Mathematical Modeling in Experimental Nutrition: Vitamins, Proteins, Methods

ADVANCES IN FOOD AND NUTRITION RESEARCH VOLUME 40 Mathematical Modeling in Experimental Nutrition Vitamins, Proteins,...

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ADVANCES IN

FOOD AND NUTRITION RESEARCH VOLUME 40

Mathematical Modeling in Experimental Nutrition Vitamins, Proteins, Methods

ADVISORY BOARD DOUGLAS ARCHER Gainesville, Florida

JESSE F. GREGORY I11 Gainesville, Florida

SUSAN K. HARLANDER Minneapolb, Minnesota

DARYL B. LUND New Brunswick, New Jersey

BARBARA 0. SCHNEEMAN Davb, California

SERIES EDITORS GEORGE F. STEWART

(1948-1982)

EMIL M. MRAK

(1948-1987)

C . 0. CHICHESTER

(1959-1988)

BERNARD S. SCHWEIGERT (1984-1988) JOHN E. KINSELLA

(1989-1993)

STEVE L. TAYLOR

(1995-

)

ADVANCES IN

FOOD AND NUTRITION RESEARCH VOLUME 40

Mathematical Modeling in Experimenta1 Nutrition Vitamins, Proteins, Methods Edited by

STEPHEN P. COBURN Department of Biochemistry Fort Wayne State Developmental Center Fort Wayne, Indiana

DOUGLAS W. TOWNSEND Department of Mathematical Sciences Indiana University/Purdue University at Fort Wayne Fort Wayne, Indiana

ACADEMIC PRESS San Diego

London

Boston

New York

Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

Copyright 'G 1996 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval sys tem, without permission in writing from the publisher.

Academic Press, Inc . 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www .apnet.com

Academic Press Limited 24-28 Oval Road, London N W 1 7DX, UK http://www .hbuk.co.uk/ap/ International Standard Serial Number: 1043-4526 International Standard Book Number: 0-12-016440-X PRINTED IN THE UNITED STATES OF AMERJCA 96 97 9 8 9 9 00 01 QW 9 8 7 6 5

4

3 2

1

CONTENTS

CONTRIBUTORS TO VOLUME 40 .............................. PREFACE ................................................

EDITOR'S NOTE ...........................................

... xiii xix xxi

Part I VITAMIN METABOLISM Chapter 1 Quantitative and Conceptual Contributions of Mathematical Modeling to Current Views on Vitamin A Metabolism, Biochemistry, and Nutrition

Michael H. Green and Joanne Balmer Green I. Introduction ...................................... 11. Historical Perspective and Early Studies .............. 111. Experimental Considerations ........................ N. Whole-Body Models for Vitamin A Metabolism ....... V. Empirical Compartmental Analysis of Vitamin A Metabolism ....................................... VI. Liver Vitamin A Metabolism ....................... VII. Other Tissues ..................................... VIII. Vitamin A Disposal Rate ........................... References .......................................

3 4 5 8

11 14 19 21 22

V

vi

CONTENTS

Chapter 2 Mathematical Modeling in Nutrition: Constructing a Physiologic Compartmental Model of the Dynamics of &Carotene Metabolism

Janet A. Novotny, Loren A. Zech, Harold C. Furr, Stephen R,Dueker, and Andrew J. Clifford Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . 11. Materials, Methods, and Model Constraints . . . . . . . . . . . 111. The Process of Constructing a Compartmental Model . . IV. Intermediate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... .. . . . .... .. . .. . . . . . . V. Statistical Considerations VI. The Final Model . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . VII. System Behavior Proposed by the Model . . . . . . . . . . . . . VIII. Unobservabie System Behavior Proposed by the Model IX. Empirical Description of the Experimental Observations . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . Final Encouraging Words . . . . . . . . . . . . . . . . . . . . . . . . . . . X. References ....................................... I.

26 29 32 35 40 43 43 45 49 51 52

Chapter 3 Experimental Approaches to the Study of &Carotene Metabolism: Potential of a Tracer Approach to Modeling &Carotene Kinetics in Humans

Joy E. Swanson, Keith J. Goodman, Robert S. Parker, and Yen-yi Wang I. 11.

Ill. IV.

Introduction ............................. . . . . . . . . . Methods .......................................... Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . References .......................................

56 65 68 74 75

Chapter 4 Modeling of Folate Metabolism

Jesse F. Gregory I11 and Karen C. Scott I. 11.

HI.

Introduction ............. ....... .. ..... ...... ..... Key Elements of Folate Metabolism Relevant to Modeling ... . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . Stabie-Isotopic Studies in Humans . . . . . .. . . . . . . . . . . ..

81 83 86

IV.

CONTENTS

vii

Summary and Conclusions . . . . . . . . . . . . . . . . . . .. . . . . . . ....................................... References

91 91

Chapter 5 Molecular Biology in Nutrition Research: Modeling of Folate Metabolism

Bi-Fong Lin, Jong-Sang Kim, Juei-Chuan Hsu, Charles Osborne, Karen Lowe, Timothy Garrow, and Barry Shane I. General Approaches . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Modeling of Folate Metabolism . . . . . . . . . . . . . .. . . . . . . References .........................................

95 96 105

Chapter 6 Modeling Vitamin B6 Metabolism

Stephen P. Coburn I. Metabolism . . .. , . . . . . . , . . . . . . . . . . . . . . . . . . .. . . . . . . 11. Kinetics _.... ... ..... ......... . .. ..... ........... . 111. Refining Models of Vitamin B6 Kinetics . . . . . . , . . . . . . . IV. Conclusions . . . . . . . . . .. . . . . . . . . . . . . . . . . , . . . . . . . . . . . References ....................................... I

107 113 117 127 127

Part II

PROTEIN AND AMINO ACID METABOLISM Chapter 7 Interrelationshipsbetween Metabolism of Glycogen Phosphorylase and Pyridoxal Phosphate-Implications in McArdle’s Disease

Robert J. Beynon, Clare Bartram, Angela Flannery, Richard P. Evershed, Deborah Leyland, Pamela Hopkins, Veronica Toescu, Joanne Phoenix, and Richard H. T. Edwards Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , Role of Cofactor in Phosphorylase Turnover . . . . . . . . . . Labeling Methods to Monitor Phosphorylase Turnover .......................................... IV. Model Systems for Phosphorylase Expression . . , . . . . . . I. 11. 111.

136 136 137 140

viii

CONTENTS

V.

Vitamin B6 and McArdle’s Disease . . . . . . . . . . . . . . . . . . ....................................... References

142 146

Chapter 8 Metabolism of Normal and Met30 Transthyretin

Denise Hanes, Loren A. Zech, Jill Murrell, and Merrill D. Benson I. 11.

111. IV.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods ...,. . . ... ............ .. .. .. ... . . . . , . ... .. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . .. . , . .. . . . . . . .. . ... . . . . . . .. . . . . . . . . References .......................................

150 150 152 154 155

Chapter 9 Use of a Four Parameter Logistic Equation and Parameter Sharing to Evaluate Animal Responses to Graded Levels of Nitrogen or Amino Acids

M. J. Gahl, T. D. Crenshaw, N. J. Benevenga, and M. D. Finke 1. 11. 111.

Diminishing Returns and Dose-Response ..................................... Relationships Diminishing Returns and Protein Quality . . . . . . . . . . . . . Response of Rats to Each Indispensable Amino Acid . , ....................................... References

157 160 162 166

Part 111 ENERGY METABOLISM Chapter 10 Total Energy Expenditure of Free-Living Humans Can Be Estimated with the Doubly Labeled Water Method

William W. Wong Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of the 2H2180 Method . . . . . . . . . . . . . . . . . . . . . . . Ill. Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Validations of the *H2l80 Method . . . . . . . . . . . . . . . . . . . 1.

11.

171 172 175 178

CONTENTS

V. Conclusion References

....................................... .......................................

ix 179 179

Part IV METHODS FOR OBTAINING KINETIC DATA Chapter 11 Microdialysis and Ultrafiltration

Elsa M. Janle and Peter T. Kissinger I. Introduction ...................................... I1. Comparison of Microdialysis and Ultrafiltration ....... I11. Examples of Studies Using Microdialysis and Ultrafiltration ..................................... IV. Summary ......................................... References .......................................

183 184 191 195 195

Chapter 12 Membrane Vesicles

Pierre Proulx I. Introduction ...................................... I1. Methods .......................................... 111. Discussion ........................................ References .......................................

197 198 203 204

Chapter 13 Culture of Mammary Tissue: Glucose Transport Processes

Jeffrey D . Turner. Annick Delaquis. and Christiane Malo I. I1. I11. IV .

Introduction ...................................... Materials and Methods ............................. Results and Discussion ............................. Conclusions ....................................... References .......................................

207 208 209 212 213

CONTENTS

X

Part V SIMULATING COMPLEX METABOLIC PROCESSES

Chapter 14 Analysis of Bioperiodicity in Physiological Responses

L. Preston Mercer and Danita Saxon Kelley I . Bioperiodicity ..................................... 217 I1. Characterization of Biological Rhythms .............. 218 111. Experimental Conditions ........................... 220 IV . Evaluation of Results .............................. 220 V . Discussion ........................................ 224 References ....................................... 226 Chapter 15 Nutrient-Response: A “Top Down” Approach to Metabolic Control

Arthur R . Schulz I. I1. 111. IV .

Lntroduction ...................................... 227 Mathematical Treatment ........................... 228 Analysis of Data on Three Dietary Proteins .......... 238 Conclusions ....................................... 240 References ....................................... 241 Chapter 16 Modeling Membrane Transport

Richard €3 . King I . Introduction ...................................... II . A Multiregion. Distributed Exchange Model .......... 111. Passive Diffusion .................................. IV . Carrier-Mediated Transport ......................... V . Building Complex Models .......................... VI . Summary ......................................... References .......................................

243 245 248 251 258 260 261

xi

CONTENTS

Part VI COMPUTATIONAL ASPECTS OF MODELING Chapter 17 Estimation and Use of Kinetic Parameter Distributions in Metabolism and Nutrition

William F. Beltz I. Introduction ...................................... 11. Definitions and Theory ............................. 111. Uses for Prior Parameter Distributions ............... IV . Applications to Metabolism and Nutrition ............ V . Estimation of Prior Parameter Distributions .......... VI . Identifiability Issues ................................ VII . Conclusions ....................................... References .......................................

265 266 269 271 273 276 278 278

Chapter 18 Essential Numerical Supports for Kinetic Modeling Software: Linear Integrators

R . C. Boston. T. McNabb. P. C. Greif. and L. A. Zech I. Introduction ...................................... II . Linear Systems of Differential Equations ............. 111. Modeling Software and Linear Integrators ............ IV. Conclusion ....................................... References .......................................

281 283 290 300 303

Chapter 19 Identifiability

John A . Jacquez I . Introduction ...................................... 11. Examples ......................................... 111. Classification of Parameters ......................... IV. Parameter Identifiability and Estimation .............. V . Identifiability: Definitions ........................... VI. Methods of Checking Identifiability ..................

305 307 310 311 313 314

xii

CONTENTS

VII . Local Identifiability at a Point ....................... ....................................... VIII . Conclusion References .......................................

318 320 321

Chapter 20 Dynamic Systems and Neural Networks: Modeling in Physiology and Medicine

Samir I . Sayegh 1. 11. 111.

Introduction ...................................... Linear Systems Modeling ........................... Nonlinear Systems. Chaos. and Fractional Dimensions ....................................... VI. Geometric Interpretation of Models and Dynamical Systems .......................................... V . Neural Networks .................................. VI . Conclusion ....................................... References .......................................

323 324

324 328 333 337 337

Chapter 21 Graph Theoretical Methods for Physiologically Based Modeling

Hong Zhang and Zhen Zhang I . Introduction ...................................... I1. The Graph Model ................................. 111. Computer Implementation ..........................

IV . Analysis of Models ................................ ....................................... References

339 340 348 348 351

...................................................

353

INDEX

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

Clare Bartram, Department of Biochemistry and Applied Molecular Biology, UMIST, Manchester M60 1QD, United Kingdom (135) William F. Beltz,’ Department of Medicine, University of California at San Diego, La Jolla, California 92093 (265) N. J. Benevenga, Departments of Nutritional Sciences and Meat and Animal Science, University of Wisconsin, Madison, Wisconsin 53706 (157) Merrill D. Benson, Division of Rheumatology, Department of Medicine, Indiana University School of Medicine, Indianapolis, Indiana 46202; Rheumatology Section, Richard L. Roudebush - Veterans Afjcairs Medical Center, Indianapolis, Indiana 46202 (149) Robert J. Benyon, Department of Biochemistry and Applied Molecular Biology, UMIST, Manchester MM) IQD, United Kingdom (135) R. C. Boston, Clinical Studies, NBC, School of Veterinary Medicine, University of Pennsylvania, Kennett Square, Pennsylvania 19348 (281 1 Andrew J. Clifford, Departmeni of Nutrition, University of California, Davis, California 95616 (25) Stephen P. Coburn, Department of Biochemistry, Fort Wayne State Developmental Center, Fort Wayne, Indiana 46835 (107) T. D. Crenshaw, Departmena of Nutritional Sciences and Meat and Animal Science, University of Wisconsin, Madison, Wisconsin 53706 (157) Present Address: Biocomp, San Diego, California 92103.

xiii

xiv

CONTRIBUTORS

Annick Delaquis: Department of Animal Science, McGill University, Montreal, Quebec H9X 3V9, Canada (207) Stephen R. Dueker, Department of Nutrition, University of California, Davis, California 95616 (25) Richard H. T. Edwards, Muscle Research Centre, Department of Medicine, University of Liverpool, Liverpool L69 3BX, United Kingdom (135) Richard P. Evershed, Department of Biochemistry, University of Liverpool, Liverpool L69 3BX, United Kingdom (135)

M. D. Finke, Departments of Nutritional Sciences and Meat and Animal Science, University of Wisconsin, Madison, Wisconsin 53706 (157) Angela Flannery, Department of Biochemistry, University of Liverpool, Liverpool L69 3BX, United Kingdom (135) Harold C. Furr, Department of Nutritional Sciences, University of Connecticut, Storrs, Connecticut 06269 (25)

M. J. Gahl,3 Departments of Nutritional Sciences and Meat and Animal Science, University of Wisconsin, Madison, Wisconsin 53706 (157) Timothy Garrow, Department of Nutritional Studies, University of California, Berkeley, California 94720 (95) Keith J. Goodman, Division of Nutritional Sciences, Cornell University, Ithaca, New York I4853 (55) Joanne Balmer Green, Nutrition Department, Pennsylvania State University, University Park, Pennsylvania I6802 (3) Michael H . Green, Nutrition Department, Pennsylvania State University, University Park, Pennsylvania I6802 (3) Jesse F. Gregory 111, Food Science and Human Nutrition Department, Universiry of Florida, Gainesville, Florida 32611 (81)

P. C. Greif, Laboratory of Mathematical Biology, National Institutes of Health, Bethesda, Maryland 20892 (281) Present Address: Nexia Biotechnologies, Inc., Ste. Anne de Bellevue, Quebec H9X 3V9, Canada. Present Address: Farmland Industries, Inc., Kansas City, Kansas 66109.

CONTRIBUTORS

xv

Denise Hanes, Division of Rheumatology, Department of Medicine, Indiana University School of Medicine, Indianapolis, Indiana 46202 (149) Pamela Hopkins, Department of Biochemistry and Applied Molecular Biology, UMIST, Manchester M60 l Q D , United Kingdom (135) Juei-Chuan Hsu, Department of Nutritional Studies, University of California, Berkeley, California 94720 (95) John A. Jacquez, Departments of Physiology and Biostatistics, University of Michigan, Ann Arbor, Michigan 48109 (305) Elsa M . Janle, Bioanalytical System, Inc., West Lafayette, Indiana 47609 (183) Danita Saxon Kelley? Department of Nutrition and Food Science, University of Kentucky, Lexington, Kentucky 40506 (217) Jong-Sang Kim, Department of Nutritional Studies, University of California, Berkeley, California 94720 (95) Richard B. King, Centerfor Bioengineering, University of Washington, Seattle, Washington 98195 (243) Peter T. Kissinger, Bioanalytical Systems, Inc., West Lafayette, indiana, 47609 (183) Deborah Leyland, Department of Biochemistry, University of Liverpool, Liverpool L69 3BX, United Kingdom (135) Bi-Fong Lin, Department of Nutritional Studies, Universityof California, Berkeley, California 94720 (95) Karen Lowe, Department of Nutritional Studies, Universityof California, Berkeley, California 94720 (95) Christiane Malo, Membrane Transport Research Group, Department of Physiology, Faculty of Medicine, Universityof Montreal, Montreal, Quebec H3C 3.77, Canada (207) T. McNabb, Clinical Studies, NBC, School of Veterinary Medicine, University of Pennsylvania, Kennett Square, Pennsylvania 19348 (281) Present Address: Department of Consumer and Family Sciences, Western Kentucky University, Bowling Green, Kentucky 42101-3576.

xvi

CONTRIBUTORS

L. Preston Mercer, Department of Nutrition and Food Science, University of Kentucky, Lexington, Kentucky 40506 (217)

Jill Murrell, Division of Rheumatology, Department of Medicine, Indiana University School of Medicine, Indianapolis, Indiana 46202 (149) Janet A. Novotny, Diet and Human Performance Laboratory, Beltsville Human Nutrition Research Center, U.S. Department of Agriculture, Beltsville, Maryland 20705 (25) Charles Osborne, Department of Nutritional Studies, University of California, Berkeley, California 94720 (95) Robert S. Parker, Division of Nutritional Sciences, Cornell University, Ithaca, New York 14853 (55) Joanne Phoenix, Muscle Research Centre, Department of Medicine, University of Liverpool, Liverpool L69 3BX, United Kingdom (135) Pierre Proulx, Department of Biochemistry, Faculty of Medicine, University of Ottawa, Ottawa, Ontario K l H 8M.5, Canada (197) Samir I . Sayegh, Physics Department, Purdue University, Fort Wayne, Indiana 46805 (323) Arthur R. Schulz, Department of Biochemistry and Molecular Biology, School of Medicine, Indiana University, Indianapolis, Indiana 46202 (227) Karen C. Scott, Food Science and Human Nutrition Department, University of Florida, Gainesville, Florida 32611 (81) Barry Shane, Department of Nutritional Studies, University of California, Berkeley, California 94720 (95) Joy E. Swanson, Division of Nutritional Sciences, Cornell University, Ithaca, New York 14853 (55) Veronica Toescu, Muscle Research Centre, Department of Medicine, University of Liverpool, Liverpool L69 3BX, United Kingdom (135) Jeffrey D. Turner? Department of Animal Science, McCilf Universiv, Montreal, Quebec H9X 3V9, Canada (207) Present Address: Nexia Biotechnologies, Inc., Ste. Anne de Bellevue, Quebec H9X 3V9, Canada.

CONTRIBUTORS

xvii

Yen-yi Wang, Taiwan, No. 17 Li-Yuan, Miao-Li 360, Republic of China (55) William W. Wong, Department of Pediatrics, USDMARS Children’s Nutrition Research Center, Baylor College of Medicine, Houston, Texas 77030 (171) Loren A. Zech, Laboratory of Mathematical Biology, National Cancer Institute, and Molecular Disease Branch, National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland 20892 (25,149,281) Hong Zhang, Department of Mathematical Sciences, Indiana University - Purdue University, Fort Wayne, Indiana 46805; and Department of Biometry and Epidemiology, Medical University of South Carolina, Charleston, North Carolina 29425 (339) Zhen Zhang, Department of Biometry and Epidemiology, Medical University of South Carolina, Charleston, South Carolina 29425 (339)

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PREFACE

The chapters in this volume stem from the fifth in a series of conferences designed to assist nutritionists in employing mathematical modeling in their research. There is increasing controversy over whether nutrient requirements should be based solely on the minimum intakes needed to avoid overt deficiency diseases. Resolution of such questions is critical to determining the optimum nutrient intake at the various stages of life and under various physiological and/or pathological stresses. Mathematical modeling offers a powerful tool for evaluating and simulating the functioning of complex metabolic systems. In addition to presenting general information on modeling (Canolty and Cain, 1985,1988), past conferences in this series have focused on the application to amino acid (Abumrad, 1991), carbohydrate (Abumrad, 1991), and mineral metabolism (Siva Subrarnanian and Wastney, 1995). The goals for this conference were to provide a workshop on the use of the Simulation, Analysis, and Modeling program (SAAM) developed at NIH plus presentations on application of modeling to vitamins and proteins, methods that might be useful for manipulating metabolic systems and obtaining the kinetic data needed for modeling, and mathematical theory and procedures relevant to modeling. ACKNOWLEDGMENTS We express our thanks to the organizationslisted below,whose support made the conference possible, and to the participants, whose contributions made it successful. Financial support for this conference was provided by USDAlNRICGP Grant 93-372008815; NIH Grant 1-R13-DWHD47826-01; Indiana University-Purdue University at Fort Wayne; Hoffman-La Roche Inc., Campbell Soup Co.;Fort Wayne Center for Medical Education, Indiana University School of Medicine; Department of Foods and Nutrition, Purdue University; and John W.Ellis, M.D.

REFERENCES 1988. Mathematical models in experimental nutrition. Prog. Food Nurr. Sci 12,211-338. Abumrad, N. (ed.) 1991. Mathematical models in experimental nutrition. J. Parenrer. Enreral Nurr. l5,44S-98S. xix

xx

PREFACE

Canolty, N. L., and Cain, T. P. (eds.) 1985. “Mathematical Models in Experimental Nutrition.” University of Georgia, Athens, GA. Siva Subramanian, K. N., and Wastney, M. E. (eds.) 1995.“Kinetic Models of Trace Elements and Mineral Metabolism During Development.” CRC Press, Boca Raton, FL.

EDITORS NOTE: AVAILABILITY OF THE SIMULATION, ANALYSIS, AND MODELING PROGRAM

The Simulation, Analysis, and Modeling (SAAM) programs were developed specificallyto allow life scientists to model complex metabolic systems by describing the characteristics of the system without requiring the user to specify the differential equations. SAAM is the original batch version of the program while CONSAM (conversational SAAM) is an interactive version. The SAAM and CONSAM programs for Intel-based computers are made available without cost from the Laboratory of Mathematical Biology, National Cancer Institute, National Institutes of Health through the following mechanisms: World Wide Web: Anonymous FIR E-mail request for diskettes: Written requests for diskettes:

Telephone for assistance:

http:llwww-saam.nci.nih.gov ftp-saam.nci.nih.gov [email protected] Dr. Peter C. Greif Bldg. 10, Room 6B13 National Institutes of Health Bethesda, MD 20892 (301) 496-8914

xxi


Part I

VITAMIN METABOLISM


ADVANCES IN FOOD AND NUTRITION RESEARCH. VOL. 40

Chapter 1 QUANTITATIVE AND CONCEPTUAL CONTRIBUTIONS OF MATHEMATICAL MODELING TO CURRENT VIEWS ON VITAMIN A METABOLISM, BIOCHEMISTRY, AND NUTRITION MICHAEL H. GREEN AND JOANNE BALMER GREEN Nutrition Department Pennsylvania State University University Park, Pennsylvania 16802

I. Introduction 11. Historical Perspective and Early Studies

111. Experimental Considerations A. Experimental Design for In Vivo Studies B. Physiological Doses IV. Whole-Body Models for Vitamin A Metabolism V. Empirical Compartmental Analysis of Vitamin A Metabolism VI . Liver Vitamin A Metabolism A. Compartmental Model B. Stores VII. Other Tissues VIII. Vitamin A Disposal Rate References

I.

INTRODUCTION

About 15 years ago, our colleague B. A. Underwood wondered whether the mathematical modeling methods we were applyingto cholesterol metabolism might be used as a different approach to determine vitamin A utilization (i.e., disposal rate) and dietary requirements. Her interest, and our initial collaboration, led to subsequent research in our lab on vitamin A dynamics in the rat. Here we integrate information and ideas that have been generated by our application of mathematical modeling to vitamin A kinetic data as a way of illustrating how this approach has advanced and complicated our understanding of retinol metabolism. In most instances, 3 Copyright 0 1996 by Academic Press. Inc. All rights of reproduction in any form reserved.

4

PART I VITAMIN METABOLISM

our paper does not review the many contributions of other investigators to this field. Rather our goal is to demonstrate that the application of a variety of modeling techniques to the vitamin A system has resulted in many different levels of insight into the metabolism, biochemistry, and nutrition of this fascinating essential nutrient.

II.

HISTORICAL PERSPECTIVE AND EARLY STUDIES

Fifteen years ago, whole-body vitamin A metabolism was described as a rather simple process (for a review, see Goodman, 1980). It was known that the lipid-soluble vitamin was absorbed from the small intestine and that retinyl esters were packaged as a component of triglyceride-richabsorptive lipoproteins (chylomicrons) (Fig. 1).Triglyceride-depleted, vitamin Acontaining chylomicron remnants were thought to be quantitatively cleared by the liver; there the vitamin was processed and secreted into plasma bound to its specific plasma transport protein, retinol-binding protein (RBP). In plasma, RBP binds to a larger protein, transthyretin (TTR), in a 1:1molar ratio. After secretion from the liver, the retinolRBP complex was assumed to deliver retinol to vitamin A-dependent peripheral tissues where it was used for vitamin A action. Metabolites, possibly retinoic acid and other oxidized derivatives, were assumed to be excreted in both urine and feces. In addition to the liver, kidneys were believed to be an important organ in whole-body vitamin A metabolism. After Underwood wondered in 1977 whether we could use kinetic methods to estimate vitamin A disposal rate in the rat, we collaborated on an in vivo kinetic experiment in vitamin A-deficient and control rats (Lewis er al., 1981). We measured plasma [3H]vitaminA disappearance for 48 hr after intravenous administration of [3H]retinol-labeled plasma and used graphical methods to calculate vitamin A disposal rate. The results sugDietary Vttamln A

FIG. 1. Initial conceptual model for vitamin A metabolism. SI, small intestine;RE, retinyl esters; ROH, retinol; RBP, retinol-binding protein.

CHAPTER 1 MATHEMATICAL MODELING OF VITAMIN A

5

gested that vitamin A metabolism was much more complex than previously thought. Specifically, the fact that a semilogarithmic plot of the plasma response data did not follow a single exponential function indicated either retinol recycling to plasma or kinetic heterogeneity of plasma retinol. Furthermore, recovery of much of the label in the liver at 48 hr indicated that plasma retinol was recycled to liver. The calculated disposal rate in vitamin A-sufficient rats was many fold higher than expected, indicating that the tracer had not sufficiently mixed with the endogenous vitamin A pools. Thus we realized that a much longer experiment was needed to study vitamin A dynamics and to accurately estimate vitamin A disposal rate. As a consequence, we designed several large scale kinetic studies to characterize whole-body vitamin A metabolism in rats at different levels of vitamin A nutriture and we used model-based compartmental analysis (Green and Green, 1990a) to analyze the data. Before summarizing those results, we will briefly describe our experimental approach.

111.

EXPERIMENTAL CONSIDERATIONS

A. EXPERIMENTAL, DESIGN FOR IN VIVO STUDIES

See Green and Green (1990b) for further details. To date, our studies have all been done in adult male rats at one of several levels of vitamin A status. Rats have been used extensively in studies of vitamin A metabolism and are generally believed to be a good model for vitamin A metabolism in humans. In our experiments, rats to be used as recipients of labeled vitamin A are fed purified diets (Duncan et al., 1993) containing various levels of vitamin A as retinyl palmitate to establish liver vitamin A reserves from deficient to high ( 4 0 to >3500 nmol). It is ideal for rats to be in a steady state with respect to vitamin A during turnover studies, although the modeling programs used can accommodate nonsteady state situations. A physiological tracer of vitamin A (see next section) is prepared in donor rats and administered intravenously to recipients. Serial plasma and tissue samples are collected from -10 min after injection until the end of the study (35115 days, depending on vitamin A status of recipients). Samples are extracted and analyzed for vitamin A radioactivity and, in some cases, vitamin A mass. Although care must be taken to ensure accuracy at each analytical step, it is worth emphasizing that reliable tracer data depends on adequate sample counting time. In our studies, all samples are counted twice to a 2-sigma error of 8590%) can be determined analytically. In either case, lymph preparations should be used for in vivo studies within 12 days of collection. Even when care is taken to handle the doses carefully, we have found that 215% of the tracer in the case of [3H]retinol-labeled plasma, and up to 40% in the case of isolated [3H]retinyl ester-labeled chylomicrons, acts “nonphysiologically” when preparations are injected into recipient rats. That is, a variable fraction of the dose (the “nonphysiological” component) is cleared from plasma within a few minutes. Presumably nonphysiological

8


components result from protein denaturation, chylomicron aggregation, or other physical changes caused by ultracentrifugation or by contact with air, glass, etc. In order to avoid these problems, we recommend that (1) maximal care, and minimal handling, be applied during preparation and administration of the labeled dose: (2) both a physiological and nonphysiological component of the dose be included during model development; and (3) an early plasma sample be collected (10 min or less after dose administration) so that the extent of the problem can be determined. It is not uncommon for investigators to normalize their data to the amount of tracer in the first plasma sample. Presentation of the data in this way merely masks the problem. Although we believe that our in vivo methods produce physiologically relevant tracers of vitamin A, we and others are currently exploring alternative techniques for preparing high specific activity, physiological tracers for retinol and especially RBP. As is the case for the [3H]retinol-labeledplasma and chylomicrons discussed above, such techniques are aimed at eliminating the need for protein iodination which may induce alterations in the protein that alter its metabolism in vivo. Until recently, iodination was the only economically feasible approach to studying RBP kinetics. Recombinant protein expression systems may be an ideal way to produce RBP labeled with radioactive or stable isotopes of amino acids andor retinol labeled with isotopes of hydrogen or carbon. Potentially useful Escherichiu coli secretion vector systems have recently been described (Sivaprasadarao and Findlay, 1993; Wang et af., 1993). In our own studies (R. Blomhoff, M. H. Green, and colleagues, unpublished results), recombinant labeled RBP produced in an in vitro expression system did not act the same in intact rats as in vivo-labeled plasma. Similarly we (M. H. Green, J. E. Smith, and colleagues, unpublished results) have observed altered kinetic behavior using in vim-labeled [3H]retinolRBP. Thus isolation/preparation procedures may alter kinetic behavior of RBP or [3H]retinolbound to RBP. For any of these new labeling techniques, the critical things will be to compare kinetics of the labeled moieties to some credible reference (e.g., [3H]retinollabeled plasma prepared in vivo) and then to determine whether any observed differences are due to important biological phenomena or to preparative problems. IV. WHOLE-BODY MODELS FOR VITAMIN A METABOLISM

As a follow-up to our collaborative study with Underwood’s lab (Lewis et uf., 1981), we carried out a 35-day in vivo turnover study in rats (n = 11) with marginal vitamin A status (liver vitamin A ranged from 100 to


9

415 nmol) (Green et al., 1985). [3H]Retinol-labeled plasma was administered; short- and long-term tracer and tracee data were collected for plasma (Fig. 2), liver, kidneys, eyes, adrenals, small intestine, lungs, testes, skin, and rest of carcass. For development of an initial model, we lumped all organs other than liver and kidneys with the rest of carcass. Based on prevailing conventional wisdom (Underwood et a[., 1979), we postulated a five-compartment starting model. We used the SAMCONSAM computer programs and model-based compartmental analysis to compare our data to the initial model. To fit the data, a model with eight physiological compartments was required: one each for plasma retinol and retinyl esters, and two each for liver, kidneys, and rest of carcass (Fig. 3). Many parameters were well identified (i.e., statistical uncertainties were low); others (e.g., those describing liver vitamin A dynamics) were not, indicating areas for future studies (see below). Several interesting hypotheses resulted from this model (Green et al., 1985). (1) Plasma retinol recycled 12 times before irreversible loss and its turnover rate (nmol/day) was 13 times the disposal rate (24 nmol/day). That is, in support of our previous results (Lewis et al., 1981), an average plasma retinol molecule apparently recycles many times before irreversible utilization. (2) In contrast to the belief that the liver is the sole source of plasma retinol/RBP, our model predicted that 55% of plasma retinol input was from the liver and 45% was from extrahepatic tissues. (3) The model predicted that, in these rats that had marginal liver vitamin A stores and that were in slight negative vitamin A balance, almost half of the whole-

0

5

10

15

20

25

30

35

(Davr) FIG. 2. Observed data and model-simulated values for fraction of dose in plasma retinol (a)and retinyl esters (A) vs time after intravenous administration of [3H]retinol-labeled plasma. The model is shown in Fig. 3; data are from Green er al. (1985).

a

LIVER

PLASMA

EXTRAHEPATIC TISSUES

b

LIVER

PLASMA

EXVWHEPATIC TISSUES

FIG. 3. (a) Modcl proposed by Green et al. (1985) for whole-body vitamin A metabolism in rats. Circles represent compartments, large triangles are functions that sum multiple compartments, small triangles indicate sites of sampling, asterisk shows site of tracer introduction, m d wide arrow indicates site of dietary vitamin A input. Compartment 11 is plasma retinol and compartment 21 is plasma retinyl esters. Parameters shown are model-derived fractional transfer coefficients [ L ( f , J or ) the fraction of compartment /’s mass transferred to compartment I per day] and estimated fractional standard deviation in parentheses. Irreversible loss, both drgradative and/or excretory, was modeled as exiting the faster tuming-over compartment in liver, kidneys, and carcass. Since the data were insensitive to changes in magnitude of loss from each site, the three fractional transfer coefficients were set equal [i.e., L(0.3) = L(O.13) = L(0.16) = 0.46 d-’1. (b) Calculated pool sizes (nmol) and mass transfer rates (nmoU day) for the model shown in a.


