that measures the increased availability of activated carrier when phosphorylation by ATP is coupled into the activation process, can be constructed from the activation ratios given by the two preceding equations: *w

cXI / cYI K A K D cRI . (c XI / cYI )no coupling cPI

(5.29)

Since the coupling drives the activation reaction Y o X to the right, the activation ratio given by Eq. (5.28) is larger than that given by Eq. (5.27), and * > 1. When there is no coupling, * = 1, by definition.

5.3.5. Flux Equations for Primary Scalar Transport As noted earlier, the only difference between the governing equations for the preceding models of primary and secondary scalar transport lies in the description of the activation/deactivation equilibria at the membrane–solution interfaces. This difference can be dealt with by an algebraic trick that allows us to write the primary transport flux equations directly from those for secondary transport. Table 5.1 presents the activation ratios at each side of the membrane, for primary and secondary scalar transport, obtained from Eqs. (5.17a), (5.26b), (5.27), and (5.29). The expressions for the activation ratio in secondary transport can be converted into those for primary transport by replacing cBI by Ka*/KD and cBII by Ka /KD. Among the seven equations that describe the simple symport model, the only ones that contain the cosolute concentrations are the pair given by Eq. (5.17a). Thus we can make the substitutions indicated above without altering any of the other equations, and the flux equation for primary transport can be obtained directly from Eq. (5.19) by making the same substitutions:

JA

D aXT KK D (*c AI c AII ) . (*c AI KK D K *)(c AII KK D ) (c AII KK D K )(*c AI KK D )

(5.30)

At level flow, JA

D aXT KK D c A (* 1) , den

(5.31)


215

where den is the denominator in Eq. (5.30). When there is no coupling, * = 1, and JA = 0, as it should.

Table 5.1. Activation Ratios for Scalar Active Transport Driven by Carrier Activation Side I Secondary transport

Primary transport

cXI I Y

c c

I X I Y

c

Side II

cBI

cXII

Ka

II Y

c

II X II Y

*

c

KD

c

cBII Ka 1 KD

5.3.6. Relation between the Coupling Parameter * and the Affinity of the Metabolic Reaction The relation between * and the affinity, A, follows from the thermodynamic equations in Chapter 1. Since symmetry with respect to the X UY equilibrium is assumed, the superscripts that designate each side of the membrane are not needed here and will be 0 omitted. From Eq. (1.65) and the definition of 'G , RT ln K D PY0 P0X ,

(5.32a)

RT ln K A P0X P 0P PY0 P0R .

(5.32b)

RT ln( K D K A ) P0P P0R .

(5.33)

Adding these equations,

Next, Eq. (1.43) is used to express the right-hand side of Eq. (5.33) in terms of the chemical potential change accompanying the metabolic reaction. Equation (1.43) is first simplified by (1) neglecting the effect of pressure on the chemical potential and (2) assuming that the activity coefficients of P and R are unity; it is then rearranged to give Pi0 Pi RT ln ci .

(5.34)

Substituting Eq. (5.34) into (5.33), and rearranging: P R P P RT ln( K D K A ) RT ln

cR , cP

(5.35)

216


where cP replaces the product of the ADP and phosphate concentrations. From Eq. (5.2), the left-hand side of Eq. (5.35) is precisely the affinity, A; from Eq. (5.29), the right-hand side is RT ln *. Thus, * = eA/RT.

(5.36)

When the affinity is positive, * is greater than one.

5.3.7. Vectorial Active Transport and the Nature of Na–K Exchange When energy from ATP is coupled into the transition of the carrier from one side of the membrane to the other, the active transport process is regarded as vectorial. Such transport is necessarily primary, since ATP is directly involved in the transport process. The Na–K exchange pump described earlier is a good model for examining the vectorial aspect of active transport. The Post–Albers scheme shown in Eqs. (5.15) is typical of P-type ATPases: ATP phosphorylates the enzyme on one side of the membrane [Eq. (5.15b)], and then that energy is used to drive the conformational change of the carrier that exposes its binding sites to the other side [Eq. (5.15c)]. Such a process has both scalar and vectorial aspects. Equation (5.15b) clearly describes a scalar activation process, which is preceded by binding of one of the substrates (Na) to the pump. However, the transition of the product of the phosphorylation reaction to Side II is not entirely passive (as it would be in pure scalar transport), and relies on the energy in the terminal phosphate bond to effect the conformational change. The energy in the ATP first activates the carrier, and then is expended in driving it across the membrane. The two steps described above can be combined: Na • E1 • ATP U Na • E2 – P + ADP.

(5.37)

This equation fits the definition of vectorial transport and can readily be modeled as I an extension of passive carrier transport. Referring to Eq. (5.14), AX is the Na • E1 II complex, and AX is Na • E2 – P. In the simple model of passive carrier transport that we developed in Chapter 4, the rate constants for the transition of AX from either side of the membrane to the other are the same. However, the exothermic hydrolysis of ATP drives the reaction given by Eq. (5.37) to the right; that is, the rate constant for the transition of the ATPase from E1 to E2 is greater than that in the opposite direction: DcAX > Dc–AX. Thus this portion of the Post–Albers scheme can be described in the same fashion as facilitated diffusion in the presence of rate constant asymmetry [cf. Eq. (4.49)]. A crucial difference is that the passive transport condition given by Eq. (4.49a) does not apply here.

5.3.8. Pumps and Leaks In many experimental studies of active transport, a level flow condition is sought by short-circuiting the experimental preparation and bathing it on both surfaces by identical solutions at similar pressures. This is done to eliminate driving forces for addi-


217

tional, passive fluxes that might confuse the interpretation of the results. In short, the data are easier to analyze and to interpret when they are taken at level flow. In vivo, the pump drives the system away from level flow: cAII becomes greater than c AI . Indeed, the purpose of most active transport systems is to accumulate selected solutes at the trans side of the membrane, which is often the interior of a cell. Under these conditions, the desirable features of level flow experiments are absent. The electrochemical potential gradient created by the pump drives two kinds of leakage fluxes, which are discussed below and illustrated in Figure 5.12.

Figure 5.12. (a) Slippage of a pump. Fluid leaks back through the pump, driven by the pressure head. (b) A shunt. Fluid leaks back through an external path in parallel with the pump. The pumps in this figure transport water against a hydrostatic pressure difference; the analogous pumps in cell membranes transport substrate against an electrochemical potential difference.

Slippage (Fig. 5.12a). For a fixed value of c AI , the substrate flux given by Eqs. (5.19) and (5.30) decreases monotonically as c AII is raised. This decrease in pump rate is not the result of transport paths in parallel with the pump. Rather, it reflects a slippage in the pump itself, analogous to the slippage in a centrifugal pump when it is operating against too high a pressure head. When c AII increases, the dissociation of AX at Side II is suppressed. As a result, the carrier more often returns to Side I with its binding site(s) still occupied by substrate. Similarly, as c AI is reduced, the rate at which activated carrier binds substrate becomes less, increasing the likelihood that the cosolute will dissociate from the carrier, allowing it to cross to Side II without substrate. At static head, the rate at which AX crosses from Side II to Side I becomes equal to the rate at which it crosses from Side I to Side II, and the net substrate flux is zero.

218


From Eqs. (5.19) and (5.30), it is seen that the static head condition (JA = 0) occurs when cAII/cAI = cBI/cBII (secondary transport) = *(primary transport).

(5.38)

The left-hand side of Eq. (5.38) is the maximum accumulation ratio of the pump. Note that a static head condition is reached even when the pump is the only path by which substrate crosses the membrane. Another kind of internal leak, seen in cotransport, occurs when the activated carrier can cross the membrane without the substrate, or when the side reaction is present. In both cases, the coupling between the cosolute gradient and the substrate flux is diminished. Shunts (Fig. 5.12b). Once the pump raises the electrochemical potential of its substrate in Phase II above that in Phase I, a driving force for passive transport in opposition to the pump has been created. If there are passive pathways in parallel with the pump, solute will leak through them from the trans side of the pump to the cis side. Such leakage paths are called shunts. When Phase II is a closed compartment, such as the inside of a cell, shunts reduce the accumulation ratio attainable by the pump. In the steady state, the net flux (active plus passive) of substrate into the cell must equal the rate at which the solute is consumed. Denoting the active flux of A by the superscript “a” and assuming that the leak rate follows the equations of free diffusion, [ J Aa k A (c AI c AII )] Acell mVcell ,

(5.39)

where Acell and Vcell are the surface area and volume of the cell, respectively, kA is the permeability of A in the cell membrane, and m is the rate at which the substrate is consumed, per unit volume. Since cAII > cAI, the passive flux is oppositely directed to the II pump flux. Solving for cA , c AII c AI

J Aa mVcell / Acell . kA

(5.39a)

For a given pump rate, the intracellular concentration is reduced by increasing either the permeability of the cell membrane or the rate of consumption of substrate within the cell. When the substrate is not consumed by the cell (as is the case for most ions) or the consumption rate is sufficiently small, the right-hand side of Eq. (5.39) can be set equal to zero. Then, in the steady state, the pump rate into the cell equals the leak rate out: J Aa k A (c AII c AI ) .

(5.40)

When the leak rate is defined by a different expression — for instance, if a carrier, not at its diffusion limit, is involved — the right-hand side of Eq. (5.40) is changed accordingly.


219

Figure 5.13 illustrates the relationship between the fluxes through the pump and leak pathways. In the figure, the solid line describes the variation in the active flux II from Phase I to Phase II as a function of the trans concentration cA for a fixed value of I the cis concentration cA . The active transport rate decreases as the opposing concenII I II tration difference cA – cA increases, falling to zero when cA reaches the static head II value, c A,s.h. . The maximum accumulation ratio of the pump is c AII,s.h. / c AI .

Figure 5.13. Dependence of pump rate (JAa, solid line) and leak flux magnitude (dashed lines) on the trans concentration of substrate, cAII, for a fixed cis concentration, cAI. See text for description.

The flux through the leak is directed toward Phase I when cAII > cAI. The magnitude II of the leak flux is plotted against cA in Figure 5.13, using dashed lines, for two differII I ent leak permeabilities, kA. Naturally, the leakage flux is zero when cA = cA , and it inII creases in proportion to the difference between the two concentrations as cA becomes larger. The magnitudes of the (oppositely directed) pump and leak rates are equal where the dashed lines intersect the operating curve of the pump; the abscissas at the intersection points, cA* and cA**, are the steady-state values of cAII corresponding to I each shunt permeability. This steady-state value decreases from c AII,s.h. toward cA as the shunt permeability increases. Thus, shunts act to reduce the maximum accumulation ratio generated by active transport systems. A sketch of the dependence of this ratio on shunt permeability is shown in Figure 5.14, for secondary scalar transport. When kA = 0, there is no shunt, and the ratio is given by Eq. (5.38). The pump becomes less and less capable of concentrating A in Phase II (i.e., the accumulation ratio approaches unity) as the shunt permeability increases.

220


Figure 5.14. Dependence of the accumulation ratio of a secondary scalar transport system on shunt permeability, kA. At each point on the curve, the passive flux through the shunt is equal and opposite to the active flux through the pump.


The operating curve of an active transport system that pumps A from Phase I to a Phase II is approximated by a simple linear expression JA = a + b'cA. (a) In terms of a and b, what is the flux at level flow? (b) In terms of a and b, what is the concentration difference across the membrane at static head?

2.

The pump in the previous problem is sited in a cell membrane and pumps into the –11 2 –12 cell. The parameters of the pump are a = 4 q 10 mol/cm -s and b = 1 q 10 2 mol/cm -s-mM. The substrate A is not metabolized by the cell, but it does leak –11 2 out via a simple carrier system whose parameters are DcXT /2 = 9 q 10 mol/cm s and K = 40 mM. Write an equation that can be solved for the steady-state concentration of A in the cell when the external concentration is 10 mM.

3.

The layer of cells that line the intestine absorb glucose (S) and can be modeled as two membranes in series. The portion of the cell membrane that faces the inside of the intestine (the luminal membrane) contains sodium/glucose symporters that pump the sugar from the digestive tract into the cell. The rest of the membrane (the abluminal membrane) contains glucose carriers that transport intracellular sugars into the interstitium and thence to nearby capillaries. The sodium ion gradient that drives the pump in the luminal membrane is created by a primary transport mechanism. Assume that the actively transported flux of glucose across the luminal membrane can be written JS = LSS'PS + P, where LSS and P are constants, and that the


221

flux of S across the abluminal membrane can be described by the simple carrier of Chapter 4 in the diffusion limit. These are the only pathways for glucose transport. Glucose is consumed negligibly by the cell. (a) Write an equation that can be solved for the intracellular concentration of glucose in the steady state, in terms of the glucose concentrations in the intestine (cSI) and interstitium (cSII), and the properties of the two transport systems, You need not solve the equation. Neglect the effect of pressure on the chemical potential and assume the system is isothermal. (b) What is the initial effect on the rate of the symport system (does the rate change continuously or discontinuously; does it rise or fall?) if: (i) The primary transport ATPase is inhibited? (ii) cSI is raised? II (iii) cS is raised? 4.

Can the coupling coefficient for symport be less than 1? If so, how? If not, why not?

5.

Show that the expression for the electrochemical potential difference of the coso B that follows Eq. (5.8) is correct. lute 'P

6.

If the potential in the interior of a cell at 37ºC is 60 mV negative relative to the surroundings, and the extracellular and intracellular sodium concentrations are 110 and 5 mM, respectively, and the extracellular glucose level is 90 mg%, how high could the intracellular glucose concentration be raised by sodium symport were it not consumed in the cell? Assume one Na ion crosses the cell membrane with each glucose molecule, that there is no glucose leak, and that all activity coefficients are unity.

7.

Derive an equation analogous to Eq. (5.10) that describes the maximum accumulation ratio achievable by an antiport system.

8.

(a) Using Eq. (5.11), derive an expression for the reversal potential of the Na– Ca exchanger in terms of the Nernst potentials of the two ions across the cell membrane. (b) Typical intracellular and extracellular concentrations of Na and Ca are: intracellular Na = 14 mM, intracellular Ca = 0.1 PM, extracellular Na = 139 mM, extracellular Ca = 1.2 mM. What is the reversal potential of the exchanger under these conditions? (c) The extracellular Na concentration in (b) is reduced gradually from 139 to 14 mM while the other three concentrations remain the same. Plot the reversal potential of the exchanger over this sodium concentration range and explain the shape of the curve.

222

9.


The active transport system in Figure 5.7 is in steady state, with sodium being pumped out of the cell and the two amino acids in. The Na–K ATPase is suddenly inhibited by the addition of ouabain, which binds to its active sites. Plot and explain the time course of the fluxes of Na and the two amino acids through the transport system, subsequent to inhibition. Assume for simplicity that there are no other pathways for any of the species to cross the cell membrane.

10. The cell membranes of a particular portion of the kidney (the thick ascending limbs of the loops of Henle) contain about 104 Na–K ATPase molecules per μm2. One enzyme molecule can hydrolyze 100 ATP molecules/sec. What is the current density produced across the renal cell membrane by the pump? Express the current density in electrical units. 11. The electrical potential inside the human red blood cell is –9 mV relative to plasma. Using literature values for the plasma and intracellular sodium and potassium levels, determine whether either ion is actively transported across the red cell membrane, and in which direction. What kind of transport system would explain your result? 12. Using the stoichiometry in Eq. (5.16), show that the free energy available from the hydrolysis of ATP is sufficient to drive the Na–K pump for normal values of intracellular and extracellular cation concentrations, and a typical cell potential of –60 mV. 13. Starting with Eqs. (5.17) and (5.18), determine whether reciprocal stimulation can take place when B can cross the membrane either as X or as AX. 14. Derive a flux equation for antiport analogous to Eq. (5.19). 15. Use Eq. (5.19) to derive an expression for substrate flux across a symporter when the cis concentration of substrate is zero. Show that the dependence of JA on cAI has the same form as the corresponding equation for a simple carrier. Explain in terms of the mechanism of the pump why the substrate flux is independent of cBII under these conditions. 16. Show that the substrate flux for secondary scalar transport at level flow is symmetric with respect to cosolute concentration. Show that this symmetry does not generally hold when the substrate concentrations on the two sides of the membrane differ, and explain why. 17. Derive Eq. (5.21). 18. Specify the changes that need to be made to Eqs. (5.18a–c) to include the side reaction in which Y binds A.