11

body vitamin A was in extrahepatic tissues. This finding contradicted the prevailing belief that essentially all of the body’s vitamin A is stored in the liver. (4) Although it had been previously speculated that vitamin A may recycle from tissues as retinyl esters in lipoproteins, our analyses showed, and our model predicted, that only a minor amount (0.6%) of the vitamin A recycling through plasma was in the form of retinyl esters. Thus we hypothesized that nearly all of the vitamin A recycles as retinol bound to RBP. This led us and others to wonder what the source of RBP is for retinol recycling: previously it was thought that all of the circulating RBP came from liver parenchymal cells. Since plasma retinol acted kinetically as a single homogeneous compartment, we conclude that both newly secreted retinoVRBP and recycling retinolRBP behave similarly. After developing the initial model, we carried out similar but more extensive studies in rats that were vitamin A-deficient (Lewis et al., 1990) or -sufficient (Green and Green, 1987) to investigate how vitamin A status influences vitamin A dynamics. Tracer and tracee data for plasma, liver, kidneys, eyes, testes, lungs, adrenals, small intestine, and rest of carcass were collected for 35 or 115 days (deficient and sufficient rats, respectively) after injection of [3H]retinol-labeledplasma. Tracer data were also collected for urine and feces. Although parts of the data from vitamin A-sufficient rats have been published (Green and Green, 1987;Green etal., 1987,1992), we continue to work on the development of a complete whole-body model for vitamin A kinetics in these rats. For the rats with low vitamin A status (liver vitamin A, 3000 nmol), the modelpredicted total traced mass underestimates measured liver vitamin A levels, as if some of the vitamin is in a nonexchangeable pool in stellate cells. Even when turnover studies were carried out for 115 days in vitamin Asufficient rats, tracer did not fully equilibrate with this pool.

I

,

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I

I

1

I

I

1

10

loo

lo00

loo00

Liver Vitamin A (nmol) FIG. 8. Observed values for liver vitamin A mass vs model-predicted total traced mass minus observed liver vitamin A mass.


VII.

19

OTHER TISSUES

In order to develop a whole-body model for vitamin A metabolism which incorporates dynamics in many tissues, a large number of data points from plasma, urine, feces, and individual organs must be modeled simultaneously. Needless to say, this juggling has proven to be a challenging exercise. An alternate modeling approachthe “forcing function”has proven useful and more straightforward for describing vitamin A dynamics in individual organs (Green et al., 1992). The forcing function approach has been used by other investigators to model zinc metabolism in humans (Foster et al., 1979) and is applicable to many different systems. This technique makes use of the fact that, since plasma is the transport medium for vitamin A (and likewise many other nutrients), it provides the vitamin A input to individual tissues. This fact allows us to “uncouple” individual organs from the whole system and makes model development more manageable. To use the forcing function approach, long-term data for plasma [3H]retinol turnover are fit to a multiexponential equation to define the temporal plasma tracer response. This function is then used to “drive” tracer input into individual tissues. The output from the tissue or organ being modeled is to unknown site(s) outside the organ. Although one cannot determine how much of the vitamin is recycled to plasma vs irreversibly metabolized, useful information is obtained on fractional transfer coefficients describing uptake of vitamin A by the tissue, within the tissue and out of the tissue. When we applied the forcing function approach to our data on vitamin A dynamics in organs of rats at three levels of vitamin A status, several interesting results were obtained (Green et al., 1992). For the eyes, a one compartment model was adequate to fit data at all three levels of vitamin A status (Fig. 9). The model indicated that only -0.1% of plasma vitamin A turnover goes to the eyes. If all of this vitamin A is used irreversibly for visual, neuronal, epithelial tissue and other functions of vitamin A, it represents only -1% of whole-body disposal rate, even in vitamin Adeficient rats. In such rats, the 0.06% of plasma retinol turnover that went to the eyes represented almost a 90%reduction in the rate of retinol transfer to the eyes. However, the eyes conserved -50% of normal eye vitamin A content and the transit time (75 days) was six times longer than in the other two groups. Thus, it appears that, in vitamin A-depleted rats, the eye can down-regulate its uptake and turnover of vitamin A. When the forcing function approach was used to describe vitamin A metabolism in the small intestine, a two-compartment model was required. The model predicted that vitamin A transfer to this organ is appreciable: 14 nmollday in normal rats or 3.4% of plasma retinol turnover. This high rate of transfer, which


20

r-

32 nmol

n 0.001256

(nn)d'

0.40 nmol/d

0.0798 C'

R = 406 m u d DR = 41 nmoVd

(.ow)

-1.6

(0.6)

( ).O..z:d-' Q wEyes . 1 nmol

nmol (n-19)

I

Ipc 5.0 nmol

R = 71 mnoUd DR = 5.8 d d 0*0133 d-l

FIG. 9. One-compartment model for vitamin A metabolism in eyes of rats with high (HI), marginal (MAR), or low vitamin A status (LO). On the right are the plasma [3H]retinol forcing function (box), the plasma retinol pool size (above box), plasma retinol transfer rate (R), and the whole-body vitamin A disposal rate (DR). Data shown with interconnectivities are model-predicted fractional transfer coefficients and their estimated fractional standard deviation (FSD) and retinol transfer rates (nmoUday). No FSD is given for the low group because turnover of vitamin A from eyes of rats in this group was so slow that the output rate constant could not be predicted with confidence. Also shown are observed (QO) and model-predicted total retinol masses (QC). Reprinted from Green er af. (1992) by courtesy of Marcel Dekker, Inc.

is surpassed only by that to kidneys and liver, may reflect the importance of vitamin A in maintenance of epithelial tissue. Modeling has also revealed some interesting ideas about the contribution of the kidneys to whole-body vitamin A metabolism. In our first model


21

(Green el at., 1985), kidneys were sampled 1, 2, 15, and 35 days after administration of [3H]retinol-labeled plasma. The model developed to fit the data predicted that only 7% of plasma retinol turnover went to the kidneys and that the average retinol molecule did not pass through kidney tubules. Since this finding was unexpected in view of work by other investigators, we simulated kidney filtration and reabsorption using CONSAM and speculated that earlier samples were needed to adequately model the role of the kidneys in vitamin A metabolism. Thus in the study of rats with low vitamin A status (Lewis et al., 1990), kidneys and other organs were first sampled 12 min after dose administration. The model predicted that 44% of the plasma retinol turnover was transferred to the kidneys. If all of this were lost, it would be seven times the whole-body disposal rate. Thus essentially all of the filtered retinol must be reabsorbed and recycled to plasma. Analyses done with our first modeling study (Green et al., 1985) indicated that plasma lipoproteins were not the vehicle for retinol recycling. Work in other labs (Soprano et al., 1986) has shown that the S3 segment of the kidney contains messenger RNA for RBP, but that is not the correct anatomic site for reabsorption. Perhaps the proximal tubule reabsorbs retino1 bound to RBP instead of degrading the filtered RBP. VIII. VITAMIN A DISPOSAL RATE

A final, more philosophical issue has been raised by our modeling studies and is one of the topics we are currently pursuing. As we have quantified vitamin A disposal rate at various levels of vitamin A status, we have been led to wonder why the body disposes of so much of a valuable essential nutrient such as vitamin A. Perhaps this is an evolutionary response to the fact that we are an open system. That is, since we consume essential nutrients such as vitamin A in the diet, we need to have physiological or biochemical means to eliminate them. To look at the determinants of vitamin A disposal rate, we used multiple regression analysis to relate our data on disposal rate for 62 rats to vitamin A intake, liver vitamin A levels, and plasma retinol pool size (Fig. 10) (Kelley el af., 1994). These three variables predict 91% of the variation in disposal rate; 68% of the reduction in sum of squares comes from plasma retinol, 18% from liver vitamin A, and 14% from vitamin A intake. Our data indicate that disposal rate does not fall until liver vitamin A levels are essentially depleted. That is, as long as plasma retinol Ievels are normal, degradation rate is high, implying a nonfunctional utilization of the vitamin. We call this “degradative preservation.” We conclude that plasma retinol is the major determinant of vitamin A disposal rate and hypothesize that

22

PART I VITAMIN METABOLISM DR = -0.39 + 0.888ROH + 0.1491N + 0.00203LNA

6o

v

50

z Y

40

10

0 0 II 0

I

i

I

I

I

I

I

t

10

20

30

40

50

60

70

80

Disposal Rate (nmoVd)

FIG. 10. Relationship between vitamin A disposal rate (DR; nmol/day), plasma retinol pool size (ROH: nmol), vitamin A intake (IN; nmoyday), and liver vitamin A (LIVA; nmol).

degradation may be driven by an intracellular pool of vitamin A in equilibrium with plasma retinol. Assuming availability of research funds, we will continue to use compartmental analysis to study vitamin A metabolism and to validate the hypotheses presented here. ACKNOWLEDGMENTS The modeling studies described in this paper were supported by grants to M.H.G. from the U.S. Department of Agriculture (81-CRCR-1-0702 and 88-37200-3537). Studies in Oslo were supported by grants from Anden Jahres Fond, Nansenfondet, the Norwegian Cancer SoLiety, and the Nordic Insulin Foundation, by a Fulbright Research Scholar Award to M.H. G., by a grant from the National Science Foundation (1NT-8419!955), and by a NATO Grant for International Collaboration.

REFERENCES Adams, W. R., and Green, M. H. (1994). Prediction of liver vitamin A in rats by an oral isotope dilution technique. 1.Nutr. lZ4, 1265-1270.


23

Adams, W. R., Smith, J. E., and Green, M. H. (1995). Effects of N-(4-hydroxyphenyl)retinamide on vitamin A metabolism in rats. Proc. Soc. Exp. Biol. Med. 2438, 178-185. Berman, M., and Weiss, M. F. (1978). “SAAM Manual,” DHEW Publ. No. (NIH) 78-180. US.Govt. Printing Office, Washington, DC. Berman, M., Beltz, W. F., Greif, P. C., Chabay, R., and Boston, R.C. (1983). “CONSAM User’s Guide,” PHS Publ. No. 1983-421. U.S. Govt. Printing Office, Washington, DC. Blomhoff, R., Berg, T., and Norum, K. R. (1988). Transfer of retinol from parenchymal to stellate cells in liver is mediated by retinol-binding protein. Proc. Nafl. Acad. Sci. U.S.A. 85,3455-3458. Blomhoff, R., Green, M. H., Green, J. B., Berg, T., and Norum, K.R. (1991). Vitamin A metabolism: New perspectives on absorption, transport, and storage. PhysioL Rev. 71, 951-990. Brownell, G. L., Berman, M., and Robertson, J. S. (1968). Nomenclature for tracer kinetics. Int. J. Appl. Radiat. Isor. 19,249-262. Duncan, T. E., Green, J. B., and Green, M. H. (1993). Liver vitamin A levels in rats are predicted by a modified isotope dilution technique. 1.Nurr. l23,933-939. Foster, D. M., and Boston, R. C. (1983). The use of computers in compartmental analysis: The SAAM and CONSAM programs. In “Compartmental Distribution of Radiotracers” (J.S. Robertson, ed.), Chapter 5, pp. 73-142. CRC Press, Boca Raton, FL. Foster, D. M., Aamodt, R. L., Henkin, R. I., and Berman, M. (1979). Zinc metabolism in humans: A kinetic model. Am. J. Physiol. 237, R340-R349. Goodman, D. S. (1980). Vitamin A metabolism. Fed. Proc., Fed. Am. SOC. Exp. Biol. 39, 2716-2722. Green, M. H., and Green, J. B. (1987). Multicompartmental analysis of whole body retinol dynamics in vitamin A-sufficient rats. Fed. Proc., Fed. Am. SOC. Exp. Biol. 46, IOll(Abstr. 4047). Green, M. H., and Green, J. B. (1990a). The application of compartmental analysis to research in nutrition. Annu. Rev. Nurr. 10,41-61. Green, M. H., and Green, J. B. (1990b). Experimental and kinetic methods for studying vitamin A dynamics in vivo. I n “Methods in Enzymology” (L. Packer,ed.), Vol. 190, pp. 304-317. Academic Press, San Diego, CA. Green, M. H., and Green, J. B. (1994). Vitamin A intake and status influence retinol balance, utilization and dynamics in the rat. 1. Nutr. l24,2477-2485. Green, M. H., Uhl, L., and Green, J. B. (1985). A multicompartmental model of vitamin A kinetics in rats with marginal liver vitamin A stores. J. Lipid Res. 26,806-818. Green, M. H., Green, 1. B., and Lewis, K. C. (1987). Variation in retinol utilization rate with vitamin A status in the rat. J. Nurr. 117,694-703. Green, M. H., Green, J. B., and Lewis, K. C. (1992). Model-based compartmental analysis of retinol kinetics in organs of rats at different levels of vitamin A status. In “Retinoids: Progress in Research and Clinical Applications” (M. A. Livrea and L. Packer, eds.), pp. 185-204. Dekker, New York. Green, M. H., Green, J. B., Berg, T., Norum, K. R., and Blomhoff, R. (1993). Vitamin A metabolism in rat liver: A kinetic model. Am. J. Physiol. 264,G509-G521. Gurpide, E. (1975). Tracer methods in hormone research. In “Monographs on Endocrinology” (F. Gross, A. Labhart, M. B. Lipsett, T. Mann, L. T. Samuels, and J. Zander, eds.), Vol. 8, pp. 1-188. Springer-Verlag,New York. Jacquez, J. A. (1985). “Compartmental Analysis in Biology and Medicine,” 2nd ed. Univ. of Michigan Press, Ann Arbor. Kelley, S. K., Green, J. B., and Green, M. H. (1994). Plasma retinol (ROH): Main determinant of vitamin A (vit A) disposal rate (DR) in vit A-sufficient rats during negative vit A balance. FASEB J. 8, A444 (Abstr. 2569).

24


Lewis, K. C., Green, M. H., and Underwood, B. A. (1981). Vitamin A turnover in rats as influenced by vitamin A status. 1. Nurr. 111, 1135-1144. Lewis, K. C., Green, M.H., Green, J. B., and Zech, L. A. (1990). Retinol metabolism in rats with low vitamin A status: A compartmental model. J. Lipid Res. 31, 1535-1548. Shipley, R. A,, and Clark, R. E. (1972). “Tracer Methods for In Vivo Kinetics.’’ Academic Press, New York. Sivaprasadarao, A., and Findlay, 1. B. (1993). Expression of functional human retinol-binding protein in Escherichiu coli using a secretion vector. Biochem. J. 296,209-215. Soprano, D. R., Soprano, K. J., and Goodman, D. S. (1986). Retinol-bindingprotein messenger RNA levels in the liver and in extrahepatic tissues of the rat. 1. Lipid Res. 27, 166-171. Underwood, B. A,, Loerch, J. D., and Lewis, K. C. (1979). Effects of dietary vitamin A deficiency, retinoic acid and protein quantity and quality on serially obtained plasma and liver levels of vitamin A in rats. 1. Nurr. 109, 7%-806. Wang. T. T.. Lewis, K. C., and Phang, J. M. (1993). Production of human plasma retinolbinding protein in Escherichia coli. Gene 133,291-294.

ADVANCES IN FOOD AND NUTRITION RESEARCH, VOL. 40

Chapter 2 MATHEMATICAL MODELING IN NUTRITION: CONSTRUCTING A PHYSIOLOGIC COMPARTMENTAL MODEL OF THE DYNAMICS OF &CAROTENE METABOLISM JANET A. NOVOTNY Diet and Human Performance Laboratory Beltsville Human Nutrition Research Center U. S. Department of Agriculture Beltsville, Maryland 20705

LOREN A. ZECH Laboratory of Mathematical Biology, National Cancer Institute Bethesda, Maryland 20892

HAROLD C. FURR Departmenr of Nutritional Sciences, University of Connecticut Storrs, Connecticut 06269

STEPHEN R. DUEKER AND ANDREW J. CLIFFORD' Department of Nutrition, University of California Davis, California 95616

I. Introduction and Background Materials, Methods, and Model Constraints 111. The Process of Constructing a Compartmental Model IV. Intermediate Models V. Statistical Considerations VI . The Final Model VII. System Behavior Proposed by the Model VIII. Unobservable System Behavior Proposed by the Model IX. Empirical Description of the Experimental Observations X. Final Encouraging Words References 11.

' Corresplonding author.

25 Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

26


1.

INTRODUCTION AND BACKGROUND

To understand the health implication of current and recommended dietary practices, i.e., increased intakes of some foods and/or nutrients such as fruits and vegetables and antioxidant vitamins and reduced intakes of others such as calories and saturated fats, an investigation of variations in health status produced by these practices is required. Tracer studies are often used to characterize the health status of individuals because their response patterns are consistent and such studies can be interpreted in a standardized way. Tracer kinetics are usually modeled with differential equations that are mapped to metabolic spaces and the exchanges (analyte flows) between them in the domain of an individual’s (a system’s) metabolism. The characteristics of these spaces and the exchanges (of nutrients/ analytes) that take place between them provide much useful information about an individual’s physiologic status. Computer hardware and modeling software capable of solving (and manipulating) differential equations efficiently and accurately are now available. Therefore, mathematical modeling has become an attractive tool for collecting and processing the research data and information needed to discover those optimal combinations of nutrients that promote health and prevent and/or minimize disease. While-many researchers have focused on the tools of molecular biology and genetics to determine biochemical mechanisms of nutrient action in animal models, a few have focused on mathematical modeling of kinetic data to achieve a quantitative understanding of the dynamics of nutrient metabolism in vivo (for recent symposia, see Abumrad, 1991; Coburn, 1992).Three recent developments stimulated interest in mathematical modeling. First, there is an opportunity to integrate quantitative characteristics of the dynamics of nutrient metabolism with knowledge of nutrient action mechanisms and health status. Second, it appears that some animal models do not mimic nutrient metabolism and health status of humans. Third, stable isotope tracers and reliable methods to measure minute amounts of them in human tissues have become more readily available. Stable isotopes are advantageous both because there is no radiation exposure to study subjects and the problems of disposing of radionuclides are avoided. The combined use of such tools as mathematical models and stable isotopes is a powerful approach for understanding the dynamics of nutrient metabolism and for tailoring their requirements to physiologic state and age. Compartmental modeling has been successfully employed to gain new integrated and quantitative insights into several biological and physical systems currently under investigation. Compartmental modeling papers have generally only detailed the theoretical basis for modeling and then

CHAPTER 2 MATHEMATICALMODELING IN NUTRITION

27

summarized the final results of their studies; details concerning model development usually haye been omitted. Because the rationale, philosophy, and details for developing a compartmental model are not well described in the scientific literature, some investigators have limited their approach to empirical descriptions of their research data. Consequently they have been unable to maximize the information potentially obtainable from their studies. This paper aims to describe the practical aspects of the process of developing a physiologically based compartmental model using a recently constructed model of the dynamics of &carotene metabolism. Also, we discuss how compartmental modeling itself has advanced understanding of P-carotene and retinol metabolism. The reason for building a physiologic compartmental model is to realize as complete a description as possible of a metabolic system under investigation. The model is built to develop an analogy of the system under investigation and to obtain values for critical parameters of the model so that unobserved portions of the dynamic and kinetic behavior of the system under investigation can be predicted. Specific information obtained about the system under investigation includes the number of pools and their sizes, how they are connected, and how their masses change over time. Physiologic compartmental models are built by analogy with a specific physiologic system under investigation in order to rigorously describe assumptions about the system and to subsequently test these assumptions. This is often accomplished by a comparison of a plasma analyte concentration-time curve predicted by the model to the actual experimental observations. Physiologic compartmental models are chosen because they can be assumed to be a suitable analogy to the physiologic system under investigation which, in turn, is assumed to consist of pools and flows of analytes. On the other hand, empirical descriptions of the system under investigation have limited use because they are less able to describe complex metabolic relationships. Therefore, by building a compartmental model of the system under investigation, it is possible to gain a greater understanding of the system, to extract considerably more information about it from the experimental observations, and to identify gaps in the existing knowledge of the system. Relationships among the system under investigation, the model, and the experimental observations are depicted in Fig. 1. The goal of the modeling process is the full realization of the system under investigation by relating elements of the model with the experimental observations. Model structure specification (connectivity) and identification (defining model parameters and estimating numerical values for them and their precision from the noisy data) are mapped to the system’s metabolic spaces and the transfer of analyte among them. The system under investigation is fully understood or realized by an analogy (point by point compari-

28


MODEL

SYSTEM

Dose

EXPERIMENTM OBSERVATIONS 0.8

-

0.6

-

0.4

-

0.2

-

0 0

o

0

o

pmoltotalROWL

0

pmol total BCarL

ROH-OH 1.0

0

0.5

aQU

0

0

0.0

0

5

10

15

20

25

0

1 0 2 0 3 0 4 0 5 0 6 0

Days after ingesting fl-CarOtened8 FIG. 1. Relationshipsamong the model, the system under investigation and the experimental observations. EHT = extrahepatic tissue; ROH = retinol; @-Car= p-Carotene. Isotopomer ratios of p-carotene-d$p-carotene and retinol-d&etinol and the concentrations of total pcarotene and total retinol in plasma by time since ingesting p-carotene-da (the experimental measurements) are shown in the bottom left and right panels, respectively. Rudiments of the model iriclude the compartment; the fractional transfer coefficient and flow from a donor to a recipien1compartment;theinitialconditionsofthe system;andasetof rulesfor relatingelements of the model with the experimental observations made on the system under investigation.

son) of the experimental observations with the model’s predicted curve; this comparison is called the fit. Building a model can facilitate the realization of a system under investigation because the model is an analogy of that system. Thus, by identification and analogy, model building allows the system under investigation to be realized and evaluated by how well the model predicts the experimental observations.


29

Practical aspects of the process of developing a physiologic compartmental model are illustrated using a recently constructed model of the dynamics of @carotene metabolism (Novotny et aL., 1995). P-Carotene is not only a significant source of vitamin A for humans worldwide, but is also reported to protect against oxidative stress, heart disease, and cancer and to enhance the immune response (Krinsky, 1989; Olson, 1992; Bendich, 1993). pCarotene is used in large amounts both as a dietary supplement and in many clinical trials. However, despite its key physiologic effects and widespread intake, the dynamics of the metabolism of this common diet constituent in humans as well as the factors that affect it are largely unknown. This encouraged us to build a physiologic compartmental model of the dynamics of @carotene metabolism in a healthy adult in viva

II. MATERIALS, METHODS, AND MODEL CONSTRAINTS An informed, consenting, and healthy 53-year-old male, weighing 94 kg, ingested a gelatin capsule containing 73 pmol (40 mg) of all-trans-& carotene-10,10’,19,19,19,19’,19‘,19’-d~ (P-carotene-d8) dispersed in -2g 01ive oil. This material was synthesized and its chemical and isotopic purities were determined as previously described (Bergen, 1992; Dueker et aL., 1994). The capsule was ingested with a light breakfast, and the subject ate a lunch and dinner at 3 and 8 hr later, respectively, that were very low in carotene and vitamin A. Blood samples were drawn just before (0 hr) and at 0.5, 1,2,5, 7,9,12, and 24 hr and 2, 3, 4,6,8,10,12,16,20,24,36,43, 57,71,85,99, and 113 days after ingestion of the p-CarOtene-ds containing capsule. Only data for the first 57 days are used, because the tracer concentrations were too low to analyze accurately after that point. The protocol was approved by the University of California, Davis, Human Subjects Review Committee. Plasma was separated by centrifugation and the concentrations of total (deuterated and protonated) @-carotene and total retinol in all plasma specimens (after saponification)were measured by high-performance liquid chromatography (HPLC); the fractional standard deviation (FSD) of the analytical method was

*

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-

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Fast turnover liver retinoid

-

0

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-

Slow turnover liver retinoid I

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FIG. 7. Compartmental model predicted masses and concentrations of retinol and retinyl ester in plasma, liver, and extrahepatic tissue of a healthy adult who ingested a single 73-wmoI dose of B-carotene-ds orally. The “Fast turnover liver retinoid” (bottom left) and “Slow turnover liver retinoid” (bottom right) each include the protio and deuterated species.

the slow turnover liver retinyl ester pool and the recycling of retinol by the liver. The fast turnover liver retinoid-d4 curve exhibits a narrow peak after the fkarOtene-d8 dose, while the slow turnover liver retinoid-d4 peak rises and remains elevated. This indicates that the retinoid-d4 is expected to have remained stored in the subject’s liver for an extended time after ingesting the fl-~arotene-dg.The slow turnover liver retinoid mass seems not to have been influenced by the ingested dose of the /3-carotene-d8. This is probably related to the abundant liver stores of vitamin A already present in this

CHAPTER 2 MATHEMATICAL MODELING IN NUTRITION

49

experimental subject; the slow turnover liver vitamin A stores alone exceeded 300 pmol. The -7 pmol total retinoid-d4 formed from the 73-pmol dose of p-carotene-dg was very small in comparison to the subject’s liver vitamin A stores. Such a dose of p-carotene might be anticipated to have a much greater influence on the slow turnover liver retinoid if the subject’s total body reserve of vitamin A were minimal (i.e., liver stores 150 pmol). Future studies with subjects of varying vitamin A status will clarify the effect (ability) of p-carotene intake on (to replenish and/or sustain) tissue retinoid concentrations. The compartmental model was also able to predict the efficiency of conversion of p-carotene to vitamin A in our subject. The model predicted that 1 pg dietary p-carotene yielded 0.054 pg retinol (the same as 0.101 pmol retinoVpmo1 p-carotene). The 0.054-pg value is considerably lower than the 0.167 pg retinoupg p-carotene which is widely accepted. However, the 0.167-pg value was established in growing rats with low reserves of retinol who were adapted to maximizing the retinol yield (Brubacher and Weiser, 1985). If our subject had been in marginal or deficient vitamin A status, the predicted yield would probably have exceeded 0.054 pg retinollpg @-carotene. Further studies are needed to determine the influence of vitamin A status on conversion of p-carotene to vitamin A and the ability of dietary carotene to maintain tissue retinoid. Figures 6 and 7 and the preceding paragraphs exemplify the usefulness of modeling in predicting metabolite concentrations in tissues and biologic spaces that are difficult to observe experimentally. Once predictions have been made, the modeling process proceeds by testing these and other predictions and further modifying the model to fit with new experimental observations. In this way, the modeling process is open-ended as the model is continually refined as more and more experimental observations become available. IX. EMPIRICAL DESCRIPTION OF THE EXPERIMENTAL OBSERVATIONS

To allow the information from the compartmental model of the dynamics of p-carotene metabolism to be compared with that from an empirical description of the same experimental observations, polyexponential fits (empirical descriptions) of our experimental observations were also made. The plasma p-carotene-dg and retinol-d4 concentration-time data were described using an empirical multiexponential description of the data using weighted, nonlinear least squares regression and the SAS NLIN procedure. Each observation was weighted by the reciprocal of its predicted value.

50

PART I VITAMlN METABOLISM

The plasma p-carotene-dg concentration-time curve was described by the five-term exponential equation y ( t ) = -30.5e-'2.3' + 17.4e-5.7' - 7.2e-3.3'+ 0.65e-0.27'+ 0 . 0 8 4 ~ - ~ Figure . ~ ~ ' . 8 shows the observed plasma p-carotenedgconcentration-time data along with the concentration-time curve specified by the five-term exponential equation. The area munder the concentration-time curve (AUC) was calculated as AUC = y(t)dt, and the area under the moment curve (AUMC) was calculated as AUMC = t X

1,

\;

y(t)dt. The AUC, AUMC, and AUMClAUC for p-carotene-d8 were calcu-

0.5 0.4

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Days after ingesting pcarotene-d8 FIG. 8. Experimentallymeasured values (circles) and best-fit line using the empirical description of the concentrations of &carotene-d8 in plasma as a function of time after a 94-kgadult male ingested 73-pmol &carotene-ds (top). Experimentally determined values (circles) and best-fit line using the empirical description of the concentrations of retinol-d4 (derived from ingested &carotene-d8) by time after ingesting B-carotene-d8(bottom). Predicted values that were negative were set to equal zero. Reprinted with permission from Novotny et al. (1995).

CHAPTER 2 MATHEMATICAL MODELING IN NUTRITION

51

lated to be 2.09 pmol X dayhiter plasma, 27.31 pmol X day2hiter plasma, and 13.05 days, respectively. The AUMCIAUC is commonly used to estimate the mean sojourn time (MST). The MST of &carotene obtained with the empirical description is much shorter than that obtained with the compartmental model of the dynamics of p-carotene metabolism. The plasma retinol-d4 concentration-time data were described with the three-term exponential equation y(t) = -1.338e-2.751r+ 0.883e-1.467r + 0.07578e-0.03642 using weighted, nonlinear least squares regression and the SAS NLIN procedure. Each observation was weighted and the AUC, AUMC, and MST were calculated as for plasma P-carotene-d8. Figure 8 shows the concentration-time data along with the concentration curve specified by the three-term exponential equation for retinol-d4. From this equation the AUC, AUMC, and MST for retinol-d4 were calculated to be 2.20 pmol X dayhiter plasma, 57.36 pmol X day2/liter plasma, and 26.12 days, respectively. The MST of retinol obtained with the empirical description is much shorter than that obtained with the compartmental model of the dynamics of &carotene metabolism. The concentration-time curves generated with the empirical models both display nonzero y-intercepts. Because calculation of AUC and AUMC with a lower integration limit of x = 0 might introduce error into the values for AUC, AUMC, and MST, these parameters were also calculated by starting the integration at the x-intercept. When the integration limit was the xintercept, the AUC, AUMC, and MST for P-carotene-ds were 3.17 pmol X dayfliter plasma, 27.36 pmol X day2hiterplasma, and 8.64 days, respectively. Under these conditions, the AUC, AUMC, and MST for retinol-d4 were 2.23 pmol x day/liter plasma, 57.37 pmol X day2/liter plasma, and 25.67 days, respectively. These results demonstrate that the error introduced into the AUC, AUMC, and MST values was small when the integration was performed from zero to infinity, and it was concluded that the discrepancy in residence time for P-carotene between the empirical description and the compartmental model was not due to the limits of integration. X.

FINAL ENCOURAGING WORDS

Modeling is an exciting and challenging means of investigating biological and physical systems. While the model presented here is specific for pcarotene, the general techniques and rationale used are typical of the compartmental modeling process. Many of the issues encountered in developing a compartmental model have been described in detail so that beginning modelers may have a greater understanding of how to proceed. We hope that our description of the development of the P-carotene model will pro-

52


vide a meaningful reference for the reader in future encounters with modeling, and we encourage the reader to embark on a modeling project of her/ his own.

ACKNOWLEDGMENTS The authors thank Meryl Wastney, Chris Calvert, Jan Peerson, and Peg Hardaway for reviewing the paper and Ali Arjomand for help in drawing some of the diagrams and figures. This research was supported by NIH (Grant ROl-DK-48307), USDA Regional Research (W-143), and Bridging Funds, Office of Vice Chancellor for Research at UCD.

REFERENCES Abumrad, N. (1991). Mathematical models in experimental nutrition. JPEN, J. Parenter. Enreral Nutr. 15,44S-98S. Bendich, A. (1993). Biological functions of dietary carotenoids. A m N. Y. Acad. Sci. 691, 61 -61. Bergen, H. R. (1992). Synthesis of deuterated P-carotene. I n “Methods in Enzymology” (L. Packer, ed.), Vol. 213, pp. 49-53. Academic Press, San Diego, CA. Berman, M., and Weiss, M. F. (1978). “SAAM Manual,’’ DHEW Publ. No. (NIH) 78-180. US. Govt. Printing Office, Washington, DUNational Institutes of Health, Bethesda, MD. Bowen, P. E., Mobarhan, S., and Smith, J. C., Jr. (1993). Carotenoid absorption in humans. I n “Methods in Enzymology” (L.Packer, ed.), Vol. 214, pp. 3-17. Academic Press, San Diego, CA. Brubacher, G. B., and Weiser, H. (1985). The vitamin A activity of p-carotene. Int. 1. Vitam. Nutr. Res. 55, 5-15. Clevidence, B. A., and Bieri. J. G. (1993). Association of carotenoids with human plasma lipoproteins.In “Methods in Enzymology” (L. Packer, ed.), Vol. 214, pp. 33-46. Academic Press, San Diego, CA. Cobelli, C., and Saccomani,M. P. (1992). Accessible pool and system parameters: Assumptions and models. JPEN, J. Parenter. Enteral Nurr. 15,45S-50S. Coburn, S. P. (1992). Application of models to the determination of nutrient requirements. 1. Nutr. 122,681S-714S. Cornwell, D. G., Kruger, F. A,, and Robinson, H. B. (1962). Studies on the absorption of beta-carotene and the distribution of total carotenoid in human serum lipoproteins after oral administration. J. Lipid Res. 3, 65-70. Cortner, J. A., Coates, P. M., Le, N. A., Cryer, D. R., Ragni, M. C., Faulkner, A., and Langer, T. (1987). Kinetics of chylomicron remnant clearance in normal and in hyperlipoproteinemic subjects. J. Lipid Res. 28, 195-206. Dimitrov, N. V.. Meyer, C., UUrey, D. E., Chenoweth, W., Michelakis,A., Malone, W., Boone, C., and Fink, G. (1988). Bioavailability of p-carotene in humans. Am. J. Clin. Nutr. 48,298-304. Dixon, Z. R., Burri, B. J., Clifford, A. J., Frankel, E. N., Schneeman, B. O., Parks, E., Keim, N. L., Barbieri, T., Wu, M. M., Fong, A. K. H., Kretsch, M. J., Sowell, A. L., and Erdman, J. W., Jr. (1994). Effects of a carotene-deficientdiet on measures of oxidative susceptibility and superoxide dismutase activity in adult women. Free Radicals Bwl. Med. 17,537-544.


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Dueker, S. R., Lunetta, J. M., Jones, A. D., and Clifford, A. J. (1993). Solid-phase extraction protocol for isolating retinol-d4and retinol from plasma for parallel processing for epidemiologic studies. Clin. Chem. (Winston-Salem,N.C.) 39,2318-2322. Dueker, S. R., Jones, A. D., and Clifford, A. J. (1994). Stable isotope methods for the study of pcarotene-ds metabolism in humans utilizing tandem mass spectrometry and high performance liquid chromatography. Anal. Chem. 66,4177-4185. Goodman, D. S., and Huang, H. S. (1965). Biosynthesis of vitamin A with rat intestinal enzymes. Science 149,879-880. Green, M.H., and Green, J. B. (1994). Dynamics and control of plasma retinol. In “Vitamin A in Health and Disease” (R. Blumhoff, ed.), pp. 119-133. Dekker, New York. Green, M. H., Uhl, L., and Green, J. B. (1985). A multicompartmental model of vitamin A kinetics in rats with marginal liver vitamin A stores. J. Lipid Res. 26,806-818. Handelman, G. L., Haskell, M. J., Jones, A. D., and Clifford, A. J. (1993). Improved GUMS determination of d4-retinolhetinol in human plasma. Anal. Chem. 65,2024-2028. Krinsky, N. I. (1989). Carotenoids and cancer in animal models. J. Nu@. 119,123-126. Krinsky, N. I., Cornwell, D. G., and Oncley, J. L. (1958). The transport of vitamin A and carotenoids in human plasma. Arch. Biochem. Biophys. 73,233-246. Krinsky, N. I., Wang, X.-D., Tang, G., and Russell, R. M. (1994). Mechanism of carotenoid cleavage to retinoids. Ann. N. Y.Acad. Sci. 691, 167-176. Langer, T., Strober, W., and Levy, R. I. (1972). The metabolism of low density lipoprotein in familial type I1 hyperlipoproteinemia. J. Clin. Invest. 51, 1528-1536. Lewis, K. C., Green, M. H., Green, J. B., and Zech, L. A. (1990). Retinol metabolism in rats with low vitamin A status: A compartmental model. J. Lipid Res. 31, 1535-1548. Novotny,J. A., Dueker, S. R., Zech, L. A., and Clifford,A. J. (1995). Compartmental analysis of the dynamics of &carotene metabolism in an adult volunteer./. Lipid Res. 36,1825-1838. Olson, J. A. (1989). Provitamin A function of carotenoids: The conversion of @carotene into vitamin A. J. Nurr. 119,105-108. Olson, J. A., and Hayaishi, 0. (1965). The enzymatic cleavage of beta-carotene into vitamin A by soluble enzymes of rat liver and intestine, Proc. Narl. Acad. Sci. U.S.A.54,1364-1369, Olson, R. E. (1992). Vitamins and carcinogenesis: An overview. In “Proc. First International Congress on Vitamins and Biofactors in Life Science in Kobe, 1991” (T. Kobayashi, ed.), pp 313-316. Center for Acad. Publications Japan. Tokyo, Japan. Sauberlich, H. E., Hodges, R. E., Wallace, D. L., Kolder, H., Canham, J. E., Hood, J., Raica, N., Jr., and Lowry, L. K. (1974). Vitamin A metabolism and requirements in the human studied with the use of labeled retinol. Vitam. Horm. (N.Y.)32, 251-275. Schmitz, H. H., Poor, C. L., Wellman, R. B., and Erdman, J. W., Jr. (1991). Concentrations of selected carotenoids and vitamin A in human liver, kidney and lung tissue. J. Nutr. U1,1613-1621. Schmitz,H. H., Poor, C. L., Gugger, E. T., and Erdman, J. W. Jr. (1993). Analysis of carotenoids in human and animal tissues. In “Methods in Enzymology” (L. Packer, ed.), Vol. 214, pp. 102-116. Scita, G., Aponte, G. W., and Wolf, G. (1993). Uptake and cleavage of &carotene by cultures of rat small intestinal cells and human lung fibroblasts. In “Methods in Enzymology” (L. Packer, ed.), Vol. 214, pp. 21-32. Academic Press, San Diego, CA. Snyder, W. S., Cook,M. J., Nasset, E. S., Karhausen, L. R., Howells, G. P., and Tipton, I. H, (1975). “Report of the Task Group on Reference Man,” Int. Comm. Radio]. Prot. No. 23, pp. 273-334. Pergamon, New York. Song, K. S., Muller, H. G., Clifford, A. J., Furr, H. C., and Olson, J. A. (1995). Estimating derivatives of pharmacokinetic response curves with varying bandwidths. Biomefrics 51,12-20.

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van Vliet. T., van Scheik, F., and van den Berg, H. (1992). &Carotene metabolism: The enzymatic cleavage to retinal. Nerh. J. Nurr. 53, 186-190. Wang, X.-D., Tang, G. W., Fox, J. G.. Krinsky, N. I., and Russell, R. M. (1991). Enzymatic conversion of @-carotene into apo 6-apo-carotenals and retinoids by human, monkey, ferret, and rat tissues. Arch. Biochem. Biophys. 285, 8-16. Wang, X.-D., Krinsky, N. I., Marini, R. P., Tang, G., Yu,J., Hurley, R., Fox,J. G., and Russell, R. M. (1992). Intestinal uptake and lymphatic absorption of @-carotene in ferrets: A model for human @-carotenemetabolism. Am. J. Physiol. 263, G4804486. Zackman, R. D., Dunagin. P. E., and Olson, J. A. (1966). Formation and enterohepatic circulation of metabolites of retinol and retinoic acid in bile duct-canulated rats. J. Lipid Res. 7,3-9. Zulim. R. A., Lunetta, J. M., Corso, F. A., Dueker, S. R., Schneider, P. D., Joyce, V., Rippon, M. B.,Wolfe, B. M., and Clifford, A. J. (1995). Retinol and p-carotene concentrations in tisues of patients with and without breast or colon cancer. Cancer (Philadelphia) (submitted for publication).


Chapter 3 EXPERIMENTAL APPROACHES TO THE STUDY OF @-CAROTENEMETABOLISM: POTENTIAL OF A 13C TRACER APPROACH TO MODELING @-CAROTENE KINETICS IN HUMANS JOY E. SWANSON Division of Nutritional Sciences Cornell University Ithaca, New York 14853

YEN-YI WANG No. 17 Li-Yuan Miao-Li 360 Taiwan, Republic of China

KEITH J. GOODMAN AND ROBERT S. PARKER Division of Nutritional Sciences Cornell University Ithaca, New York I4853

1. Introduction A. Biological Effects of p-Carotene in Humans B. Aspects of p-Carotene Metabolism Requiring Clarification C. Limitations of Animal Models in the Study @-CaroteneMetabolism D. Use of Nontracer Venus Tracer Methods for Modeling Purposes in Humans 11. Methods A. Dose Preparation and Administration B. Plasma Collection, Storage, and Extraction C. Plasma Analyte Quantification D. Sample Preparation for Gas Chromatography-Combustion-Isotope Ratio Mass Spectrometry (GC-C-IRMS) E. GC-C-IRMS Analysis 55 Copyright Q 1996 by Academic Press, Inc. All righu of reproduction in any form IeSeNed.

56

PART I VITAMIN METABOLISM I l l . Results and Discussion A. Examples of ['3C]/3C-GC-C-IRMSData B. Utility and Potential Advantages of the ('3C]/3CTracer Approach C. Additional Considerations of the ['3C]pCApproach IV. Conclusions References

I. INTRODUCTION

The purpose of this chapter is to briefly review the biological relevance of @carotene (PC) to human health, previous approaches to model or study quantitative aspects of PC absorption and metabolism, and the potential merits of a I3C-based stable tracer approach recently developed at Cornell and how it might be useful for PC modeling in humans. A preliminary report of this approach has been published (Parker et al., 1993). While the provitamin A activity and therapeutic usage of &carotene in the treatment of photosensitivity disorders is well documented, a therapeutic or preventative role of PC in cancer, cardiovascular disease (CVD), or other degenerative diseases, as well as a modulator of immune function and oxidative balance, is speculative and currently an active area of investigation. Several fundamental aspects of p C metabolism in humans are poorly understood. These include, quantitative information on the absolute absorption efficiency of PC, the magnitude of effect of factors which influence BC absorption or biotransformation of PC to vitamin A, kinetics of PC in various lipoprotein fractions, tissue sites of PC storage and biotransformation, and quantitative estimates of rates of specific tissue or total body elimination of PC. This lack of detailed knowledge of PC metabolism makes it more difficult to design and interpret appropriate clinical trials to test the role of PC in human health, predict the efficacy of dietary sources of fiC in ameliorating vitamin A deficiency, and propose plausible mechanisms for reported biological effects of /3C in humans and animals. It is speculated that the causative agent in many of the reported biological effects is intact PC rather than a retinoid metabolite, however the studies conducted to date cannot confirm this hypothesis. The development and application of mathematical and compartmental kinetic models, using stable isotope tracer techniques, offers the advantages of safely studying PC metabolism in humans, of providing predictive information regarding P C biokinetics, and possibly of aiding in the evaluation of the role(s) of PC in various biological responses.

CHAPTER 3 THE STUDY OF P-CAROTENE METABOLISM

57

A. BIOLOGICAL EFFECTS OF P-CAROTENE IN HUMANS

1. Cancer Numerous epidemiological studies have consistently shown and continue to show that consumption of diets high in PC or high in PC-rich fruits and vegetables is correlated with a lower risk of developing some types of cancers (Ziegler, 1989; van Poppel, 1993; Gerster, 1993). Cancer clinical trials, which have been completed in exclusively high-risk subjects, indicate the j3C is probably not a powerful therapeutic agent late in the process of carcinogenesis (Greenberg et al., 1990, 1994; Heinonen et al., 1994). Lung cancer risk may actually be increased in smokers receiving PC. Because the effects of smoking on PC metabolism are largely unknown, a mechanism for the observed increased risk is difficult to propose. If this effect is actually due to PC, a follow-up after cessation of supplementation should show a return of risk to that of the non-PC group. Information on the kinetics of total body PC elimination would be most helpful in aiding the design of such a follow-up. However, there have been many reports showing the efficacy of supplemental PC in reversing and reducing the incidence of oral leukoplakia, a preneoplastic lesion associated with oral cancer, in high-risk subjects (Stich et al., 1988; Garewal et al., 1990, 1993). In addition, recent observations also suggest PC may reduce the risk of gastric cancer via an interaction with ornithine decarboxylase activity in the stomach mucosa (Bukin et al., 1993, 1995). These latter findings suggest that early intervention with PC may yield greater health benefits than interventions occurring later in the progression of a disease. 2. Cardiovascular Disease A preliminary study of Gaziano and co-workers (1990) reported a 44% reduction in all major coronary events and a 49% reduction in all major vascular events in subjects receiving 30 mg of PC every other day for 60 months compared to a placebo group. This preliminary report has created speculation that PC may have mitigating effects in CVD. However, studies investigating several possible biochemical mechanism of action for PC in reducing CVD have produced inconsistent results. Numerous small-scale clinical trials have found an increase in serum HDL-cholesterol levels following PC supplementation (Mathews-Roth and Gulbrandsen, 1974; Bencich et al., 1989; Gaffney ef al., 1990; Manago et al., 1992), while others have reported no such effect (Ringer et al., 1991; Allard et al., 1994). PC

58


has also been shown to have little protective effect on the formation of oxidized LDL, a risk factor in CVD (Princen er al., 1992; Reaven er aL, 1993). Consequently, the role of PC as a preventative or curative agent in CVD remains speculative. 3. Modulation of Immune Function and Antioxidant Status

Currently, the available evidence for PC as a modulator of immune function and oxidative balance is also inconsistent. Supplemental PC has been shown to ameliorate ultraviolet light-induced suppression of cellular immune response as measured by delayed-type hypersensitivity tests in both young and older men (Fuller er al., 1992; Herraiz et al., 1994), alter population characteristics of immune cells by increasing the number of Thelper cells, natural killer cells, andlor cells with activation markers in healthy subjects (Alexander et al., 1985; Watson et al., 1994; Prabhala et al., 1991), and inhibit the respiratory burst reaction in granulocytes isolated from human subjects (Clausen, 1992). However, other investigators using similar PC doses have found no effect of supplemental /3C on lymphocyte subpopulations (Ringer et al., 1991; van Poppel ef al., 1993) or neutrophil superoxide formation (Mobarhan et al., 1990). While PC has been proposed to perform an antioxidant function, analogous to that of a-tocopherol, its effects on indicators of antioxidant status have little relationship to PC dose. For example, the magnitude of reduction in serum thiobarbituric acidreactive substances (Mobarhan et al., 1990) and breath pentane (Gottlieb et al., 1993) was essentially equivalent between subjects receiving 120 mg/day and those receiving 15 mg/day, even though an 8-fold increase in dose and 2.6-fold greater serum PC enrichment occurred in the group receiving 120 mg/day. Lack of information on PC metabolism over these wide dose differences or the effect of repeated elevated doses on /3C metabolism and biokinetics hampers interpretation of these results. B. ASPECTS O F @CAROTENE METABOLISM REQUIRING CLARIFICATION 1. Absorption and Biotransformation

Quantitative information on the absolute absorption efficiency of PC in humans is sparse. The effect of matrix, magnitude of dose, or coingested foods on absorption efficiency are poorly understood, if at all. Matrices such as naturally occurring complexes in plant foods, or pharmaceutical matrices such as water disperable beadlets, crystalline powders, or oil suspensions of PC, are likely to have a large impact on absorption efficiency.

CHAPTER 3 THE STUDY OF &CAROTENE METABOLISM

59

Typical dietary intakes of j3C range from 1to 6 mg from foods to pharmacological supplementations of 30 to 300 mg and encompass a range over which absorption efficiency probably varies considerably. Absorption efficiency, to the extent reflected by plasma PC levels, has been shown to be affected by several factors. PC absorption has been shown to be facilitated by coconsumption of fat andlor a meal (Dimitrov et al., 1988; Prince and Frisoli, 1993). Rock and Swendseid (1992) reported lower plasma j3C levels in subjects consuming 12 g of citrus pectin with 25 mg of j3C compared to subjects receiving the same PC dose without pectin. Recently, Weststrate and van het Hof (1995) reported a reduction in plasma j3C and lycopene concentrations in healthy subjects after consumption of a relatively high dose (12.4 g/day) and a lower dose (3 g/day) of sucrose polyester, a synthetic fat substitute, relative to controls. In addition, there are numerous reports showing the extreme interindividual variation in plasma response following PC supplementation (Johnson and Russell, 1992;Dimitrov et al., 1988; Brown et al., 1989).Dietary factors, such as fat, fiber, or fat substitute content, or physiological factors, such as bile salt composition or secretion, mixed micelle formation, or general health, may have interactive and independent roles in the observed variation in plasma j3C enrichment levels. The above factors and possibly other factors which are currently unknown may contribute to this variation through involvement in mechanisms which occur during absorption and intestinal conversion of j3C to vitamin A. However, the relative importance of each factor to the observed variation in final plasma PC levels achieved following supplementation remains to be determined. At present the only experimental studies concerning efficiency of absorption and intestinal bioconversion of j3C to vitamin A in humans stems from two reports from the mid 1960s using radiolabeled j3C. These studies were conducted in elderly hospitalized cancer patients with cannulated thoracic lymph ducts using 14C-or 3H-j3C and reported that the amount of intact j3C absorbed varied considerably (9 to 30%of dose) and that retinyl esters were the major lymphatic product of PC metabolism, representing 61 to 88% of recovered 3H or 14C(Goodman et aZ., 1966;Blomstrand and Werner, 1967).However, controversy regarding postabsorptive bioconversion of j3C to vitamin A and retinoic acid exists because of a lack of appropriate techniques to study such a process. While in vitro animal tissue studies strongly indicate central cleavage of j3C to two molecules of retinal as the primary mechanism of conversion of j3-carotene to vitamin A, the stoichiometry of this reaction in humans remains unconfirmed. Zeng and co-workers (1992) employed synthetic carotenoid analogs to evaluate carotenoid metabolism in humans. The advantage of this approach is that low doses can be used because such carotenoids are not endogenous

60


to human plasma, unlike PC and other carotenoids of dietary origin. EthylP-apo-8'-carotenoate, 4,4'-dimethoxy-P-carotene, and P-apo-8'-carotenal (100 pmol) dissolved in peanut oil were administered to healthy human subjects as part of a light breakfast. Serial venous blood samples were drawn and concentration versus time curves of the carotenoid analogs or their metabolites were obtained by HPLC. The data suggested that these carotenoids differed with respect to sites and mechanisms of biotransformation, rates of absorption and elimination, and maximum serum concentration and time to maximum concentration. Mathematical modeling was utilized to determine absorption rate by area under curve (AUC) values and elimination rate using estimated mean sojourn time. A 270-fold difference in the rate of absorption between carotenoids was reported, with ethyl-@-apo8'-carotenoate being the fastest (1130 pmol - hr-l - liter-'), 4,4'-dimethoxyP-carotene intermediate (159 pmol * hr-' - liter-') and the P-apo-8'-carotenal metabolite,P-apo-8'-carotenylpalmitate, the slowest (4.2 pmol hr-' - liter'). Of the three analogs, only P-apo-8'-carotenal exhibited significant biotransformation to retinoid metabolites. However, the mechanism for conversion could not be determined. The large differences in plasma kinetic behavior between the carotenoid analogs employed in this study suggest structure-specific mechanisms of elimination and emphasize the need to apply stable isotope techniques to the study of uptake and metabolism of naturally occurring carotenoids. Questions regarding the extent of postabsorptive bioconversion of pC to vitamin A persist. Animal data indicate the liver possesses this capability, but the relative importance of intestinal mucosa versus liver is unknown. Novotny and co-workers (1995) reported a compartmental model which predicted that both liver and intestinal mucosa were important sites for biotransformation of PC in the human, with 43% of total conversion occurring in the liver and 56% in the intestinal mucosa. However, the model assumed a stoichiometry of 1 mol retinol per mole PC, and the effect on the model assuming a 2 :1 ratio was not discussed. +

2. Transport and Storage

The carotenoid composition of various lipoprotein classes has been well described, as have the changes in PC concentration in such classes following large single oral doses of PC (Johnson and Russell, 1992; Traber et al., 1994). However, other aspects of plasma carotenoid transport are poorly understood. These include: (1) the rates of turnover of PC in LDL and HDL, (2) the physiological significance of HDL-PC (reverse transport?), (3) the extent and rate of exchange between lipoproteins of PC in comparison with other carotenoids, and (4) the extent to which recycling of plasma PC into and out of liver, adipose, or other tissues occurs.

CHAPTER 3 THE STUDY OF 8-CAROTENE METABOLISM

61

/3C supplementation studies in humans have shown sustained elevated levels for several days to weeks following PC cessation (Dimitrov er al., 1988; Johnson and Russell, 1992). These observations suggest considerable recycling of PC between extrahepatic tissues and plasma. Reports of PC concentration in human tissues indicate that adipose tissue is a major storage site (Dagadu, 1967; Parker, 1988). Parker (1988) reported an average adipose PC concentration of 0.62 pg/g in adults. This concentration would yield a total body adipose PC content of about 11 mg for a 75-kg adult of 24% body fat. Also, an average plasma concentration of 0.3 pM would yield an average total plasma pool size of roughly 0.5 mg. Thus oral doses of 15 to 100 mg, commonly employed in the study of /3Cmetabolism, would clearly perturb steady-state kinetics in these two body pools. Using smaller oral doses of carotenoid analogs, Zeng et al. (1992) reported a 50-fold difference in serum concentration between the carotenoid analogs 4,4’-dimethoxy-/?-carotene and ethyl-P-apo-8’-carotenoate and the metabolite of P-apo-8’-carotenal, P-apo-8’-carotenyl palmitate. It was also shown that the time post dosing to reach this maximum concentration range between 5.5 and 27.1 hr. This variation suggests marked differences in transport or volume of distribution between carotenoids of different structures.

3. Elimination Few estimates of the rates of elimination of carotenoids from the bloodstream in the human have been published. Zeng and associates (1992) estimated mean sojourn time of the three carotenoid analogs using the simulation, analysis, and modeling (SAAM) computer program to be 144 hr for 4,4’-dimethoxy-&carotene, 209 hr for ethyl-/3-apo-8’-carotenoate and for the P-apo-8’-carotenal metabolites, 124 hr for P-apo-8’-carotenol and 43 hr for P-apo-8’-carotenyl palmitate. These values, with the exception of that for P-apo-8’-carotenyl palmitate, are much longer than the mean residence time of 51 hr estimated for d8+-carotene by Novotny etal. (1995) using SAAM. To date, the report of Notovny er af. (1995) represents the only attempt to quantitatively estimate kinetic parameters of PC metabolism such as mean transit time, mean residence time, fractional catabolic rate, or rate of total body elimination. Moreover, the effects of physiological conditions such as pregnancy, lactation, malnutrition, infection, inflammation, or chronic illness on PC kinetics have yet to be explored.

C. LIMITATIONS OF ANIMAL MODELS IN THE STUDY &CAROTENE METABOLISM Animals used for studies of pC metabolism include the rat, mouse, hamster, guinea pig, chicken, ferret, preruminant calf, monkey, and baboon.

62


There are substantial and well-characterized differences between species in PC metabolism. Rodents, including the rat, mouse, and guinea pig, exhibit highly efficient intestinal conversion of @C to vitamin A with the consequence that litttle PC is absorbed intact (Thompson etaf.,1950;Huang and Goodman, 1965). Moreover, VLDL and HDLs are the major lipoproteins involved in the transport of lipids in most rodents, whereas in the human LDL is the predominant lipid carrier, especially of PC. Studies with ferrets reveal a very unique handling of vitamin A by this animal such that serum retinyl ester levels are greatly elevated relative to the human even in the fasting state (Ribaya-Mercado et af., 1992; Wang et al., 1992). It was clearly shown in the PC radiotracer studies that humans are moderate absorbers of intact PC and bioconvert flC primarily to retinyl esters without elevated serum levels of retinyl esters (Goodman et al., 1966; Blomstrand and Werner, 1967). In addition, Gugger et al. (1992) reported the complete elimination of PC from ferret plasma within 76 hr of a single oral dose of 10 mg PCkg body wt. This result is at variance with many human studies which show elevated serum pC for several weeks following a single oral dose (Novotny ef al., 1995; van Vliet et al., 1995). Rabbits are also “white fat” species, in that carotenoid deposition in adipose tissue is well below that observed in humans and insufficient to result in the typical yellow hue of human fat. At the present time an animal model which parallels the human with respect to known aspects of PC absorption, metabolism, transport, and tissue incorporation has not been developed. Consequently, extrapolation of data derived from animal models to humans is fraught with uncertainity and limitations. Animal models do offer the advantage of the capability to obtain data on solid tissue kinetics of PC assimilation and elimination. For obvious reasons this cannot be done in humans, in which under most circumstances only plasma, blood cells, and adipose tissue can be repeatedly sampled from the same subject. D. USE OF NONTRACER VERSUS TRACER METHODS FOR MODELING PURPOSES IN HUMANS

I. Nontracer Methods Studies involving unlabeled PC depend on detection and measurement of concentration changes of PC in plasma or plasma fractions such as chylomicrons. While chylomicron response may be useful for the study of issues related to PC absorption, this fraction is not useful for examination of postabsorptive events. A carefully designed study by van Vliet er al. (1995) compared the utility of a chylomicron (TGR) fraction with that of


63

plasma to follow the absorption of a single oral dose of PC. The TGR fraction exhibited a well-defined peak in PC concentration at 5 hr postdose, coinciding with the absorption of triglyceride, while plasma PC concentration did not change significantly over this period. The dose was 15 mg PC, of which 11%was estimated to be absorbed, using several assumptions. Thus, the effective (absorbed) dose in this model may have been about 1.6 mg. The insensitivity of plasma PC to single doses of PC was further illustrated by the results of Johnson and Russell (19922). These authors reported that 7 of 11subjects showed no increase in plasma PC concentration following a single oral dose of 120 mg PC. In contrast to the findings of van Vliet et al. (1995), who reported chylomicron PC responses in all subjects studied, these 7 subjects exhibited no significant PC enrichment in the chylomicron fraction, suggestingimpairment of absorption or efficient conversion to retinyl esters in these subjects. While effective doses of 1 to 2 mg seem small, the total body pool of pC can be estimated at 15 to 20 mg, with a total plasma pool of about 0.5 to 1 mg. In comparison with these pool sizes, a 2-mg effective dose is relatively large. In addition, the average daily effective dose of PC from food sources is likely to be less than 1 mg, since matrix effects probably impair bioavailability of PC to a larger extent than with supplements such as that used by van Vliet et al. (1995). The minimum effective dose required to yield a significant TGR fraction response, taking into account the error associated with measurement, has not been determined, but is not likely to be much less than 1mg. Consequently, use of unlabeled PC, even coupled with use of the TGR fraction, is probably not well suited to study PC uptake and metabolism at effective doses typically derived from dietary sources. 2. Tracer Methods

A tracer method by definition is one which utilizes a trace or minute amount of labeled material, a sensitive and precise detection device, an administration protocol which ensures physiological and metabolic processing similar to that of the tracee, and does not perturb the mass or underlying kinetics of the tracee (Green and Green, 1990, Wolfe, 1992). An ideal tracer is chemically and physically identical to the tracee, but distinguishable from the tracee via substitution of one or more atoms within its structure. The location of the substituted atom(s) should minimize atom exchange and physiological discrimination of the tracer from the tracee. pC tracers are potentially useful in the study of many aspects of PC metabolism including absorption, biotransformation to retinoids, transport, storage, and elimination. There are four methods of labeling PC,

64


two radioactive forms (3H and 14C) and two stable isotope forms ('H and I3C). Goodman et al. (1966) employed 3H-/3C and Blomstrand and Werner (1967) utilized both 3H- and I4C$3C to follow the steady-state kinetics of PC over a 22-hr time period. These two radiotracer studies utilized approximately 0.05 mg of 3H-f3C and 0.4 to 1.3 mg of 14C-@C (Goodman ef al., 1966; Blomstrand and Werner, 1967). The effective dose, assuming 30% of dose was absorbed, for 3H-PC is about 15 p g and about 390 pg for 14C-f3C.Both values are upper end estimates and are well below the total body PC pool size (16 to 20 mg), average plasma pool (0.5 mg), and adipose storage site (11 mg). The use of radioactive isotopes also involves health risk to the subjects and possible isotopic effects with 3H have been reported (Bell ef al., 1986; Argound et al., 1987). Stable isotopes, on the other hand, have no known physiological or health risks when given in small doses (amounts that do not perturb mass of tracee) and have been shown not to induce isotopic effects on chemical reaction. Under acidic conditions, deuterium is known to exhibit very large dissociation constants, which must be taken into consideration when using this stable isotope. Two approaches using stable tracers to study PC metabolism in humans have been reported (Parker ef al., 1993; Dueker et al., 1994). At Cornell, we have developed a "C-based stable tracer approach utilizing biosynthetic PC completely substituted with I3C (per-labeled PC) and isotope ratio mass spectrometry. This approach is described below. Dueker er a/. (1994) have used synthetic d&C and conventional tandem mass spectrometry, and a compartmental model based on data derived from a single subject has been proposed (Novotny et al., 1995). These data were obtained following a single dose of 40 mg @C,of which the model predicted 22% (about 9 mg) was absorbed. While this effective dose may represent up to one-half the estimated total body pool of PC, Novotny ef af. (1995) argued that steady state kinetics prevailed throughout the period of observation. A stable tracer method which couples the use of PC highly enriched in 13C with high-precision isotope ratio mass spectrometry offers several advantages over approaches requiring conventional mass spectrometry. The most salient feature is the ability to use low doses representative of true tracer conditions (e.g., 0.015 to 1.0 mg in the case of ['3C]flC). Because small changes in carbon isotope ratio can be detected with high precision of measurement, long-term plasma kinetics (including terminal half-life estimates) can be obtained even for analytes with slow rates of elimination (c.g., PC). These features are clearly advantageous for the collection of data for mathematical modeling.