223

19. The affinity of ATP hydrolysis under normal intracellular conditions is 11–14 kcal/mol. Using a value of 11 kcal/mol, calculate the coupling parameter * for a primary scalar system in which the carrier is activated by the hydrolysis of a single molecule of ATP. 20. In a pump–leak system in which the pump substrate is not metabolized, how does the pump rate change as the permeability of the leak is increased? Does it approach a limit? If so, what is the limit? If not, why not? 21. A primitive cell has only two pumps. Pump #1 is a primary pump that uses ATP to accumulate B in the cell from the outside medium. Pump #2 is a symport system that uses the gradient of B to extrude C from the cell. C enters the cell via a passive leak, but B crosses the cell membrane only via the pumps. (a) The cell is functioning normally. Then all ATPases are suddenly inhibited. Sketch the time course of the transport rates through Pumps #1 and #2, starting just prior to inhibition, and explain the difference between them. (b) The cell is functioning normally. Then the extracellular concentration of C is raised. Sketch and explain the time course of the intracellular concentration of B, starting just prior to the change in the extracellular concentration of C.

6 NONEQUILIBRIUM THERMODYNAMICS

INTRODUCTION Chapter 5 described a class of transport processes in which the substrate flux is driven by either the affinity of a metabolic reaction or the electrochemical potential difference of a cosolute. This coupling of a flux to a force that is not its conjugate driving force is often seen in biological transport. Nonequilibrium thermodynamics, also called irreversible thermodynamics, is the formalism that describes the relationships among the many forces and fluxes that are simultaneously present in living systems. Our interest is primarily in linear nonequilibrium thermodynamics, where the fluxes depend linearly on the several driving forces in the system. This linear relation is presented in the first section of this chapter, following some important definitions. In subsequent sections, the basic equations of linear nonequilibrium thermodynamics are applied to passive and active transport. In the section on passive transport, we deal for the first time with the thermodynamics of solvent flow, and introduce the process of osmosis. The chapter concludes with a discussion of the limitations of nonequilibrium thermodynamics.

6.1.

THE BASIC PHENOMENOLOGICAL EQUATIONS

6.1.1. Conjugate Forces and Fluxes As indicated above, nonequilibrium thermodynamics concerns itself with the transport phenomena that occur when multiple forces are acting simultaneously on the species in the system, causing a number of simultaneous fluxes. In such cases, the fluxes and the forces can be enumerated in a number of equivalent ways. For instance, when a voltage is applied across a membrane bounded by solutions of a single electrolyte, the resulting ionic fluxes can be described in terms of cation flux and anion flux, or in terms of cation (or anion) flux and current density. Similarly, when a membrane separates two nonelectrolyte solutions whose pressures and concentrations differ, the driving forces can be expressed in terms of the concentration and pressure differences, or equivalently in terms of the chemical potential differences of solvent and solute. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_6, © Springer Science+Business Media, LLC 2008

225

226

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Among these options are particular sets of conjugate forces and fluxes which, we are taught by nonequilibrium thermodynamics, possess certain useful properties. These properties will be described in more detail later in this section. The identification of conjugate forces and fluxes is based on a thermodynamic quantity called the dissipation function, I: IT

di S . dt

(6.1)

The quantity di S/dt is the rate at which entropy is locally generated in a unit volume of membrane as a result of irreversible transport processes taking place in it; di S/dt and the dissipation function are necessarily positive. The local variable I can be integrated across the membrane to give a

a

) ¨ Idx T ¨ 0

0

di S dx . dt

(6.2)

From the thermodynamic equations in Chapter 1, it can be shown that I and ) are the rates at which free energy is dissipated during transport. The units of I and ) are 3 2 2 cal/cc-sec (in SI units, J/m -s) and cal/cm -s (in SI units, J/m -s), respectively. Each flux Ji contributes to the dissipation function. Paired with each flux is a corresponding conjugate force Xi, such that

I Ji Xi .

(6.3a)

i

The {Xi} can also be defined such that

) J i Xi .

(6.3b)

i

The passage of the ith species across the membrane dissipates free energy at a rate i / dx ) cal/cc-sec. Therefore, by Eq. (6.3a), the electrochemical potenequal to J i (dP tial gradient is conjugate to the species flux. The integrals across the membrane of the conjugate forces defined by Eq. (6.3a) are also conjugate forces, because their products with the {Ji} satisfy Eq. (6.3b). The integral of the electrochemical potential gradient is the electrochemical potential difference, which accordingly is also an acceptable conjugate. Equation (6.3b) also identifies the affinity as the force that is conjugate to the progress of a chemical reaction. As remarked earlier, the coupled transport processes taking place in a complicated system can often be described in terms of alternative sets of fluxes. Corresponding to each of these sets is a matching set of conjugate driving forces. Any set of force–flux pairs whose pairwise products add up to the dissipation function is equally amenable to nonequilibrium thermodynamic analysis. As we will see, some sets are better for some purposes than others, and sometimes different sets are used to examine different aspects of the same transport system. Because the dissipation function can be written in either the differential or integral form, both the gradient and the difference of the electrochemical potential are usable


227

as conjugate driving forces. The gradient would seem to have a better claim to the title of “force” since a force is, after all, the gradient of a potential. However, for reasons of simplicity and convenience (and ignorance, because we know much more about the conditions at the membrane surfaces than we do about the gradients inside it), we will use the integrated forms almost exclusively. The differential forms find more use in free solution studies or experiments on inanimate systems, which can be more thoroughly instrumented.

6.1.2. Phenomenological Coefficients and Linear Thermodynamics The relationships among the forces and fluxes in a nonequilibrium system are described in nonequilibrium thermodynamics by a set of phenomenological coefficients. A convenient starting point for introducing these coefficients is the expression presented in Chapter 2 for the free diffusion of a single nonelectrolyte when the only driving force for solute flux is the chemical potential gradient of the solute itself:

dP ¬ J s U s cs s . dx ®

(2.4)

In Chapter 2, we used the expression for the chemical potential in terms of solution variables [Eq. (1.41)] to replace cs(–dPs/dx) in Eq. (2.4) by –RTdcs /dx, from which we obtained Fick's First Law of Diffusion, which was subsequently integrated across the membrane. Here, for reasons that will become clear later, we set up the integral differently: P II

a

s J ¨ s dx ¨ (dP s ) . 0 U c P Is s s

(6.4)

Since the flux is independent of position in the membrane, it can be brought out in front of the integral sign; in addition, the integration of the right-hand side is trivial, and Eq. (6.4) becomes a

Js ¨ 0

dx P Is P IIs . U s cs

(6.5)

By our earlier definition of the ' operator, the right-hand side of Eq. (6.5) is 'Ps. The integral on the left is called the integral or generalized resistance of the membrane, Rss. Equation (6.5) becomes

JsRss = 'Ps.

(6.6)

Corresponding to this version of the integrated flux equation is a similar version of the original differential form Eq. (2.4): J s rss

dP s , dx

(6.7)

228


where rss = 1/(Uscs). The two resistances, Rss and rss, are related by a

a

Rss ¨ rss dx ¨ 0

0

dx . U s cs

(6.8)

Continuing this seemingly trivial exercise, we note that the integrated form (6.6) can also be written: Js = Lss'Ps,

(6.9)

where Lss is the integral or generalized conductance of the system; for this particular case, the conductance is simply the reciprocal of the resistance. These generalized resistances and conductances, which define the dependence of a flux on its conjugate driving force, are referred to as straight phenomenological coefficients. It is not apparent that Eq. (6.6) or (6.9) should be preferable in any way to the proportionality between flux and concentration difference that was developed in Chapter 2. For one thing, the chemical potential difference is clearly more difficult to express in terms of measurable quantities than is the concentration difference. In addition, it is found experimentally that the solute permeability is generally a weak function of solute concentration. Since the permeability is proportional to the solute mobility, the latter quantity is also insensitive to concentration. Thus, referring to Eq. (6.8), it is clear that Rss and Lss are considerably more concentration dependent than is the permeability, since concentration appears in the defining integral. What, then, do the new equations have to offer? When there is only one force and one flux in the system, as in the example above, they have nothing to offer; to describe the flux of a single solute, the concentration difference is clearly the preferred driving force. But, when there are multiple forces and fluxes, Eqs. (6.6) and (6.9) can be generalized to a relatively simple set of equations that includes all the interactions among the forces and fluxes in the system. Suppose n pairs of conjugate fluxes and driving forces are present; then the flux of the ith species is given by n

J i Lij X j (i = 1,….,n).

(6.10)

j 1

More complete theory allows for higher powers of {Xj} on the right-hand side, but the linear expression above is sufficient for our purposes. The {Lij} are the phenomenological coefficients of the system. The straight coefficients are those for which i = j; those coefficients for which i z j are termed the cross-phenomenological coefficients. As an example, consider the coupled fluxes of a nonelectrolyte and solvent (subscript “w”) across a membrane. By Eq. (6.10), the fluxes of the two species are Js = Lss'Ps + Lsw'Pw,

(6.11a)

Jw = Lws'Ps + Lww'Pw.

(6.11b)


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The number of phenomenological coefficients needed to describe all the fluxes in a system rises rapidly as the number of species increases. This rise is not as rapid as it might at first appear, because the phenomenological coefficients are not independent (we will see another reason in §6.2.7). In particular, the matrix of the phenomenological coefficients is symmetric; that is, Lij = Lji.

(6.12)

For example, in Eqs. (6.11), Lsw = Lws. Equation (6.12) is the Onsager reciprocal relation, one of the most powerful principles of nonequilibrium thermodynamics. Since each flux can be written as a linear sum of all the forces, each force can be written as a linear sum of all the fluxes: n

X j R ji J i (j = 1,…,n).

(6.13)

i 1

Reciprocity also holds for the {Rji}:

Rji = Rij.

(6.14)

The R’s are related to the L’s by a general expression that is easily derived using linear algebra. The equation for a specific R in terms of the L-matrix is R ji

L ji L

,

(6.15)

where |L| is the determinant of the L-matrix and |Lji| is the minor determinant of the matrix corresponding to Lji. In most practical applications of nonequilibrium thermodynamics to biological transport, n does not exceed three, and it is easy to move between the L-representation and the R-representation without using matrix algebra. As we have seen, when n = 1, L11 = 1/R11. The choice of representation to use depends on the problem being addressed. The two formulations are equivalent, but usually they are not equally convenient. In this chapter, we will use the L-representation to describe passive transport processes, and the R-representation to examine active transport.

6.1.3. Frictional Interpretation of the Phenomenological Equations In 1961, Kedem and Katchalsky (1961) rewrote the phenomenological equations of the previous subsection in yet another form, in which the phenomenological coefficients were replaced by a set of frictional coefficients. According to the frictional model, the flux of each species is determined by a balance between (1) its conjugate driving force and (2) the frictional drag on the species by all of the other components of the system, including the membrane. The frictional drag on the jth species by the ith component was given by Fji = fji(vi – vj),

(6.16)

230


where fji is a frictional coefficient and vi is the nominal velocity of the ith component, defined as Ji /ci. The membrane is chosen as the frame of reference, so its velocity is zero. The force balance that determines the fluxes is similar to Eq. (6.13): n

X j Fji Fjm (j = 1, ..., n).

(6.17)

i 1 iv j

For instance, when a single nonelectrolyte is present, Eqs. (6.16) and (6.17) give J J ¬ f J X s fsw w s sm s , cw cs ® cs

(6.18a)

J J ¬ f J X w fws s w wm w . cs cw ® cw

(6.18b)

The relationship between the frictional coefficients of Eqs. (6.18) and the generalized resistances of Eq. (6.13) is straightforward. The reciprocal relation Rsw = Rws becomes fsw f ws . cw cs

(6.19)

Thus, the number of independent frictional coefficients required to describe the transport system is the same as the number of phenomenological coefficients. An advantage of the frictional representation is that the frictional coefficients are generally less sensitive to concentration than are the phenomenological coefficients; also, when the transport mechanism is known, the frictional coefficients are easier to interpret in terms of that mechanism and the properties of the constituents of the system. However, unless the transport mechanism is well understood — and this requires information beyond that provided by nonequilibrium thermodynamics — it is unwise to assign any more physical content to measured frictional coefficients than to the phenomenological coefficients to which they are related.

6.1.4. A Cautionary Note Before Proceeding In the last section of this chapter, we will discuss in some detail the limitations of the nonequilibrium thermodynamic description of coupled transport processes; one of these, alluded to above, is worth stressing now. No direct information regarding mechanism is provided by nonequilibrium thermodynamics. As a result, one cannot predict the values of the phenomenological coefficients (or the frictional coefficients to which they are related) using only this methodology. The coefficients are, indeed, phenomenological: even though the relationships among them must satisfy the principles of nonequilibrium thermodynamics, they are experimental quantities, whose values are ultimately dependent on mechanisms to which nonequilibrium thermodynamics is blind.


231

6.2. NONEQUILIBRIUM THERMODYNAMIC DESCRIPTION OF PASSIVE TRANSPORT As in Chapter 2, we begin our discussion of passive transport by considering a system consisting of a single nonelectrolyte dissolved in a solvent, say water. Most of this section will deal with the application of nonequilibrium thermodynamics to passive transport in this attractively simple system. The description of the coupled transport of solvent and a single nonelectrolyte is followed by extensions to electrolytes and multiple solutes.

6.2.1. Setting the Stage Two solutions of a nonelectrolyte are separated by a membrane; the concentrations and/or the hydrostatic pressures of the solutions are unequal. The only necessary difference between this system and that in Chapter 2 is that the membrane in this case is permeable to water. We have already seen the equations that describe the resulting fluxes: Js = Lss'Ps + Lsw'Pw,

(6.11a)

Jw = Lsw'Ps + Lww'Pw,

(6.11b)

where the Onsager reciprocal relation has been used to replace Lws by Lsw. The chemical potentials whose differences appear in Eqs. (6.11) are those just inside each face of the membrane. Chemical potential is a continuous function of position across the membrane–solution interface; if it were discontinuous, then infinite gradients would exist, a condition inconsistent with the presence of a finite flux. Thus, an arbitrarily small error is introduced by basing the driving forces on the chemical potentials in the solutions immediately adjacent to the membrane (see Fig. 6.1). The driving forces can then be written in terms of the concentrations and pressures in the solutions, which are more accessible than the inside of the membrane. When the solutions are assumed to be well stirred, the chemical potentials can be written in terms of the bulk concentrations of the solutions. The membrane properties are reflected in only the phenomenological coefficients: Lss, Lsw, and Lww. From a practical point of view, Eqs. (6.11) can be made much easier to use by rewriting the chemical potential differences 'Ps and 'Pw in terms of the more readily measured concentrations and pressures in the two solutions. The first of these differences is easy to translate, using Eq. (1.41): 'P s Vs 'P RT ' ln cs .

(6.20)

For now, we assume the solutions to be ideal (Js = 1) and dilute; the effects of nonideality and nondiluteness will be dealt with later. We also assume that there are no temperature gradients across the membrane; this is why no 'P 0s term appears in Eq. (6.20). Indeed, if there were such gradients, a third force/flux pair reflecting the dissipation due to the flow of heat would have to be included in the phenomenological equations.

232


Figure 6.1. Chemical potential profile across a membrane. The chemical potential difference 'P is measured immediately inside the faces of the membrane. The difference 'Pc is based on solution properties at a distance H from the faces. Because P is continuous, lim 'Pa 'P . The chemical poHl 0 tentials in the bulk solutions are denoted by the subscript b.

Another assumption that will prove very useful is that the concentration difference across the membrane is small relative to the average concentration in the bathing solutions. When this is the case, 'ln cs can be approximated as follows: ' ln cs

'cs , cs

(6.21)

where cs (csI csII ) / 2 . The error of this approximation is a function of the concentration ratio r csI / csII . Both sides of Eq. (6.21) can be written in terms of r, and are plotted in Figure 6.2. The error of the approximation does not reach 10% until r = 3.