65

It. METHODS A. DOSE PREPARATION AND ADMINISTRATION Unicellular green algae was grown in a closed growth chamber with 13C02as the sole carbon source by Martek, Inc. (Columbia, MD). This resulted in the biosynthesis of per-labeled (>98% 13C) ['3C]P-carotene, as determined by electron ionization mass spectrometry and by isotope ratio mass spectrometry of the corresponding perhydro-P-carotene analog following serial dilution with unlabeled PC. All-tr~ns-['~C]/3C was purified from a crude hexane extract of algal lipids by repeated crystallization from petroleum ether. The purified [13C]j3C used in subsequent experiments consisted of 95 to 97% all-rruns-j3Cwith a-carotene making up most of the remaining 5 to 3 %. One milligram of ["C]PC was completely dissolved in 1 g of high oleic acid safflower oil (HOASO; Stepan Co., Maywood, NJ) and emulsified into 70 ml of non-vitamin-fortified skim milk, 30 g mashed banana, and an additional 19 g of HOASO using a hand held homogenizer. The banana was added for emulsion stability and taste. Male subjects ranging in age from 27 to 41 years were placed on a lowcarotenoid diet 48 hr prior to the [13C]PCdose to allow for the clearance of gastric and intestinal carotenoids. On the morning of the dose, subjects were fitted with an indwelling catheter with a three-way stopcock in a forearm vein. A baseline blood sample was taken, followed by the consumption of the ['3C]/3C-containing banana milk drink. A standardized light breakfast (one-half bagel) was provided with an additional 100 ml of nonvitamin-fortified skim milk in order to rinse both the glass which contained the dose and the subject's mouth. The low-carotenoid diet was continued 24 hr post dose. A standard lunch and evening meal were consumed 3 and 9 hr post dose, respectively. Subsequent blood samples were taken at the times shown in Figs. 2-5. B. PLASMA COLLECTION, STORAGE, AND EXTRACTION Blood collected at each time point was allowed to remain on ice for 15 min prior to centrifugation at 1800g. Plasma was transferred to 5-ml cryogenic vials and stored at -80°C until analyzed. Plasma lipids were extracted from duplicate 2.2-g plasma aliquots for isotope ratio analysis or from 0.25-g aliquots for HPLC quantification of retinol and j3C after addition of internal standard (retinyl acetate). Plasma aliquots were deproteinized with 1 vol of ethanol and lipids extracted with 3 vol of hexane (Optima Grade, Fisher Scientific, Rochester, NY).

66


C. PLASMA ANALYTE QUANTIFICATION The method of Thurnham etal. (1988) was modified for the quantification of plasma PC and retinol. Plasma extracts were dissolved in 40 pl of dimethylforamide and vortexed and then 210 pl of acetonitrile/methanol/chloroform (47/47/6,v/v/v) was added. Reconstituted samples were vortexed and sonicated for 40 sec prior to being transferred to autosampler vials and sealed under nitrogen. The HPLC system consisted of a photodiode array detector (Waters 996, Millipore Corp., Milford, MA) with Millennium software, a Waters 717 plus autosampler, and a Hewlett-Packard Model 1050 pump. Analytes of interest were separated using acetonitrile/methanol/ chloroform (47/47/6, v/v/v), with 0.05 M of ammonium acetate and 1% triethylamine at a flow rate of 1.2 mumin and a 4.6 X 15-cm Spherisorb ODs-2 column (LKB Instruments Ltd., Surrey, UK) maintained at 26°C using a column heater (Timberline Instruments Ltd., Boulder, CO). This analysis does not discriminate between I3C-enrichedand nonenriched analytes, but rather measures the total concentration of each isotopomer. The retention times of retinol, retinyl acetate (internal standard), and &carotene were 2.1,2.6, and 16.9 min, respectively. Plasma concentrations of retinol and PC were calculated using a standard curve for each analyte and an internal standard to correct for volume recovery.

D. SAMPLE PREPARATION FOR GAS CHROMATOGRAPHY-COMBUSTION-ISOTOPE RATIO MASS SPECTROMETRY (GC-C-IRMS) Unesterified retinol was separated from retinyl esters and PC using reverse-phase semipreparative HPLC (Vydac TP201 column, 10 mm X 25 cm,Separations Group, Hesperia, CA) using methanol/dichloromethane (76/24, v/v) as the mobile phase at a flow rate of 1.2 ml/min. Eluant was monitored at 325 nm and two fractions were collected: (1) retinol, retention time of 9.7 min. and (2) retinyl esters plus carotenes, collection interval was 16.5 to 18 min. The retinyl ester-carotene fraction was saponified with absolute ethanol and saturated aqueous potassium hydroxide at 45°C for 25 min and the resulting retinol and PC were extracted with hexane. Retinyl ester-retinol and PC were separated by liquid-liquid partitioning using hexane and dimethylformamide (DMF) (2/5, v/v). The PC partitioned into hexane (>95%) and the retinol into DMF (>95%). Further purification of all-truns-PC from other isomers was performed using analytical reversephase HPLC (Vydac TP201, 4.6 mm X 15 cm, Separations Group) and methanol/dichloromethane (95/5, v/v) at 0.9 ml/min. The purified all-rrans-


67

PC fraction was dissolved in DCM and hydrogenated to its thermally stable perhydro-/3Canalog using platinum oxide (Aldrich Chemical Co., Milwaukee, WI) under hydrogen gas overnight at 65°C. The hydrogenated PC samples were filtered to remove the platinum oxide, redissolved in hexane, and subjected to isotope ratio analysis as described below.

E. GC-C-IRMS ANALYSIS Methodologic issues concerning use of GC-C-IRMS have been discussed by Goodman and Brenna (1992). The carbon isotopic composition of perhydro-/3C was determined using a 5880A Hewlett-Packard GC interfaced to a Finnigan MAT 252 high-precision GIRMS via a ceramic combustion furnace maintained at 850°C. Perhydro-/3C was injected onto a DB-1 capillary column (0.32 pm id. X 15 m, J&W Scientific, Folsom, CA) using cool on-column injection. The linear velocity of the carrier gas (helium) was 20 cm/sec and the GC programmed from 60 to 265°C at 25"C/min, from 265 to 300°C at 10"C/min, and held at 300°C for 4 min and then increased to 325°C at 30"C/min and held for 10 min to ensure elution of any remaining compounds. Perhydro-PC eluted at about 13min (approximately 303°C). The column eluant was continuously and quantitatively combusted to C02, which was swept into the ionization chamber. In this particular application the masses 44 ("C02) and 45 (13C02) are of most interest and were continuously monitored. The mass 46, used to calibrate for 1 7 0 contamination in the mass 45 channel, has negligible effect on 13C/'2C carbon isotope ratios for the experimental data that is presented below; however, it was also continuously monitored. The computer-generated delta units (S13C), reflecting the 13C/'2Cisotope ratio of the sample relative to an international carbonate standard (PeeDee Belemnite, PDB), were converted to atom percentage 13C using the Atom % 13C =

(100 X R45) X ( 613C/1000 + 1) 1 + (R45) X (S'3C/1000 + 1)

R45 represents the ratio of the signal intensities in the m/z 45 channel to the m/z 44 channel for the PDB C 0 2standard and is defined as 0.0112372. The atom percentage excess (APE) 13C in the perhydro-PC peak was obtained by subtracting the baseline atom percentage 13Cfrom that of all subsequent time points. Thus, APE represents the proportion of plasma PC which is labeled at any point in time and takes into account natural abundance13Cor 13Cenrichment persisting from previous doses of [13C]/3C in a given subject.

68


RESULTS AND DISCUSSION

111.

A. EXAMPLES O F [I3C]/3C-GC-C-IRMS DATA The following figures include selected data to illustrate the types of kinetic curves that can be generated using this approach and which may subsequently be subjected to empirical or compartmental modeling. The qualitative changes in I3Cenrichment in plasma PC (APE) over the initial 50 hr and over the entire course of data collection, after dosing with 1 mg '3C-labeled PC, are shown in Figs. 1 and 2. The reproducibility of APE measurement and the purification procedure is illustrated by the fact that error bars, representing isotope ratio measurement of two independent plasma samples, are usually within the figure symbols. Two general features common to all I3Cenrichment versus time curves are the peak in enrichment at 5 hr, and a second, broader peak between 24 and 48 hr. These two peaks of 13C appearance and elimination represent the movement of labeled PC into and out of the lipoproteins that are known to be involved in the transport of PC during absorption and distribution. Earlier studies using large doses of unlabeled PC and lipoprotein separation by ultracentrifuga-

15 I

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26

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52

Hours FIG. 1 . A comparison of the short-term plasma kinetic patterns of atom 96 excess (APE) '"C: in plasma gC fractions versus time, from 0 to 51 hr, in three subjects following an oral dose of 1 mg ['3C]@C.Each symbol represents the following: subject 1 (+), subject 2 (A), subject 3 (0).Each point represents the mean 2 SD of two independent determinations.


69

0 100200300400500600700800900

Hours FIG. 2. A comparison of the long-term plasma kinetic patterns of atom % excess (APE) I3Cin plasma fiC fractions versus time in three subjects followingan oral dose of 1 rng [13C]fiC. Each symbol represents the following:subject 1 (+), subject 2 (A), subject 3 (0).Each point represent the mean % SD of two independent determinations.

tion indicate that the 5-hr peak represents the secretion and clearance of chylomicron-associated [‘3C]PC (Krinsky et al., 1958; Cornwell et al., 1962; Johnson and Russell, 1992; van Vliet et al., 1995). P-Carotene appears to be absorbed exclusively via this route. This 5-hr peak also coincides exactly with the peak in [‘3C]retinyl esters (data not shown) or with unlabeled retinyl ester in the case of dosing with unlabeled PC (van W e t et al., 1995). Retinyl esters are a commonly used marker for chylomicrons. The second broad peak between 24 and 48 hr represents labeled PC secreted by the liver associated with very low density lipoproteins (VLDL) and very likely encompasses the period during which these VLDL particles undergo lipolysis to IDL and LDL. As this lipolyticprocess occurs relatively rapidly, the broad and extented nature of this second peak probably reflects recycling of labeled PC into and out of the liver, i.e., hepatic reprocessing of VLDL or LDL particles. This phenomenon may be investigated further using modeling approaches and is reflected in the compartmental model recently proposed by Novotny et al. (1995). The “absorption” peak (5 hr) and “distribution” peak (2448 hr) are incompletely separated, indicating that hepatic secretion occurs shortly after hepatic uptake of chylomicron remnants containing labeled PC. This

70


overlap of absorption and distribution processes has also been evident in studies using high doses of unlabeled PC. In subjects absorbing only small amounts of PC (labeled or unlabeled), the 5-hr peak may represent only a shoulder on the leading side of the distribution peak. In such instances, use of absorption phase AUC or similar approaches to estimate absorption efficiency would be of limited use. This further underscores the need for development of mathematical models that would be more independent of observation of a distinct absorption peak. B. UTILITY AND POTENTIAL ADVANTAGES OF THE [13C]PC TRACER APPROACH

In general, relatively large single oral doses are needed to produce measurable increases in plasma PC concentration, during either absorption or distribution phases (Dimitrov er al., 1987). A single dose of 1 mg would not be expected to result in such peaks. In fact, no such peaks in plasma total PC corresponding with absorption or distribution were observed in these studies, as illustrated by the plots shown in Fig. 3. The lack of an absorption peak at doses of 1 mg or below probably reflects both the

0

0

8 162432404856647280

Hours PIG. 3. Total plasma &carotene (labeled plus unlabeled f3C) concentration, from 0 to 75 hr, in three subjects followingan oral dose of 1 rng ['3C]/3-carotene.Each symbol represents the following: subject 1 (+), subject 2 (A), subject 3 (0).Each point represents the mean 2 SD of two independent determinations.


71

duration of chylomicron secretion of lymph into plasma (2 to 3 hr) and the rapid rate of removal of chylomicrons from plasma by the liver. The halflife of chylomicrons in the human has been estimated to be roughly 11min (Berr el af.,1985; Cortner e? at., 1987; Redgrave e? af., 1993). The plots in Fig. 3 also indicate that the changes in I3Cenrichment in plasma lipoproteins seen with 1-mg doses occur under steady-state conditions. That is, labeled PC is replacing unlabeled PC in the absence of a measurable change in total PC concentration. Such conditions are desirable from a modeling perspective, as discussed further below. Conventional mass spectrometry generally requires use of larger PC doses, as reported by Novotny et af., 1995, in order to observe details of absorption kinetics. 13C-LabeledPC is clearly evident in plasma by 3 hr after a dose of 1 mg [13C]PC,illustrating the enhanced sensitivity of GC-C-IRMS to detect and measure PC early in the process of absorption. In contrast, no measurable increase in concentration of labeled PC was observed prior to 5 hr post dosing with 40 mg of d&C (Novotny et al., 1995). The approach described here is clearly suitable for obtaining data on the long-term kinetics of plasma PC elimination in humans as well. Figures 2 (linear plot) and 4 (log plot) illustrate data obtained over 800 hr following a dose of 1 mg [13C] PC. Even at such prolonged times, changes in carbon isotope ratio in the plasma PC pool can be observed when measuring at 100-hr intervals. Such capability is needed in order estimate the terminal half-life of plasma components with slow turnover rates, such as PC or other lipids transported in the core of LDL particles. In general, an observation period spanning at least five half-lives is desirable for such estimations. In the examples shown here, PC elimination appears roughly biphasic over the period from 48 hr through the end of data collection, with a relatively rapid phase (through approximately 150 hr) followed by a slower phase. More substantive analyses will be required to yield a more detailed description of these kinetics, and such efforts are underway. Published reports of the rate of disappearance of plasma PC following its removal from the diet also suggest a biphasic elimination curve (e.g., Rock et af., 1992), but no efforts have been made to model such data, which suffer from the drawback of changing pool size. In our model, subjects are allowed to resume self-selected diets after 36 hr postdose. This may result in changes in plasma PC concentrations as a function of occasional large fluctuations in dietary intake of several milligrams. Since APE represents the proportion of plasma PC molecules that are labeled with 13C,addition of unlabeled PC to the plasma pool from dietary sources will decrease APE even though the concentration of labeled PC remains unaffected. In such instances, the data can be expressed as the plasma concentration of labeled PC (e.g., nmoliliter) by multiplying APE

72


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HOUrS FIG. 4. Presentation of the terminal elimination kinetics of atom 9% excess (APE) 13C in plasma gC fractions versus time in three subjects following an oral dose of 1 mg ['3C]flCas semi-log plots. Each symbol represents the following: subject 1 (+), subject 2 (A), subject 3 (0).Each point represents the mean 2 SD of two independent determinations.

by the plasma total PC concentration (labeled plus unlabeled, as determined by HPLC) and correcting for the extent of I3C labeling in the PC dose. A comparison of ['3C]pC data expressed as APE with that expressed as plasma concentration (nmol [13C]pC/liter)is shown in Fig. 5. The stable tracer approach described above offers several advantages over other tracer techniques employed for the study of PC metabolism in humans. First, the health risks involved with the use of radioactive tracer methods are completely avoided. The radiotracer studies conducted in the mid 1960s (Goodman er al, 1966; Blomstrand and Werner, 1967) both involved hospitalized patients, many with serious illnesses such as cancer. In such cases, the added risk associated with small doses of tritium or I4C are small in comparison with existing deleterious conditions. However, in otherwise healthy persons, such an argument cannot be made. It is often assumed that the sensitivity associated with radiotracer techniques far exceeds that of stable tracer approaches. While in some cases, depending on the specific instrumental analyses employed, that may be true. In this case, however, the doses of PC needed to obtain high-quality data are of the same order as those of radiotracer methods. The dose used to generate the

CHAPTER 3 THE STUDY OF @CAROTENE METABOLISM

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1, ,

,

O o

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,

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FIG. 5. A comparison of the terminal elimination kinetics of plasma PC expressed as atom open circles) or as nmoles per liter (filled circles) following a single oral dose of 1 mg ["CIPC. Symbols represents the mean values for subject 3. % excess (APE;

data shown here (1 mg) is of the same order (or less) than those of the I4C studies of Blomstrand and Werner (1967). In fact, doses of l3C-PC considerably less than 1mg may be employed with the GC-C-IRMS method used here, depending of the extent of I3C enrichment of the dose. This is due both to the high precision of measurement of carbon isotope ratio offered by high-precision isotope ratio mass spectrometers and to the relatively low total body pool of PC estimated to be on the order of 1525 mg in most individuals (Parker, 1988). The detection limits and precision of GC-C-IRMS are orders of magnitude improved compared to conventional quadrupole mass spectrometry (Brenna, 1994). Fractions of tracer/ tracee below 0.002 generally cannot be accurately measured by conventional MS, while ratios of 0.00001 can be reliably quantified by high-precision IRMS (Goodman and Brenna, 1992; Brenna, 1994). Consequently, approaches involving conventional MS generally require higher doses of labeled PC. For example, the &-PC method of Dueker et d. (1994), using tandem MSMS, employed a dose of 40 mg labeled PC, of which 9 mg was estimated to have been absorbed (Novotny et al., 1995). The precision (and consequently sensitivity) of the GC-C-IRMS approach is manifested in the ability to measure small changes in isotope ratio in PC over long periods of time, as discussed above. In addition to the advantages of safety, sensitivity, and precision of measurement, the current approach is unlikely to be complicated by chemical exchange or isotope effects on the metabolism or disposition of PC in

74


biological systems. Deuterium is susceptible to exchange with hydrogen, depending on the acidity of the hydrogen atom involved, although the likelihood of such exchange occurring with p C is small. On the other hand, octadeuterated pC can be baseline-resolved from unlabeled p C by reversephase HPLC (Deuker er d.,1994),suggesting that interaction with lipophilic environments (e.g., cell membranes or hydrophobic regions of enzymes) may be affected by deuteration. In contrast, we have been unable to resolve ~er-'~C-labeled p C from natural abundance p C by HPLC, suggesting that the risk of isotope effects in vivo are very small. C. ADDITIONAL CONSIDERATIONS OF T H E ["C]pC APPROACH

The GC-C-IRMS approach employed here cannot be used without caution. The isotope ratio of eluting G C peaks can be substantially reduced by coeluting or partially resolved substances of different isotope ratio. Recently, Goodman and Brenna (1994) have published techniques using exponentially modified Gaussian and Haarhoff-VanderLinde functions to integrate overlapping fatty acid methyl ester peaks possessing different isotope ratios. Second, the relationship between delta"CpDB and fraction I3C (13C as proportion of total carbon) is not linear over the entire range of enrichment. Consequently, the actual relationship must be determined using appropriate calibration curves to account for any nonlinearity within the range of "C enrichment encountered. Last, measurement precision is limited by analyte concentation in sample. Therefore, analytes present in low concentration in plasma may require the collection of larger blood samples. Alternatively, curve-fitting algorithms may produce satisfactory precision and accuracy for samples of low signal intensity (Goodman and Brenna, 1995). IV. CONCLUSIONS

Stable tracer approaches to the study of pC metabolism in humans offer several advantages over the use of unlabeled or radioactive pC. p-Carotene highly enriched with I3C, coupled with the use of gas chromatographycombustion-high precision isotope ratio mass spectrometry, constitutes a safe and sensitive approach which requires only very small doses of p C typical of (or less than) daily dietary intake and which do not perturb endogenous pool sizes. Subtle changes in isotope ratio can be measured over long periods of time with a low risk of isotope effects, permitting estimation of terminal half-lives of pC or its metabolites in plasma. Conse-


75

quently, it appears that such an approach will be useful in obtaining data needed to model the kinetics of plasma PC and its retinoid metabolites in humans, providing insights into aspects of absorption and metabolism of PC which have been difficult to address through direct experimental means. ACKNOWLEDGMENTS J.E.S. and R.S.P. thank Bonnie Marmor and Amy Spielman for assisting with sample preparation and analysis, Dr. Wesley Canfield for technical advice and clinical assistance,Dr. J. T. Brenna for helpful comments and advice regarding this work and manuscript, Thomas Corso for adapting and calibrating the HPLC method used to quantitate total plasma pcarotene and retinol, and Dr. Richard Caimi for technical assistance with the Finnigan 252 IRMS. This research was supported by USDA Competitive Reseach Grant 92-37200-758, NIH-NIDDKD Grant DK 43729-01A1, and a grant from Hoffmann-LaRoche, Inc.

REFERENCES Alexander, M., Newmark, H., and Miller, R. G. (1985). Oral beta carotene can increase the number of OKT4+ cells in human blood. Immunol. Len. 9,221-224. Allard, J. P., Royall, D., Kurian, R., Muggli, R., and Jeejeebhoy, K. N. (1994). Effects of p-carotene supplementation on lipid peroxidation in humans. Am. J. Clin. Nutr. 59,884890. Argound, G. M., Shade, D. S., and Eaton, R.P. (1987). Underestimation of hepatic glucose production of radioactive and stable tracers. Am. J. Physiol. 252, E S E 6 1 2 . Bell, P. M., Firth, R. G., and Rizza, R. A. (1986). Assessment of insulin action in insulindependent diabetes mellitus using 6-'4C-glucose,3-3H-glucoseand 2-3H-glucose: Differences in the apparent pattern of insulin resistance depending on the isotope used. J. Clin. Invest. 78, 1479-1486. Bencich, J., Schmeisser, D., Bowen P., and Mobrahan, S. (1989). Beta-carotene increases HDL levels in young men fed a high polyunsaturated fat, very low cholesterol liquid formula diet. FASEB L 3, A955. [Abstract 42371. Berr, F., Eckel, F., and Kern, F., Jr. (1985). Plasma decay of chylomicron remnants is not affected by heparin-stimulated plasma lipolytic activity in normal fasting man. J. Lipid Res. 26,852-859. Blomstrand, R., and Werner, B. (1967). Studies on the intestinal absorption of radioactive p-carotene and vitamin A in man: Conversion of p-carotene into vitamin A. Scud. J. Clin. Lab. Invest. l9,33%345. Brenna, J. T. (1994). High-precision gas isotope ratio mass spectrometry: Recent advances in instrumentation and biomedical applications. Acc. Chem. Res. 27, 340-346. Brown, E. D., Micozzi, M. S., Craft, N. E., Bieri, J. G., Beecher, G., Edwards, B. K., Rose, A., Taylor, P. R., and Smith, J. C. (1989). Plasma carotenoids in normal men after a single ingestion of vegetables or purified pcarotene. Am. J. Clin.Nutr. 49, 1258-1265. Bukin, Yu. V., Zaridze, D. G., Draudin-Krylenko, V. A., Orlov, E. N., Sigacheva, N. A., Fu Dawei, F., Kurtzman, M. Ya., Schlenskaya, I. N., Gorbacheva, 0.N., Nechipai, A. M., Kuvshinov, Yu. P., Poddubny, B.K., and Maximovitch,D. M. (1993). Effect of p-carotene

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supplementation on the activity of ornithine decarboxylase (ODC) in stomach mucosa of patients with chronic atrophic gastritis. Eur. J. Cancer Prev. 2,61-68. Bukin, Yu.V.,Draudin-Krylenko, V. A.. Orlov, E. N., Kuvshinov, Yu.P., Poddubny, B. K., Vorobyeva, 0. V., and Shabanov, M. A. (1995). Effect of prolonged p-carotene or DL-a-tocopheryl acetate supplementation on ornithine decarboxylase activity in human atrophic stomach mucosa. Cancer Epidemiol. Biomarkers Prev. 4,865-870. Clausen, J. (1992). The influence of antioxidants on the enhanced respiratory burst reaction in smokers. A n n N.Y. Acad Sci 669,337-341. Comwell, D. G., Gruger, F. A., and Robinson, H. B. (1%2). Studies on the absorption of beta-carotene and the distribution of total carotenoid in human serum lipoproteins after oral administration. J. Lipid Res. 3, 65-70. Cortner, J. A., Coates, P. M., Le,N-A., Cryer, D. R., Ragni, M. C., Faulkner, A., and Langer, T. (1987). Kinetics of chylomicron remnant clearance in normal and in hyperlipoproteinemic subjects. 1. Lipid Res. 28, 195-206. Dagadu, J. M. (1967). Distribution of carotene and vitamin A in liver, pancreas, and body fat of Ghanaians. Br. 1. Nurr. 21,453-456. Dimitrov, N. V., Boone, C. W., Hay, M. B., Whetter, P., Pins, M., Kelloff, G. J., and Malone, W. (1987). Plasma beta-carotene levels-kinetic patterns during administration of various doses of beta-carotene. J. Nurr. Growth Cancer 3,227-237. Dimitrov, N . V., Meyer, C., Ullrey, D. E., Chenoweth, W., Michelakis, A., Malone, W., Boone, C., and Fink, C. (1988). Bioavailability of pcarotene in humans. Am. 1. Clin. Nurr. 48,298-304. Dueker, S. R., Jones, A. D., Smith, G. M., and Clifford, A. J. (1994). Stable isotope methods for the study of p-carotene-ds metabolism in humans utilizing tandem mass spectrometry and high-performance liquid chromatography. Anal. Chem. 66,4177-4185. Fuller, C. J., Faulkner, H., Bendich, A., Parker, R. S., and Roe, D. A. (1992). Effect of 8carotene supplementation on photosuppression of delayed-typehypersensitivity in normal young men. A m J. Clin Nurr. 56,684-690. Gaffney, P. T., Buttenshaw, R. L., Lovell, G. A., Kenwill, W. J., and Ward, M. (1990). &carotene supplementation raises serum HDL-cholesterol. Aust. N. Z. 1. Med. 20, 365. Garewal, H. S., Meyskens, F. L., and Killen, D. (1990). Response of oral leukoplakia to betacarotene. 1. Clin. OncoL 8, 1715-1720. Garewal, H. S., Meysken, F., Friedman, S., Alberts, D., and Ransey, L. (1993). Oral cancer prevention: The case for carotenoids and anti-oxidant nutrients. Prev. Med. 22, 701711. Gaziano. J. M., Manson, J. E., Ridker, P. M., Buring, J. E., and Hennekens, C. H.(1990). Beta-carotene therapy for chronic stable angina. Circularion 82, 111-120. [Abstract 07%]

Cierster, H. (1993). Anticarcinogenic effect of common carotenoids. Inr. 1. Viram. Nutr. Res. 63, 9.3-121. Goodman, K. J., and Brenna, J . T. (1992). High sensitivity tracer detection using high-precision gas chromatography-combustion isotope ratio mass spectrometry and highly enriched [U-13C]-labeledprecursors. A d Chem. 64, 1088-1095. Goodman K.J., and Brenna. J. T (1994). Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope rato mass spectrometry. Anal. Chem 66,1294-1301. Goodman, K. J., and Brenna, J. T. (1995). High-precision gas chromatography-combustion isotope ratio mass spectrometry at low signal levels. J. Chromarogr. A 689, 63-68.

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Goodman, D. S., Blomstrand, R., Werner, B., Huang, H. S., and Shiratori, T. (1966). The intestinal absorption and metabolism of vitamin A and 8-carotene in man. J. Clin. Invesr. 45,1615-1623.

Gottlieb, K., Zariing, E. J., Mobarhan, S., Bowen, P., and Sugerman, S. (1993). p-carotene decreases markers of lipid peroxidation in healthy volunteers. Nurr. Cancer. 19,140-141. Green, M. H., and Green, J. B. (1990). The application of compartmental analysis to research in nutrition. Anna Rev. Nutr. 10, 41-61. Greenberg, E. R., Baron, J. A,, Stukel, T. A., Stevens, M. M., Mandel, J. S., Spencer, S. K., Elias, P. M., Lowe, N., Nierenberg, D. W., Bayrd, G., Vance, J. C., Freeman, D. H., Clendenning, W.E., Kwan, T., and the Skin Cancer Prevention Study Group (1990). A clinical trial of beta carotene to prevent basal-cell and squamous-cell cancers of the skin. N. EngL J. Med 323,789-795. Greenberg, E. R., Baron, J. A., Tosteson, R. D., Freeman, D. H., Beck, G. J., Bond, J. H., Colacchio, T. A., Coller, J. A., Frankl, H. D., Haile, R. W., Mandel, J. S., Nierenberg, D. W.,Rothstein, R., Snover, D. C., Stevens, M. M., Summers, R. W., and Van Stolk, R. U. (1994). A clinical trial of antioxidant vitamins to prevent colorectal adenoma. N. Engl. J. Med 331,141-147. Gugger, E. T., Bierer, T. L., Heme, T. M., White, W. S., and Erdman, J.W. (1992). p-carotene uptake and tissue distribution in ferrets (Mustela putorius furo).J. Nutr. 122, 115-119. Heinonen, 0. P., Albanes, D., and the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study Group (1994). The effect of vitamin E and beta carotene on the incidence of lung cancer and other cancers in male smokers. N. Engl. J. Med 330,1029-1035. Herraiz, L., Rahman, A., Parker, R. S., and Roe,D.A. (1994). The role of p-carotene supplementation in prevention of photosuppression of cellular immunity in elderly men. FASEB J. 8, A423. Huang, H. S., and Goodman, D. S. (1%5). Vitamin A and carotenoids. 1. Intestinal absorption and metabolism of ''C-labelled vitamin A alcohol and p-carotene in the rat. J. Biol. Chem. 240, 2839-2844. Johnson, E. J., and Russell, R. M. (1992). Distribution of orally administered @-carotene among lipoproteins in healthy mean. Am. J. Clin. Nutr. 56, 128-135. Krinsky, N. I., Cornwell, D. G., and Oncley, J. L. (1958). The transport of vitamin A and carotenoids in human plasma. Arch. Biochem. Biophys. 73,233-246. Manago, M., Tamai, H., Ogihara, T., and Mino, M. (1992). Distribution of circulating pcarotene in human plasma lipoproteins. J. Nub: Sci. Vitaminol. 38,405-414. Mathews-Roth, M. M., and Gulbrandsen, C. L. (1974). Transport of beta-carotene in serum of individuals with carotenemia. Clin. Chem. 20,1578-1579. Mobarhan, S., Bowen, P., Anderson, B., Evans, M., Stacewicz-Sapuntzakis,M., Sugerman, S., S i m , P., Lucchesi, D., and Friedman, H. (1990). Effects of p-carotene repletion on p-carotene absorption, lipid peroxidation, and neutrophi superoxide formation in young men. Nutr. Cancer 14, 195-206. Novotny, J. A., Dueker, S. R., Zech, L. A., and Clifford, A. J. (1995). Compartmental analysis of the dynamics of p-carotene in an adult volunteer. J. Lipid Res. 36, 1825-1838. Parker, R. S. (1988). Carotenoid and tocopherol composition of human adipose tissue. Am J. Clin N W . 47,33-36. Parker, R. S., Swanson, J. E., Marmor, B., Goodman, K. J., Spielman, A. B., Brenna, J. T., Vierick, S. M., and Canfield, W. K. (1993). Study of p-carotene metabolism in humans using ['3C]p-carotene and high precision isotope ratio mass spectrometry. Ann N.Y. Acad. Sci. 691,8695. Prabhala, R. H., Garewal, H. S., Hicks, M. J., Sampliner, R. E., and Watson, R. R. (1991). The effects of 13-cb-retinoic acid and beta-carotene on cellular immunity in humans. Cancer 67,1556-1560.