6.2.2. The Chemical Potential of the Solvent No equation comparable to Eq. (1.41) was derived for the solvent in Chapter 1; now is the time to do so. To obtain a general expression for solvent chemical potential, we consider a system identical to that used in Chapter 1, consisting of k solutes, indexed by i, in solvent water. Begin by taking the differential of Eq. (1.26), which holds for solutes and solvent alike: dP w Vw dP dP cw .

(6.22)

Equations (1.31) (the Gibbs–Duhem equation) and (1.35) (from the van 't Hoff Law in differential form) are used to express dP cw in terms of measurable solution properties. First, Eq. (1.31) is rearranged to give dPcw

1 k ni dPic . nw i1

(1.31)


233

Figure 6.2. % ln cs and %cs / cs as functions of the concentration ratio, r.

Equation (1.35) is then used to replace the right-hand side of Eq. (1.31). In this case, the result is simpler if we do not replace nwVw by V (as we did in Chap. 1): k

dPcw Vw RT dci .

(6.23)

i 1

Equations (6.22) and (6.23) prescribe the chemical potential of the solvent in differential form. Since Vw is essentially constant, dPw can easily be integrated across the membrane to give the required chemical potential difference:

'P w Vw 'P Vw RT 'ci .

(6.24)

When there is only one solute,

'P w Vw 'P Vw RT 'cs .

(6.25)

As was the case when we derived the expression for the chemical potential of the solute, the use of the van 't Hoff relation limits the applicability of the preceding equations to solutions that can be treated as ideal and dilute.

234


6.2.3. A New Set of Forces and Fluxes; Osmotic Pressure Now that the chemical potential differences can both be written in terms of pressures and concentrations, we can rewrite Eqs. (6.11) accordingly. If we linearize the concentration dependence of 'Ps according to Eq. (6.21), then Js and Jw become linear functions of 'P and 'cs. However, these functions are rather awkward; for instance,

L ¬ J s ( LssVs LswVw )'P ss LswVw RT 'cs . cs ®

(6.26)

This complexity arises because the solute and solvent fluxes are not conjugate to 'P and 'cs. The dependence of the transport rates on 'P and 'cs can be expressed more conveniently by finding a new set of fluxes whose conjugate driving forces are at least proportional to the pressure and concentration differences. To do this, we return to the dissipation function for the system. Using Eqs. (6.3b), (6.20) with (6.21), and (6.24), and regroupingҏ, ) J s 'P s J w 'P w RT 'cs ¬ J wVw ( 'P RT 'cs ) J s Vs 'P cs ®

(6.27)

J ¬ ( J wVw J sVs )'P s J wVw RT 'cs . cs ®

Equation (6.27) shows that the coupled fluxes of solute and solvent, driven by 'Ps and 'Pw, can be represented by an alternative set of fluxes and forces. The first flux, J wVw J sVs , is simply the volume flux across the membrane, denoted Jv. The units of 2 Jv are cc/cm -sec. The volume flux is conjugate to 'P, the hydrostatic pressure difference across the membrane. The second flux, J s / cs J wVw , is termed the exchange flux, JD; it is conjugate to RT'cs. This latter driving force also has the units of pressure, and is the osmotic pressure difference across the membrane. Osmotic pressure will be discussed in more detail later in this section. A physical meaning for JD can be derived, as follows. We saw earlier that the nominal velocity of solute or solvent in the membrane, vi, is defined as Ji /ci . Accordingly, J s / cs approximates the mean velocity of the solute. We also saw in Chapter 1 that, in dilute solutions, Vw x V / nw 1/ cw ; it follows that J wVw x J w / cw vw . Hence the exchange flux is the difference between the velocities of the two species in the membrane; put another way, it is the nominal velocity of the solute relative to the solvent.

Corresponding to these new conjugate pairs is a new set of phenomenological equations, equivalent to Eqs. (6.11): Jv = Lvv'P + LvD'S

(6.28a)

JD = LvD'P + LDD'S,

(6.28b)


235

where 'S = RT'cs. We should note that, in most texts, the three phenomenological coefficients in Eqs. (6.28) are denoted Lp (not Lvv), LpD (not LvD), and LD (not LDD). The subscripts used here are more consistent with the general notation used earlier in this chapter; in any case, these coefficients will shortly be replaced by a different set whose notation is standard.

6.2.4. The Kedem–Katchalsky Equations The next step in the derivation of the final working equations for transmembrane flux is to manipulate Eqs. (6.28) and obtain a pair of expressions that give Jv and Js in terms of the hydrostatic and osmotic pressure differences. Although the solute flux is not given explicitly by Eqs. (6.28), it can easily be found by adding the volume and exchange fluxes: J v J D J sVs

Js J s (1 Vs cs ) . cs cs

(6.29)

The product Vs cs is the volume of solute per unit volume of solution (cc solute/mol solute q mols solute/cc solution), and is much less than unity in reasonably dilute solutions. Neglect it in Eq. (6.29) and solve for Js: J s cs ( J v J D ) .

(6.30)

The solute flux is seen to be a linear function of the volume and exchange fluxes; therefore, its dependence on the hydrostatic and osmotic pressure differences across the membrane should also be linear and expressible in terms of the three phenomenological coefficients in Eqs. (6.28). In practice, these coefficients are replaced by three related quantities: Hydraulic conductivity or flow conductivity: J ¬ L p w v Lvv ; 'P ® 'S0

(6.31a)

Reflection coefficient: Vw

LvD ; Lvv

(6.31b)

Solute permeability: J ¬ Z w s . 'S ® J v 0

(6.31c)

These three transport coefficients characterize the passive flux of a given solute and a given solvent through a given membrane as comprehensively as did the phenomenological coefficients in the earlier equations. The physical significance of two of the

236


new coefficients is rather clear: the hydraulic conductivity measures the volume flow induced by a hydrostatic pressure difference, and Z measures the solute flux induced by a concentration difference (like the permeability in Chap. 2). The meaning of the reflection coefficient is less obvious and will be discussed later, once the final flux equations have been obtained. It remains to write Jv and Js in terms of these new coefficients. The easiest way to do this is to use Eqs. (6.31) to express the phenomenological coefficients of Eqs. (6.28) in terms of the new coefficients; then Eqs. (6.28) and (6.30) will provide the results we seek. Replacing Lvv by Lp and LvD by –VLp in Eqs. (6.28) and (6.30): J v L p 'P VL p 'S ,

(6.32a)

J s cs ( J v VL p 'P LDD 'S) .

(6.32b)

Next, Eq. (6.31c) is used to replace LDD in Eq. (6.32b) by the solute permeability. Equation (6.31c) defines Z in terms of the solute flux at zero volume flux. From Eq. (6.32a), when the volume flux is zero, 'P = V'S.

Substituting this expression into Eq. (6.32b), again at zero volume flux,

J s cs (V2 L p LDD )'S . By substituting this expression for the solute flux into Eq. (6.31c), the following relation between Z and LDD is obtained: Z cs (V2 L p LDD ) .

(6.33)

Equation (6.33) is solved for LDD: LDD

Z V2 L p . cs

(6.34)

Substituting Eq. (6.34) into (6.32b), ¬ Z'S J s cs J v VL p 'P V2 L p 'S cs ® cs [ J v V( L p 'P VL p 'S)] Z'S

(6.35)

The parenthesized quantity multiplying V in Eq. (6.35) is simply Jv; thus, J s cs J v (1 V) Z'S .

(6.36)

Equations (6.32a) and (6.36) are the equations most commonly used to characterize the coupled flux of solute and solvent through a membrane permeable to both species. They are often referred to as the Kedem–Katchalsky equations, named after the au-


237

thors who introduced them to the biological community (Kedem and Katchalsky, 1958). A numerical example illustrating their use is given in Table 6.1. Table 6.1. Numerical Illustration of the Kedem–Katchalsky Equations Volume flux, J v L p 'P VL p 'S Solute flux, J s cs J v (1 V) Z'S Consider the transport of sucrose across a Cuprophan dialysis membrane at 37ºC when: csI = 155 mM = 1.55 q 10–4 mols/cc csII = 125 mM = 1.25 q 10–4 mols/cc 'P = 100 torr = 100 q (1333 dynes/cm2-torr) = 1.333 q 105 dynes/cm2 The transport coefficients for this system are (Meyer and Friedman, 1977): Lp = 4 q 10–11 cc/dyne-s V= 0.2 Z = 6 q 10–15 mols/dyne-s From csI and csII , cs = 1.4 q 10–4 mols/cc 'S= (1.987 cal/mol-K) q (310 K) q (3 q 10–5 mols/cc) q (4.186 q 107 dyne-cm/cal) = 7.74 q 105 dynes/cm2 Then, Jv = 4 q 10–11 q (1.333 q 105 – 0.2 q 7.74 q 105) cc/cm2-s = –8.60 q 10–7 cc/cm2-s Js = 1.4 q 10–4 q (–8.60 q 10–7) q (1 – 0.2) + 6 q 10–15 q 7.74 q 105 mols/cm2-s = 4.55 q 10–9 mols/cm2-s

Each term in the Kedem–Katchalsky equations can be associated with a specific physical process. The first term in the volume flow equation can be regarded as the hydraulic flow induced by the hydrostatic pressure difference, and the second term, called osmotic flow or osmosis, is the contribution to the volume flux resulting from the osmotic pressure difference across the membrane. Similarly, the first term in the solute flux equation can be regarded as the rate at which solute is carried across the membrane (i.e., convected) by the volume flux (this is often termed solvent drag), while the second term has the form of a diffusional component driven by the solute concentration difference. However intuitively satisfying this decomposition of the Kedem–Katchalsky equations may be, it must be remembered that each molecule of solvent and solute is acted upon by all driving forces, so the processes described above are not really separable; for instance, it is meaningless to ask whether a particular solvent molecule crossed the membrane hydraulically or by osmosis.

6.2.5. Physical Significance of the Reflection Coefficient: Semipermeable Membranes and the Osmometer, Steric Effects and Sieving The physical significance of the reflection coefficient might be expected to be somewhat less clear than that of the other two transport coefficients, if only because the

238


reflection coefficient appears in both Kedem–Katchalsky equations. It measures the effectiveness with which osmosis is driven by a given osmotic pressure difference [Eq. (6.32a)], and also influences the rate of convection of solute by the volume flux [Eq. (6.36)]. Like solute permeability, the value of the reflection coefficient depends on the properties of both the solute and the transport barrier.

Figure 6.3. (a) Pure osmosis. Only solvent crosses the membrane. (b) (top to bottom) Hydrostatic pressure, concentration, and solvent chemical potential profiles between the two phases in (a). The dashed line shows the hydrostatic pressure profile at osmotic equilibrium, where there is no water flow. The entire pore is inaccessible to solute, and unstirred layers are ignored.

We begin by examining the dependence of volume flux on osmotic pressure. Consider the system shown in Figure 6.3a; the membrane passes water but is impermeable to solute (such a membrane is termed semipermeable), and the pressures in both phases are the same. Since the concentrations of solute are different in each phase, there is a corresponding interphase difference in the chemical potential of the solvent, which can cross the membrane. Since 'cs > 0 and 'P = 0, it follows from Eq. (6.25) that 'Pw < 0; thus, solvent flows from Phase II, where its chemical potential is higher, to Phase I, where it is less. This flow is pure osmosis, driven by only the osmotic pressure difference across the membrane, and is directed toward the phase having the higher osmotic pressure. Also note that, contrary to statements that still appear in the


239

transport literature, the driving force in this case is not the concentration gradient of the solvent. The most plausible physical driving force for osmosis is a hydrostatic pressure gradient in the membrane pores. Since the membrane is semipermeable, there must be a portion of each pathway across it that is inaccessible to solute and is therefore occupied by water alone. Assume for clarity that this is the case for the entire pore. As noted in the context of Figure 6.1, the chemical potential of the solvent must be a continuous function; a discontinuity would correspond to an infinite gradient and thus an infinite driving force for water flow. This means (see Fig. 6.3b) that μw is continuous at the membrane faces x = 0 and x = a, where a is the length of the pore. Thus, from Eq. (6.25), the discontinuous drop in solute concentration at the left end of the pore (x I = 0), from cs to zero, must be compensated for by a corresponding drop in hydrostatic pressure. The situation is the same at the other end of the pore (x = a), but the decrease II I in pressure is less because cs < cs . Thus, a hydrostatic pressure gradient, driving hydraulic flow toward Phase I, develops in the pore. The profiles of hydrostatic pressure, solute concentration and solvent chemical potential, in and adjacent to the pore, are shown in Figure 6.3b. Returning to Figure 6.3a, the flow of solvent can be stopped by raising the pressure in Phase I until 'Pw becomes zero; then the solvent will be at equilibrium across the membrane and flow will cease. From Eq. (6.25) and the definition of osmotic pressure, it is apparent that the hydrostatic pressure difference necessary to bring the solvent flow to a halt is equal to the osmotic pressure difference across the membrane; when 'P = 'S'Pw = 0. This relationship between the hydrostatic and osmotic pressure differences at zero solvent flow (and zero volume flow, since the membrane does not pass solute) can be used to design an osmometer, as shown in Figure 6.4. Water crosses the semipermeable membrane, raising the hydrostatic pressure in Phase I. The difference between the hydrostatic pressures in Phase I and Phase II is proportional to the difference, h, between the heights of the two fluid surfaces: 'P = Ugh,

(6.37)

where U is the density of the solution and g is the gravitational constant. At equilibrium, this pressure head is equal to the osmotic pressure difference across the membrane. The hydrostatic pressure profile across the osmometer membrane at equilibrium is shown by the dashed lines in Figure 6.3b. Osmosis across a semipermeable membrane is described by the Kedem– Katchalsky equations. We have already seen that the volume flux across any membrane is zero when 'P = V'S. Thus, an alternative definition of the reflection coefficient is 'P ¬ . V 'S ® J v 0

From the preceding, it is evident that V = 1 for a semipermeable membrane.

(6.38)

240


Figure 6.4. An osmometer. The osmotic pressure is higher in Phase I than in Phase II. 'P = PI – PII is equal to 'S = SI – SII. If SI is known, then SII can be calculated from 'P.

When the reflection coefficient is unity, the osmotic pressure difference has as great an influence on volume flow as does the hydrostatic pressure difference [see Eq. (6.32a)]. When the substitution V = 1 is made in Eq. (6.32a), it takes the form of Starling's Law, originally developed to describe the volume flux across the capillary wall under combined differences in hydrostatic and osmotic pressure. In this context, it is worth noting that relatively small concentration differences generate considerable osmotic pressure differences: at 37ºC, when 'cs = 1 mM, RT'cs = 'S= 19 torr. Since the hydrostatic pressure in the capillaries is only about 40 torr more than that in the interstitial fluid outside, the osmotic pressure difference across the capillary wall necessary to reduce the volume flow to a modest level corresponds to a concentration difference of only a few millimoles per liter. If the membrane is leaky and passes solute, then the hydrostatic pressure difference needed to stop osmosis is less than 'S; that is, V < 1. This behavior can also be understood in terms of the profiles in Figure 6.3. When the membrane is semipermeable, the solute cannot enter the pore, and a hydrostatic pressure difference develops at each end of the pore to maintain continuity of the solvent chemical potential; that is, to compensate for the difference between the osmotic pressure of the ambient solution and that in the solute-free pore. When solute can enter the pore, a solute-free region remains, but it does not include the entire cross-section of the pore. Assume that the solute is a sphere of radius s, and let the center of the sphere define the radial position,


241

U, of the solute molecule in the pore. Clearly, if the pore radius is r, the radial position of the solute cannot exceed r – s (see Fig. 7.3 in the next chapter). The fraction of the 2 2 2 pore cross-section accessible to the solute is )s = [(r – s) /r ] = (1 – s/r) . This exclusion of the the solute from a portion of the pore is known as the steric effect, and we will discuss it more in the next chapter. Anderson and Malone (1974) derived an expression for the reflection coefficient in terms of a radial potential energy profile that described the partial or complete exclusion of solute from the pore; when applied to a profile representative of steric exclusion, they obtained 2

s ¬2 ¯ V = (1 – )s) = ¡¡1 1 °° . ¡¢ r ® °± 2

(6.39)

As the solute radius approaches that of the pore, the reflection coefficient approaches unity, in agreement with the result for a semipermeable membrane. Leakier membranes have smaller reflection coefficients. The reflection coefficient approaches zero for nonselective barriers, such as membranes whose transport pathways are much larger than the solute. This can also be seen from Eq. (6.39), as s/r o 0. A solute concentration difference across such a barrier will induce very little osmosis, even when the osmotic pressure difference is high. Sometimes the reflection coefficient is described as the ratio of the “apparent” or “effective” osmotic pressure difference to the true difference. An alternative interpretation of the reflection coefficient can be arrived at from the solute flux equation, Eq. (6.36). As the reflection coefficient rises from zero to unity, the flux of convected solute given by the first term in the equation falls from cs J v to zero. Noting that the convected solute flux when V = 0 is the flux that would be expected if solute were simply carried along in the solvent, the reflection coefficient has been interpreted as a measure of the retarding effect of the membrane on the solute relative to its resistance to solvent flow; that is, as a measure of the sieving effect of the barrier. The sieving effect of membranes is used in two important separation processes: ultrafiltration and reverse osmosis. In ultrafiltration, a solution of a relatively large membrane-impermeable species (often protein) is concentrated by raising its hydrostatic pressure to drive solvent across the membrane while the solute stays behind. The pore size in the membrane is typically 2 nm. A similar process takes place in reverse osmosis, which is used for desalination. Water is forced under hydrostatic pressure across a membrane whose pores, in contrast to those of ultrafiltration membranes, are so narrow (ca. 0.5 nm) that its reflection coefficient for sodium chloride and other small solutes is very close to unity. The osmotic pressure of the impure water is much higher than that of the desalted product, and the volume flow is driven against the osmotic pressure gradient. This is why the process is called reverse osmosis.