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Prince, M.R., and Frisoli, J. K. (1993). Beta-carotene accumulation in serum and skin. Am. J. Clin. Nurr. 57, 175-181. Princen, H. M. G., van Poppel, G., Vogelezang, C., Buytenhek, R., and Kok, J. (1992). Supplementation with vitamin E but not @-carotenein vivo protects low density lipoprotein from lipid peroxidation in vitro: Effect of cigarette smoking. Arrerioscler. Thromb. 12,554-562. Reaven, P. D., Khouw, A., Beltz, W. F., Parthasarathy, S., and Witztum, J. L. (1993). Effect of dietary antioxidant combination in humans: Protection of LDL by vitamin E but not by @-carotene.Arrerioscler. Thromb. U,59CMOO. Redgrave, T. G., Ly, H. L., Quintao, E. C. R., Ramberg, C. F., and Boston, R. C. (1993). Clearance from plasma of triacylglycerol and cholesteryl ester after intravenous injection of chylomicron-like emulsions in rats and man. Biochem. A 290, 843-847. Ribaya-Mecado,J. D., Fox, J. G., Rosenblad, W.D., Blanco, M.C., and Russell, R. M. (1992). @carotene, retinol, retinyl ester concentrations in serum and selected tissues of ferrets fed p-carotene. J. Nufr. l22, 1898-1903. Ringer, T.V., DeLoof, M.J., Winterrowd, G. E., Francom, S. F., Gaylor, D. K., Ryan, J. A., Sanders, M.E., and Hughes, G. S. (1991). Beta-carotene’s effects on serum lipoproteins and immunologic indices in humans. Am. J. Clin. Nurr. 53,688-694. Rock, C. L., and Swendseid, M.E. (1992).Plasma p-carotene response in humans after meals supplemented with dietary pectin. Am. 1. Clin. Nufr. 55, 96-99. Rock, C.L.,Swendseid, M. E., Jacob, R.A., and McKee, R. W. (1992). Plasma carotenoid levels in human subjects fed a low carotenoid diet. J. Nurr. 122, 96-100. Stich, H. F., Rosin, M. P., Hornby, A. P., Mathew, B., Sankaranarayanan, R.,and Nair, M. K. (1988).Remission of oral leukoplakias and micronuclei in tobaccobetel quid chewers treated with beta-carotene and with beta-carotene plus vitamin A. Int. J. Cancer. 42, 195-199. Thompson, S.Y.,Braude, R., Coates, M. E., Cowie, A. T., Ganguly, J., and Kon, S. K. (1950). Further studies of the conversion of p-carotene to vitamin A in the intestine. Br. J. Nurr. 4,398-420. Thurnham, D. I., Smith, E., and Flora, P. S. (1988). Concurrent liquid-chromatography assay of retinol, a-tocopherol, @-carotene,a-carotene, lycopene, and p-cryptoxanthin in plasma, with tocopherol acetate as internal standard. Cfin. Ckm. 34,377-381. Traber, M.G., Diamond, S. R.,Lane, J. C., Brody, R.I., and Kayden, H. J. (1994). p-Carotene transport in human lipoproteins. Comparisons with a-tocopherol. Lipids 29,665-669. van Poppel, G . (1993). Carotenoids and cancer: An update with emphasis on human intervention studies. Eur. 1. Cancer 29A, 1335-1344. van Poppel, G . ,Spanhaak, S., and Ockhuizen,T. (1993).Effect of beta-carotene in immunological indices in healthy male smokers. Am. J. Clin Nurr. 57,404-407. van Vliet,T., Schreurs, W. H. P., and van den Berg, H. (1995). Intestinal &carotene absorption and cleavage in men: Response of @-caroteneand retinyl esters in the triglyceride-rich lipoprotein fraction after a single dose of p-carotene. Am. J. Clin. Nufr. 62, 110-116. Wang, X-D., Krinsky, N.I., Marini, R.P., Tang, G., Yu,J., Hurley, R., Fox, J. G.. and Russell, R. M. (1992). Intestinal uptake and lymphatic absorption of p-carotene in ferrets: A model for human p-carotene metabolism. Am J. Physiol. 263, G48ffi486. Watson, R. R.,Prabhala, R. H., Plezia, P. M., and Alberts, D. S. (1994).Effect of @carotene on lymphocyte subpopulations in elderly humans: Evidence for a dose-response relationship. Am. 1. Clin. Nurr. 53, 90-94. Weststrate, J. A., and van het Hof, K. H. (1995). Sucrose polyester and plasma carotenoid concentrations in healthy subjects. Am. J, Clin Nurr. 62,591-597.


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Wolfe, R. R. (1992). “Radioactive and Stable Isotope Tracers in Biomedicine: Principles and Practices of Kinetic Analysis.” Wiley, New York. Zeng, S., Furr, H. C., and Olson, J. A. (1992). Metabolism of carotenoid analogs in humans. Am J. Clia Nutr. 56,43H39. Ziegler, R. G. (1989). A review of epidemiologic evidence that carotenoids reduce the risk of cancer. J. Nurr. 119,116122.


ADVANCES IN FOOD AND NUTRITION RESEARCH.VOL. 40

Chapter 4 MODELING OF FOLATE METABOLISM’ JESSE F. GREGORY 111 AND KAREN C. SCOTI Food Science and Human Nutrition Departmenr University of Florida Gainesville, Florida 32611

I. Introduction 11. Key Elements of Folate Metabolism Relevant to Modeling 111. Stable-Isotopic Studies in Humans 1V. Summary and Conclusions

References

1.

INTRODUCTION

Folate is a generic term for the group of structurally related pteroylglutamates and related compounds that exhibit similar vitamin activity as substrates and coenzymes involved in one carbon metabolism. Through this metabolic function, folate is involved, either directly or indirectly, in essential cellular processes, including the synthesis of nucleic acids, regeneration of methionine from homocysteine, and methylation of proteins, nucleic acids, and other compounds with S-adenosyl-methionine as methyl donor. Thus, many diverse cellular functions depend on an adequate supply of this vitamin and are impaired during periods of inadequate nutritional intake. Because of the role of folate in nucleic acid synthesis, the replication and maturation of various cell types (e.g., erythrocytes and intestinal mucosal epithelium) is impaired in folate deficiency. The metabolism and function also involves other nutrients, including zinc, riboflavin, vitamin B12, and vitamin B6, consequently, folate metabolism and utilization can be impaired as a secondary effect of other nutrient deficiencies. Presented in part at the conference entitled “Mathetmatical Modeling in Experimental Nutrition Conference V,” Fort Wayne, IN, May 8-11,1994. 81 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reSeNed.

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Folic acid (i.e., pteroylglutamic acid) is the synthetic form of the vitamin employed in food fortification and in nutritional supplements. Naturally occurring forms of folate exist mainly as tetrahydrofolate species with variable length of a polyglutamyl side chain (Fig. 1). The intestinal absorption of folic acid and reduced folates occurs in the jejunum via specific carrier mediated transport of the vitamin in monoglutamyl form. Deconjugation of dietary polyglutamyl folates is catalyzed mainly by intestinal pteroylpolyglutamate hydrolases (conjugases) and must occur prior to absorption. Most studies comparing the bioavailability of monoglutamyl versus polyglutamyl folates have indicated superior utilization of the monoglutamyl form. This provides strong evidence of the rate limiting nature of the in vivo action of the intestinal conjugase in folate absorption. Food composition appears to have considerable influence on the bioavailability of dietary folates. Folic acid added to various cereal grain-based foods exhibits approximately 50% bioavailability relative to folic acid administered as a supplement in water (Colman, 1982; Colman et al., 1975). In addition, unidentified components of many foods have the ability to inhibit intestinal conjugase activity in vitro (Bhandari and Gregory, 1990) and, thus, may inhibit the deconjugation of dietary folates in viva

9 7

FOOH

C-N--F-CHrCHrCOOH H

H2N

Folic (Ptemyl-LGlutamic) Acid

PdyglutamyiTetrahydrofotabr

H2N

H

Substitumt (R) -CH3 (mew) -CHO (formyl)

-CH=NH

Position

I 5orlO

-CH2-

(fonnimino) (methylone)

5 land10

-CH=

(meth.nyl)

5 and 10

FIG. 1. Chemical structures of folic acid and tetrahydrofolates. Tetrahydrofoiates are shown as the polyglutarnyl form, n = 2-9. R represents the one-carbon substituent for the folates shown, and R = H for unsubstituted tetrahydrofolates.

CHAPTER 4 MODELING OF FOLATE METABOLISM

83

Considerable uncertainty and controversy exists concerning the folate requirement for humans. The review of data concerning the human folate requirement by the Food and Nutrition Board (1989) suggests that the daily maintenance requirement is 100-200 pg of available folic acid equivalents. The 1989 RDAs were reduced to 200 and 180 pg for adult men and women, respectively, from the previous RDA of 400 on the basis of such evidence (Food and Nutrition Board, 1989). Similarly, the Canadian RDA for folate was set at 3 pgkg body wt or 210 pg for a 70-kg individual. These lower RDAs may be inadequate for certain population groups, however (Sauberlich, 1990;Bailey, 1992;McPartlin et al., 1993).It is currently difficult or impossible to predict the quantitative effect on folate nutritional status of factors such as: (a) changes in folate intake, (b) differences in folate bioavailability, (c) effects of pregnancy and lactation on folate requirements, and (d) pharmaceuticals with antifolate properties. In addition, the development of mathematical models would improve our ability to evaluate methods of nutritional status assessment for this vitamin. II. KEY ELEMENTS OF FOLATE METABOLISM RELEVANT TO MODELING

Although many studies of the in vivo kinetics of folate have been conducted, none has fully represented all aspects of folate metabolism in any single animal species. The documented differences in certain aspects of folate digestion and metabolism among certain animal species may further complicate modeling, as does the influence of the dietary folate intake before and during the kinetic study. However, since the basic processes of absorption, metabolic function, catabolism, and excretion are quite similar, one would expect that similar models would apply to most species. At its simplest, folate metabolism can be described as two kinetically detectable pools: (a) one small pool with rapid turnover kinetics, which is composed mainly of the monoglutamyl forms of folates in plasma, and (b) a larger, slow-turnover pool of folates functionally trapped in tissues as polyglutamyl forms, many of which are noncovalently bound to proteins. A schematic overview of whole body folate metabolism is shown in Fig. 2. The commercial availability of [3H]folic acid at high-specific activity has made feasible many studies of folate kinetics. The studies involving bolus dose administration of the labeled folate consistently have shown a biphasic curve for disappearance of folate by urinary and fecal routes. Representative of this approach is the well-designed study by Tamura and Halsted (1983) of folate metabolism in monkeys. Following a bolus dose of [3H]folic acid, the isotope was excreted by fecal and urinary routes. The extent of

84


Dietary Folate P8iElB8uC

u b I

FIG. 2. Schematic diagram of whole-body folate metabolism. Dietary forms of polyglutamyl folates are deconjugated in the jejunum (vertical rectangle on left) by mucosal and, to a lesser extent, pancreatic hydrolases. Secretion of folate in bile accounts for an enterohepatic circulation that has not been incorporated into modeling. Tissues behave, at least superficially, as a large, kinetically slow, pool.

fecal loss was quite pronounced and approached 30% of daily turnover. On the basis of urinary excretion of total tritium, and assuming a two-pool model with output from the rapid turnover pool (Fig. 3), half times of 1.8 t 0.3 and 119 2 11 days were reported for rapid and slow phases of folate turnover, respectively. Another important finding was that chronically alcoholic monkeys exhibited substantially altered short-term kinetics; half times for folate turnover in alcoholic monkeys were 1.1 2 0.2 and 85 ? 26 days Folate Input

Irreversible Losses urinary folate fecal folate catabolism

'3

-

FIG. 3. Two-pool model of folate metabolism with output only from the rapid turnover

pool. Output represents the sum of urinary, fecal, and catabolic losses.


85

for rapid and slow phases, respectively. Whether this difference reflected the nutritional status of the monkeys, effects of alcohol on folate catabolism, or other metabolic factors was not determined. The cumulative excretion curves showed markedly reduced in vivo retention of the isotope in alcoholic animals, with increased excretion by both urinary and fecal routes. The excretion of folate and its metabolites by both urinary and fecal routes was demonstrated in a study with [2-14C]folicacid administered to a single adult woman as several daily doses followed by periodic urine and fecal collection over several months (Krumdieck et al., 1978). Fecal excretion accounted for nearly half of the isotope elimination. From excretion of urinary 14C, two pools were readily apparent with half times of -31.5 hr and -100 days. Periodic administration of diphenylhydantoin to this subject (a Hodgkin’s disease patient in remission) did not notably alter the turnover kinetics. A similar study with radiotracers was conducted in our laboratory to examine the short-term kinetics of folate metabolism in rats. Three different folates were separately employed to assess any differences in bioavailability among these forms of the vitamin (Bhandari and Gregory, 1992). Similar to the study with monkeys, cumulative excretion via the fecal route comprised approximately one-third of the folate turnover, with the remainder as urinary excretion of intact folates and catabolic products. There were no detectable differences among these forms of the vitamin with respect to excretion kinetics by urinary or fecal routes and only minor short-term differences in tissue distribution in relation to the form of folate administered. Assuming a two-pool model with one output, half times of -0.1 and 13-16 days were calculated. Further kinetic studies in rats are in progress using controlled dietary folate intake and longer collection periods to permit long-term modeling (Scott and Gregory, 1994). Other previous studies with radiolabeled folates in rats also have shown biphasic patterns of urinary excretion (Murphy and Scott, 1979). Many short-term kinetic studies have been reported on the basis of plasma and serum folate levels in humans. Typical protocols involve the administration of a folate tracer dose to folate-saturated subjects or else a large folate dose given to subjects of more normal folate status (Reich and Gonczy, 1979; Bunni et al., 1989; Wolfrom et al., 1990; Priest et al., 1991; von der Porten et al., 1992). Although these studies have clinical relevance with respect to the short-term kinetics of acute doses of the vitamin (e.g., postchemotherapy 5-formyl-tetrahydrofolate), they often provide little or no information about the major body folate pool that exhibits much slower kinetics. A key aspect of folate metabolism is the catabolism. Folate catabolism occurs through oxidative cleavage of tetrahydrofolates at the C9-NlO bond

86


to yield a pterin (e.g., pterin-6-carboxylic acid or pterin-6-carboxaldehyde) and para-aminobenzoylglutamate (pABG), or its polyglutamate form. This presumably occurs within tissue folate pools through nonspecific oxidation or interaction with metabolically generated free radicals (Shaw et al., 1989). Following enzymatic hydrolysis of polyglutamyl forms of pABG to monoglutamyl pABG, this catabolite is acetylated in the liver to para-acetamidobenzoylglutamate (apABG), which is rapidly excreted in urine (Murphy et al., 1976; Murphy and Scott, 1979; McPartlin er al., 1992, 1993). Although folate catabolism has been known for many years (Dinning et al., 1957), its quantitative significance has only recently been recognized. McPartlin and associates (1986) have shown that apABG excretion is much greater than urinary folate excretion in adult humans. Significantly, folate catabolism increases during the second trimester of pregnancy and accounts for much of the increased folate requirement (McPartlin et al., 1993). The excretion of total pABG (free and acetamido) in rats and humans far exceeds urinary folate (Murphy er al., 1976; Murphy and Scott, 1979; McNulty et al., 1993; Kownacki-Brown et af., 1993; Wang et al., 1994). It must be recognized in modeling that folate catabolism, which occurs by mainly oxidative cleavage of tetrahydrofolates by free radicals and various oxidants (primarily in liver and other tissues), represents a major irretrievable loss from the body. Most previous studies of the in vivo kinetics of folate have employed a two-pool model with output only from the rapid turnover pool representing the sum of urinary, fecal, and catabolic losses (Fig. 3). Our current working hypothesis is that kinetic models, for greatest physiological accuracy, should include a provision for irretrievable losses from the tissue pool as well as from the rapid turnover (i.e., plasma) pool.

111.

STABLE-ISOTOPIC STUDIES IN HUMANS

Stable-isotopic labeling methods permit the evaluation of micronutrient kinetics with the high specificityof isotopic procedures but with no radiation exposure to the subjects. Another advantage is the ability to conduct studies in which two or more labeled forms are administered simultaneously (e.g., monoglutamyl and polyglutamyl folates, oral and iv routes, etc.). The synthesis, use, and mass spectral analysis of folates labeled with deuterium or other stable isotopes has been reviewed (Gregory and Toth, 1990; Gregory, 1989). Folates labeled with deuterium in either the 3',5'-positions of the central benzoyl moiety (2Hz-folate or d2-folate; Gregory and Toth, 1988a; Gregory, 1990) or the p,p,-y,-y-positions of the glutamate moiety ('h-folate or d4-folate; Gregory and Toth, 1988b) may be readily prepared and are

CHAFTER 4 MODELING OF FOLATE METABOLISM

87

fully suitable for in vivo use (Fig. 4). Although initially developed for use in the study of folate bioavailability, these labeled folates permit the study of long-term folate kinetics through protocols that involve chronic administration for extended periods of time. These protocols and modeling procedures are in their early stages of development at this time. Studies conducted to date have been based primarily on measurement of changes in the isotopic enrichment of urinary folate (isolated by affinity chromatography). Von der Porten et al. (1992) conducted a study of the changes in folate nutritional status during chronic (4 week) supplementation with a relatively large (1.6 mg/day) dose of folic acid. The use of d2-folk acid permitted initial observations of in vivo labeling, although the protocol was not designed for isotopic kinetics. In addition, the protocol of 4-week supplementation followed by total withdrawal of the supplement was, by design, not a steadystate situation in terms of folate status of the subjects and, presumably, the size of body folate pools. After only 7 days of supplementation and continuing over the 28-day period, mean enrichment of urinary folate was 69%, in comparison to the fact that the isotopic enrichment of total ingested folate (dietary + supplemental) was approximately 89%. Thus initial labeling of body folate occurred rapidly, but isotopic equilibrium was not obtained even after 28 day with an intake of -9 times the RDA value. Analysis of erythrocyte folate over the course of supplementation indicated a maximum isotopic enrichment of only -10%. This is consistent with the fact that folate is deposited in the erythrocyte primarily at the time of erythropoiesis;

H

H

[Glu-2HJFoilc Acid

[3',5'-2HdFoilc Acid FIG. 4. Structure of deuterium-labeled forms of folk acid.

88


thus, labeling of erythrocyte folate lags behind other body pools as a result of the long life time of preexisting unlabeled erythrocytes. Several studies have been conducted to examine in vivo kinetics of folate under conditions approximating more normal intake levels. We have recently conducted a study in which a single adult female was maintained on a protocol with controlled diet for 10 weeks with a total folate intake of approximately 202 pg/day, composed of -12 pg/day from a low-folate diet and 190 &day from synthetic folic acid in apple juice (Gregory ef al., 1994). The first 2 weeks of the study were an equilibration period in which the 190 pg/day folic acid was not isotopically labeled, while for the next 8-week period the subject consumed d2-folic acid. This protocol maintained the subject in steady state as judged by serum and urinary folate concentration. As shown in Fig. 5, enrichment of urinary folate increased progressively over the experimental period, although the final enrichment (-10%) was far less than that calculated for total ingested folate (-28.4%). Fitting these data to the two-pool model with output only from pool 1 (e.g., Fig. 4), a fractional catabolic rate (FCR) for total body folate was estimated to be -0.0055 day-', corresponding to a system mean residence time (MRT) of -182 days. Preliminary analysis of these data with an expanded model to incorporate catabolic losses primarily from the tissue pool (Fig. 6) yielded a similar fit to that shown in Fig. 5. The estimated FCR was 0.00342 loo

1

I

1

i

I

ingested folate = 0.284

_______.___________.--------.~.~~~----~h

0

3w

lo-' .r

*

*

.'. * ".

-

a

z w 10-3,

1


89

Folate Input

intestine

FIG. 6. Expanded model of in vivo folate metabolism. The pools are defined as follows: 1, rapid turnover folate; 6, slow turnover folate (tissues);2, irretrievable losses by fecal excretion and catabolism; 3, cumulative excretion of urinary folate; 4, fractional (daily) excretion of urinary folate. Analysis was performed with parallel models for labeled and nonlabeled folate.

day-', equivalent to an MRT of -292 for total body folate. Estimated masses for pool 1 (rapidly exchanging folate) and pool 6 (slow-turnover tissue folates) were 10.9 and 76.8 mg, respectively. These preliminary results indicate the very slow turnover of body folate and the experimental impracticality of attempting to achieve isotopic steady state with this protocol. An unanswered question in this type of protocol is the bioavailability of the ingested folate. If one assumes a fractional absorption of 0.67 for total ingested folate (i.e., labeled + dietary), this model yields estimates for the masses of pools 1 and 6 of 2.95 and 36.5 mg, respectively. Thus, the bioavailability of ingested folate is a critically important factor in controlling the in vivo pool sizes and, thus, the nutritional status of the individual. As an alternative to the long-term controlled dietary protocol, a similar study was conducted with free-living men (n = 4) consuming self-selected diets (Stites er aL, 1994). After a 2-week equilibration period with 200 pg/ day unlabeled folate supplement, the subjects consumed 200 pg/day of a 1:1 mixture of dCfolic acid and unlabeled folic acid for an 8-week period, followed by a switch back to 200 pg/day of unlabeled folate. Dietary folate intake was estimated from food records using a computerized data base. Typical results (Fig. 7) indicate slow labeling of body folate, as discussed above. Analysis of these data using the expanded model (Fig. 6) yielded

90


-

0.125:'

9

0.100~

0

' '

I

I

'

' ' I

I

-I 1 0

88 0.075

-

r--

3

0.050

a

0.025

z9 Z

W

O.OO0

TIME (d)

FIG. 7. Isotopic enrichment of urinary folate in an adult male subject. In addition to dietary folate, the subjects received the following daily supplement: Days 15-75,100 pg d4-folic acid + 100 pg nonlabeled folic acid; Days 76-126,200 pg nonlabeled folic acid (no d4-folk acid) (Stites er al., 1994). The solid line representsenrichment as predicted using the model shown in Fig. 6.

FCR and MRT values of approximately 0.0088 day-' and 114 days, respectively. The mean masses of folate in pool 1 (rapidly exchanging folate) and pool 6 (slow turnover tissue folates) were 10.9 and 76.8 mg, respectively. These masses were greater than those estimated from the previous study with one female subject, which may reflect the larger body mass of the male subjects as well as their greater folate intake in this study. On the basis of analysis of these results, it has become apparent that reliance on measurements of enrichment of urinary folate leaves excessive uncertainty in all aspects of modeling and kinetic calculations. Since urinary folate accounts for only several percentages of the intake, the fate of the remaining >98% cannot be determined with certainty solely from the enrichment of urinary folate. It is unlikely that determination of the enrichment of fecal folate would provide useful information for modeling because colonic bacteria can both synthesize and utilize the endogenous folate that bypasses reabsorption in the enterohepatic circulation. Catabolism to form pABG and its acetylated derivative, which are primarily excreted in urine, may account for half or more of folate turnover from the body. Thus, measurement of isotopic enrichment of these catabolites will be included in further kinetic protocols to provide a more complete accounting of the labeled folate. The use of animal models will be of help in this area as various tissues may be sampled over time to determine folate stores and turnover. This information can then be extrapolated to a human kinetic model based on known similarities and differences in metabolism between the species.


91

IV. SUMMARY AND CONCLUSIONS

It can be concluded that developing, then improving, a model of folate metabolism is a long-term project. Not only must one include the multiple routes of excretion of folic acid and its catabolites (pABG, apABG, pterins), but the interconversion of the various forms of folate must also be considered, including polyglutamylation within tissues. Many factors affect folate metabolism, including dietary folate level, nutritional status of vitamins B6, B12, and riboflavin, zinc status, alcoholism, and physical states such as pregnancy and lactation. In many cases, the effects of these factors are seen in altered excretion rates of intact folates and metabolites, but the effects on tissue levels of the various folates and transfer rates between tissues are not well understood. Preliminary human and animal kinetic models are being developed in our laboratory based on studies conducted under controlled dietary conditions. These models will provide a base from which to study the effects of altered folate nutriture as well as the influence of other factors such as pregnancy and aging on folate metabolism. ACKNOWLEDGMENTS This research was supported by Grants 88-37200-3620.91-37200-6305,and 92-37200-7466 from the USDA National Research Initiative-Competitive Grants Program and by Clinical Research Center Grant RR00082. The authors extend thanks to Drs. Loren Zech and Waldo Fisher for their efforts in the development of the kinetic model. The long-term collaborations of Drs. John Toth and Lynn Bailey are gratefully acknowledged as are the contributions to this research of Sneh Bhandari, Elizabeth Donaldson, Susan Hillier, Sarah Hofler, Caroline O’Keefe, Tracy Stites, Elizabeth Thomas, and Jerry Williamson.

REFERENCES Bailey, L. B. (1992). Folate-Evaluation

of new recommended dietary allowance. J. Am.

Diet. Assoc. 92,463.

Bhandari, S. D., and Gregory, J. F. (1990). Inhibition by selected food components of human and porcine intestinal pteroylpolyglutamate hydrolase activity. Am. J. Cfin. Nutr. 51, 87-94. Bhandari, S. D., and Gregory, J. F. (1992). Folk acid, 5-methyl-tetrahydrofolateand S-formyltetrahydrofolate exhibit equivalent intestinal absorption, metabolism and in vivo kinetics in rats. J. Nutr. 122,1847-1854. Bunni, M. A., Rembiesa, B. M., Priest, D. G., Sahovif, E., and Stuart, R. (1989).Accumulation of tetrahydrofolates in human plasma after leucovorin administration. Cancer Chemorher. PharmacoL 23,353-357. Colman, N. (1982).Addition of folk acid to staple foods as a selective nutrition intervention strategy. Nutr. Rev. 40,225-233.

92


Colman, N.,Green, R., and Metz, J. (1975). Prevention of folate deficiencyby food fortification. 11. Absorption of folic acid from fortified staple foods. Am. J. Clin. Nutr. 28,459-464. Dinning, J. S., Sime, J. T., Work, P. S., Allen, B., and Day, P. L. (1957). The metabolic conversion of folic acid and citrovorum factor to a diazotizable amine. Arch. Biochem. Biophys. 66,114-119. Food and Nutrition Board (1989). “Recommended Dietary Allowances,” 10th ed. National Research Council, National Academy of Sciences, Washington, DC. Gregory, J. F. (1989). Chemical and nutritional aspects of folate research: Analytical procedures, methods of folate synthesis, stability, and bioavailability of dietary folates. Adv. Food Nutr. Res. 33,l-101. Gregory, J. F. (1990). Improved synthesis of [3’,5’-2H2]folicacid: Extent and specificity of deuterium labeling. J. Agric. Food Chem. 38, 1073-1076. Gregory, J. F., and Toth, J. P. (1988a). Chemical synthesis of deuterated folate monoglutamate and in vivo assessment of urinary excretion of deuterated folates in man. Anal. Biochem. 170,94-104. Gregory, J. F., and Toth, J. P. (1988b). Preparation of folic acid specifically labelled with deuterium at the 3‘,5‘-positions. J. Labelled Compd. Radiopharm. 25,1349-1359. Gregory, J. F., and Toth, J. P. (1990). Stable-isotopic methods for in vivo investigation of folate absorption and metabolism. In “Folic Acid Metabolism in Health and Disease’’ (M. F. Picciano, E. L. R. Stokstad, and J. F. Gregory, eds.), pp. 151-170. Wiley-Liss, New York. Gregory, J. F., Bailey, L. B., Thomas, E. A,, Toth, J. P., Cerda, J. J., and Fisher, W.R. (1994). Stable-isotopic study of long-term folate metabolism in a human subject. FASEB J. 8, A920. Kownacki-Brown,P. A., Wang, C., Bailey, L. B., Toth, J. P., and Gregory, J. F.(1993). Urinary excretion of deuterium-labeled folate and the metabolite p-aminobenzoylglutamate in humans. J. Nutr. l23,1101-1108. Krumdieck, C . L., Fukushima, K., Fukushima, T., Shiota, T., and Butterworth, C. E. (1978). A long-term study of the excretion of folate and pterins in a human subject after ingestion of 14C folic acid, with observations on the effect of diphenylhydantoin administration. Am. J. Clin. Nutr. 31,88-93. McNulty, H., McPartlin, J. M., Weir, D. G., and Scott, J. M. (1993). Reversed-phase high performance liquid chromatographic method for the quantitation of endogenous folate catabolites in rat urine. J. Chromatogr. 61459-66. McPartlin, 1. M., Weir, D. G., Courtney, G., McNulty, H., and Scott, J. M. (1986). The level of folate catabolism in normal human populations. I n “Pteridines and Folic Acid Derivatives” (B. Cooper and V. Whitehead, eds.), pp. 513-516. de Gruyter, New York. McPartlin, J. M.,Courtney, G., McNulty, H., Weir, D. G., and Scott, J. M. (1992). The quantitative analysis of endogenous folate catabolites in human urine. Anal. Biochem. 246,256-261. McPartlin, J. M., Halligan, A., Scott, J. M., Darling, M., and Weir, D. G. (1993). Accelerated folate breakdown during pregnancy. Lancer 341,148-149. Murphy, M. and Scott, J. M.(1979). The turnover, catabolism, and excretion of folate administered at physiological concentrations in the rat. Biochim. Biophys. Acta 583,535-539. Murphy, M., Keating, M., Boyle, P., Weir, D. G., and Scott, J. M. (1976). The elucidation of the mechanism of folate catabolism in the rat. Biochem. Biophys. Res. Commun. 7& 1017-1024. Priest, D. G., Schmitz, J. C., Bunni, M. A., and Stuart, R. K. (1991). Pharmacokinetics of leucovorin metabolites in human plasma as a function of dose administered orally and intravenously.J. Nail. Cancer Inst. 83, 1806-1812.


93

Reich, S. D., and Gonczy,C. (1979). Mathematicalmodeling-guide to high-dosemethotrexate infusion therapy. Cancer Chemother. PharmacoL 3,25-31. Sauberlich, H. E. (1990). Evaluation of folate nutrition in population groups. In “Folic Acid Metabolism in Health and Disease” (M. F. Picciano, E.L. R. Stokstad, and J. F. Gregory, eds.), pp. 211-236, Wiley-Liss, New York. Scott, K. C., and Gregory, J. F. (1994). Long term folate metabolism in the growing rat. FASEB J. 8, A704. Shaw,S., Jayatilleke, E., Herbert, V., and Colman, N. (1989). Cleavage of folatesduringethanol metabolism. Role of acetaldehyde/xanthine oxidase-generated superoxide. Biochem. J. 257,277-280.

Stites, T., Gregory, J. F., Bailey, L., Fisher, W., and Toth, J. (1994). Kinetic modeling of deuterium labeled folate in human subjects. FASEB J. 8, A704. Tamura, T., and Halsted, C. H. (1983). Folate turnover in chronically alcoholic monkeys. J. Lab. Clin Med. 101,623-628. von der Porten, A. E., Gregory, J. F., Toth, J. P., Cerda, J. J., Curry, S. H., and Bailey, L. B. (1992). In vivo folate kinetics during chronic supplementation of human subjects with deuterium-labeled folic acid. J. Nutr. 122, 1293-1299. Wang, C., Song, S., Bailey, L. B., and Gregory, J. F. (1994). Relationship between urinary excretion of p-aminobenzoylglutamate and folate status of growing rats. Nutr. Res. 14, 15-884.

Wolfrom, C., Hepp, R., Hartmann, R., Breithaupt, H., and Heme, G. (1990). Pharmacokinetic study of methotrexate, folinic acid and their serum metabolites in children treated with high-dose methotrexate and leucovorin rescue. Eur. J. Clin. Pharmacol. 39,377-383.


ADVANCES IN FOOD AND NUTRITION RESEARCH, VOL.40

Chapter 5 MOLECULAR BIOLOGY IN NUTRITION RESEARCH: MODELING OF FOLATE METABOLISM BI-FONG LIN, JONG-SANG KIM,JUEI-CHUAN HSU, CHARLES OSBORNE, KAREN LOWE, TIMOTHY GARROW, AND BARRY SHANE Department of Nutritional Sciences University of California Berkeley, Carifomin 94720

1. General Approaches 11. Modeling of Folate Metabolism

A. Introduction B. Role of Folylpolyglutamate Chain Length C. Compartmentationof Cellular Folate D. Role of Folylpolyglutamate Synthetase Level in Folate Accumulation E. Role of Folylpolyglutamate Synthetase in Anti-folate Cytotoxicity F. Summary References

1.

GENERAL APPROACHES

The application of molecular biological techniques to the study of nutrient control of metabolism has received increased attention over the past 10 years. The isolation and characterization of genes encoding key enzymes in metabolic pathways has revealed new insights into the processes by which nutrient status influences metabolic fluxes through these pathways. The ability to manipulate enzyme levels by genetic means has allowed the development of various metabolic models for assessing whether particular enzymes exert key kinetic control over metabolic pathways and for assessing the role of particular enzymes in the compartmentalization of nutrients and metabolites in cells and tissues. Transgenic animals expressing elevated enzyme levels in specific tissues can be developed and the effects of reduction or elimination of a specific enzyme can be ascertained using gene knockout techniques. 95 Copyright 0 19% by Academic Res, Inc. AU rights of reproduction in any form reserved.

96


Introduction of genes or cDNAs into mammalian cells by transfection allows the development of kinetic models expressing various levels of the protein of interest. A major advantage of this system is that it is possible to modify the level or specificity of a single enzyme in a common background so theoretically all other enzyme levels in the cell are unchanged. Care must be taken to ensure that the "modified" system truly represents the normal physiology of the cell. Many enzymes that catalyze steps in a metabolic pathway are associated together as multiprotein complexes. The protein introduced via an expression system may not form part of such a complex, due to insufficient associated proteins for forming the complex or, if a cDNA from a different species is used, because the protein may differ from the endogenous protein in residues required for complex formation. Kinetic data obtained in such cases may not reflect the true kinetics and flux of the pathway that the enzyme is normally associated with. The protein may also have to be directed to the correct localation in the cell and techniques are available to target proteins to subcellular compartments. The cDNA is normally under the control of a eukaryotic promoter. This promoter can be inducible, which allows regulated expression of the encoded protein, or can be constitutive. Depending on the transfection conditions, the expression of the cDNA of interest can be transient or permanent. The vectors used in these studies normally also express an antibiotic resistance gene to allow a simple selection for cells that express the transfected DNA. In this report, we describe the application of some of these techniques to the study of folate metabolism and its homeostasis. It. MODELING OF FOLATE METABOLISM

A.

"RODUCTION

Tissue folates are metabolized to poly-y-glutamyl coenzymes (Fig. 1) that function in metabolic cycles involved in amino acid and DNA precursor H

OH

/

FIG. 1. Structure of folylpoly-y-glutamate.


97

synthesis (Shane, 1989). Folate coenzymes are involved in serine-glycine interconversion, methionine synthesis, and choline degradation, histidine catabolism, de novo purine synthesis, and thymidylate synthesis. While monoglutamates are the forms absorped across the intestinal mucosa and are the circulating and transport forms of folate, tissue retention and accumulation requires their conversion to the polyglutamate species. The polyglutamates are also the active coenzymatic forms, usually with greatly reduced K,,, values for their respective enzymes and affinities that are up to three orders of magnitude greater than the pteroylmonoglutamate forms (Shane, 1989). Folylpolyglutamate synthetase catalyzes the addition of glutamate residues to cellular folates and antifolates to form the physiological active coenzymatic forms of the vitamin and more potent anti-folate agents (Shane, 1989). Bacterial folylpolyglutamate synthetase enzymes normally only metabolize folate to the triglutamate or tetraglutamate forms while mammalian folylpolyglutamate synthetase will metabolize folate to longer derivatives. Chinese hamster ovary (CHO) cell mutants that lack folylpolyglutamate synthetase activity (AUXB1) have greatly reduced folate pools, due to an inability to retain folates, and are auxotrophic for methionine, glycine, purines, and thymidine (Taylor and Hanna, 1977; Osborne et al., 1993; Lowe et al., 1993). The auxotrophy can not be relieved by elevating intracellular pteroylmonoglutamate to folate levels normally found in wild type CHO cells (Osborne et al., 1993), demonstrating that the auxotrophy is not due to low folate levels but is due to an inability to synthesize folylpolyglutamates.Wild-type CHO cells normally contain hexa- and heptaglutamates and mammalian tissues contain a range of polyglutamates varying in glutamate chain length from the pentaglutamate to the decaglutamate. To understand why mammalian tissues synthesis these long chain polyglutamate species, and to define the role of folylpolyglutamate synthetase in cellular folate accumulation, we have developed a number of mammalian cell models transfected with various folylpolyglutamate synthetase genes (human and Escherichia coli) and containing altered folate coenzyme distributions. These cells have also been used to study the role of different folate derivatives in the various metabolic cycles of one-carbon metabolism. B. ROLE OF FOLYLPOLYGLUTAMATE CHAIN LENGTH 1. Folate Retention

CHO AUXBl mutants transfected with the E. coli folylpolyglutamate synthetase gene (AUX-coli) express the E. coli protein in the cytosol and metabolize folates primarily to triglutamates rather than the hexa- and


heptaglutamates normally found in wild-type CHO cells (CHO-WT) (Osborne et al., 1993). In pulse chase studies with cells preincubated in medium containing labeled folinate, or the anti-folate methotrexate, the trace amounts of pteroylmono- and diglutamate in CHO-WT and AUX-coli cells are rapidly lost from the cells, while the triglutamates that accumulate in AUX-coli are retained approximately as effectively as the longer polyglutamate derivatives in CHO-WT cells. Metabolism of folate or antifolates to the triglutamate appears to be sufficient for effective intracellular retention and accumulation of the vitamin. 2. Metabolic Effectiveness

To evaluate the metabolic effectiveness of different polyglutamate chain length folates in the various metabolic cycles of one carbon metabolism, CHO cells were cultured in medium lacking products of one carbon metabolism, such as purines, thymidine or glycine, and the levels of intracellular folate that supported half maximal growth rates were assessed (Lowe et aL, 1993). No significant differences were found in folate requirements between CHO-WT cells and AUX-human transfectants expressing various levels of human folylpolyglutamate synthetase activity and containing predominant folylpolyglutamates of chain length varying from the tetra- to the octaglutamate (Lowe et al., 1993). In AUX-coli cells, which contain triglutamates, similar intracellular folate concentrations to CHO-WT supported growth in thymidine and purine-free medium, but the folate requirement for growth in medium lacking glycine was increased about 100-fold (Table I). TABLE I SUBCELLULAR FOLATE CONCENTRATIONS SUPPORTING HALF-MAXIMAL GROWTH RATES IN MEDIA LACKING PURINES, THYMIDINE, OR GLYCINE

-Glycine Cell line WTI2 AUXBl AUX-coli AUX-mcoli AUXcoli-mcoli a

-Purines

-Thymidine

Cytosol Mitochondria Cytosol Mitochondria Cytosol Mitochondria 0.81 -b -100 1.1 4.8

0.49

-

0.45 0.66

pmol/W ceha 0.43 0.60 0.56 0.60

0.27 0.24 0.10

0.12 -20 0.20 0.14 0.17

0.08 0.06 0.03

1 pmoU106 cells is equivalent to an intracellular concentration of approximately 1 pM. No growth or mitochondrial pool negligible or absent.

CHAF'TER 5 MODELING OF FOLATE METABOLISM

99

Glycine is synthesized from serine via the serine hydroxymethyltransferase reaction. Mammalian cells contain two isozymes of this enzyme, one cytosolic and one mitochondrial. Cells defective in the mitochondrial isozyme require glycine for growth (Garrow et al., 1993). Although the data shown in Table I suggested that longer polyglutamates than pteroyltriglutamates were required for glycine synthesis,further studies showed that AUXcoli, which expresses E. coli folylpolyglutamate synthetase in the cytosol, lacked mitochondrial folates despite possessing normal cytosolic folate pools (Lin et al., 1993). The inability of these transfectants to grow in the absence of glycine could have been due to lack of mitochondrial folate rather than a metabolic ineffectiveness of pteroyltriglutamates. Additional studies indicated that CHO-WT and AUX-human transfectants expressed folylpolyglutamatesynthetase activity in both the cytosol and the mitochondria and accumulated normal folate pools in these subcellular compartments (Lin et al., 1993). An E. coli protein would not be expected to enter the mitochondrion due to lack of a mitochondrial leader sequence (Fig. 2). To further investigate the effectiveness of pteroyltriglutamates in glycine synthesis, E. coli folylpolyglutamate synthetase was targeted to the mitochondria of AUXBl and AUX-coli cells using a modified E. coli folylpolyglutamate synthetase gene construct preceded by a mammalian mitochondrial leader sequence (Lin and Shane, 1994). The leader sequence was obtained from a human ornithine transcarbamoylase cDNA. Cells expressing the E. coli enzyme in their mitochondria (AUX-coli-mcofi and AUXmcoli, Table I) accumulated mitochondrial folate. In these cells, pteroyltriglutamates functioned as effectively as the longer glutamate chain length

mP

pSV2-hyg-mfolC 7.9 kb

FIG. 2. Mammalian expression plasmid. This plasmid contains a modified E. coli FPGS gene (mfoK) and a bacterial hygromycin resistance gene bordered by mammalian expression sequences. Each is preceded by a SV40 promoter and followed by a poly(A) signal and a splice region from t antigen. The vector also contains an ori region from pBR322 and an ampicillin resistance gene to allow for growth and selection in bacteria.

100


folates found in wild type CHO cells in the metabolic cycle of glycine synthesis provided they were located in the mitochondria (Table I). Preliminary studies have been carried out on the ability of these transfectants to synthesize methionine via the BIZ-dependent 5-methyltetrahydrofo1ate:homocysteine transmethylase (methionine synthase) reaction. CHO-WT cells will grow in the absence of methionine provided sufficient homocysteine and vitamin BIZare provided. CHO cell transfectants expressing E. coli folylpolyglutamate synthetase grow more slowly or do not grow under these conditions, suggesting that the longer glutamate chain length folates typically found in mammalian tissues are required for methionine synthesis (Lowe et ai., 1993). C. COMPARTMENTATION OF CELLULAR FOLATE AUX-coti cells lacked mitochondrial folate despite possessing high levels of cytosolic folate and folylpolyglutamates can not enter the mitochondria of mammalian cells. As indicated above, targeting of the E. coli folylpolyglutamate synthetase to the mitochondria of these cells (AUX-coli-mcoli)

restored mitochondrial folate pools, indicating that mitochondrial folate accumulation is dependent on mitochondrial folylpolyglutamate synthetase activity. However, AUX-mcoli cells, which express E. coli folylpolyglutamate synthetase activity in their mitochondria but not in their cytosol, also possessed normal cytosolic and mitochondrial folate pools. Pulse chase experiments indicated that mitochondrial folylpolyglutamates can be released without prior hydrolysis and CHO transfectants expressing E. coli folylpolyglutamate synthetase activity solely in the mitochondria possessed normal cytosolic folylpolyglutamate pools (Fig. 3). In these cells mitochondrial folate accumulation did not appear to be limited by mitochondrial folate transport but was governed by competition between mitochondrial

FIG. 3. Model for mitochondrial and cytosolic accumulation of folylpolyglutamates.


101

and cytosolic folylpolyglutamate synthetase activity. Elevated expression of cytosolic folylpolyglutamate synthetase activity led to a mitochondrial folate deficiency which could only be overcome by expression of very high levels of mitochondrial folylpolyglutamate synthetase. Similar results were observed when AUXBl cells were transfected with a human folylpolyglutamate synthetase cDNA (Garrow et al., 1992). Expression of the human cDNA encoding a mature folylpolyglutamatesynthetase protein restored cytosolic folylpolyglutamate synthetase activity in these cells and overcame the cell’s requirement for thymidine and purines but the cells remained auxotrophic for glycine, reflecting the absence of a folate pool in the mitochondria (Table 11). Expression of a human cDNA encoding a folylpolyglutamate synthetase with a mitochondrial leader sequence restored folylpolyglutamate synthetase activity in the mitochondria and the cells contained normal mitochondrial folate pools and were prototrophic for glycine (Table 11). Although cells expressing human folylpolyglutamate synthetase solely in the mitochondria are prototrophic for thymidine and purines, which are synthesized in the cytosol, their cytosolic foate pools are quite small, which contrasts to the normal cytosolic folate pools observed in cells expressing E. coli folylpolyglutamate synthetase solely in the mitochondria. This probably reflects the slower movement of longer chain length folylpolyglutamatesout of the mitochondria. Although a cytosolic folylpolyglutamate synthetase isozyme is not absolutely required, it is needed for establishment of normal cytosolic folate pools. D. ROLE OF FOLYLPOLYGLUTAMATE SYNTHETASE LEVEL IN FOLATE ACCUMULATION Table 111 shows the effect of folylpolyglutamate synthetase activity on the accumulation of folate and methotrexate, an antifolate drug, by CHO cell transfectants expressing levels of human folylpolyglutamate synthetase ranging from 2 to 1400%of wild-type CHO cells. Tissue culture cells express TABLE I1 FOLATE AND FPCS DISITUBUTION AND GLYCME REQUIREMENT IN CHO TRANSFECTANTS

FPGS activity in

Folate in

FPGS DNA transfected

Cytosol

Mitochondria

Cytosol

Mitochondria

Glycine auxotrophy

None Human cDNA Human cDNA(mito)

No Yes No

No No Yes

No Yes Yes

No No Yes

Yes Yes No

102

PART I VITAMIN METABOLISM TABLE 111 FOLATE A N D METHOTREXATE ACCUMULATION BY CHO CELL TRANSFECTANTS

Folate or analog accumulation (pmoY1O6 cells) FPGS activity (pmoVhr/106cells)

cell

90 CHO-WT >2 AUX-h~man-0.02~ 6 AUX-hum-0.07 19 AUX-humn-0.21 71 AUX-h~m~-0.79 A U X - ~ U ~ M - ~ ~ X 1400

5 nM folinate

20 pM folinate

5 PM methotrexate

2.3 0.4 1.3 1.7 2.2 3.7

22.9 5.5 11 19 44 430

4.5 3.7 3.9 4.4 9.7 135

a AUX-humun-x are CHO cells expressing human FPGS activity, x representing activity relative 10 wild-type CHO cells.

higher levels of folylpolyglutamate synthetase than are normally found in mammalian tissue. The lower range of activity levels shown in this study are similar to enzyme levels found in mammalian tisues while the higher levels are more typical of some of the levels that have been observed in leukemia cells. At low to physiological medium folate concentrations (5 nM folinate), there was little effect of folylpolyglutamate synthetase activity on accumulation except in cells expressing very low levels of activity. Kinetic constants obtained with purified human folylpolyglutamate synthetase (unpublished data) were used to model folate accumulation and the derived data are compared with actual accumulation rates in Table IV. The data assume no competition between folate mono- and diglutamate TABLE IV FOUNATE

(2 nM)

ACCUMULATION BY CHO CELLS AND THEORETICAL TRANSPORT AND

POLYGLUTAMYLATION RATES

cell

CHO-WT CHO-h~m-0.02 CHO-human-O.07 CHO-hm-0.21 CHO-hwMn-0.79

Influx rate (calc) a

FPGS (pmoVhr)

Folate uptake (actual) (fmoVhr)

Polyglutamylation rate" (calculated) (ftnoYhr)

90 >2 6 19 71

35 12 30 47 40

640 >27 81 256 957

30 fmolhr

Calculated from kinetic constants of purified human FPGS.


103

substrates for chain elongation, and none would be expected at the intracellular folate concentrations obtained under these conditions (low intracellular folate). Triglutamate synthesis is required for folate retention and, at low intracellular folate concentrations, rates of conversion of pteroylmonoto diglutamate and diglutamate to triglutamate are similar. Pteroylmonoglutamate concentrations were calculated from that found in AUXBl cells or were analyzed directly. Theoretical polyglutamylation rates (cellular capacity to synthesize pteroyltriglutamates) exceeded accumulation rates except in cells expressing very low levels of folylpolyglutamate synthetase, and the actual accumulation rates were similar to the calculated folate influx rates under these conditions. Folate accumulation was limited by influx and was not responsive to folylpolyglutamate synthetase activity except in cells expressing lower levels of folylpolyglutamate synthetase, i.e., enzyme levels found in some mamalian tissues. As folate accumulation rates mirrored calculated influx rates, essentially all transported folate was metabolized to retained polyglutamate derivatives (Lowe et al., 1993). When cells were incubated with pharmacological levels of folinate (20 mM), which mimics the dosage used in some chemotherapeutic treatments, or with the antifolate methotrexate, which is a poor substrate for folylpolyglutamate synthetase, cellular accumulation was highly dependent on folylpolyglutamate synthetase activity (Table 111). CHO-WT cells accumulated about 3 to 5% as much methotrexate and folinate as cells overexpressing human folylpolyglutamate synthetase, indicating that, at a maximum, only 5% of transported folate or methotrexate was retained by the wild type cell. Modeling of folate accumulation under these conditions is shown in Table V. Although folinate accumulation was dependent on folylpolyglutamate synthetase activity, accumulation was significantly lower than the calculated polyglutamylation rates for conversion of monoglutamate to diglutamate and diglutamate to triglutamate. Folate accumulation requires conversion to the triglutamate and with high intracellular levels of folate, competition between entering pteroylmonoglutamate and cellular polyglutamates and diglutamate formed should decrease the rate of triglutamate synthesis, and this competition would be expected to increase as cellular folate levels increase. When this competition is modeled by using kinetic parameters for human folylpolyglutamate synthetase and cellular folate content at the start and end of the incubation with folinate, a range of rates for conversion of diglutamate to triglutamate, which should represent cellular accumulation rates, is obtained (Table V). For example, for cells expressing folylpolyglutamate synthetase at 20% of that in CHO-WT cells (CHO-humn-0.21, Table V), the initial pteroyltriglutamate formation rate would be 2.4 pmol/hr/106cells and would drop to 0.5 pmol/hr/106cells after the 22-hr incubation. This compares with an average accumulation rate of

104

PART I VITAMIN METABOLISM TABLE V

FOLINATE

(20 ph'f)ACCUMULATION

BY CHO CELLS AND THEORETICAL TRANSPORT AND

POLYGLUTAMYL.ATION RATES

Cell CHO-WT CH0-h~t~~t1-0.2 CHO-human-0.7 CHO-hum-0.21 CHO-human-0.79 CHO-~UPTUW-~~

Influx rate (calc)

FPGS (pmollhr)

90 >2 6 19 71

1600

Folate uptake (actual) (pmollhr) 0.58 0.03 0.21 1.10 2.18 22.2

Theoretical polyglutamylation rate (pmoVhr) 1 to 2"

2 to 3'

2 to 3 (inhib)*

60 >1.6 4.8 15.3 57.3

30 >0.16

0.04-0.03

1.4 7.3 37.2

0.39-0.14 2.4-0.5 14.5-2.0

28 pmollhr

'Calculated from kinetic constants of purified human FPGS. Triglutamate synthesisrate obtained from kinetic parameters if rate modified by inhibition by entering pteroylmonoglutamate and cellular folylpolyglutamate stores.

1.1 pmol/hr/106 cells over this period. Folate accumulation under these conditions can be modeled entirely by using kinetic parameters for folylpolyglutamate synthetase and is totally dependent on the level of folylpolyglutamate synthetase activity. Influx rates under these conditions are greater than 28 pmolhrll06 cells, and transport does not limit accumulation with high doses of folinate.

E. ROLE OF FOLYLPOLYGLUTAMATE SYNTHETASE IN ANTI-FOLATE CYTOTOXICITY Methotrexate accumulation was also dependent on folylpolyglutamate synthetase activity in these cells (Table 111). The cytotoxicity of methotrexate does not require its conversion to polyglutamates but cellular accumulation of the drug is highly dependent on folylpolyglutamate synthetase activity (Kim et al., 1993). There was no difference in sensitivity of the different transfectants to the drug when the cells were continuously exposed to the drug. However, cells expressing higher levels of folylpolyglutamate synthetase were much more sensitive to pulse exposure (4 hr) to methotrexate than cells with lower levels (Table VI). These studies clearly show that the level of folylpolyglutamate synthetase can be a major determinant of methotrexate cytotoxicity even though its target in the cell is dihydrofolate reductase and effective inhibition of the reductase does not require polyglutamylation of methotrexate.


105

TABLE VI EFFECT OF FPGS ACTIVITY ON SENSITIVITY OF CELLS TO METHOTREXATE

EDw, cells

72-hr methotrexate exposure (nM) 4-hr methotrexate exposure ( p M ) ~

CHO-WT AUX-human-0.02 AUX-human-O.07 AUX-human-0.21 AUX-humn-0.79

3-10 1-3 3-10 3-10 3-1 0

10-33 33-100 10-33 3-1 0 1-3

F. SUMMARY Model CHO cells obtained by transfecting CHO mutants with the E. coli and human folylpolyglutamate synthetase genes have proven useful for assessing the role of folylpolyglutamates in one carbon metabolism and for delineating how folate intracellular stores are regulated. Cells expressing enzymes in specific subcellular compartments, expressing enzymes with different substrate specificity’s, and expressing enzyme activity at different levels, all in a common background, in this case the CHO cell, has allowed the development of kinetic models for assessing the role of folylpolyglutamate synthetase in folate retention and in the cytotoxicity of antifolates. REFERENCES Garrow, T. A., Admon, A., and Shane, B. (1992). Expression cloning of a human cDNA encoding folylpoly-y-glutamate synthetase and determination of its primary structure. PrOC. Narl. ACUd. SC~.U.S.A. 89,9151-9155. Garrow, T. A., Brenner, A., Whitehead, V. M., Chen, X.-N., Duncan, R. G., Korenberg, J. R. and Shane, B. (1993). Cloning of human cDNAs encoding mitochondrial and cytosolic serine hydroxymethyltransferasesand chromosomal localization. I. Biol. Chem 268, 11910-11916. Kim, J.-S., Lowe, K. E., and Shane, B. (1993). Regulation of folate and one carbon metabolism in mammalian cells. 4. Role of folylpolyglutamate synthetase in antifolate metabolism and cytotoxicity. J. Bwl. Chem 268,21680-21685. Lin, B.-F., Huang, R.-F. S., and Shane, B. (1993). Regulation of folate and one carbon metabolism in mammalian cells. 3. Role of mitochondria1folylpolyglutamate synthetase. 1. Bwl. Chem. 268,21674-21679. Lin, B.-F., and Shane, B. (1994). Expression of Escherichia coli folylpolyglutamate synthetase in the Chinese hamster ovary cell mitochondrion. J. BioL Chem. 269,9705-9713. Lowe, K. E., Osborne, C. B., Lin, B.-F., Kim, J.-S., Hsu J.-C.,and Shane, B. (1993). Regulation of folate and one carbon metabolism in mammalian cells. 2. Effect of folylpolyglutamate

106


synthetase substrate specificity and level on folate metabolism and folylpolyglutamate specificity of metabolic cycles of one carbon metabolism. J. BioL Chem. 268,21665-21673. Osborne, C. B., Lowe, K. E., and Shane, B. (1993). Regulation of folate and one carbon metabolism in mammalian cells. 1. Folate metabolism in Chinese hamster ovary cells expressing Escherichia coli or human folylpoly-y-glutamate synthetase activity. J. Biol. Chem. 268,21657-21664. Shane, B. (1989). Folylpolyglutamate synthesis and role in the regulation of one carbon metabolism. Vitam. Horm. (N.Y.)45,263-335. ‘Taylor, R. T., and Hanna, M. L. (1977). Folate-dependent enzymes in cultured Chinese hamster ovary cells: Folylpolyglutamatesynthetase and its absence in mutants auxotrophic for glycine + adenosine + thymidine. Arch. Biochem. Bwphys. 181,331-344.

ADVANCES IN FOOD AND NUTRITION RESEARCH,VOL. 40

Chapter 6 MODELING VITAMIN B6 METABOLISM STEPHEN P. COBURN Department of Biochemistry Fort Wayne State Developmental Center Fort Wayne, Indiana 46835

1. Metabolism A. Metabolic Pathways B. Plasma C. Erythrocytes D. Tissue Distribution E. Excretion F. Pregnancy and Lactation 11. Kinetics A. Studies Using Radioactive Tracers B. Studies Using Stable Isotope Tracers C. Calculation of Vitamin B6 Requirement for Growth 111. Refining Models of Vitamin B6 Kinetics A. Improving the Agreement of the Parameter Estimates with Physiological Considerations B. Comparison of Single Bolus vs Continuous Dosing Protocols C. Influence of Altered Metabolic States Due to Fasting and Microgravity D. Comparisons between Metabolism of Pyridoxine, Pyridoxal, and Pyridoxamine E. Incorporation of Protein-Binding Calculations 1V. Conclusions References

1.

METABOLISM

A. METABOLIC PATHWAYS Vitamin B6 is one of the most versatile enzyme cofactors. Pyridoxal phosphate-containing proteins are found in each IUB enzyme category except ligases (category 6). [Tong and Davis (1995) reported that 2-amino3-ketobutyrate-CoA ligase is a pyridoxal phosphate enzyme. However, the 107 Copyright 8 1996 by Academic Press. Inc. All rights of reproduction in any form reserved.

108


EC number for the enzyme is 2.3.1.29,which is in category 2, transferases.] The 1992 edition of Enzyme Nomenclature (Nomenclature Committee of the International Union of Biochemistry and Molecular Biology, 1992) lists almost 120 pyridoxal phosphate-containing enzymes. Zubay (1988) has suggested that vitamin B6 catalyzes the rupture of a wider variety of chemical bonds than most other cofactors. The metabolic interconversions of vitamin B6 (Fig. 1) complicate the assessment of requirements and status as well as the development of models of vitamin B6 metabolism. Therefore, we will briefly review our current understanding of the biochemical and physiological pathways before discussing the models. In studies of B6 vitamer concentrations it is essential that the tissue be homogenized in a protein denaturing solution. Otherwise, there can be rapid shifts in the distribution of vitamers, particularly in liver (Coburn et al., 1988b). The metabolic sequence in liver (Colombini and McCoy, 1970; Johansson et al., 1974) appears to be that pyridoxine is phosphorylated to pyridoxine 5’-phosphate ( k l ) and then oxidized to pyridoxal 5’-phosphate (k2).It can then interchange with pyridoxamine 5’-phosphate ( l ~ 3 , k - ~ )Similarly, . pyridoxamine can be phosphorylated (k4) and then equilibrated with pyridoxal5‘phosphate (Johansson ef aZ., 1974). The rate constants (fraction/minute) proposed for these reactions in mouse liver are (Johansson ef al., 1974):

H

O CYOHO

~

T

CHaNH2 ~

~

~

HsC

CYOH

‘

/ pi:2

41

k:L::w CHaNHa

CYOH

H y c o H

PM

n

~~~~~

PN

N

N

PLP

PMP

{I H

O

O

T

~~~~

H3C PL

PA

FIG. 1. Proposed major pathway of vitamin B6 metabolism. PN, pyridoxine;PNP, pyridoxine 5’-phosphate;PM, pyridoxamine; PMP, pyridoxamine 5’-phosphate;PL, pyridoxal; PLP, pyridoxal 5’-phosphate; PA, 4-pyridoxic acid.

CHAPTER 6 MODELING VITAMIN B6 METABOLISM

109

kl = 0.07, kz = 0.11, k3 = 0.03, k-3 = 0.07 and k4 = 0.04. This study did not examine the hydrolysis of pyridoxal5'-phosphate to pyridoxal and the oxidation of pyridoxal to 4-pyridoxic acid. There was limited conversion of pyridoxine or pyridoxine 5'-phosphate to other compounds in perfused rat muscle (Buss et af.,1980), raising the possibility that earlier data suggesting significant direct conversion of pyridoxine to pyridoxal in the muscle of intact mice (Colombini and McCoy, 1970) might be an artifact. Hydrolysis of pyridoxal phosphate to pyridoxal followed by oxidation to 4-pyridoxic acid is the major catabolic pathway for vitamin B6 in most mammalian species. In cats, however, the major urinary metabolites are pyridoxine 3-sulfate and N-methylpyridoxine (Coburn and Mahuren, 1987). Also, in humans receiving very large vitamin B6 intakes excretion of 5pyridoxic acid may become significant (Mahuren et af., 1991). Modeling vitamin B6 metabolism is further complicated by the fact that the activity of the kinase, oxidase, and phosphatase enzymes varies between organs and species. A very simplified'diagram of vitamin B6 metabolism is shown in Fig. 2. In the intestine any phosphorylated forms are hydrolyzed. The free vitamers are readily taken up by diffusion into the intestinal wall where significant phosphorylation (Middleton, 1979) and other metabolism (Middleton, 1985) occurs. In mice small doses (up to 14nmol) of pyridoxine (Sakurai ef af., 1988) and pyridoxamine (Sakurai et al., 1992) were converted almost completely to pyridoxal before being released into the portal circulation. While it is clear that the intestinal microflora produce vitamin B6,

I t PERIPHERALnssuEs

PUSMA

FIG. 2. Schematic illustration of whole-body metabolism of vitamin B6. For clarity, the reaction sequences within each tissue are not shown.

110


tracer studies found little difference in isotope dilution between conventional and germ-free rats, suggesting that animals receiving adequate vitamin B6 intake do not utilize significant amounts of microbially produced vitamin B6 (Coburn et al., 1989a). B. PLASMA Because the polar nature of pyridoxal phosphate prevents it from readily crossing membranes, the primary source of pyridoxal phosphate in plasma appears to be secretion by the liver, presumably as a protein complex (Lumeng et al., 1974). After a small oral dose the vitamer concentration in human plasma usually peaks in 1-2 hr (Benson ef al., 1994; Contractor and Shane, 1970). With large doses the pyridoxal phosphate concentration may not peak until much later (Ubbink et al., 1987). In normal individuals plasma pyridoxal phosphate concentrations do not usually exceed lo00 nmolfliter even at high vitamin B6 intakes (Coburn et al., 1983; Bhagavan er al., 1975). However, they can go higher under conditions such as Down’s syndrome (Coburn et al., 1983; Bhagavan et al., 1975) or hypophosphatasia (Whyte et al., 1985) in which the regulatory processes are modified. There appears to be no limit to the concentrations of pyridoxal and pyridoxic acid in plasma (Coburn et al., 1983). The distribution volume of intravenously administered pyridoxal phosphate is about twice the plasma volume (Lumeng et al., 1974). This could reflect binding to the walls of the vascular system or equilibration of protein bound pyridoxal phosphate with an interstitial pool. Clearance of pyridoxal phosphate from plasma has been examined in the rat (Bode and van den Berg, 1991), dog (Lumeng er al., 1984), pig (Coburn et al., 1992a), goat (Coburn et al., 1992a), and human (Lui et al., 1985). Bode et al. (1987) found that although plasma pyridoxal phosphate kinetics in the rat followed a biexponential curve, the kinetics could be described better by a three-compartment system with a saturable reentry process than by a two-compartment open model. Perhaps more relevant than the clearance values is the amount of pyridoxal phosphate removed per unit time. Based on data from our laboratory plus other sources (Coburn ef al., 1984b; Coburn and Mahuren, 1983) the values were estimated to be 3.6 (human), 3.8 (pig), 5 (goat), 25 (dog), and 123 (rat) nmolkg body wt/ hr. The values for the dog are subject to considerable error because the estimated plasma pyridoxal phosphate value (Coburn et al., 1984b) had a very large standard deviation, probably due to variations in vitamin B-6 intake. Therefore, the value might well be comparable to that for humans, pigs, and goats. The reason for the higher value for rats probably reflects a higher metabolic rate. Veitch et al. (1975) reported that perfused rat liver


111

released pyridoxal phosphate at a rate of 2.4 nmol/hr/g wet wt. Assuming that liver is 3% of the body wt (Coburn et al., 1988b), that would amount to 72 nmol/hr/kg body wt, which is reasonably comparable to the clearance reported above. Sorrell et al. (1974) reported that vitamin B6 was released from perfused rat liver at a rate of only 0.1-0.2 nmol/hr/g. The lower value may be due in part to the fact that Sorrell er al. (1974) deliberately avoided including albumin in the perfusing medium because of its ability to complex vitamins while Veitch et af. (1975) did use albumin. Lack of albumin may have allowed some of the secreted pyridoxal phosphate to be removed by the liver since both groups recirculated the perfusing medium during the experiments. The role of protein binding in protecting pyridoxal phosphate from hydrolysis is well recognized (Li et al., 1974; Lumeng et al., 1974). The higher release value is more in line with the observed clearance. In addition, assuming that formation of new tissue requires 15 nmol vitamin B6/g (Coburn, 1990,1994) and that rats grow about 14%/day (Lumeng et af., 1978), the new tissue in a 50-g rat would require 105 nmol/day. If pyridoxal phosphate were the only source of vitamin B-6 and liver were about 3% of the body wt, it would have to release about 3 nmol/hr/g to meet the growth requirement. Therefore, the high release and clearance values observed in the rat appear compatible with other metabolic measures. This is another example of the marked interspecies differences in vitamin B-6 metabolism. The importance of alkaline phosphatase in regulating pyridoxal phosphate concentrations in plasma is indicated by the decreased pyridoxal phosphate observed in liver disease characterized by increased alkaline phosphatase (Labadarios et al., 1977) and the high concentrations of pyridoxal phosphate found in hypophosphatasia (low alkaline phosphatase) (Whyte et al., 1985). The relative importance of pyridoxal and pyridoxal phosphate in vitamin B6 transport to peripheral tissues remains uncertain. Limited evidence in pigs and goats suggests that tissues take up approximately equal amounts of pyridoxal and pyridoxal phosphate (Coburn et al., 1992a). C. ERYTHROCYTES The role of erythrocytes in vitamin B6 metabolism remains uncertain. Mouse and human erythrocytes have higher oxidase activity and, therefore, convert pyridoxine to pyridoxal phosphate appreciably faster than erythrocytes from rat, hamster, and rabbit (Fonda, 1988). Anemic rats showed increased urinary loss of label administered as pyridoxal, suggesting that uptake by erythrocytes may conserve pyridoxal (Ink and Henderson, 1984).