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6.2.6. Osmotic Pressure of Solutions; Donnan Osmotic Pressure; Osmotic Effects on Cells Osmotic Pressure of Solutions. In the preceding derivations, the concept of osmotic pressure arose from the expression for the chemical potential of the solvent. If the solution is sufficiently dilute, Equation (6.24) applies, and each solute contributes separately to the ideal osmotic pressure difference: 'Sideal = RT i 'ci . The summation includes all solutes, ions as well as nonelectrolytes. This expression can also be I II j written as 'Sideal = Sideal – Sideal , where Sideal = RT i cij , the ideal osmotic pressure of the jth phase. Since the osmotic pressure of an ideal solution depends on only its total concentration N = 6ci , irrespective of the identity of the solutes, it can be obtained simply by adding up the “particles” of solute per unit volume of solution. Thus, the total concentration of a solution containing 100 mM NaCl and 100 mM sucrose is 300 mM (100 mM each from Na, Cl, and sucrose). The osmolarity of a real solution is defined as the total concentration of an ideal solution that has the same osmotic pressure as the real solution in question. By this definition, if a solution is sufficiently dilute, its total concentration and osmolarity are essentially identical. Osmolarity is generally measured in milliosmols/liter, mOsm/l, to distinguish it from total concentration, which is measured in millimoles/liter. Two solutions having the same osmolarity are called isosmolar. For instance, in a world in which all solutions are ideal, the NaCl/sucrose solution described above, and a 100 mM solution of Na2SO4, would be isosmolar. Their osmolarity would be 300 mOsm/l. The molality of a solution is defined similarly to the molarity, except that the concentration is referred to 1 kg of solvent rather than 1 liter of solution. When molality is used instead of molarity, the associated terms described above are altered accordingly; for instance, the number of milliosmols/kg solvent is termed the “osmolality,” and two solutions having the same osmolality are isosmolal. For dilute solutions, molarity and molality are virtually identical. Most solutions encountered in living systems are not ideal, primarily because ion– ion and ion–solvent interactions reduce the effective number of independent particles in solution. For instance, the osmolarity of a 155-mM solution of NaCl at 25ºC is only 288 mOsm/l even though its total concentration is 310 mM; its osmotic pressure is 7% less than it would be were the solution ideal. The osmotic coefficient of a solution, M, is the ratio of the osmolarity of the solution to its total concentration (which is the same as the ratio of its osmotic pressure to the osmotic pressure it would exert if it were ideal); for the 155-mM NaCl solution, M 0.93. The osmotic coefficient of most physiological solutions (plasma, intracellular and interstitial fluid) is near this value. With this definition of the osmotic coefficient, the osmolarity, O, is equal to the product of the osmotic coefficient and the total concentration, so that the true osmotic pressure of a solution, Sreal = RTO. Other authors identify O with the total concentration, in which case Sreal = RTMO; the product MO is then referred to as the effective or corrected osmolarity of the solution.


243

Figure 6.5. Dependence of the osmotic coefficient on molar concentration (M) for (1) D-glucose, (2) NaCl, and (3) KCl at 20ºC. From data in Lide (2004).

In general, the osmotic coefficient depends on temperature, concentration and the identity of the solute. Although it is less than unity for electrolyte solutions at physiological concentrations, it can be greater than unity for more concentrated ionic and nonelectrolyte solutions and for protein solutions. For instance, the osmotic coefficient of the highly concentrated hemoglobin inside red blood cells is near 4 (Baumgarten and Feher, 2001). The osmotic coefficient of plasma protein solutions is a nonlinear function of concentration that can be described by a second-degree polynomial with positive coefficients. The variation of osmotic coefficient with concentration is shown in Figure 6.5 for several common solutes. Another term used to describe osmotic pressure is tonicity. The tonicity of a solution measures the osmotic flow it can induce across a specific membrane; thus, in contrast to osmolarity, which depends on only the composition of the solution, the tonicity depends on the reflection coefficients of the solutes in the membrane under consideration. For ideal solutions, the tonicity, T i Vi ci . Solutions having the same tonicity are said to be isotonic; when the solutions on the two sides of a membrane are isotonic, osmosis is absent. The terms hypotonic and hypertonic specify whether the solution in question has a lesser or greater tonicity than a reference solution, often the intracellular fluid. When the membrane in question is impermeable to all of the solutes in the solution, all of the reflection coefficients are unity, and the tonicity and osmolarity are identical.

244


Figure 6.6. Normalized Donnan osmotic pressure vs. cII / c II .

Donnan Osmotic Pressure. As shown in Chapter 1, when charges are trapped in a phase that can still exchange some electrolyte with a second phase, a Donnan equilibrium develops across the interface between the two phases. It is easy to show from Eq. (1.55) that the total concentration of mobile ions in the first phase [Phase I in Eq. (1.55)] exceeds that in the electrolyte solution (Phase II) with which it has equilibrated:

'c c AI cCI c AII cCII (cII )2 4(c II )2 2c II 0 .

(6.40)

This concentration difference gives rise to an osmotic pressure difference SD = RT'c between the two phases; the osmotic pressure in the charged phase is always higher. The dependence of SD (the Donnan osmotic pressure) on cII is plotted in nondimensional form as Figure 6.6.


245

Equation (6.40) does not include the contribution of the impermeant charged molecules in Phase I to the osmotic pressure of the phase. In most cases, the impermeant charges are carried on large macromolecules, which are relatively few in number, and whose contribution to the osmolarity of Phase I can safely be ignored (recall that cII is the concentration of impermeant charge in Phase I, not the concentration of charge-bearing molecules). The Donnan osmotic pressure promotes a volume flux into Phase I and is one of the reasons that charged gels tend to swell. Osmotic Effects on Cells. We noted in Chapter 2 that the total concentrations of the intracellular and extracellular fluids are similar, and they are indeed very close to isosmolar. This is a good thing: for the cell to maintain its normal volume, there cannot be any net flux of water into or out of it. Thus, any difference in osmolarity across the cell membrane must be balanced by a difference in hydrostatic pressure, the magnitude of which would depend on the reflection coefficients of the solutes in the cell and its environment. Because the osmotic pressure difference between the intracellular and extracellular fluids is very small, the hydrostatic pressure difference that arises to balance it is within the load bearing capabilities of the cytoskeleton and plasma membrane. When, in nonphysiological experiments, the extracellular osmotic pressure is suddenly increased or decreased from normal, the driving force for water flow rises markedly and the major initial flux is that of the solvent. The cell shrinks or swells, depending respectively on whether the external solution has been made hypertonic or hypotonic. The deformation of the cell alters the internal hydrostatic pressure, but this change is minor compared to the imposed change of external osmotic pressure; accordingly, the change in cell volume continues until the intracellular fluid becomes nearly isotonic with the external solution. During the initial rush of water flow, the solute fluxes into or out of the cell are not sufficient to alter substantially the intracellular solute content (as opposed to intracellular concentration). If, furthermore, the intracellular and extracellular fluids are essentially isotonic when water flow subsides — that is, if we neglect the small hydrostatic pressure difference across the cell membrane — it is easy to calculate the dependence of the final cell volume on the experimentally imposed extracellular osmotic pressure. The cell is considered to consist of two compartments: an aqueous compartment of variable volume V a that contains N moles of dissolved solute, and an inert phase of volume Vs that consists of fixed volume organelles, solid matter, and water that is bound to intracellular macromolecules and thus is osmotically inactive. Assume that the cell has achieved osmotic equilibrium with a solution whose osmotic pressure is Sc. Since we are neglecting hydrostatic pressure differences, the internal and external fluids are isotonic at equilibrium, so the intracellular osmotic pressure is also ScThe intracellular osmotic pressure is related to N and V a by Sa

RT MN . Va

(6.41)

246


Solving for V a , and adding in the volume of the inert phase, the cell volume, Vc, is given by Vc

RT MN Vs . Sa

(6.42)

Thus, a plot of cell volume vs. the reciprocal of the extracellular osmotic pressure (or extracellular concentration) should be a straight line with a y-intercept equal to the volume of the inert phase. This behavior has been demonstrated for a variety of cells, including red blood cells (LeFevre, 1964). When the osmotic pressure ambient to a suspension of erythrocytes is reduced, the cells can swell considerably, since they are normally disc shaped and can increase their volume substantially without increasing their surface area; acute increases in surface area are accompanied by tensile stresses in the cell membrane that can lead to rupture. Naturally, the cell cannot enlarge indefinitely as the external osmotic pressure is reduced, and when the concentration of the external solution becomes less than a threshold value, the cell membrane ruptures, causing hemolysis.

6.2.7. Passive Transport of Multiple Nonelectrolytes The Kedem–Katchalsky equations are only slightly more complicated when more than one solute is present. Of course, there are more phenomenological coefficients; referring to Eq. (6.10), the rank of the matrix of coefficients is n, where n is the number of 2 species (the solvent and n – 1 solutes). Thus, there are n coefficients, of which n are straight coefficients. Thanks to the Onsager reciprocal relation, only half of the re2 maining n – n cross coefficients are independent, so the total number of independent 2 coefficients is n + (n – n)/2 = n(n + 1)/2. Thus, for a single solute, n = 2, and (as we have seen) three coefficients are sufficient to describe the transport behavior of the system. When there are two solutes (n = 3), six coefficients are needed. This number rises quadratically as the number of solutes increases. To reduce the number of transport coefficients needed to describe a system containing multiple solutes, it is usually assumed that there is no direct coupling between the chemical potential difference of any given solute and the fluxes of all the other solutes. Then, the only cross coefficients are of the form Lsi w , where the subscript “i” denotes the ith solute. There are n – 1 of these, because there are n – 1 solutes. Since there are n straight coefficients, a total of 2n – 1 phenomenological coefficients are sufficient to describe the system; these can be translated into the hydraulic conductivity, n – 1 reflection coefficients (one for each solute), and n – 1 solute permeabilities. The Kedem–Katchalsky equations become n1

J v L p 'P L p Vi 'Si ,

(6.43a)

J i J v ci (1 Vi ) Zi 'Si (i 1,..., n 1) ,

(6.43b)

i 1

where 'Si RT 'ci .


247

6.2.8. Passive Transport of Electrolytes: Electrokinetic Phenomena When the solutes crossing a barrier are charged, the nonequilibrium thermodynamic description of the transport processes is more difficult, and the transport system exhibits some new features. Some of these new features can be identified simply by examining the dissipation function, without attempting the derivation of the transport equations, which are described fully in Kedem and Katchalsky (1963). Assume that the system contains only ions (indexed by i) and solvent; the extension to a combination of charged and uncharged solutes is straightforward. The driving force conjugate to ionic flux is the electrochemical potential difference, so the dissipation function is i . ) J w 'P w J i 'P

(6.44)

i

Proceeding in a fashion analogous to that used in Eq. (6.27), but using Eq. (6.24) instead of (6.25) because multiple solutes are involved, ¬ RT 'ci ) J wVw ('P RT 'ci ) J i Vi 'P ziᑠ'\ ci ® J ¬ ( J wVw J iVi )'P i J wVw RT 'ci (ᑠ J i zi )'\ ci ®

(6.45)

The dissipation function given above is similar in some ways to that for a single nonelectrolyte [Eq. (6.27)], and different in others. As before, the first product shows that the volume flux (which now includes contributions from each solute) is conjugate to the hydrostatic pressure difference. This product is followed by the sum of the contributions of the exchange fluxes to the dissipation rate; the exchange flux of each solute is conjugate to RT'ci, which is that solute's contribution to the osmotic pressure difference across the membrane. The last product in Eq. (6.45) did not appear in Eq. (6.27), and arises specifically because charged solutes are present. The driving force is the electrostatic potential difference across the membrane; the flux to which that force is conjugate is the current density. Nonequilibrium thermodynamics tells us that the members of this last force–flux pair can be coupled to the other forces and fluxes in Eq. (6.45). Such interactions are known as electrokinetic phenomena. Among these are: (1) electroosmosis, in which a volume flow is driven by an imposed potential difference, and (2) streaming effects, which arise when a component of the transmembrane current is driven, or a component of the transmembrane potential is generated, by a hydrostatic pressure-driven flow. The phenomenological coefficients defining these phenomena are derived in the paper by Kedem and Katchalsky (1963) referred to earlier.

248


6.3. NONEQUILIBRIUM THERMODYNAMIC DESCRIPTION OF ACTIVE TRANSPORT Active transport is incorporated into the nonequilibrium thermodynamic description of coupled transport by introducing the affinity, A, of the driving metabolic reaction and its conjugate flux, the rate of the reaction per unit membrane area, Jr. The reaction rate and the affinity of the reaction must be defined such that the reaction dissipates free energy at a rate equal to Jr A. Often, Jr is taken to be the mols of ATP hydrolyzed per unit area per unit time, and A is then the affinity of the hydrolysis reaction. The flux equations for active transport given by nonequilibrium thermodynamics are considerably simpler than those derived from kinetic analysis in Chapter 5, but they tell us nothing about the mechanism of the process. They do have their place, however, and we will use them to try (unsuccessfully) to construct an objective definition of active transport. Furthermore, since the forte of nonequilibrium thermodynamics is the description of coupled processes, we will also use the phenomenological equations to examine the several proposed measures of the “tightness” with which the metabolic reaction is coupled to the flux of substrate.

6.3.1. Definition of Active Transport For the simplest case of primary transport of a single uncharged solute, A, the phenomenological equations are JA = LAA 'PA + LAr A,

(6.46a)

Jr = LrA 'PA + Lrr A.

(6.46b)

Because of the notation adopted throughout the text, A in Eqs. (6.46) designates both the pump substrate and the affinity of the driving reaction. In context, there should be little confusion regarding the meaning of A at any point in the discussion to follow. The terms “substrate” and “affinity” will be used instead of the symbol wherever possible. For a system described by Eqs. (6.46), the requirement for active transport of substrate is that LAr be nonzero. This criterion is independent of the direction of either the flux of substrate or its chemical potential gradient; “downhill” active transport in the same direction as the passive component of the flux is fully within this definition. The definition of active transport is more difficult in systems containing multiple solutes and more coupling of forces and fluxes. In such systems, it is convenient to regard active transport as a coupling of the substrate flux to Jr (rather than to the affinity), and to write the phenomenological equation for JA in terms of resistances. To do this, we start with Eq. (6.13): X j R ji J i . i

The sum includes all species fluxes, and Jr. Partition the sum as follows:

(6.13)


X j R jj J j R jr J r

249

R ji J i .