112


D. TISSUE DISTRIBUTION Seven days or more after administration of labeled pyridoxine to mice (Colombini and McCoy, 1970 Dahlkvist et al., 1969), rats (Coburn et al., 1988b), and miniature pigs (Coburn et al., 1985) 70-80% of the total label in the body was located in muscle, 10-20% in liver, and the remainder in other tissues. The percentage in the liver was highest in mice. The uptake and turnover of label were quite high in liver and low in brain (Colombini and McCoy, 1970; Dahlkvist et af., 1969).

E. EXCRETION While significant amounts of vitamin B6 appear in the feces, label from B6 vitamers is excreted almost exclusively in the urine (Cox et al., 1962; Tillotson et al., 1967). Pyridoxic acid is the major urinary metabolite in many species. Excretion of other vitamin B6 compounds is minimal with normal vitamin B-6 intakes but increases rapidly with larger intakes. Biliary excretion is minimal in both the rat (Lui er af., 1983) and the chicken (Heard and Annison, 1986). Metabolic balance studies in humans also suggested that urinary excretion was almost the sole route of excretion (Lui et al., 1985). Therefore, it appears that urinary excretion of pyridoxic acid may be a good indicator of absorbed vitamin B6. However, with natural diets urinary pyridoxic acid may account for only 50% of the vitamin B-6 intake (Lindberg et al., 1983), suggesting decreased bioavailability in natural foods possibly as a result of the occurrence of glycosidic derivatives (Gregory et al., 1991). F. PREGNANCY AND LACTATION Pregnancy and lactation pose some difficult challenges for modeling vitamin B6 metabolism. Pyridoxal phosphate concentrations in plasma decline during pregnancy. Much of this decline appears to be correlated with the increased activity of placental alkaline phosphatase. Barnard et al. (1987) found that pyridoxal concentrations in pregnant women increased to compensate for the decline in pyridoxal phosphate. In the course of testing our chromatographic method for measuring vitamin B6 on a variety of samples, we happened to include goat milk, which contains a very high concentration of pyridoxal phosphate (Coburn and Mahuren, 1983). We then noticed that bovine milk contained about the same total vitamin B-6 content but had a lower concentration of pyridoxal phosphate. Further study revealed a strong inverse relationship between the pyridoxal phosphate content of milk and the alkaline phosphatase


113

activity in goats, cattle, swine, dogs, and rats (Coburn et af., 1992b). Since it is usually assumed that free pyridoxal phosphate cannot cross membranes because it is charged, the assumption is that in these species significant amounts of pyridoxal phosphate are transported into the milk, probably bound to protein. Surprisingly, human milk fails to show any relationship between alkaline phosphatase and pyridoxal phosphate content. Both compounds are found only in small amounts in human milk. This suggests that human mammary tissue cannot transport pyridoxal phosphate efficiently. Further work is needed to clarify the mechanisms involved in these processes. While human mammary tissue apparently has limited ability to transport pyridoxal phosphate, about 23% of the label appearing in the fetal compartment was in the form of pyridoxal phosphate when human placenta was perfused with labeled pyridoxal (Schenker et al., 1992). Perfusion with pyridoxal phosphate resulted in limited transport unless the perfusate was recirculated, thus allowing hydrolysis to pyridoxal. The appearance of pyridoxal phosphate on the fetal side during perfusion with pyridoxal suggests that the pyridoxal was phosphorylated and secreted into the fetal circulation presumably bound to protein. Another interesting aspect of vitamin B6 secretion in ruminant milk is the large quantity of vitamin B6 involved. If we assume that a cow can produce 40 kg milk/day and that the total vitamin B6 content is about 2 pmolfliter, then the total daily output of vitamin B6 would be about 80 pmol or 16 mg. Further work is needed to determine whether this large influx of vitamin B6 is uniformly distributed or selectively processed by the mammary tissue. 11.

KINETICS

Our interest in vitamin B6 was stimulated by reports that vitamin B6 metabolism was altered in Down’s syndrome (McCoy ef af., 1969). Since Down’s syndrome is associated with trisomy of chromosome 21, it seemed most likely that we would be dealing with altered rates of metabolism. Therefore, we have been attempting to examine the kinetic aspects of vitamin B6 metabolism. There have been three basic approaches to examining the kinetics of vitamin B6 metabolism. One has been to examine individual enzymes (Merrill and Henderson, 1990) or tissues (Middleton, 1985; Mehansho and Henderson, 1980; Mehansho et af., 1979, 1980; Hamm et al., 1979, 1980; Buss et al., 1980). A second has been to examine the changes with time after administration of unlabeled (Ubbink et al., 1987; Hamaker et af., 1990;

114


Chang and Kirksey, 1990; Wozenski et al., 1980; Lui et al., 1985; Spannuth er al., 1977; Contractor and Shane, 1970; Bode and van den Berg, 1991; Ubbink and Serfontein, 1988; Speitling et aL, 1990; Kant et al., 1988; Bode er al., 1987) or labeled vitamin B-6 (Butler et al., 1985; Fonda and Eggers, 1980; Fonda er al., 1980; Bode et al., 1992) to intact animals or humans. In some of these cases noncompartmental analysis was used to estimate clearances and distribution volumes. A discussion of the assumptions and calculations for noncompartmental analysis is presented by Wolfe (1992). The third approach has been to gain some insights into the metabolic state of vitamin B6 through the use of compartmental analysis. A. STUDIES USING RADIOACTIVE TRACERS

The first attempts at compartmental analysis were based on following the excretion of label in the urine after administration of a single dose of labeled pyridoxine ( Johansson er al., 1966a,b). The excretion curves were biphasic, suggesting that at least two pools were involved. Johansson et al. (1966b) proposed a two-pool model with a small, rapid turnover pool in equilibrium with a large, slow turnover pool (Fig. 3). They assumed that excretion occurred only from the small pool and that the specific activity of the urinary pyridoxic acid was equal to the specific activity of the small pool. These assumptions allowed them to make some estimates of the rate constants and pool sizes. The predicted total pool in 250-g rats after an intraperitoneal dose was about 3-5 mg (Johansson er al., 1966a; Johansson and Tiselius, 1973) or 50-100 nmoYg. This is much higher than the 16 nmol/ g found by direct measurement (Coburn et al., 1988b). At least part of the discrepancy probably results from the fact that the line used by Johansson er al. to estimate the initial specific activity value appears to decline more slowly than the data. This would lead to an overestimate of the pool size.

i

kg

FIG. 3. Two-pool model of vitamin B6 metabolism.


115

Also, the isotope retention curves and estimates of the rate constants appear to be very sensitive to the dose. It is usually assumed, particularly in radioisotope studies, that the amount of tracer is much smaller than the pool size. Johansson et al. (1966a; Johansson and Tiselius, 1973) used doses of 5 to 200 pg of pyridoxine, which they acknowledged would constitute a significant fraction of the pool (Johansson et al., 1966a). This may partially explain why the rate constant(k3) for excretion from the small pool averages 0.60 when calculated from their data for the 20-pg dose compared with 2.3-4.1 based on the 200-pg dose (Johansson and Tiselius, 1973). The two-pool model was also applied to humans (Johansson et al., 1966b; Tillotson et al., 1967; Shane, 1970). The predicted total body pools ranged from 107 to 190 pmol following oral administration of label (Johansson et al., 1966b; Tillotson et al., 1966) compared with 345 to 725 pmol following intravenous administration ( Johansson et al., 1966b; Shane, 1970). The marked influence of the route of administration results from the difference in the distribution of oral and intravenous doses. One hour after placing pyridoxine in a jejunal loop in rats 81% of the dose was recovered in the liver (Serebro et al., 1966) while only about 3% of an intravenous dose in mice was found in the liver at that time (Colombini and McCoy, 1970). Similarly, in rats the half-lives for the fast and slow portions of the isotope retention curves were about 30 min and 10 hr, respectively, after an intraperitoneal dose compared with 80 min and 9 days after intramuscular administration. The difference between the oral and intravenous results reflects the importance of the intestinal tract and liver in metabolizing vitamin B6. Label administered orally or intraperitoneally enters the portal circulation where it can be taken up and oxidized by the liver. As noted above, oxidation may also occur in the intestinal wall. Only a portion of the label injected into the muscle or peripheral circulation will reach the liver on the first pass. However, the fact that even the highest predicted values for the body pool were less than the lo00 pmol which we had found through muscle biopsies (Coburn et aL, 1988a) suggested that the two-pool model was not completely appropriate. The underestimation of the human pool suggested the existence of a pool with a turnover too slow to be detected by the short-term, single dose protocol. Further evidence of such a pool was the inability to achieve uniform labeling of vitamin B6 pools after daily administration of labeled pyridoxine for 73 days in adult rats (Coburn et al., 1989a), 120 days in guinea pigs (Coburn et al., 1984a), and 230 days in miniature swine (Coburn et al., 1985). This large, slow turnover pool is most likely in muscle, which contains about 70-80% of the vitamin B6 in the body (Coburn et al., 1985,1988b).The muscle pool is associated primarily with glycogen phosphorylase (Butler et al., 1985). Using pyridoxal phosphate as an indicator of glycogen phospho-

116


rylase turnover in muscle Butler et al. (1985) estimated the half-life at 11.9 days in mice. Turnover in humans consuming normal diets was estimated at 1-4% of the body pool per day (Johansson et af., 1966b; Tillotson el af., 1967). However, when men were restricted to a vitamin B6 intake of only 1.76 pmoVday, pyridoxic acid excretion declined to an amount approximately equal to the intake, thus establishing a new steady state with a net loss of only about 4% of the body pool and no statistically significant decline in vitamin B6 in muscle (Coburn et al., 1991).

B. STUDIES USING STABLE ISOTOPE TRACERS Although the previous tracer studies in humans utilized radioactive compounds, we felt that stable isotopes would allow us to conduct a wider variety of studies. Deuterium was chosen over carbon-13 because of its lower cost. Methods for incorporating deuterium into the metabolically stable 2-methyl and/or the 5-methylene groups were developed (Coburn et al., 1982). This gave us the capability of labeling vitamin B-6 compounds with two, three, or five deuterium atoms. By selecting routes which utilized inexpensive sources of deuterium, the compounds can be synthesized at a cost for labeled materials of less than $lO/g and $250/g for the D2 and D5 forms, respectively. Using these compounds demonstrated that the halflife for labeling of urinary pyridoxic acid in the subjects receiving a restricted vitamin B-6 intake was about 75 days and the half-life of the large pool was estimated at about 1700 days (Pauly et al., 1991). The vitamin B-6 content of muscle in vitamin B-6 deficient rats was also conserved until the deficiency became so severe that the rats started to lose weight (Black et af., 1978). The studies with 14C-pyridoxinementioned above led Johansson and Tiselius (1973) to conclude that the elimination of vitamin B6 from the body reservoir is slow and is not increased by omitting pyridoxine from the diet. Of particular interest is the observation that retention of isotope was increased to a similar degree whether rats had received a vitamin B6 deficient diet for 24 hr or 6 months prior to administration of tracer ( Johansson et af.. 1966a). Therefore, in contrast to the common assumption that water-soluble vitamins are readily flushed from the body, vitamin B6 can be conserved during periods of low intake. C. CALCULATION OF VITAMIN B6 REQUIREMENT FOR GROWTH If tissues can efficiently conserve vitamin B6, we postulated that the main requirement of vitamin B6 during growth might be to supply the new tissue. As noted above, the overall body vitamin B6 content of several species


117

averages about 15 nmollg. We found that multiplying 15 nmoVg times the grams of daily gain yielded a reasonable agreement with experimentally observed requirement data in species ranging from fish to man (Coburn, 1990,1994). The major exceptions were primates, where data are limited, and prawn, where there may be significant losses due to leaching during feeding. The calculation described above does not include any compensation for protein intake. Protein intake does influence some measures of vitamin B6 requirement in some circumstances. However, it is interesting to note that the calculation does give good results for carnivores such as cats and fish without any adjustments for their high protein intake. We suspect that in at least some circumstances large increases in protein intake in noncarnivores may force the animal to metabolize an unusually large amount of amino acids for energy and/or other purposes, thus requiring increased activity of aminotransferases and other amino acid processing enzymes with a resulting increase in the vitamin B-6 requirement. The fact that growth in animals receiving high protein intakes is often below maximal levels even with vitamin B-6 supplementation suggests that in some cases the high protein intake significantly alters processes in addition to vitamin B-6 requirement. More study of these interactions is needed. 111.

REFINING MODELS OF VITAMIN B6 KINETICS

The conservation of vitamin B6 during low intake has several consequences for modeling vitamin B6 metabolism. First, only 4% of the body pool was lost; and yet, urinary excretion of pyridoxic acid declined to about 10% of baseline values (Coburn et af., 1991). It is usually assumed that pyridoxic acid arises from the oxidation of pyridoxal. While pyridoxal phosphate concentrations in plasma did decline to about 10%of baseline values, pyridoxal concentrations declined only to 50% of baseline values (Coburn et af., 1991). Michaelis-Menten kinetics cannot explain such a large drop in pyridoxic acid excretion without a comparable drop in pyridoxal concentrations. However, a large drop in reaction rate with a relatively small change in total substrate could be achieved if there is binding or compartmentalization of the substrate. The role of protein binding in the metabolism of pyridoxal phosphate has long been recognized (Lumeng et af., 1974). These data suggest that some type of binding may also be involved in pyridoxal metabolism. A second consequence of these data is that the turnover of the vitamin B6 pool appears to be dependent on vitamin B6 intake. The original turnover estimates of l%/day were obtained with normal vitamin B6 intakes, which

118


are about 1%of the body pool/day. A half-life of 1700 days with an intake of 1.76 pmoYday is not unreasonable when one considers that it would take at least 284 days to replace half of the loo0 pmol pool, assuming 100% retention of label. A third consequence is the question of whether glycogen phosphorylase conserves vitamin B6 because it has a very low rate of degradation or because it recycles the pyridoxal phosphate very efficiently. Based on the rapid decay of the free vitamin B6 pool in muscle, the slow turnover of the protein-bound vitamin B6 pool, the failure to detect apo-phosphorylase, and the agreement between turnover rates based on amino acid or vitamin B6 data, Beynon et nl. (1986) concluded that recycling would be minimal. We feel that the effect of vitamin B6 intake on turnover rates tends to favor the recycling option under conditions of limited intake. However, more data are needed before a final decision can be made. The model of Johansson er al. (1966b) assumes that excretion occurs only from the small pool and that the specific activity of the urinary pyridoxic acid is equal to the specific activity of the small pool. One assumption of this model which may not be valid is that the input pool is identical to the output pool. The model also requires recycling. It could be written to exclude recycling by allowing excretion directly from the large pool in addition to excretion from the small pool. (The concentration of isotope appearing in the urine is too high to limit excretion solely to the large pool.) The disadvantage of allowing excretion from both pools is that the model becomes nonidentifiable if only urine data are available. Therefore, it is important to recognize that mathematical models are usually imperfect attempts to describe complex metabolic systems. Certain compromises may be made in order to obtain preliminary estimates of selected parameters. A. IMPROVING THE AGREEMENT OF THE PARAMETER ESTIMATES WITH PHYSIOLOGICAL OBSERVATIONS

Experimental data never fit the model precisely. Often a variety of parameter estimates may give similarly good fits to the model and some of the predicted characteristics may be inconsistent with the physiological observations (Coburn et nl., 1985). In the two-pool model used by Johansson et al., the urinary excretion of isotope after a single bolus will follow an equation of the general form

Y Johansson et al. showed that

=

Me-mT+ Ne-"T.


119

kl=m+n-kz-kjor kl + k2 = m + n - k3

(1)

k2 = mn/k3 or m = k2k3/n. Substituting Eq. (2) in Eq. (1) and rearranging yields

kl

+ k2 = (k2k3 + n2 - kga)/n.

(3)

The fractional turnover rate of the total pool equals k2k3/(kl + k2) or k A / ( A + B ) , where A and B represent the pool sizes. Therefore,

k2k3/(kl+ k2) = k&(A + B ) or k2/(kl + k2) = A/(A + B ) .

(4)

Substituting Eq. (3) for (k, + k2) in Eq. (4)and solving for k2 yields

k2 = (n2A - nkgl)/(n(A+ B ) - k d ) .

(5)

At the steady state the excretion, k d , must equal the input (I).Therefore, Eq. ( 5 ) can be written as k2

=

(n2A- nZ)/(n(A + B ) - I ) .

(6)

Equation (6) defines the fractional turnover rate(k2) for the large pool (B) in terms of the exponent (n)for the slow phase of the curve, the small pool ( A ) , the large pool (B), and the intake ( I ) . In the case of normal vitamin B6 intake, 15 nmol/g times g body wt usually yields a reasonable estimate of the total pool. Since muscle usually contains 70-80% of the pool, muscle biopsies provide a means of verifying the pool size. The input and/or excretion can usually be measured. The exponent can be obtained by fitting a curve to the data. The size of pool A must be estimated. However, assuming that A is small relative to B and that n is small, variations in A have relatively little effect on the estimates of k2.Therefore, Eq. (6) provides a means for making kz more consistent with physiological observations. B. COMPARISON OF SINGLE BOLUS VS CONTINUOUS DOSING PROTOCOLS Since we had been unable to achieve uniform labeling of vitamin B6 pools by daily administration of labeled pyridoxine to adult animals, we

120


started with weanling rats on the assumption that if they were fed only I4C-labeled pyridoxine, by the time the rats reached adult size the vitamin B6 pools would be over 90% labeled even if none of the vitamin B6 present at weaning were lost. After 130days the specific activity of urinary pyridoxic acid was essentially equal to the specific activity of the ingested material (Coburn et al., 1988b). At that time half of the animals were continued on the same vitamin B6 intake except that it was labeled with ' H instead of 14C. The other half were given a single l-pmol dose of ['HI-pyridoxine followed by a daily dose of unlabeled pyridoxine. The animals were sacrificed at intervals over the following 120 days. This allowed us to determine whether the decline in the 'H content of the muscle after a single dose of label accurately reflected the decline of the 14Cwhich had been incorporated over 130 days. The curves for the decline in 14C,decline in 3H in the single dose rats, and the rise in 3H in the daily dose rats could all be fit by the same exponents, indicating that the daily dose and single dose protocols both accurately reflected the turnover of the vitamin B6 pool in muscle (Coburn et af., 1989b). In addition, after the initial washout of the rapid turnover pools the decline in the specific activity of pyridoxic acid in the urine of the single dose rats was parallel to the decline of the 3H content of the muscle of those rats. This suggests that a single bolus of labeled pyridoxine followed by examining urinary pyridoxic acid can be used to evaluate the turnover of vitamin B-6 in muscle, noninvasively. C. INFLUENCE OF ALTERED METABOLIC STATES DUE TO FASTING AND MICROGRAVITY As noted above, the body is able to conserve vitamin B-6 quite efficiently when vitamin B6 is the only limiting nutrient. What would happen during fasting? We had the opportunity to analyze plasma and urine samples collected during a 21-day fast during which volunteers consumed only water (Coburn el al., 1995). Plasma pyridoxal phosphate and urinary excretion of pyridoxic acid for the fasting group were similar to the low vitamin B6 group for about 10 days. Then the values in the fasting group tended to return to normal while those in the low vitamin B-6 group continued to decline. The most likely explanation is that as the fasting group started to lose weight vitamin B-6 was released from the muscle. The above discussion clearly illustrates the importance of muscle in vitamin B6 metabolism under altered nutrient intake. What would happen to vitamin B6 metabolism if muscle metabolism were altered without altering vitamin B6 intake. Such circumstances exist in a microgravity environment or under conditions of prolonged bed rest. Muscle mass decreases even though nutrient intake is adequate. We had an opportunity to measure


121

pyridoxic acid excretion in subjects undergoing 17 weeks of bed rest (Coburn er al., 1995). In this case there was lean tissue loss of about 4% but no loss of body weight (LeBlanc ef al., 1992). Pyridoxic acid excretion increased about 10% during the bed rest period. The loss of a higher percentage of pyridoxic acid than lean tissue raised the possibility of selective loss of vitamin B-6-containingproteins such as glycogen phosphorylase. In fact, based on fiber length glycogen phosphorylase activity did decline about 50% in fast glycolytic fibers of rats subjected to space flight compared with ground-based controls, leading Manchester et af. (1990) to speculate that the fibers might be shifting to a low glycogenolytic type. Although increased pyridoxic acid excretion was detected in the first week of bed rest in the 17-week study, no increase in pyridoxic acid excretion was observed in a second 14-daybed rest study (Coburn ef al., 1995). The reason for the discrepancy between the two studies is not known at this time but may be related to differences in vitamin B-6 intake and/or protein metabolism. D. COMPARISONS BETWEEN METABOLISM OF PYRIDOXINE, PYRIDOXAL, AND PYRIDOXAMINE There is little information on the relative rates of pyridoxine, pyridoxamine, and pyridoxal metabolism. Wozenski et al. (1980) concluded that pyridoxamine might be metabolized slower than pyridoxine. Pyridoxal may be metabolized faster than pyridoxine (Shane, 1970). Having access to three labeled forms of vitamin B6 allowed us to compare the metabolism of pyridoxine, pyridoxal, and pyridoxamine, simultaneously, in humans (Benson ef d.,1994). The protocol involved drinking a solution containing 5 pmol each of D2-pyridoxamine, D3-pyridoxal, and D5-pyridoxine. Urine samples were collected at intervals over the following 8 hr and the appearance of the three vitamers in pyridoxic acid was determined. About 44% of the pyridoxal dose was excreted compared with about 14%of the pyridoxine and pyridoxamine. These data have two important implications for modeling vitamin B6 in humans. First, while small doses of pyridoxine and pyridoxamine in mice (Sakurai ef al., 1988,1991, 1992) and pigs (Coburn et af., 1994) appear to be converted completely to pyridoxal (presumably via phosphorylation, oxidation, and hydrolysis) in the intestinal wall before entering the portal circulation, to date we have been unable to develop a model which can explain our human data simply by delaying the appearance of pyridoxine and pyridoxamine in the portal circulation as pyridoxal. If these two compounds were converted completely to pyridoxal in the intestinal wall and if pyridoxic acid is formed primarily in the liver, then pyridoxine and pyridoxamine would have to be lost to the same extent as pyridoxal

122


although at a slightly later time due to the extra oxidation steps. The fact that they did not follow the pyridoxal excretion curve suggests that in humans pyridoxine and pyridoxamine had an opportunity to enter pools that were not immediately equilibrated with pyridoxal. We interpret these data to suggest that pyridoxal enters the liver where much of it is immediately oxidized to pyridoxic acid while pyridoxine and pyridoxamine are phosphorylated and oxidized to pyridoxal phosphate which can enter a variety of pools and thus is somewhat protected from immediate oxidation to pyridoxic acid.

E. INCORPORATION OF PROTEIN-BINDING CALCULATIONS The second important feature of the triple label studies is the increase in endogenous pyridoxic acid excretion from 3 to 6 hr compared with 2 to 3 hr (Fig. 4). (The larger excretions after 6 hr reflect longer collection periods.) If there is no limitation on pool sizes, increasing the pool size by adding isotope will increase total loss from the pools because labeled as well as unlabeled material is being lost. However, loss of unlabeled material will increase only if there is a limit on pool sizes. This flushing effect has been recognized since the earliest tracer studies of vitamin B6 metabolism (Brain and Booth, 1964; Johansson et al., 1966b). One way to cause such

-

A

0

E,

1.00-1 D2 PA (from PM) - .I 03 PA (from PL) Q

1

;0.75 : -

.c

-

0

P)

D5 PA (from PN)

0

a

A+' -

0

.., ......

...

0.50

w

-------------------------_---........................

- * DOPA(endog.)

'.-..A

q..

,

0

0

0

............................................

t

2

3

4

5

6

7

'

t a

'

' 9

'

?

-

10

Time of Collection (hours after breakfast) FIG. 4. Appearance of label from pyridoxine, pyridoxamine, and pyridoxal in urinary pyridoxic acid after simultaneous oral administration of 5 pmol of a deuterated form of each

vitamer to a healthy man. Lines are predictions of the model described in Fig. 5 and Table I. (Analysis and figure were produced by SAAM31 from the Laboratory of Mathematical Biology, National Cancer institute, Bethesda, MD 20892.)


123

an effect is through the use of binding sites. If the binding sites are normally almost filled and if losses occur only from the unbound material, then a small increase in the total pool can produce a large increase in the free pool with a consequent increase in excretion. For example, the intrinsic dissociation constant, Ki, for binding of a ligand to a protein assuming no interaction between sites is (Stenesh, 1993):

Ki = [vacant sites][free ligand]/[bound sites]. Rearranging, [free ligand] = Ki [bound sites]/[vacant sites], since [vacant sites] = [total sites] - [bound sites], then [free ligand] = Ki [bound sites]/([total sites] - [bound sites]) [bound sites] = [bound ligand] [total ligand] = [free ligand] + [bound ligand].

(7)

Therefore, [bound ligand] = [total ligand] - [free ligand]

(8)

Substituting Eq. (8) into Eq. (7) and solving for [free ligand] yields a quadratic equation in terms of Ki, [total binding sites], and [total ligand]; all of which it may be possible to measure or estimate. The solution to the equation is [free ligand] = ([total ligand] - Ki - [total binding sites] 2 ((([total binding sites] + Ki - [total ligand])**2) + 4*Ki*[total ligand]) **0.5)/2. One of the simplest models which gives a reasonable approximation of our data is shown in Fig. 5. As noted above, pyridoxamine and pyridoxine must enter at a location different from pyridoxal. For simplicity, the path from pool 3 to 4 to 2 is shown as unidirectional. While the model is not identifiable from urine data alone, there are some constraints. The size of

124


Iniest.PM

FIG. 5. Preliminary model to fit appearance of label from pyridoxal. pyridoxine, and pyridoxamine in urine as pyridoxic acid (Figure produced by SAAM I1 from the SAAM Institute, IJniversity of Washington, Seattle, W A 98195.)

pool 2, which is controlled by k(7,2), is restricted by the highest D 2 D 0 ratio observed. The turnover rates and steady-state pool sizes are shown in Table I. k(2,4) incorporates the binding equations described above. This requires estimates of the binding constant and the number of binding sites. A wide variety of combinations of these values will fit the model as long as the resulting flux through pool 4 is about 0.4-0.7 pmolhr. We have arbitrarily utilized the smallest number of binding sites (30 pmol) which gave a reasonably good fit to the data (Fig. 4). The dissociation constant was 0.15. These values yield a steady-state size of 32.6 pmol for pool 4. The fact that a slow pool of 32.6 pmol is adequate to model this shortterm experiment does not limit the total body pool to that size. It has proven difficult to get an optimal fit to both the exogenous and endogenous data simultaneously, probably as a result of the complexities of vitamin B6 metabolism. Our next goal is to add plasma and the large, slow turnover pool so that the model will be able to simulate both short-term and longterm metabolism as well as distinguish between oral and intravenous dosing. As noted earlier, our data suggested that a 50% decrease in pyridoxal concentration might result in a 90% decrease in pyridoxic acid excretion (Coburn el al., 1991).The pyridoxal data suggested that the maximum pool


125

TABLE I PARAMETERS AND STEADY-STATE POOL SIZES

FOR THE MODEL SHOWN IN FIG. 5 AS APPLIED TO THE DATA

Path

IN FIG. 4

Fractional turnover rates (pmoUhr) 1.6 0.23 0.21 1.3 0.98 0.98 1.3 0.84

Steady-state pool sizes with a vitamin 8 6 intake of 0.357 pmoVhr Pool 2 3 4

pnol 0.4

0.6 32.6

that ingested pyridoxal encounters before some of it is oxidized is about 2.5 pmol. These observations allow the following series of equations to be developed. Let

Fl = [free ligand] under normal vitamin B6 intake, B1 = [bound sites] = [bound ligand] under normal vitamin B-6 intake, T = [total binding sites], Kd = dissociation constant, F2 = [free ligand] under low vitamin B-6 intake, and B2 = [bound sites] = [bound ligand] under low vitamin B6 intake. Then,

and

F2 = KdB2/(T-B2). Since we are assuming that low vitamin B-6 intake reduces the pyridoxal pool by half

126


Since we are assuming that low B6 intake reduces free pyridoxal by 90%

Using these equations we can solve for Fl in terms of Kd and T yielding a quadratic equation with the solution Fi = (-0.84Kd + 0.15T f. (((0.84Kd - 0.15T)**2) - (4*0.04*0.4*&*T))**0.5)/(2*0.04).

Assuming that Kd is 0.1 and T is 2.5, one set of solutions is Fi = 0.361 B1 = 2.215

F2 = 0.0361 B2

=

1.252

These results meet the stipulation that a 50% decline in the total pool causes a 90% decline in the free ligand. These equations can easily be incorporated into the model and updated as more accurate data on binding constants and pool sizes become available. One key question which must be considered when incorporating such equations into models is the rate of equilibration. The above equations assume instantaneous equilibration of bound and free ligand. One alternative is that equilibration requires a significant amount of time. Another possibility is that sites become available only with the turnover of protein. The latter situation would be more likely with pyridoxal phosphate than pyridoxal. If equilibration is not instantaneous, the model may have to include separate compartments for free and bound material with rate constants governing the exchange. In the case of slow equilibration, the binding rate might be set equal to X and the release rate set equal to X*F/B, where F is [free ligand] and B is [bound ligand] calculated as above. This assures that the relative pool sizes are compatible with the binding equation. Modeling programs such as SAAM (available from the Laboratory of Mathematical Biology, National Cancer Institute, Bethesda, MD 20892) can be written so that these values are recalculated continuously throughout the experiment to adjust for changing conditions. This allows simulation of a variety of circumstances in addition to steady-state conditions. If it is desired to limit turnover of ligand to turnover of protein, input into the bound pool can be limited to a specified number of binding sites. If it is assumed that the binding sites have fmt priority on the supply pool, then the rate to the bound pool could be expressed as SIP, where S is new binding sites available/ unit time and P equals the size of the supply pool. If binding were less


127

tight, the amount bound could be calculated as described above. Again, SAAM can recalculate the rates, continuously. Therefore, a variety of techniques can be used to incorporate the effects of protein binding into models of vitamin B-6 metabolism. A key point, particularly for modeling short time periods, is that the initial pool sizes should normally be set to steady-state conditions. If other values are chosen, upon starting the model the pools will move toward steady-state sizes, and this may complicate the interpretation of the performance of the model. Iterating to improve the fit of the model may alter the steady-state pool sizes and therefore require resetting the initial conditions. One way to establish new steady-state conditions automatically within the SAAM program is to run the model for a sufficient time to reach steady state before starting the experiment. In the case of vitamin B6 in humans, we routinely use 10,OOO hr to establish a steady state. The experiment can then be started using the QO or T-interrupt procedures in the SAAM program. IV. CONCLUSIONS

1. Models of vitamin B6 metabolism should allow for differencesbetween oral and intravenous administration of label. 2. At least some tissues have the ability to conserve vitamin B6 during periods of low intake, making turnover dependent on intake. 3. The conservation and flushing effects observed in vitamin B6 metabolism can be simulated by binding relationships. 4. Monitoring isotope excretion in pyridoxic acid after simultaneous administration of D2-pyridoxamine, D3-pyridoxal, and D5-pyridoxine provides a practical way to examine short-term metabolism of these compounds. 5. Monitoring isotope excretion starting 2 weeks after a single, large, oral dose of label may provide information about the turnover of vitamin B-6 in slow turnover pools such as muscle.

ACKNOWLEDGMENTS Our studies of vitamin 86 kineticshave been supportedby grants from the USDA/NRICGP, the most recent being 95-37200-1703.

REFERENCES Bamard, H. C., de Kock, J. J., Vermaak, W. J. H., and Potgieter, G.M.(1987). A new perspective in the assessment of vitamin B-6 nutritional status during pregnancy in humans. J. Nutr. 117,1303-1306.