(6.47)

other species

Solve Eq. (6.47) for Jj , and let the jth species be the substrate. Then, the substrate flux is given by

¬ 1 JA X A RAr J r RAi J i . RAA other ® species

(6.48)

In Eq. (6.48), the first term on the right-hand side is the contribution to the flux of substrate that is driven by the substrate's own chemical (or electrochemical) potential difference, XA; the straight resistance, RAA, is positive. The second term measures the contribution of the metabolic reaction, and the third term can be regarded as the drag or entrainment of the substrate by the fluxes of other species. For instance, the convection of substrate in moving solvent would be included in the third term, with i = w and RAw < 0 (since solvent drag carries the solute in the same direction as the solvent). The sum in Eq. (6.48) also includes inhibitory effects. If species B is a competitive inhibitor of the flux of A, then RAB is positive, since JA is diminished when more B uses the carrier to cross the membrane; i.e., when JB increases. Equation (6.48) takes on a particularly simple form when the substrate is in thermodynamic equilibrium across a membrane (i.e., XA = 'PA = 0) and the substrate flux is not coupled to that of any other species. In that case, JA = –RArJr /RAA. For an active A , and thermodytransport system whose substrate is an ion, 'PA is replaced by 'P namic equilibrium is achieved by short-circuiting the membrane (so that '\ = 0) and maintaining identical concentrations of substrate on the two sides. Under these conditions, the substrate flux should be directly proportional to the rate of ATP hydrolysis. Primary active transport is the direct coupling of substrate flux to Jr; that is, RAr z 0. Similarly, secondary transport via the cosolute B would be characterized as follows: RAr = 0 (primary transport of A is absent), RBr z 0 (B is the substrate of a primary active transport system), and RAB z 0 (the flux of A is coupled to that of B). The preceding description suggests a means for deciding whether any particular flux (e.g., of substrate A) has an active component: if RAr is nonzero, A is actively transported; if RAr is zero, examine those fluxes (call them {JAc}) to which JA is coupled (i.e., RAAc z 0), and see if any of these fluxes are coupled to a Jr (i.e., RAcr z 0). If none are, look for metabolic coupling to any species fluxes to which the {JAc} are coupled, and so forth. This is an attractively straightforward and objective algorithm; the problem in using it is that every flux in a living organism can ultimately be traced to a metabolic reaction of some kind. The arbitrariness of such a definition of active transport has prompted some to suggest that only primary active transport should bear that title. At the very least, the claim that a particular flux has an active component should be accompanied by an indication of the process to which that component is thought to be coupled.

250


Some investigators have proposed that active transport be defined as the flux of substrate against its electrochemical potential gradient. Although uphill transport is one indication of an active process, this definition does have some problems. One difficulty is that it makes no mention of metabolic coupling, which underlies all active transport. Furthermore, active transport systems that pump uphill are also capable of supporting a downhill flux when the electrochemical potential of the substrate is lower at the trans side of the membrane. In this case, the pump and leak fluxes proceed in the same direction. This mode of operation is more common when the coupling between ATP hydrolysis and transport is weak.

6.3.2. Coupling Between Nonconjugate Forces and Fluxes Nonequilibrium thermodynamics provides a formal means for characterizing, through the phenomenological coefficients, the degree to which a flux is coupled to a force that is not its conjugate. Thus, it is perfectly suited to describe coupling in active transport, in which the substrate flux is coupled to a nonconjugate driving force, the affinity of the metabolic reaction. The degree of coupling, q, between the ith flux and the jth force is given by q

Lij ( Lii L jj )1/ 2

.

(6.49)

Since Lij = Lji, the same value of q holds for the coupling of the jth flux to the ith force. The coupling can be positive or negative, depending on the sign of Lij. Thermodynamic considerations dictate that L2ij dLiiLjj; thus, the absolute value of q lies between zero and one. When q = 0, there is no coupling; when coupling is complete, |q| = 1. In applying Eq. (6.49) to active transport, i is set equal to A, the substrate. For primary transport, j = r, the driving reaction; for secondary transport, j = B, the cosolute. In cotransport, A travels down B's (electro)chemical potential gradient, so JA and B have the same sign. Therefore, LAB > 0, and q is positive. In countertransport, the 'P cross coefficient is negative, and so is q. When two force–flux pairs are completely coupled, the ratio of the fluxes is fixed — one cannot proceed without the other — and is independent of either driving force. 1/2 This can be seen by substituting Lij = (LiiLjj) into the flux equations: 1/2

Ji = LiiXi + (LiiLjj) Xj, 1/2

Jj = LjjXj + (LiiLjj) Xi. 1/2

Multiply Ji by (Ljj /Lii) : 1/ 2

L jj ¬ J i L ® ii

( Lii L jj )1/ 2 X i L jj X j = Jj ( | q | = 1).

(6.50)


251

The stoichiometry of the coupled fluxes (i.e., Ji /Jj) is determined by the ratio of the straight phenomenological coefficients. The fluxes may be similarly or oppositely directed, since the sign of the square root of Ljj /Lii can be positive or negative. We previously referred to such transport as obligatory. Values of |q| less than unity reflect incomplete coupling between the driving process — the dephosphorylation of ATP or the flux of cosolute — and the flux of the substrate. One class of processes that diminish the degree of coupling are the leakage fluxes described in Chapter 5. Recall that two kinds of leak can be identified: internal and external. An internal leak (called “slippage” in Chap. 5) occurs within the pump itself; examples include the transition of unloaded carrier from the cis side of the membrane to the trans side, or of loaded carrier in the opposite direction. External leaks include transport pathways in parallel with — but physically separate from — the pump. The degree of coupling can be measured in several ways. It can be written in terms of: 1. the dependence of the fluxes on one another: sJ ¬ q 2 i < sJ j ® Xi

¬ sJ j , sJ i ® Xj

(6.51a)

or 2. the driving forces at which each flux is zero: q2

( X i ) Ji 0 ( X i ) J j 0

,

(6.51b)

or 3. the flux of one species when either the flux or driving force of the other is zero: q2 1

( J j ) J i 0 ( J j ) Xi 0

.

(6.51c)

Equation (6.51c) shows that, when coupling is incomplete, ( J j )Ji 0 z 0; that is, the jth flux is nonzero even when the ith flux is zero. Consider the case in which the subscript i denotes the substrate and j denotes the chemical reaction that drives the active substrate flux. With this designation of i and j, we see that, when the coupling is incomplete, metabolic activity is necessary ( J j v 0) to maintain the nonequilibrium state created by the pump, even in the absence of a substrate flux ( J i 0) . This is a result of the leaks referred to above. Two other thermodynamically based measures of flux coupling have been applied to active transport. The first of these is the efficacy, which is based on the driving forces at static head. The efficacy with which the jth force drives the ith flux is

252


X ¬ H i X j ®

.

(6.52)

J i 0

Of greater interest is the efficiency, which is the power associated with the active substrate flux, divided by the rate at which energy is dissipated by the driving process. Consider a secondary scalar transport process in which the substrate is an uncharged amino acid and the cosolute is sodium ion. The work of transporting nA mols of the amino acid A against a chemical potential difference P IIA P IA is

nA (P IIA P IA ) nA 'P A ; when the substrate flux is JA, the rate of transport work is equal to –JA'PA. Similarly, the rate at which energy is dissipated by the sodium ion B B (for a chemical passing down its own electrochemical potential gradient is J B 'P reaction, the dissipation rate is Jr A). The efficiency with which the jth process drives the ith flux is K

JX J 'P power out i i A A . B power in JjXj J B 'P

(6.53)

The efficiency depends strongly on the experimental conditions. Consider a pump whose operating curve is given by Figure 6.7. The pump normally operates in the second quadrant, pumping substrate from Phase I to Phase II (JA > 0) against its chemical potential gradient ('PA < 0). From Eq. (6.53), its efficiency in doing so is generally greater than zero. But at the two intercepts that bound that segment of the operating curve, the efficiency is zero! At level flow, XA = 'PA is zero, and at static head, JA is zero. Thus, as the operating point moves from the static head in the direction of increasing flux, the efficiency rises from zero, passes through a maximum, and then falls as the level flow condition is approached. It can be shown that the maximum efficiency depends on the degree of coupling, and is unity when |q| = 1.

6.3.3. Nonequilibrium Thermodynamics of Motor Proteins and ATP Synthase Motor proteins, such as the kinesins of Chapter 3 and the actin and myosin of the myofibril in Chapter 9, employ the energy in ATP to create motion against a resisting force, such as the viscous drag on secretory vesicles being carried along a microtubule or a force opposing the contraction of muscle. These proteins convert chemical energy into mechanical work rather than solute flux, but the nonequilibrium thermodynamic treatment of both conversions is similar. The conjugate force–flux pair for motor proteins consists of the force, F, against which the motor acts, and the velocity, v, of the protein. These are coupled with the progress of the metabolic reaction and its affinity; using the linear formulation,

v = LvvF + LvrA,

(6.54a)

Jr = LvrF + LrrA,

(6.54b)

where the subscript “v” in the generalized conductances denotes the protein.


253

Figure 6.7. Operating curve of a pump in the absence of external leaks. The variation of efficiency between static head and level flow is also shown.

Since the straight phenomenological coefficients are always positive, the linear formulation implies that the velocity vector of the motor is in the same direction as the force. And so it would be, if the protein responded passively to the force in the absence of ATP. The energy provided by ATP hydrolysis lifts the system to a decidedly nonequilibrium state, driving the motor in a direction opposite to the force. We will see below that when the system is far from equilibrium, additional, higher-order terms must be added to the linear formulation to describe properly the dependence of the fluxes on the forces. Similar considerations apply to active ion transport. We saw in Chapter 3 that the molecular motor ATP synthase is driven by the electrochemical potential difference of hydrogen ions across the inner mitochondrial membrane. The synthase, in turn (and in turning), drives the phosphorylation of ADP. Using the formalism of Eq. (6.48), 1. The rotation of the synthase rotor (JA) is conjugate to the torque it experiences (XA) and is also coupled to the flux of hydrogen ions across the membrane [Ji in Eq. (6.48)]. 2. The phosphorylation reaction (Jr ) is conjugate to its affinity and is also coupled to the motion of the rotor. The first of these processes could be close to equilibrium since the torque is derived from Brownian motion, but the second can be far from equilibrium, since the phosphorylation reaction is highly endothermic.

254


6.4. LIMITATIONS OF NONEQUILIBRIUM THERMODYNAMICS Although nonequilibrium thermodynamics is a powerful tool for analyzing and characterizing coupled processes, it is not without its limitations. Some of these have already been noted. In this section, we will examine the impact of these limitations in a more organized way, with particular emphasis on the usefulness and range of applicability of the flux equations. The applicability of these equations is further limited by the approximations made in deriving them, and we will also deal with these restrictions.

6.4.1. Closeness to Equilibrium: A Limitation Intrinsic to Linearized Nonequilibrium Thermodynamics The basic equation underlying the applications of nonequilibrium thermodynamics described in this chapter is the linear relationship between the fluxes and the forces given by Eq. (6.10). We will first consider the limitations on the use of this equation. When Eq. (6.10) was first presented, it was noted that a more complete expression would include higher-order terms in the driving forces. Indeed, the linear relation can be regarded as the first terms in a Taylor series expansion for the ith flux: n

n

n

J i Lij X j Lijk X j X k ... . J 1

(6.55)

j 1 k 1

The justification for truncating the series after the linear terms is that the expansion variables — in this case, the driving forces — are small. Since the driving forces are zero at equilibrium, the requirement that the forces be small is equivalent to a requirement that the system be close to equilibrium. Such a requirement is not very useful from a quantitative point of view. How close to equilibrium is close enough? How small a driving force is small enough? The answer depends on the problem at hand. If one is trying to measure a particular transport property, then “small enough” might mean: sufficiently small that the error in the final answer introduced by neglecting higher-order terms is less than the experimental error. If nonequilibrium thermodynamics is being used in a more qualitative way — say, to identify the existence of a particular interaction — “small enough” might mean: sufficiently small that the neglect of higher-order terms does not cause us to conclude that the interaction exists when it doesn't, or vice versa. Since the phenomenological coefficients are experimental quantities, it is in most cases impossible to estimate, a priori, the errors in flux prediction resulting from the departure of the transport system from equilibrium (ironically, such departure is necessary for transport to take place at all). Some insight into the factors that affect these errors can be gained by considering the free diffusion of a single nonelectrolyte; for this process, the flux equation derived in Chapter 2 can be used as a gold standard against which the nonequilibrium thermodynamic result can be compared. That gold standard is Eq. (2.20): Js = ks'cs ks (csI csII ) .

(2.20)


255

In the nonequilibrium thermodynamic formulation, the flux equation for the same system is Eq. (6.9):

Js = Lss'Ps.

(6.9)

For an isothermal, isobaric system, Eq. (6.9) becomes

Js = LssRT 'ln cs Lss RT ln

csI . csII

(6.56)

Equating the right-hand sides of Eqs. (2.20) and (6.56), and solving for Lss:

ks c I c II (6.57) ¸ s I sII . RT ln(cs / cs ) If the solute flux is proportional to its conjugate driving force 'Ps, then Lss is constant. Assume that Lss approaches a limiting value, Leqss , when the driving force approaches Lss

zero. Then the deviation of Lss from Leqss as 'Ps increases from zero is a measure of the departure of the nonequilibrium thermodynamic description of the system from linearity. Alternatively, the constancy of Lss measures the validity of the linear thermodynamic description of the diffusion process. Returning to Eq. (6.57), we see that Lss is the product of two terms, the first involving the permeability and the second the concentrations at the two sides of the membrane. The permeability depends on these concentrations, too, and we will approximate that dependence here by assuming that the permeability is a function of only the mean concentration, cs . Now consider an experiment in which the initial concentration of solute is the same at both sides of the membrane. Then the concentration at Side I is raised, and that at Side II is lowered, such that the mean concentration is unchanged. When the boundary conditions are varied in this way, the permeability remains constant by our previous assumption, and the variation of Lss with the chemical potential difference is given by the second term in Eq. (6.57). Let r csI / csII ; Eq. (6.57) can then be written as

Lss

ks csII r 1 ¸ . RT ln r

(6.58)

The driving force is a particularly simple function of r: 'Ps = RT ln r.

(6.59)

The quantities r and csII in Eq. (6.58) are related by the requirement that the mean concentration, given by Eq. (6.60), is fixed:

cs

csI csII c II (r 1) s . 2 2

(6.60)

256


Equation (6.60) is rearranged to express csII in terms of cs ; substituting this expression into Eq. (6.58), Lss

ks cs 2(r 1) kc w s s ᑬ(r ) . ¸ RT RT (r 1) ln r

(6.61)

Figure 6.8. The function ᑬ, defined in Eq. (6.61). Note that ᑬ is the ratio of the functions in Figure 6.2; thus, the appropriateness of the linearization of 'ln cs, and the closeness to equilibrium of the transport system, go hand in hand.

Since the mean concentration and permeability are constant, the variation of Lss as the driving force rises from zero is given by the function ᑬ(r). This function is unity at equilibrium (i.e., when r = 1), so Leqss is equal to ks cs / RT . ᑬ(r) is plotted against r in Figure 6.8. Clearly, this transport system is close enough to equilibrium even when the concentration at one side of the membrane is twice that at the other, since L55 is only 4% less than its equilibrium value when r = 2. This kind of analysis can also be applied to facilitated diffusion. For the simplest carrier, Eq. (4.39) is used as the gold standard to measure the variation of LAA with 'PA, much as Eq. (2.20) was used for free diffusion; the function corresponding to ᑬ(r) in Eq. (6.61) is ᑬ

I

II

where r = cA /cA .

ln r r 1 r ( K / c AII ) 1 ( K / c AII )

,

(6.62)


257

In primary active transport, the affinity of the metabolic reaction that drives the pump must be small. “Small” in this case means: A 'Pa, and the profile looks like a corrugated well rather than a corrugated mesa.

7.2. MODELS OF TRANSPORT THROUGH PORES Toward the end of the previous chapter, we distinguished between two approaches to the description of passive transport: the nonequilibrium thermodynamic approach, which is appropriate when the solute flux is coupled to solvent flow, and which requires that the system be close to equilibrium; and the earlier analyses of free and facilitated transport, in which solutions were presented for the solute flux in systems


271

that are farther from equilibrium, but in the absence of solvent drag. The models of pore or channel transport that have been proposed can be classified similarly.

7.2.1. Classification of Pore Transport Models One class of models describes transport through ion-specific channels in the cell membrane. The approach taken can be microscopic, as in the energy barrier models treated in Chapter 4, or phenomenological. An example of the latter is Hodgkin and Huxley's (1952) pioneering work on nerve, which will be discussed in Chapter 9. When the phenomenological approach is taken, the open channel is assumed to have a particular conductance (or conductances, if the channel has multiple open states), which is found experimentally, and the rate at which ionic current passes through the channel depends on the fraction of time that its gate is open. The mechanisms that underlie the unitary conductance are not of primary concern. Since any macroscopic membrane patch contains a large number of channels, the current density can also be expressed in terms of the fraction of gates that are open at any instant. In both of these approaches — the microscopic and the phenomenological — the sole force driving an ion through the channel is the gradient of its electrochemical potential. The second class of model, which will be discussed in this section, describes a less selective pore, through which nonelectrolytes and water also pass. These models attempt to interpret the phenomenological transport coefficients of nonequilibrium thermodynamics in terms of the mechanics and kinetics of flow and diffusion through a pore. A particular application of these models is the estimation of the effective pore radius from transport data. These estimates are not quantitative; still, if the calculated radius is unrealistically high or unrealistically low, this is certainly an indication that other transport pathways are likely to be present. Similarly, a reasonable radius is evidence for (but by no means proof of) transport through pores in the membrane. Pore sizes computed from such models have also been used to decide whether solutes or water cross cell layers by passing through the cells, or between them. A calculated pore size much larger than the expected size of cell membrane pores is taken as evidence for significant transport through the wider passages between the cells. Broadly speaking, one can define three regimes of pore transport, depending on the ratio of the solute radius, s, to the pore radius, r. Examples of all three will be presented in the subsections to follow. These models of pore transport will assume that spherical, uncharged solute molecules (including tracer solvent molecules) are passing through a cylindrical pore. A reference will be given subsequently to extensions of the theory that treat nonspherical and charged particles, and pores of different shape. The three regimes are as follows: 1. If s 0.5, solute particles can no longer pass one another, and transport is single file. Single-file transport is treated using approaches similar to those used earlier for ion channels, which are similarly restrictive. Many of the features of specific ion channels, such as selectivity filters and electrostatic forces, are generally absent from single-file transport models, though the notion of binding sites is retained to enumerate the number of solute molecules that a fully occupied pore might contain.

7.2.2. Hydraulic Conductivity of a Pore The easiest phenomenological coefficient to interpret in terms of pore transport is the hydraulic conductivity. The pore is simply regarded as a straight cylindrical tube of radius r and length equal to the membrane thickness, a; the pore axis is perpendicular to the membrane face. The pressure drop, 'P, required to drive a flow, q, through such a pore is given by the Hagen–Poiseuille equation: 'P

8aKq , Sr 4

(7.8)

where K is the solvent viscosity. The volume flux is equal to the flow rate divided by the cross-sectional area of the pore: Jv

q . Sr 2

(7.9)

The hydraulic conductivity is defined by Eq. (6.31a): Lp

Jv . 'P

(6.31a)

Substituting Eqs. (7.8) and (7.9) into (6.31a), Lp

r2 . 8 aK

(7.10)

The hydraulic conductivity of the pore is directly proportional to the square of the pore radius and inversely proportional to the membrane thickness and the viscosity of the solvent. A similar expression can also be derived for a membrane that contains a specified number of pores, of specified radius, per unit area; the derivation of this expression is left as an exercise.


273

If the axis of the pore is not perpendicular to the membrane faces, its length, l, replaces the membrane thickness, a, in the denominator of Eq. (7.10). The ratio l/a p 1 is known as the tortuosity of the pore, W.

Figure 7.4. (a) Nonelectrolyte diffusion through a cylindrical pore. The solute toward the “bottom” of the pore (in the side view) cannot enter because the radial distance from its center to the pore wall is less than s. The sterically available area, Ast, is outlined by the dashed circle in the front view of the pore. (b) Solvent film with pore-sized patch. The area of the patch is A0 = Sr2.

7.2.3. Hindered Diffusion; Solute Permeability as a Probe of Pore Radius The radius of membrane pores can be estimated from the dependence of nonelectrolyte permeability on solute size. Ions are not as good for this purpose, because their effective radius depends strongly on whether they are hydrated when they are in the pore (recall Chap. 4). In addition, when ions are used, the analysis is complicated by the possibility of electrical effects arising from the membrane potential or charged sites in the pore. Nonelectrolyte diffusion through a pore is most simply modeled by considering a spherical solute, of radius s, diffusing through a straight cylindrical channel, as shown in Figure 7.4a. Our objective is to compare the rate of solute transport through the pore to that across a film of solvent whose thickness is the same as the length of the pore. Transport through the film proceeds by free diffusion and is given by Eq. (2.18):

J s0 ks0 'cs , where the superscript denotes the solvent film.

(2.18)

274

CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

The flux through the pore is less than that through the film, for two reasons. First, the walls of the pore at the membrane face impede the entry of the solute into the pore; this is termed the steric effect. Second, once the solute has entered the pore, it encounters a viscous resistance that, because of the nearby walls of the channel, exceeds the resistance in free diffusion. We will consider each of these effects in turn. The Steric Effect. As can be seen from Figure 7.4a, the center of any solute molecule that enters the pore must cross the x = 0 plane at least a distance s from the wall of the channel. Thus, the area of the pore accessible to solute is 2

r s ¬ Ast S(r s) Ao , r ® 2

(7.11)

2

where Ao is the cross-sectional area of the pore, Sr . The Viscous Effect. The viscous effect causes a further reduction in the flux through the pore. Consider a small circular patch of solvent film whose area is the same as that of 0 the pore (Fig. 7.4b). Let n be the number of mols of solute crossing the film through the patch, per unit time:

n0 J s0 Ao ks0 Ao 'cs .

(7.12)

The number of moles, n, crossing through the pore per unit time, can be described by an equation similar to Eq. (7.12):

n = ksAst'cs.

(7.13)

Equation (7.13) defines the pore permeability, ks; ks measures the ease of transport through the pore, once the solute has gotten inside. Since the patch and the pore present the same area to the solute, any differences 0 between n and n are of either steric or viscous origin. The ratio of the two transport rates is k A n 0s ¸ st . 0 n ks Ao

(7.14)

The second factor on the right-hand side of this equation is the steric contribution to 0 the difference between n and n , given by Eq. (7.11). It is the ratio of the rate at which molecules enter the pore to the rate at which they enter the patch. The viscous contribution is contained in the first factor, which is the ratio of the ease with which the particle diffuses through the pore once it has entered, to the ease with which it crosses the (wall-less) solvent film. An equation for this ratio was derived by Faxen (1922, cited in Renkin, 1954): 3

5

s¬ s¬ s¬ ks s 1 2.104 2.09 0.95 w 1 P . 0 r ® r ® r ® ks r

(7.15)


275

In deriving this equation, it was assumed for simplicity that the spherical solute molecules are centered along the pore axis and that the solution in the pore is sufficiently dilute that each such molecule crosses the membrane independently. Equation (7.15) is an approximation to the numerical solution of the governing flow equations that applies when s < 0.4r. The viscous effect given by Eq. (7.15) is plotted against s/r in Figure 7.5. The figure shows how ks decreases relative to the permeability of the solvent film as the radius of the solute approaches that of the pore.

Figure 7.5. (a) Faxen's function [Eq. (7.15)] for the viscous effect on solute diffusion through a pore. (b) Combined steric and viscous effects [Eq. (7.16)] on solute diffusion. (c) Anderson and Quinn's (1974) function [see Eq. (7.23)] for the viscous effect on solute convection through a pore. Note that the steric and viscous terms, here and in Figure 7.8, approach unity as the pore diameter becomes much larger than the solute diameter (s/r o 0).

Substituting Eqs. (7.11) and (7.15) into (7.14), 2 s ¬¯ s ¬ n ¡ ° 1 P 1 . r ®° r ® n 0 ¢¡ ±

(7.16)

This ratio is also plotted against s/r in Figure 7.5. Equation (7.16) can be used to estimate pore size from the dependence on solute radius of the permeabilities of a series of nonelectrolytes. When Eq. (7.16) is used in this way, it is assumed that these solutes all use the same pores, and that the transport channels are all of similar size. The applicability of Eq. (7.16) is also limited by the assumptions and approximations used to obtain Eq. (7.15).

276


Equations (7.12) and (7.16) can be used to predict the rate of diffusion of a solute of radius s through a pore of radius r, in terms of the free-diffusion permeability: 2

s ¬¯ s ¬ n ks0 Ao ¡1 P ° 1 'cs . r ®° r ® ¢¡ ±

(7.17)

Note that, throughout this derivation, ks0 is the permeability of a solvent film whose thickness is the same as the length of the pore. If the tortuosity of the pore, W, is greater than one, ks0 will be proportionately less than that of a film whose thickness is the same as that of the membrane: if the membrane thickness is a, the pore length is l, and the free diffusion coefficient of the solute is Ds , ks0

Ds Ds ¬ 1 W a ® l

(7.18)

The flux of solute through the pore, Js, is equal to n/Ao ; using Eq. (2.18), 2 2 s ¬¯ s ¬ s ¬¯ s ¬ 0 ¡ ° ¡ ° J s k 1 P 1 'cs J s 1 P 1 . r ®° r ® ¡¢ ¡¢ r ®°± r ® ± 0 s

(7.19)

2

Thus, the polynomial in s/r, [1 – P(s/r)](1 – s/r) , measures the ratio of the flux through the pore to that through a film of thickness, l; call this quantity )d . The ratio of the flux through the pore to that through a film of thickness, a, is )d /W. Equation (7.19) shows that the apparent permeability of a pore, ksapp , is given by ksapp w

Js ks0 ) d (s / r ) . 'cs

(7.20)

Equation (7.20) was used in an interesting way by Wang and Veenstra (1997), who employed it to estimate the pore radius of the rat gap junction protein connexin43 from measurements of the conductance of the channel to a series of six cations. They used a slightly different expression for )d (s/r) and assumed that ksapp was proportional to )d , neglecting the variation among the free diffusion coefficients of the solutes (the effect of this omission is examined in an exercise). The hydrated radii of the six ions were used and the apparent permeability of each ion normalized by that of lithium. The value of r that best fit their data was 0.63 nm; this was interpreted as the radius of the “modest” selectivity filter of the channel, and was consistent with estimates based on the size of several sugars that were too large to pass through. The fit to their data is shown in Figure 7.6.

7.2.4. Other Factors Affecting Estimated Pore Size; The Equivalent Pore The pore size of larger channels, such as those between cells, can also be estimated from solute permeabilities, using solutes much larger than those used to size pores in cell membranes. Deen and coworkers (Deen et al., 1980, 1983) estimated the intercellular pore size in the glomerular capillary wall (in the kidney; see Chap. 10) from the


277

fluxes of a series of charged and neutral macromolecules. Their model was based on the convective diffusion equation (2.29), modified to include the hindrance to diffusion and convection arising from the wall of the pore. Steric effects were included through a partition coefficient at the ends of the pore. This coefficient, like the hindrance factors, depended on s/r, but not on solute charge. When the solute was charged, a Donnan equilibrium condition was applied at the ends of the pore; the contribution of the transmural electrostatic potential gradient to the driving force for solute flux was expected to be negligible and was omitted.

Figure 7.6. Correlation of the normalized ionic permeability of rat connexin Cx43 vs. hydrated ion radius. The permeabilities are normalized by that of lithium, indicated by the open circle. The curve is the predicted dependence for a pore radius r = 0.63 nm. Reprinted with permission from Wang and Veenstra (1997). Copyright © 1997, The Rockefeller University Press.

Their results showed that pore theory can explain the effect of molecular size on glomerular permeability. The derived pore radius was found to depend on the shape of the polymers that were used. In rat, the pore radius that best described the size dependence of the fluxes of a series of dextrans was 4.7 nm, while that for a series of horseradish peroxidases was 3.7 nm. This difference reflects deviations between the assumptions of the pore transport model (e.g., simple, uniform geometry), and reality. Because of these deviations, a pore radius obtained by applying such models to transport data is often referred to as an equivalent pore radius; it is the radius of a cylindrical pore whose transport properties would match the experiments if the other assumptions of the model held. The equivalent radius that explains the permeability of one class of molecule cannot necessarily be used to predict the permeability of members of another class. When the solute bears a charge, its flux depends on the membrane potential, the charge density in the pore, ionic strength, and the variables that govern the flux of uncharged molecules. Figure 7.7, taken from Deen et al. (1983), is a prediction by

278


their model of the dependence of the filtrate/plasma concentration ratio in the glomerulus (proportional to solute flux through the glomerular capillary wall) on solute radius, solute charge, and the concentration of fixed charge in the pore. The figure shows that the solute flux is strongly dependent on these variables when r and s are comparable. This dependence is separate from the effect of molecular shape, which can add another decade of variability to the solute flux (Deen et al., 1983).

Figure 7.7. Glomerular filtrate-to-plasma concentration ratio (T) as a function of solute charge (Z) and radius (s). Results are shown for two values of membrane fixed charge concentration (Cm), and a pore radius r = 5 nm. The ratio T is proportional to solute flux. In this model, solute charge affects flux only through Donnan equilibria at the ends of the pore; when Cm = 0, Donnan effects are absent, and T is independent of Z. 1 Å = 0.1 nm. Reprinted with permission from Deen et al. (1983). Copyright © 1983, Springer-Verlag.


279

7.2.5. Hindered Convection (Sieving); the Reflection Coefficient as a Probe of Pore Radius It is possible to obtain an equivalent pore radius from the reflection coefficient of a single solute. Consider a sieving experiment, in which a solution is driven through a cylindrical pore by hydrostatic pressure, and the solute concentration is the same on both sides of the membrane. Except for a small entrance region near its mouth, the pressure-driven solvent velocity profile in the pore is parabolic:

U2 ¬ v vmax 1 2 , r ®

(7.21)

where v(U) is fluid velocity in the axial direction and U is radial distance from the axis of the pore. The maximum velocity is on the centerline, at U = 0. Because of the velocity profile, more of the solute convected through the pore does so close to the pore axis, as compared to the diffusional case discussed earlier. For this experiment, the steric ratio is (Ferry, 1936)

Ast r s ¨ 2SUvdU 0 Ao

r

2

4

s¬ s¬ ¨ 2 SUvdU 2 1 1 . r ® r ® o

(7.22)

The steric ratios given by Eqs. (7.11) and (7.22) are plotted against s/r in Figure 7.8. Owing to the velocity profile, as described above, the ratio is closer to unity — that is, the steric effect is less — for a sieving experiment than for a diffusion experiment.

Figure 7.8. Steric ratio as a function of solute radius/pore radius for (a) diffusion [Eq. (7.11)] and (b) convection (sieving) [Eq. (7.22)].

280


When transport through the pore is by convection rather than diffusion, a different polynomial (Anderson and Quinn, 1974) represents the viscous effect, subject to the same restrictions as Eq. (7.15); for the ith species: ni ¡ si ¬ si ¬ 2 1 1 r ® r ® ni0 ¡¡ ¢ 2

2

4

¯ ¯ ° ¡1 P ' si ¬° w Q* , ° i °¡ r ®± ±° ¢

(7.23)

3

where Pc(si /r) = 0.667(si /r) + 0.163(si /r) . This expression for the viscous effect in convection is also plotted against s/r in Figure 7.5. We see that viscous retardation is considerably less in convection than in diffusion. The difficulties encountered by the solute in entering and passing through the pore are shared by the solvent molecules as well. The solute is sieved by the pore if it is retarded more than the solvent. When the pore is very large, convection through it is the same as that through a solvent film, and there is no sieving: ns0 csVw nw0 ,

(7.24)

where cs is the uniform solute concentration. The sieving effect of the pore can be related to the solute reflection coefficient, using the Kedem–Katchalsky equation for solute flux [Eq. (6.36)] and approximating the volume flux through the pore by J wVw . In the absence of a concentration gradient ('S = 0, cs = cs ), Eq. (6.36) becomes J s cs J wVw (1 V) .

(7.25)

Dividing the flux terms on each side of the equation by the area of the pore, ns cs nwVw (1 V) ,

(7.26)

where ni = Ji /Ao is the rate at which the ith species crosses the pore, in mols/pore-s. Equations (7.23), (7.24), and (7.26) can be combined to express the reflection coefficient as a function of the radii of the solute, solvent, and pore: V 1

vs* . vw*

(7.27)

Equation (7.27), with Eq. (7.23), can be used to estimate the pore size from a single reflection coefficient, provided that the solute and solvent pass through the same pores. The reflection coefficient need not be measured by a sieving experiment. Goldstein and Solomon (1960) carried out osmotic equilibrium experiments to measure the reflection coefficients of the human red blood cell membrane for nine solutes whose molecular size could be estimated. They used P(si /r) given by Eq. (7.15) rather than Pc(si /r) in the definition of vi* . The equivalent pore radius that best fit their data was 0.42 nm, not an unreasonable value. The fit to their data is shown in Figure 7.9. The implications of their choice of polynomial is examined in an exercise. Equations (7.23) and (7.27) can also be used to estimate the reflection coefficient of a solute in a pore from the solute and pore radii.


281

Figure 7.9. Reflection coefficients of the nine solutes used by Goldstein and Solomon (1960) vs. solute radius. The curve is a plot of vs* / vw* (see text) vs. solute radius for pore radius r = 0.42 nm and solvent radius sw = 0.15 nm. Reprinted with permission from Goldstein and Solomon (1960). Copyright © 1960, The Rockefeller University Press.

If the flow of solvent through the pore is not appreciably retarded, vw* | 1 and V = 1 – vs* . The concentration of the solution exiting the pore, cf, is equal to the mols of solute crossing, ns, divided by the accompanying volume of solvent, Vw nw ; using Eq. (7.26) and the relation between V and vs* ,

c f cs vs* .

(7.28)

Thus, under these conditions, vs* is the ratio of the concentration of the filtrate exiting the pore to the upstream concentration of the solute. This ratio is referred to as the sieving coefficient of the solute in the membrane. Equation (7.26) is based on the assumption that the concentration in the pore is uniform and the same as that at both faces of the membrane. When this assumption holds, Eq. (7.26) describes the relative amounts of solute and solvent passing through the pore. If, however, the downstream compartment is comprised of the less concentrated filtrate, cs is no longer uniform in the pore, and these assumptions fail. The sieving coefficient can still be defined as cf /cs, as it often is, but it is no longer equal to vs* . Another treatment of sieving that is based on continuum concepts rather than nonequilibrium thermodynamics, and that includes transmembrane concentration differences, is presented in the following subsection and illuminates this problem further.

282


A variety of models of solute diffusion and convection through pores that are small enough to affect both access and transport, yet large enough that continuum modeling can still be applied, can be found in a review by Deen (1987). This review also includes alternative expressions for the viscous effect in cylindrical pores and similar equations for slit-shaped pores (which are used to model gaps between capillary endothelial cells), a model of simultaneous diffusion and convection like that described below, and modifications to the theory that apply to charged and nonspherical solutes and more concentrated solutions in which solute–solute interactions cannot be neglected.

7.2.6. Combined Diffusion and Convection through Pores We saw in Chapter 2 that, when free diffusion takes place in the presence of solvent motion, the solute flux equation contains two terms, the first for diffusion and the second for convection: J s Ds

dcs vcs , dx

(2.29)

where v is the local fluid velocity. This equation can readily be modified to describe convective diffusion through cylindrical pores in the cell membrane or slit-like gaps between cells. We saw in the previous subsections that both components of the solute flux are reduced relative to transport in free solution as a consequence of steric and viscous effects resulting from the pore geometry. Let the ratios of each flux component in the pore to that in free solution be designated ), appropriately subscripted (d = diffusion, c = convection). Furthermore, replace v by vm, the average fluid velocity in 2 the pore. With these substitutions, Js becomes the flux through the pore, in mols/cm pore area-s. Equation (2.29) becomes J s Ds ) d

dcs vm ) c cs . dx

(7.29) I

In the steady state, Js is the same throughout the length of the pore. Letting cs denote II the concentration of solute at the x = 0 face of the pore, and cs denote the concentration at the x = a face, integration of Eq. (7.29) yields csII exp(Pe p ) csI I J s vm ) c cs ¸ , 1 exp(Pe p ) 1

(7.30)

) ¬ v a ¬ where Pe p c m is the pore Peclet number, which measures the relative con Ds ® ) d ® tributions of convection and diffusion to the solute flux through the pore. The second term in the expression for Pep is the Peclet number, Pe, for convective diffusion in free solution, and the first term modifies this quantity to reflect solute and solvent interac-


283

tions with the pore. The characteristic length a is generally taken to be the distance over which diffusion takes place, usually the length (as it is here) or radius of a channel. In free solution, convection dominates when Pe >> 1, and diffusion dominates when Pe 1 indicates the presence of pores, but it cannot by itself identify the mechanism by which solute passes through them.


289

7.3. ELECTRICAL ANALOGS Many of the material properties and transport variables associated with ion flow across biological barriers have analogs in the properties and variables that describe electrical circuits. Some examples are listed in Table 7.1. In view of these similarities, it is not surprising that the methods used to describe current flow and voltage effects in electrical circuits have also been used to model and analyze ion transport and its interaction with membrane potential in biological systems.

Table 7.1. Parallels between Ion Transport and Electrical Circuits Ion transport

Electrical circuit

Ion flux Membrane potential Ion mobility Rheogenica pump Membrane capacitance

Current density Voltage difference Conductance Current source Capacitance of a capacitor

a

A rheogenic pump transfers net charge across the membrane. Such pumps have often been referred to as electrogenic, since their activity contributes to the membrane potential. However, an electrically neutral (i.e., nonrheogenic) pump can also affect the membrane potential through the concentration differences it produces. The term “rheogenic” is more precise.

The principal differences between biological ion flows and electrical circuits are: (1) biological and electrical currents are both driven by electrical potential differences, but biological currents are also driven by differences in concentration; (2) the current in biological systems is generally carried by more than one species; and (3) the biological system is rarely as linear as its electrical analog. We will deal with the first two of these differences in this section, reserving the third for Chapter 10.

7.3.1. Equivalent Circuit for the Passive Flux of a Single Ion The construction of an equivalent circuit for a single ion crossing a membrane under concentration and electrostatic potential gradients begins with the electrodiffusion equation (2.36): d ln ai d \ ¬ ziᑠ J i U i ci RT . dx dx ®

(2.36)

For this application, it is unnecessary to approximate the activity of the ion by its concentration, as was done in Chapter 2. 2 Let Ii be the current carried by the ith ion, in mols of charge/cm -s; Ii = ziJi (to 2 convert Ii to amps/cm , the usual measure of electrical current density, multiply it by the Faraday, which has the units of coul/mol). From Eq. (2.36),

290


RT d ln ai d \ ¬ . I i zi2ᑠUi ci zi ᑠ dx dx ®

(7.41)

Eq. (7.41) can be rearranged to give Ii dx RT d ln ai d \ . ¸ 2 zi ᑠ ᑠ Ui ci zi

(7.42)

Equation (7.42) is integrated across the membrane: II 1 a dx ¯ ° RT ln ai (\ I \ II ) . Ii ¡ 2 ¨ ¡ ᑠz 0 U c ° ziᑠ aiI i i± ¢ i

(7.43)

The first term on the right-hand side of Eq. (7.43) is the negative of the Nernst potential of the ith ion, Ei [see Eq. (1.47)]. The second term is the membrane potential, '\. Since the left-hand side of the equation must also be a voltage, the coefficient of Ii must be a resistance; call it ᑬi. Define the membrane conductance of the ith ion by Gi = 1/ᑬi. Then, Eq. (7.43) becomes Ii = Gi('\– Ei).

(7.44)

Equation (7.44) summarizes the dependence of the current carried by the ith ion on the potential difference across the membrane and the ionic activities at each side. The activities determine the ion's Nernst potential, and the current carried by the ionic flux is proportional to the difference between the membrane potential and the Nernst potential. The proportionality constant is the ionic conductance of the membrane. When Ei = '\, the ion is in equilibrium across the membrane, and Ii (and Ji) is zero. An analogous equation [Eq. (4.17)] was presented in Chapter 4 to describe the relation between the unitary conductance and unitary current of a single open ion channel. Equation (4.27) related the unitary conductance of a single channel to the measured (mean) conductance of a population of such channels operating in parallel, in terms of the number density of channels and the fraction that are open at any given time. Equation (7.44) predicts a linear dependence of flux on membrane potential, if the conductance is constant. This is not a bad assumption if the potential is not varied over too large a range. Since the conductance depends on the concentration profile of the ion in the membrane, it is sensitive to the ionic concentration in the external solutions. The equivalent circuit described by Eq. (7.44) is shown in Figure 7.12.

7.3.2. Equivalent Circuit for the Passive Transport of Multiple Ions The circuit shown in Figure 7.12 describes the flux of only a single ion. The passage of multiple ions across the membrane is represented by a collection of single-ion equivalent circuits in parallel, as illustrated in Figure 7.13 for the most commonly studied trio of ions — sodium, potassium, and chloride. This approach to modeling multi-ion transport implies that the ion fluxes do not interact; such would be the case if each species crossed the membrane via its own set of ion selective channels.


291

Figure 7.12. Equivalent circuit for a single ion. The orientation of the battery in the membrane depends on the sign of the Nernst potential.

Figure 7.13. Equivalent circuit for multiple ions.

The model of multi-ion transport shown in Figure 7.13 leads to a particularly useful relationship between the open-circuit potential and the ionic conductances of the membrane. The total passive current across the membrane in Figure 7.13 is p

I = GNa('\ – ENa) + GK('\ – EK) + GCl('\ – ECl),

(7.45)

where the superscript “p” indicates that only passive fluxes are considered. Solving p Eq. (7.45) for the membrane potential at open circuit (I = 0), '\ o.c.

GNa ENa GK EK GCl ECl Gi Ei p (I = 0). GNa GK GCl Gi

(7.46)

292


Equation (7.46) shows that the membrane potential is a weighted average of the Nernst potentials of the permeant ions; the weighting factors are the fractions of the total conductance contributed by each ion. Therefore, as the membrane conductance of any ion increases relative to the other conductances, the open-circuit potential of the membrane approaches the Nernst potential of that ion. Increases in a single-ion conductance can occur because of an increase in the concentration of the ion or an increase in its effective mobility, as would occur if the channels normally traversed by that ion were open a larger fraction of the time. Recall that the solutions of the electrodiffusion equation in Chapter 2 showed the same dependence of membrane potential on the relative mobilities of the ions. The relationship given by Eq. (7.46) is the basis of an electrochemical (as opposed to electronic) method for voltage clamping a membrane. The membrane whose potential is to be fixed is loaded with valinomycin, a potassium-selective ionophore, thereby increasing the potassium conductance of the membrane by up to several orders of magnitude. Consequently, the membrane potential becomes nearly equal to EK. The Nernst potential of potassium, at which the membrane is clamped, can be adjusted by changing the ambient potassium concentration. This technique is particularly helpful when voltage clamping by means of an intracellular electrode is difficult. Even in the absence of added potassium-selective ionophore, the potassium permeability of the plasma membrane is considerably greater than that of sodium, owing to a large number of potassium channels (so-called potassium leak channels) in the membrane that are open most of the time. Accordingly, the resting potentials of most cells are within 10 mV of EK. The several parallel paths in Figure 7.13 can be reduced to a single path by simple algebraic manipulation of the preceding equations. Start with the more general form of Eq. (7.45): p

I = 6Gi('\– Ei) = 6Gi('\) – 6GiEi.

(7.47)

Equation (7.46) is used to replace the second term in Eq. (7.47) by '\o..c..6Gi , yielding an expression having the same form as Eq. (7.44): p

I = (6Gi)('\ – '\o.c.).

(7.48)

Thus, the current–voltage behavior of a circuit representing the transport of multiple ions can be replaced by a Thevenin equivalent circuit with a single conductance equal to the sum of the conductances in the original circuit, and a single battery potential equal to the open circuit potential. This identity allows complex parallel transport systems to be modeled — at least as far as electrical behavior is concerned — by a simpler system consisting of a single nominal path. The general rule for constructing a Thevenin equivalent circuit is given by Thevenin's theorem. In terms of conductances, it states that any combination of batteries and conductances with two terminals can be reduced to a single battery, whose voltage is equal to the open circuit potential of the system, and a single conductance equal to the short-circuit current divided by the battery voltage.


293

If the compositions at both sides of the membrane are the same, then all Nernst p potentials are zero, and Eq. (7.45) simplifies to I = '\6Gi , which is Ohm's Law. The conductance of the membrane is the sum of the individual Gi's.

7.3.3. The Electrical Analog of a Rheogenic Pump As noted in Table 7.1, the most obvious electrical equivalent of a rheogenic pump is a current source. A current source, in turn, can always be modeled by a battery and series conductance. The latter representation is usually used when constructing equivalent circuits for actively transporting cells and tissues. The pump pathway is placed in parallel with the passive paths across the membrane in which the pump resides. When the substrate concentration is the same at both sides of the membrane, the pumpa generated current, I , is given by an equation very similar to Eq. (7.44): a

a

a

I = G ('\ – E ).

(7.49)

According to this expression, the pump rate is a linear function of the potential difference across the membrane; this is not unreasonable, since the pump moves net charge from one side of the membrane to the other, and the ease of doing so should depend on the membrane potential. a The “conductance” of the pump, G , measures the sensitivity of the pump rate to a '\. The quantity E is referred to as the electromotive force (EMF) of the pump; multiplied by the conductance, it gives the pump rate at short circuit. As '\is raised from a a a a value less than E to a value greater than E , the pump current changes sign. Since I a crosses zero when the membrane potential equals the EMF of the pump, E is also termed the reversal potential of the pump. a As in passive systems, the open-circuit potential can be driven toward E if the a experimental conditions (usually nonphysiological) are such that G is much greater than the passive conductance of the membrane; for instance, if the individual ion conductances were made very small. The EMF of the pump is the open-circuit potential of the membrane in the absence of passive flux. Since passive leaks are always present a under physiological conditions, '\ never reaches E . If the membrane potential is always much smaller than the EMF of the pump, then the fractional change in pump rate accompanying the normal variations in membrane potential will be small, and the pump may be regarded as a voltage-independent, constant-current source. Under these conditions, the equation for open-circuit potential is particularly simple. The total current across the membrane, equal to zero at open circuit, is a

p

I=I +I .

(7.50)

Substituting Eq. (7.47) into (7.50), setting the total current equal to zero, and solving for the membrane potential,

'\ o.c.

Gi Ei I a (I = 0). Gi

(7.51)

294


Equation (7.51) can be used in the same way as Eq. (7.46) was in the previous subsection, to derive the Thevenin equivalent circuit for a multi-ion system in the presence of active transport. The total current–voltage relation is identical to Eq. (7.48), with '\o.c. given by Eq. (7.51); proof of this is left as an exercise. The first term in the numerator of Eq. (7.51) measures the contribution of the passive currents to the membrane potential, and usually dominates the second term, which reflects the rheogenic contribution of active transport. Note, however, that the passive currents themselves arise from the electrochemical potential differences created by the active transport systems in the cell membrane. The overall conductance of a number of conducting paths in parallel is the sum of the individual conductances, and the overall resistance is the reciprocal of the overall a a conductance, so the second term in Eq. (7.51), I /6Gi, can be equated to I Rm, where Rm is the membrane resistance. This term has sometimes been referred to as the voltage drop of the pump current across Rm. This is clearly wrong, since the pump current does not pass through the resistive channels that are traversed by the passive ion fluxes. In fact, the current that generates the additional voltage represented by the second term in Eq. (7.51) is the additional passive current that arises to balance the pump current and restore the total current to zero. This “neutralizing” passive current is oppositely directed to the pump current, so the term is negative.

7.3.4. Some Final Remarks All the equivalent circuits described above are purely resistive. Thus, they should apply equally well to transient and steady-state processes. However, when using equivalent circuits to model biological systems that are changing with time, care must be taken when the time scale is very short, or when the experiment is long. The capacitance of the membrane must be taken into account when the membrane potential can change rapidly. Whenever the potential difference across the membrane is altered, the capacitive charge on the membrane changes. The transient current associated with the change in membrane charge is proportional to the rate at which the membrane potential varies. Equivalent circuits including membrane capacitance are used to model excitable tissue, and will be illustrated in Chapter 9. The ionic conductances and the Nernst potentials both depend on the composition of the phases adjacent to the membrane. These compositions can change during the course of an experiment, particularly if nonphysiological conditions (e.g., short circuiting) are imposed or if any of the compartments are of limited size, like the inside of a cell. In this event, purely resistive models remain appropriate, but the battery voltages and conductances are not constant. When this problem is anticipated, every effort is made to conclude the experiment before significant changes in composition have taken place. Two applications of electrical analogs will be discussed in subsequent chapters. In Chapter 9, the relation between membrane potential and ion conductances will be used to describe the mechanism by which the action potential in nerve is generated. In Chapter 10, we will discuss the use of equivalent circuits to interpret the electrical properties of membranes and tissues in terms of the EMFs and conductances of the


295

transport pathways. This discussion is reserved for the chapter on epithelia because electrical analogs have been used most frequently to interpret experiments on this class of tissues. As mentioned above, the implications of the nonlinearities that are seen in the current–voltage relations of many biological membranes and tissues are also deferred to Chapter 10.


Suppose we construct a bilayer membrane each layer of which has a different partition coefficient for a particular solute. Layer I faces Phase I and Layer II faces Phase II. The two layers meet in the middle of the membrane. Assume that I II Ks 2 molecules of cAMP per kinase molecule) has bound to the regulatory subunits, they undergo a conformational change and cleave from the catalytic subunits, which are activated. The latter subunits then diffuse in the cytosol until they find a target protein, such as a transport molecule or a regulatory protein in the nucleus, to activate. This process, far downstream from the initial binding of the hormone to its receptor, accomplishes the task that the hormone originally signaled the cell to perform. Recall that when multiple ligand molecules (cAMP in this case) are needed to activate a substrate molecule (such as PKA), the dependence of substrate activity on ligand concentration curves sharply upward when the concentration of ligand is far from saturation (Fig. 4.5). For this reason, the stoichiometry of cAMP–PKA activation (n > 2) increases the sensitivity of the cellular response to the cytosolic cAMP level. The kinase can act in several ways. Suppose the hormone signals an increase in the activity of a transport system. There are three principal ways in which this can be accomplished: by promoting the synthesis of additional transporter molecules; by activating channels and increasing the fraction of time each channel is open; and by promoting the migration of transporter-loaded storage vesicles to, and the insertion of their contents into, the intended membrane. As noted earlier, ADH uses the last of


305

these mechanisms, promoting the release of aquaporin-2-loaded vesicles into the mucosal membrane to increase its flow conductivity. The response to the agonist depends on the cell type, and its strength depends on the increase in cyclic AMP level that is induced. The amount of second messenger produced within the cell depends, in turn, on the hormone level in the blood and the density of receptors on the cell surface. An increase in either of these causes the number of receptors whose binding sites are occupied to rise, accelerating the rate of cyclic AMP synthesis, and elevating the concentration of cAMP in the cell. Once the level of stimulating hormone in the blood falls, the cellular response diminishes, since the enzyme phosphodiesterase continuously degrades the cyclic AMP to an inactive form. The activity of this enzyme, in turn, is inhibited by the drug theophylline, which consequently prolongs and strengthens the hormone-induced response. The duration of the response is also limited by the autohydrolysis of the GTP on the D subunit of the G protein, which limits the duration of cyclase activation. Additionally, intracellular protein phosphatases can remove the phosphate group from activated transport molecules, rendering them inactive. Cyclic AMP is capable of inhibiting certain protein phosphatases, thereby increasing the duration and intensity of the response of the transporter system. We can construct a simple model to relate the regulatory response to the agonist stimulus. If agonist binding and unbinding are rapid, relative to changes in agonist level, the binding process can be regarded as being in equilibrium, and the number of cell receptors occupied by agonist, Nb , is given by the simple Michaelis–Menten expression for saturable binding employed in Chapters 3 and 4: Nb NT ¸

cA , K cA

(8.1)

where NT is the total number of receptors on the cell surface, cA is the ambient agonist concentration, and K is the equilibrium constant for the dissociation of the agonistreceptor complex. If the rate of production of cyclic AMP is proportional to the number of occupied receptor sites, and the rate of degradation of the messenger is proportional to the amount that is present in the cell, then the dynamics of the intracellular messenger concentration are given by dccAMP k f N b kr ccAMP . dt

(8.2)

Here, kf reflects the kinetics of the G protein and cyclase, while kr reflects that of the phosphodiesterase. The steady-state level of cyclic AMP is obtained by setting the right-hand side of Eq. (8.2) to zero:

(ccAMP )s.s.

kf kr

NT ¸

cA . K cA

(8.3)

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CH. 8: REGULATION AND FEEDBACK

G proteins are also involved in regulatory processes that use calcium as a second messenger, and operate directly on channels as well. Table 8.1 lists several G proteins involved in transport and the actions of their subunits.

Table 8.1. Selected G Proteins Involved Directly or Indirectly in the Regulation of Transport G protein

1

Functions

Gs

Activates adenylyl cyclase Activates Ca channels Downregulates Na channels

Gi

Inhibits adenylyl cyclase Inhibits Ca channels Activates K channels

G0

Inactivates Ca channels Activates K channels 1 Activates phospholipase C-E

Gq

Activates phospholipase C-E1

See §8.1.3.

8.1.2. Direct Hormonal Regulation Without an Extracellular Receptor: Aldosterone The hormone aldosterone controls sodium reabsorption in the kidney and acts on its target cells without the help of either an extracellular receptor or a second messenger. Lipid soluble, it diffuses through the cell membrane and into the cytoplasm, where it combines with specific receptor proteins. The hormone–receptor complex then diffuses into the nucleus of the cell, where it promotes the transcription of genes to mRNA that codes for proteins that increase the sodium throughput of the cell layer. In particular, newly synthesized Na–K ATPase molecules are inserted into the serosal surface of the cell to actively transport sodium toward the renal capillaries, while additional sodium channels are inserted into the mucosal surface to increase its sodium conductance, thereby maintaining a physiological intracellular sodium level while providing additional substrate for the serosal pump. When a hormone influences both the entry and exit of a species that is transported across the cell, as is the case here, throughput can be more efficiently regulated and the intracellular concentration of the transported species can be more stable. When a hormone or other agonist regulates transport activity by promoting biosynthesis rather then by increasing the open probability of channels already in the membrane or by inserting ready-made transporters from storage vesicles, there is a time delay between the stimulus and the response. In the case of aldosterone, new mRNA appears 20–30 min after the cell has been exposed to the hormone. Fif-


307

teen to 90 min later, the effect of newly synthesized protein is seen in the sodium transport rate.

8.1.3. Calcium-Based Regulation Calcium ion is an important player in transport regulation. Increases in cytosolic calcium initiate changes in a host of transport-related processes. Thus, calcium is a regulator of transport, like cyclic AMP. Calcium ion concentrations and dynamics are also regulated by transport. In resting cells, the cytosolic calcium concentration is very low, as a result of ATPase and antiporter molecules that transport the ion into the extracellular fluid or into sequestering organelles; the increase in cytosolic calcium that constitutes the signal arises from passive transport back into the cytosol from external fluid or internal stores. Thus, calcium dynamics reflects both the regulation by transport that serves to maintain the resting low-calcium state, and the regulation of transport that follows an increase in the cytosolic calcium level. We will consider both aspects of the role of calcium in order. In the last part of this subsection, we will use the modeling of calcium transport inside the cell as a point of departure for a discussion and illustration of compartmental analysis. Maintenance of a Low Cytosolic Calcium Concentration. Calcium is pumped out of the cytosol by three active transport systems. Ca ATPases in the cell membrane and the membranes of the endoplasmic reticulum (ER) carry out primary active transport, extruding calcium from the cell or sequestering it in the ER; in muscle, where calcium dynamics plays a crucial role, the sarcoplasmic reticulum functions similarly to the endoplasmic reticulum in other cell types. As pointed out in Chapter 5, the Ca ATPase in both reticulum membranes is the sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump. Secondary active transport also plays a role here: the Na–Ca antiporter in the plasma membrane uses the transmembrane free energy difference of three sodium ions to pump a single ion of calcium out of the cell. As a consequence of these active transport systems, the intracellular calcium concentration can be drawn down to as low as 0.1 μM, while that in the extracellular fluid is around 1 mM. Calcium is also accumulated in mitochondria and by calcium binding molecules in the cytosol that further reduce the internal activity of the ion. Accumulation by itself cannot maintain a low intracellular calcium level indefinitely, since the capacity of these sequestering sites is limited; as they become saturated, calcium entering the cell passively across the cell membrane would eventually cause the cytosolic level to rise. The active transport systems in the plasma membrane, which extrude calcium ions into the effectively limitless capacity of the extracellular space, are the only mechanisms that can oppose passive entry of calcium in the long term. Similarly, prolonged elevation of the intracellular calcium level — and thus a prolonged response to the original agonist — requires influx from the extracellular space as well as the release of ions sequestered in organelles. Calcium Signaling. Although the magnitudes of the calcium ion concentrations inside and outside the cell are small relative to the concentrations of the major ions, the calcium concentration ratio across the cell membrane is much larger, leading to a calcium Nernst potential that is even more positive that that of sodium. In addition,

308


the electrostatic potential inside the cell is negative relative to that outside. As a consequence, the driving force for calcium entry into the cell is enormous. The primary route for calcium to reenter the cytosol — and initiate its signaling role — is through calcium channels in the endoplasmic reticulum (ER) or plasma membrane; mitochondrial calcium stores may also be released. When calcium reentry is signaled by agonist–receptor binding at the cell membrane, a G protein is activated, which subsequently activates the membrane-bound enzyme phospholipase C-E, which in turn acts on a membrane phospholipid to produce a messenger molecule, inositol 1,4,5-triphosphate (IP3). IP3 binds to calcium release channels in the ER, causing them to open and release stored calcium into the cytosol. The consequent increase in cytosolic calcium increases the calcium conductance of the plasma membrane, and additional calcium floods into the cytosol from the extracellular space through calcium channels and/or nonselective cation channels. Voltage-gated calcium channels in the cell membrane can also open, without requiring prior calcium release from the ER; this mechanism is particularly relevant to muscle function. Mechanical stretch can cause mechanically sensitive calcium-permeable channels in the plasma membrane to open if the cell volume increases sufficiently; we shall see below that this allows calcium activation to play a role in cell volume regulation. Once its concentration in the cytosol has risen, calcium ion can act directly on those channels for which it is an activating ligand; these include Ca-activated potassium and chloride channels. Many of the effects of calcium in the cell are mediated by the calcium-binding protein calmodulin. When calmodulin's binding sites are occupied, which occurs at a cytosolic calcium concentration of about 1 μM, it undergoes a conformational change and activates an associated kinase. The kinase can then phosphorylate target molecules, similarly to PKA. Among the transport or transport-related proteins influenced by calcium, either directly or through calmodulin, are adenylyl cyclase, cyclic AMP phosphodiesterase, cyclic GMP-gated sodium and calcium channels, the plasma membrane Ca ATPase, and at least one sodium channel. The interaction of the calcium and cyclic AMP systems indicated by this list adds an additional level of complexity to the modeling of these regulatory processes. Some of the intracellular calmodulin molecules are bound via tethers to the cytosolic tail of voltage-gated calcium channels. When these channels are opened (by membrane depolarization), calcium enters the cytosol through the pore, and some of these ions bind to the calmodulin. When its sites are occupied, the calcium-binding protein undergoes a conformational change and inactivates the channel. Since the binding sites are exposed to the higher calcium concentrations at the pore exit, they become occupied more quickly than sites on calmodulin molecules freely diffusing in the cytoplasm, thereby expediting this negative feedback process. Calcium also mediates the effect on transport of a second product of phopholipase C-E activity, diacylglycerol. Diacylglycerol activates a protein kinase, PKC, which can inhibit certain ion channels when signaled to do so by a rise in cytosolic calcium. Additional transport-related effects of alterations in intracellular calcium levels include changes in gene expression, and the stimulation of exocytosis and secretion, as described in Chapter 3. Another calcium-binding protein, troponin C, is important in muscle contraction and will be discussed in Chapter 9.


309

Figure 8.3. (a) Schematic showing the principal compartments involved in cytosolic calcium regulation and the transport mechanisms responsible for calcium fluxes between them. Included is an artificial compartment that represents calcium-binding molecules that equilibrate with unbound (“free”) calcium in the cytosol. Other organelles not in the figure that participate in calcium handling include mitochondria and certain vesicles. Open circles denote primary active transport systems and closed circles denote exchangers/cotransporters. Channels are indicated by gaps in the membrane, and mechanically activated channels have springs. (b) Representation of calcium dynamics using the compartmental analysis formalism. Note that the extracellular fluid is not regarded as a compartment, since its composition is assumed to be unaffected by calcium exchange; note also that in some cases a single pair of arrows represents more than one transport process. The paired arrows reflect the bidirectionality of intercompartmental transport, which is the case here even though several unidirectional transporters are present. Note also that compartments need not be physically discrete; the calcium-binding molecule “compartment” is distributed throughout the cytosol. Two arrows describe transport across the plasma membrane; the inward-directed arrow represents passive channel influx, and the outward-directed arrow represents primary and secondary active transport.

Calcium Handling in Cells and Compartmental Analysis. The dynamics of cytosolic calcium depends on the fluxes between four principal regions, as shown in Figure 8.3a. Proceeding from the outside in, they are the extracellular fluid, the cytosol, various anionic molecules in the cytoplasm that can bind calcium, and the endoplasmic reticulum. The major calcium transport systems that move calcium between these compartments are also indicated. We see that the figure consists of (1)

310


compartments containing solute and (2) interfaces between compartments. The interfaces contain transport systems that convey solute between the compartments on the two sides of the interface. Calcium transport between these compartments and across these interfaces can be modeled using a formalism known as compartmental analysis. We have already used some equations from compartmental analysis: when we examined, in Chapter 2, the conditions for quasisteadiness in the face of changing boundary conditions, and in Chapter 4, when we modeled tracer kinetics in the erythrocyte. Here we will be more formal and general in our presentation of the method. General formulation of compartmental analysis. Classical compartmental analysis relies on two fundamental assumptions: 1. The rate at which solute enters or leaves any given compartment is slow enough, relative to the rate of diffusion of the solute in the compartment, that the concentration gradients in the compartment are negligible; that is, the concentration in the compartment is always uniform. Then, that concentration, cij, is equal to nij /Vj, where nij is the number of moles of the ith species in the jth compartment, and Vj is the volume of the compartment. This is often referred to as the well-mixed assumption even though mechanical mixing is generally absent from living systems. 2. The rate of change of the concentrations in the two compartments at an interface is slow enough that transport across the interface can be regarded as quasisteady. The requirement on cell dimensions and membrane thickness for this condition to be met was discussed in Chapter 2. Since the equations of compartmental analysis are essentially mass balances, we add the condition, readily met, that the quantity of solute contained within the interface itself (i.e., material in the process of passing through) is negligible relative to that in the compartments. The condition that must be satisfied for the first requirement to be met is similar to that previously derived for the quasisteady state, except that tc must now be greater than the diffusional time constant for the compartment rather than for the membrane. We will derive this expression for a spherical cell (in contrast to a planar membrane) using a notation consistent with that in Chapter 2. For a sphere subject to a change in concentration at its surface, a reasonable time constant for the intracellular concentration to approach uniformity is (Truskey et al., 2004) t d ,cell

r2 , 2 Dc ,cell

(8.4)

where r is cell radius. The time constant for changing concentration is as before: tc

aV , Ds S

(8.5)


311

where Ds is the solute diffusion coefficient in the cell membrane. The volume and surface area of the cell can be written in terms of its radius, and the ratio Ds /a can be replaced by the permeability, ks; requiring td,cell

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