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Benson, A., Coburn, S. P., Mahuren, J. D., Szadkowska, Z., Schaltenbrand, W. E., Hachey, D. L., and Townsend, D.W. (1994). Kinetics of vitamin B-6 metabolism examined in humans using simultandous oral administration of D2-pyridoxamine, D3-pyridoxal and D5-pyridoxine. FASEB 1. 8, A703. Beynon, R. J., Fairhurst, D., and Cookson, E. J. (1986). Turnover of skeletal muscle glycogen phosphorylase. Biomed Biochim Acta 45,1619-1625. Hhagavan, H. N., Coleman, M., and Coursin, D. B. (1975). Distribution of pyridoxal 5‘phosphate in human blood between the cells and the plasma: effect of oral administration of pyridoxine on the ratio in Down’s and hyperactive patients. Biochem.Med. 14,201-208. Black, A. L., Guirard, B. M., and Snell, E. E. (1978). The behavior of muscle phosphorylase as a reservoir for vitamin B-6 in the rat. J. Nufr. 108,670-677. Bode, W., and van den Berg, H. (1991). Pyridoxal5‘-phosphate and pyridoxal biokinetics in male Wistar rats fed graded levels of vitamin B-6. J. Nutr. El, 1738-1745. Rode, W., Hekman, P., and van den Berg, H. (1987). Influence of vitamin B-6 nutritional status upon plasma kinetics of PLP after i.v. dosing to rats. In “Biochemistry of Vitamin 8-6” (T. Korpela, and P. Christen, eds.), pp. 407-410. Birkhaeuser, Boston. Bode, W., Mocking, J. A. J., and van den Berg, H. (1992). Retention of C-14 label is lower in old than in young Wistar rats after oral dosing with &-I42 pyridoxine. J. Nutr. l22,1462-1471. Brain. M. C., and Booth, C. C. (1964). The absorption of tritium labelled pyridoxine hydrochloride in control subjects and in patients with intestinal malabsorption. Cur 5,241-247. Buss, D. D., Hamm, M. W., Mehansho, H., and Henderson, L.M. (1980). Transport and metabolism of pyridoxine in the perfused small intestine and the hind limb of the rat. J . Nictr. 110, 1655-1663. Butler, P. E., Cookson, E. J., and Beynon, R. J. (1985). The turnover of skeletal muscle glycogen phosphorylase studied using the cofactor, pyridoxal phosphate, as a specific label. Biochim. Biophys. Acta 847,316-323. Chang, S., and Kirksey, A. (1990). Pyridoxine supplementation of lactating mothers: Relation to maternal nutrition status and vitamin B-6 concentrations in milk. Am. J. Clin.Nutr. 51,826-831. Coburn, S . P. (1990).Location and turnover of vitamin B-6 pools and vitamin B-6 requirements of humans. Ann. N. Y. Acad. Sci. 585, 76-85. Coburn, S. P. (1994). A critical review of minimal vitamin B-6 requirements for growth in various species with a proposed method of calculation. Vitam. Horm. (N. Y.)48,259-300. (:oburn, S. P.. and Mahuren. J. D. (1983). A versatile cation-exchangeprocedure for measuring the seven major forms of vitamin 8-6 in biological samples. Anal. Biochem. U9,310-317. tobum, S. P., and Mahuren, J. D. (1987). Identification of pyridoxine 3-sulfate, pyridoxal 3wlfate and N-methylpyridoxineas major urinary metabolites of vitamin B-6 in domestic ats. 1. Biol. Chem. 262,2642-2644. Coburn, S. P., Lin, C. C., Schaltenbrand, W. E., and Mahuren, J. D. (1982). Synthesis of deuterated vitamin 8-6 compounds. J. Labelled Compd. Radiopharm. 19, 703-716. roburn, S. P., Schaltenbrand, W. E., Mahuren, J. D., Clausman, R. J., and Townsend, D. W. (1983). Effect of megavitamin treatment on mental performance and plasma vitamin 8-6 concentrations in mentally retarded young adults. Am. J. Clin. Nutr. 38, 352-355. Cobum, S. P., Mahuren, J. D., Erbelding, W. F., Townsend, D. W., Hachey, D. L., and Klein, P. D. (1984a). Measurement of vitamin B-6 kinetics in vivo using chronic administration of labelled pyridoxine. In “Chemical and Biological Aspects of Vitamin 8-6 Catalysis” (A. E. Evangelopoulos, ed.), Part A, pp. 43-54. Alan R. Liss. New York. Coburn, S. P., Mahuren, J. D., and Guilarte, T. R. (1984b). Vitamin B-6 content of plasma of domestic animals determined by HPLC, enzymatic and radiometric microbiologic methods. J. Nurr. 114,2269-2273.


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Coburn, S. P., Mahuren, J. D., Szadkowska, Z., Schaltenbrand, W. E., and Townsend, D. W. (1985). Kinetics of vitamin B-6 metabolism examined in miniature swine by continuous administration of labelled pyridoxine. In “Mathematical Models in Experimental Nutrition” (N. L. Canolty and T. P. Cain, eds.), pp. 99-111. University of Georgia, Athens. Coburn, S. P., Lewis, D. N., Fink, W. J., Mahuren, J. D., Schaltenbrand, W. E., and Costill, D. L. (1988a). Human vitamin B-6 pools estimated through muscle biopsies. A m J. Clin. NuW. 48,291-294. Coburn, S. P., Mahuren, J. D., Kennedy, M. S., Schaltenbrand, W. E., Sampson, D. A., O’Connor, D. K., Snyder, D. L., and Wostmann, B. S. (1988b). B-6 vitamer content of rat tissues measured by isotope tracer and chromatographic methods. BioFactors 1,307-312. Coburn, S. P., Mahuren, J. D., Wostmann, B. S., Snyder, D. L., and Townsend, D. W. (1989a). Role of intestinal microflora in the metabolism of vitamin B-6 and 4’-deoxypyridoxine examined using germfree guinea pigs and rats. J. Nutr. 119,181-188. Coburn, S. P., Mahuren, J. D., Kennedy, M. S., and Townsend, D. W. (1989b). Validation of the single bolus protocol for measuring turnover of the slow pool of vitamin B-6 in muscle. In “Program for Mathematical Models in Experimental Nutrition: Advances in Amino Acid and Carbohydrate Metabolism” [Abstract]. Vanderbilt Univ. Med. Cent., Nashville, TN. Coburn, S. P., Ziegler, P. J., Costill, D. L., Mahuren, J. D., Fink, W. J., Schaltenbrand, W. E., Pauly, T. A., Pearson, D. R., Conn, P. S., and Guilarte, T. R. (1991). Response of vitamin-B-6 content of muscle to changes in vitamin B-6 intake in men. A m 1. Clin. Nun. 53,1436-1442. Coburn, S. P., Mahuren, J. D., Kennedy, M. S., Schaltenbrand, W. E., and Townsend, D. W. (1992a). Metabolism of [“Cl- and [3zP]pyridoxal5’-phosphate and [3H]pyridoxal administered intravenously to pigs and goats. J. Nutr. 122,393-401. Coburn, S. P., Mahuren, J. D., Pauly, T. A., Ericson, K. L., and Townsend, D. W. (1992b). Alkaline phosphatase activity and pyridoxal phosphate concentrations in the milk of various species. 1. Nutr. 122,2348-2353. Coburn, S . P., Mahuren, J. D., Schaltenbrand, W. E., Frederick, R. E., Coolman, R. A., Townsend, D. W., and Cline, T. R. (1994). Vitamin B-6 metabolism in pigs raised on low, normal or high vitamin B-6 intakes. FASEB 1. 8, A919. Coburn, S. P., Thampy, K. G., Lane, H. W., Conn, P. S., Ziegler, P. J., Costill, D. L., Mahuren, J. D., Fink, W. J., Pearson, D. R., Schaltenbrand, W. E., Pauly, T. A,, Townsend, D. W., LeBlanc, A. D., and Smith, S. M. (1995). Pyridoxic acid excretion during low vitamin B6 intake, total fasting, and bed rest. Am. J. Clin. Nun. 62,979-983. Colombini, C. E., and McCoy, E. E. (1970). Vitamin B-6 metabolism. The utilization of [14C]pyridoxineby the normal mouse. Biochemistry 9,533-538. Contractor, S. F., and Shane, B. (1970). Blood and urine levels of vitamin 8-6 in the mother and fetus before and after loading of the mother with vitamin B-6. Am. J. Obstet. Gynecof. 107,635-640. Cox, S . H., Murray, A., and Boone, I. (1962). Metabolism of tritium labelled pyridoxine in rats. Proc. SOC. Exp. Biol. Med. 109,242-244. Dahlkvist, G., Lindstedt, S., and Tiselius, H. (1%9). Studies on the distribution and elimination of [3H8] pyridoxine in mice. Acta PhysioL S c d 75,427-432. Fonda, M. L. (1988). Pyridoxamine phosphate oxidase activity in mammalian tissues. Comp. Biochem. PhysioL B 90B,731-737. Fonda, M. L., and Eggers, D. K.(1980). Vitamin B-6 metabolism in the blood of young adult and senescent mice. Exp. Gerontol. 15,465-472.

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PART I VITAMM METABOLISM

Fonda, M. L., Eggers, D. K., Auerbach, S., and Fritxh, L. (1980). Vitamin B-6 metabolism in the brains of young adult and senescent mice. Exp. Gerontol. 15,473-479. Gregory, J. F., Trumbo, P. R., Bailey, L. B., Toth, J. P., Baumgartner, T. G., and Cerda, J. J. (1991). Bioavailability of pyridoxine-5’-beta-D-glucoside determined in humans by stable-isotopic methods. J. Nurr. 121, 177-186. Hamaker, B., Kirksey, A., and Borschel, M.W.(1990). Distribution of B-6 vitamen in human milk during a 24-h period after oral supplementation with different amounts of pyridoxine. Am. J. Clin. Nutr. 51, 1062-1066. Hamm, M. W., Mehansho, H., and Henderson, L. M. (1979). Transport and metabolism of pyridoxamine and pyridoxamine phosphate in the small intestine of the rat. J. Nufr. 109,1552-1559. Hamm, M. W., Mehansho, H., and Henderson, L. M. (1980). Management of pyridoxine and pyridoxal in the isolated kidney of the rat. 1. Nutr. 110, 1597-1609. Heard, G. S., and Annison, E. F. (1986). Gastrointestinal absorption of vitamin B-6 in the chicken (Gallus domesticus). J. Nurr. 116, 107-120. Ink, S. L., and Henderson, L. M. (1984). Effect of binding to hemoglobin and albumin on pyridoxal transport and metabolism. 1. BioL Chem. 259,5833-5837. Johansson, S., and Tiselius, H. (1973). Metabolism of tritium labeled and carbon-14 labelled pyridoxine in the rat. S c a d J. Clin. Lab. Invest. 32,9-14. Johansson, S., Lindstedt, S., and Register, U. (1966a). Metabolism of labeled pyridoxine in the rat. Am. J. Physiol. 210, 1086-1095. Johansson,S., Lindstedt, S., Register, U., and Wadstrom,L. (1966b). Studieson the metabolism of labelled pyridoxine in man. Am. 1. Clin Nufr. 18, 185-196. Johansson, S., Lindstedt, S., and Tiselius, H. (1974). Metabolic interconversions of different forms of vitamin B-6.1. BioL Chem. 249,6040-6046. Kant, A. K., Moser-Veillon, P.B., and Reynolds, R. D. (1988). Effect of age on changes in plasma, erythrocyte, and urinary B-6 vitamers after an oral vitamin B-6 load. Am. J. Cfitz. Nutr. 48,1284-1290. Labadarios, D., Rossouw, J. E., McConnell. J. B., and Williams, R. (1977). Vitamin B-6 deficiency in chronic liver disease-evidence for increased degradation of pyridoxal 5’phosphate. Gut 18,23-27. LeBlanc, A. D., Schneider, V. S., Evans, H.,Pientok, C., Rowe, R., and Spector, E. (1992). Regional changes in muscle mass following 17 weeks of bed rest. 1. Appl. Physiol. 73,2172-2178. Li, T. K., Lumeng, L., and Veitch, R.L. (1974). Regulation of pyridoxal5’-phosphate metabolism in liver. Biochm. Biophys. Res. Commun. 61,677-684. Lindberg, A. S., Leklem, J. E., and Miller, L. T. (1983). The effect of wheat bran on the bioavailability of vitamin B-6 in young men. 1. Nufr. 113,2578-2586. Lui, A., Lumeng, L., and Li, T. K. (1983). Biliary excretion of C-14 labeled vitamin B-6 in rats. 1. Nutr. 1l3, 893-898. Lui. A., Lumeng, L., Aronoff, G. R., and Li, T. K. (1985). Relationship between body store of vitamin B-6 and plasma pyridoxal-P clearance: Metabolic balance studies in humans. J. Lab. Clin. Med. 106,491-497. Lumeng, L., Brashear, R. E., and Li, T. K. (1974). Pyridoxal5’-phosphate in plasma: Source, protein-binding, and cellular transport. J. Lab. CXn. Med 84, 334-343. Lumeng, L., Ryan, M. P.. and Li, T. K. (1978). Validation of the diagnostic value of plasma pyridoxal 5’-phosphate measurements in vitamin B-6 nutrition of the rat. J. Nutr. 108, 545-553. Lumeng, L., Schenker, S., Li, T. K.,Brashear, R. E., and Compton, M.C. (1984). Clearance and metabolism of plasma pyridoxal 5’-phosphate in the dog. J. Lab. Clin Med. 103,59-69.


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Mahuren, J. D., Pauly, T. A., and Coburn, S. P. (1991). Identification of 5-pyridoxic acid and 5-pyridoxic acid lactone as metabolites of vitamin B-6 in humans. J. Nutr. Biochem. 2,449-453. Manchester, J. K., Chi, M. M., Noms, B., Ferrier, B., Krasnov, I., Nemeth, P. M., McDougal, D. B., and Lowry, 0.H. (1990). Effect of microgravity on metabolic enzymes of individual muscle fibers. FASEB J. 4,55-63. McCoy, E. E., Colombini, C., and Ebadi, M. (1969). Metabolism of vitamin B-6 in Down’s syndrome. A m N. Y. Acad Sci. 166,116-125. Mehansho, H., and Henderson, L. M. (1980). Transport and accumulation of pyridoxine and pyridoxal by erythrocytes. J. Biol. Chem. 255,11901-11907. Mehansho, H.,Hamm, M. W., and Henderson, L. M. (1979). Transport and metabolism of pyridoxal and pyridoxal phosphate in the small intestine of the rat. J. Nutr. 109,1542-1551. Mehansho, H., Buss, D. D., Hamm, M. W., and Henderson, L. M. (1980). Transport and metabolism of pyridoxine in rat liver. Biochim. Biophys. Actu 631, 112-123. Memll, A. H., and Henderson, J. M. (1990). Vitamin 8-6 metabolism by human liver. Ann. N. Y. A ~ a dSci. 585, 110-117. Middleton, H. M. (1979). In vivo absorption and phosphorylation of pyridoxine hydrochloride in rat jejunum. Gastroenterology 7643-49. Middleton, H. M. (1985). Uptake of pyridoxine by in vivo perfused segments of rat small intestine: A possible role for intracellular vitamin metabolism. J. Nutr. 115, 1079-1088. Nomenclature Committee of the International Union of Biochemistry and Molecular Biology (1992). “Enzyme Nomenclature.” Academic Press, San Diego, CA. Pauly, T. A., Szadkowska, Z., Coburn, S. P., Mahuren, J. D., Schaltenbrand, W. E., Booth, L., Hachey, D. L., Ziegler, P. J., Costill, D. L., Fink, W. J., Pearson, D., Townsend, D., M i d i , R., and Guilarte, T. (1991). Kinetics of deuterated vitamin B-6 metabolism in men on a marginal vitamin B-6 intake. FASEB J. 5, A1660. Sakurai, T., Asakura, T., and Matsuda, M. (1988). Transport and metabolism of pyridoxine in the intestine of the mouse. J. Nurr. Sci. Vitaminol. 43,179-187. Sakurai, T., Asakura, T., Mizuno, A., and Matsuda, M. (1991). Absorption and metabolism of pyridoxamine in mice. 1. Pyridoxal as the only form of transport in blood. J. Nutr. Sci. Vitaminol. 31, 341-348. Sakurai, T., Asakura, T., Mizuno, A., and Matsuda, M. (1992). Absorption and metabolism of pyridoxamine in mice. 2. Transformation of pyridoxamine to pyridoxal in intestinal tissues. J. Nutr. Sci Vitaminol. 38, 227-233. Schenker, S., Johnson, R. F., Mahuren, J. D., Henderson, G . I., and Coburn, S. P. (1992). Human placental vitamin B-6 (pyridoxal) transport-normal characteristics and effects of ethanol. Am. J. Physiol. 262, R966-R974. Serebro, H. A., Solomon, H. M., Johnson, J. H., and Hendrix, T. R. (1966). The intestinal absorption of vitamin B-6 compounds by the rat and hamster. Johns Hopkinr Med. J. 119,166-171. Shane, B. (1970). Metabolism of vitamin B-6 in pregnancy. Ph.D. Thesis, University of London. Sorrell, M. F., Baker, H., Barak, A. J., and Frank, 0. (1974). Release by ethanol of vitamins into rat liver perfusates. Am. J. Clin. Nutr. 27,743-745. Spannuth, C. L., Warnock, L. G., Wagner, C., and Stone, W. J. (1977). Increased plasma clearance of pyridoxal 5’-phosphate in vitamin B-6 deficient uremic man. J. Lab. Clin. Med. 90,632-637. Speitling, A., Heseker, H., and KUbler, W. (1990). Pharmacokinetic properties of the plasma B-6 vitamers after single and chronic oral pyridoxine mega doses. Ann. N. Y. Acad. Sci. 585,557-559. Stenesh, J. (1993). “Core Topics in Biochemistry,” p. 189. Cogno Press, Kalamazoo, MI.

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Tillotson, J. A., Sauberlich, H. E., Baker, E. M., and Canham, J. E. (1967). Use of ’‘C-labeled vitamins in human nutrition studies. Pyridoxine. Proc. Inr. Congr. Nutr. 7th 1966, Vol. 5. pp. 554-557. ligase from beef liver mitochonTong, H., and Davis, L. (195). 2-amino-3-ketobutyrate-CoA dria: An NMR spectroscopic study of low-barrier hydrogen bonds of a pyridoxal 5’phosphate-dependent enzyme. Biochemistry 34,3362-3367. Ubbink, J. B., and Serfontein, W. J. (1988). The response of the plasma B-6 vitamers to a single. oral pyridoxine supplement. I n “Clinical and PhysiologicalApplications of Vitamin €3-6”( 1. E. Leklem and R. D. Reynolds, eds.), pp. 29-34. Alan R. Liss, New York. Ubbink, J. B., Serfontein, W. J., Becker, P.J., and de Villiers, L. S. (1987). Effect of different levels of oral pyridoxine supplementation on plasma pyridoxal-5“-phosphate and pyridoxal levels and urinary vitamin B-6 excretion. Am. 1. Clin. Nutr. 46, 78-85. Veitch, K. L., Lumeng. L.. and Li, T. K. (1975). Vitamin B-6 metabolism in chronic alcohol abuse: The effect of ethanol oxidation on hepatic pyridoxal 5‘-phosphate metabolism. J. Clin. Invest. 55, 1026-1032. Whyte, M. P.. Mahuren, J. D., Vrabel, L. A.,and Coburn, S. P. (1985). Markedly increased circulating pyridoxal 5‘-phosphate levels in hypophosphatasia.J. Clin. Invest. 76,752-756. Wolfe. R. R. (1992). “Radioactive and Stable Isotope Tracers in Biomedicine.” Wiley-Liss, New York. Wozenski, J. R., Leklem, J. E., and Miller, L. T. (1980). The metabolism of small doses of vitamin B-6 in men. 1. Nurr. 110,275-285. Zubay, G. (1988). “Biochemistry,” 2nd ed., p. 352. Macmillan, New York.

Part II PROTEIN AND AMINO ACID METABOLISM

ADVANCES M FOOD AND NUTRITION RESEARCH, VOL. 40

Chapter 7 INTERRELATIONSHIPS BETWEEN METABOLISM OF GLYCOGEN PHOSPHORYLASE AND PYRIDOXAL PHOSPHATE-IMPLICATIONS IN MCARDLE’S DISEASE ROBERT J. BEYNON AND CLARE BARTRAM Department of Biochemistry and Applied Molecular Biology UMIST Manchester M60 IQD, United Kingdom

ANGELA FLANNERY, RICHARD P. EVERSHED, AND DEBORAH LEYLAND Department of Biochemistry University of Liverpool Liverpool L69 3BX, United Kingdom

PAMELA HOPKINS Department of Biochemistry and Applied Molecular Biology UMIST Manchester M60 IQD, United Kingdom

VERONICA TOESCU, JOANNE PHOENIX, AND RICHARD H. T. EDWARDS Muscle Research Centre Department of Medicine University of Liverpool Liverpool L69 3BX, United Kingdom

I. Introduction 11. pole of Cofactor in Phosphorylase Turnover 111. Labeling Methods to Monitor Phosphorylase Turnover A.

. B.

Radiolabelled Cofactor Stable-Isotope-LabelledCofactor 135 Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reSeNed.

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PART I1 PROTEIN AND AMINO ACID METABOLISM 1V. Model Systems for Phosphorylase Expression A. Muscle Wasting B. Muscle Growth V. Vitamin B6 and McArdle’s Disease A. Biochemical Heterogeneity B. Molecular Genetics C. McArdle’s Disease and Vitamin B6 Metabolism References

1.

INTRODUCTION

Glycogen phosphorylase (EC 2.4.1.1.,“phosphorylase”) is the key regulator of glycogenolysis, catalyzing the phosphorolysis of glycogen to produce glucose-1-phosphate. The enzyme is particularly abundant in muscle, where it functions to provide monosaccharide units for glycolysis in response to demands mediated by endocrine signals or by rises in intracellular calcium. The cofactor of glycogen phosphorylase is pyridoxal5’-phosphate (PLP). This cofactor, linked via a Schiff base to a lysine residue (Lys680 in the rabbit sequence), is tightly bound to the enzyme and cannot be resolved from the apo-enzyme unless powerful denaturants are used. The use of PLP in the phosphorylase reaction is unusual and involves the 5’-phosphate group rather than the aldehydic group that is used more commonly in, for example, transaminases ( Johnson, 1992). The control of phosphorylase is subtle, and it is subject to regulation through phosphorylation and through allosteric inhibition (Johnson ef al., 1992). It is less clear that changes in intracellular concentration of the enzyme serve a regulatory €unction, whether mediated through changes in the rates of synthesis or degradation. As part of a study of protein degradation in normal and abnormal skeletal muscle, we have focused on phosphorylase as an abundant sarcoplasmic protein possessing several properties that make it particularly suitable for such studies (Beynon ef al., 1993). In this chapter, we review our work on the turnover of glycogen phosphorylase, with particular emphasis on the cofactor as a turnover label.

II. ROLE OF COFACTOR IN PHOSPHORYLASE TURNOVER

Two properties of the muscle phosphorylase pool associated with PLP (its large size and slow kinetics of exchange) led us to speculate that the cofactor might provide a specific label with which to monitor the turnover of the enzyme in vivo (Fig. 1).Implicit in this suggestion is the requirement that the cofactor is incorporated into the enzyme as a cosynthetic or immediately postsynthetic event. Further, reutilization of cofactor within mus-

CHAPTER 7 GLYCOGEN PHOSPHORYLASE

137

pyridoxine I

whole body pools

test

J

Biivitamers f8H

FIG. 1. Interrelationships between vitamin B6 and phosphorylase metabolism. The low rate of turnover of glycogen phosphorylase (gpb) and the lack of exchange of free and proteinbound PLP mean that exchange into the muscle pool is largely controlled by the kinetics of turnover of the enzyme. At present, it is not known whether resolution of the holo-enzyme is a prerequisite or consequence of phosphorylase degradation. Reproduced with permission of The Biochemical Journal.

cle should be minimal, which imposed a requirement for rapid hydrolysis of PLP to pyridoxal (PL) and release of PL from the muscle to the circulation. Our early experiments suggested that these criteria were met. Apo-phosphorylase cannot be detected in muscle, even under conditions of vitamin B6 (pyridoxine, PN) deficiency (R. J. Beynon and D. M.Leyland, unpublished observations). Secondly, low-molecular-weight radiolabeled B6 vitamer pools were cleared very rapidly following a pulse dose of radiolabeled pyridoxine, consistent with a very labile muscle pool of free vitamers. Finally, the rate of degradation of phosphorylase measured by cofactor labeling was the same as that measured by continuous infusion of radiolabeled amino acids (Beynon et al., 1986). 111.

LABELING METHODS TO MONITOR PHOSPHORYLASE TURNOVER

A. RADIOLABELED COFACTOR The principle of using radiolabeled vitamin B6 as a label for phosphorylase degradation is straightforward and takes the form of a classical pulse-

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PART I1 PROTEIN A N D AMINO ACID METABOLISM

chase protocol (Cookson and Beynon, 1989). Animals were injected subcutaneously with [G-3H] pyridoxine (typically 100-200 pCi for a 25-g mouse). At different times after injection, skeletal muscle was homogenized and a high-speed supernatant fraction (containing virtually all of the radioactivity) was prepared. One portion of the supernatant fraction was separated on (3-25 gel filtration to assess the low-molecular-weight pool. A second portion was applied to a column of AMP-Sepharose to bind glycogen phosphorylase, which was subsequently eluted with a solution of AMP. The eluate contained several proteins, but the only PLP-binding protein in the fraction was phosphorylase. This simple, one-step affinity procedure therefore allowed isolation of phosphorylase. Ten days after injection of label, the low-molecular-weightpool contained virtually no label, whereas label remained associated with phosphorylase. Subsequent isolation of phosphorylase from different animals over a range of time periods between 10 and 30 days defined an exponential decay, the rate constant of which was taken as the rate of degradation of phosphorylase. This was subsequently confirmed by independent measurement of the rate of turnover of the enzyme using continuous infusion of labeled amino acids-a method that is independent of reutilization artifacts. The rate of turnover of phosphorylase was the same when measured by either method (Beynon et af., 1986; Cookson and Beynon, 1989). B. STABLE ISOTOPE-LABELED COFACTOR Phosphorylase has a relatively low rate of turnover, and as such, large doses of radiolabeled pyridoxine were needed to obtain an adequate degree of labeling. In addition, the need for serial sampling of the decay curve, using individual animals, introduced substantial biological variation. Both aspects of this experimental system precluded application of the method to humans, and we considered the possibility of a different approach, based on stable-isotope-labeled pyridoxine, to monitor phosphorylase degradation. It has long been recognized that the simplest model of vitamin B6 metabolism required two pools, a small, mobile pool and a large, slow pool. Coburn and colleagues (reviewed in Coburn, 1990) have proposed that the only source of the large, slow pool is muscle glycogen phosphorylase-large because of the abundance of phosphorylase in skeletal muscle and slow because of the low rate of exchange of label into and out of this protein. This suggestion was based on calculations of the accessibility of this pool (Coburn er af., 1991) and from muscle biopsy studies (Coburn et al., 1988). We believe that the rate of exchange into and out of this pool reflects the rate of turnover of the enzyme.


139

Dideuterated pyridoxine was synthesized according to methods developed by Coburn et al. (1982) and was purified and administered to mice in drinking water. The animals consumed a vitamin B6-deficient diet for the duration of the experiment. At times throughout the experiment, urine samples were taken and the excreted 4-pyridoxic acid was analyzed as the terr-butyldimethylsilyl derivative by GCMS in selected ion monitoring mode. The tert-butyl group is lost to yield a fragment ion of mlz 224 ( m k 222 for unlabeled samples) corresponding to the monoderivatized lactone of 4-pyridoxic acid (Leyland er al., 1992). The technique of oral administration of dideuterated PN and analysis of urinary output of labeled 4-pyridoxic acid has the advantage of being noninvasive, and, of course, repeated urinalysis permits definition of the labeling curve in a single animal over an extended period. We have analyzed the labeling kinetics of mice over a 50-day period using this technique (Fig. 2). At the end of this period the isotope abundance has almost attained that of the ingested material, although the failure to reach maximal isotope abundance may reflect a low level of pyridoxine in the diet formulation.

0 0

10

20

30

40

50

Time (d)

FIG. 2. Whole animal labeling kinetics with stable-isotope-labeled pyridoxine. Mice were placed on a vitamin B6-deficient diet and labeled with dideuterated pyridoxine. At intervals, urine samples were collected and the isotope abundance of excreted 4-pyridoxic acid was measured. After 50 days, the animals have attained a plateau labeling. When analyzed as a biexponential process, the rate constant for the labeling of the slow pool is the same as that obtained by pulse labeling with radioactive pyridoxine. The symbols correspond to the data from four individuals in the study.

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PART I1 PROTEIN AND AMINO ACID METABOLISM

The kinetics of the rise to a plateau value can be analyzed by nonlinear curve fitting as a biexponential equation and yields rate constants for turnover of the large and slow components, from which the rate constant of loss of PLP from the phosphorylase pool (=rate of degradation of enzyme) can be calculated (Coburn, Chapter 6, this volume). The rate constant for turnover of the slow component was 0.13 2 0.031 day (mean _t SEM., n = 10) from which a value of turnover of the phosphorylase pool can be calculated as O.l/day (Beynon er al., 1996). This compares well with values of O.l2/day obtained for gastrocnemius muscle (Leyland et al., 1990) and 0.13/day for total hind limb and back muscle (Leyland and Reynon, 1991).The fast pool (presumed to be all labile forms of the vitamin) was turning over very quickly, with a rate constant of 1.3 5 0.4/day (a halflife of 12 hr). However, the experimental protocol that we use does not permit acquisition of a sufficiently detailed data set to acquire accurate kinetics on the fast pool. This preliminary analysis of the data also implies that the fast pool accounts for about 50% of the total vitamin B6 in the body-it is not yet clear whether this is consequential to the inability to define the fast phase with a high degree of precision or whether the muscle phosphorylase itself partitions into two pools that differ in accessibility. For example, enzyme bound to the glycogen particle might be more stable than enzyme free in the sarcoplasm. Further work is needed to resolve these issues. IV. MODEL SYSTEMS FOR PHOSPHORYLASE EXPRESSION

Muscle wasting or growth is a result of imbalance between the opposing processes of protein synthesis and degradation, and, in many conditions, the relative contributions of the two are not known. Moreover, many studies on protein turnover analyze total protein pools, which obscures the behavior of individual proteins within those pools. The pyridoxine labeling methods that we have developed have allowed us to explore phosphorylase degradation under a number of different conditions. Two models of muscle wasting processes have been studied in the mouse, and the role of degradation in muscle growth has focused on the chicken. A. MUSCLE WASTING

Two models for muscle wasting have been studied in the mouse (Table I). Section of the sciatic nerve causes a rapid denervation-induced atrophy in the lower limb muscles, and we have measured different parameters of phosphorylase expression in the gastrocnemius of the mouse after unilateral

141


TABLE I EXPRESSION OF PHOSPHORYLASE IN TWO

MOUSE MODELS FOR MUSCLE WASTING ~~

~

Phosphorylase (SAb,96 control)

mRNA (% of control)

Animal model

ha

C57BU6J denervation Control limb Operated limb

0.121day 0.2lday

-

-

30

50

C57BU6Jdyldy Control Mutant

0.13lday O.OS/day

40

a

Rate constant for degradation. Specific activity.

denervation, using the contralateral muscle as a control (Leyland et al., 1990). The response to denervation is biphasic. In the first 4-5 days after denervation, phosphorylase mRNA levels decline markedly, to about 50% of the values in the control limb. Phosphorylase content in muscle (measured by activity) declines slowly over this first phase but in the second period (5-25 days) disappears from the muscle more quickly. The specific activity of the enzyme declines during the experimental period to 30% of control values, indicating preferential loss of this enzyme relative to other muscle proteins. The acceleration in the loss of enzyme is reflected in enhanced degradation of 02/day compared to O.ll/day in the control limb. Although we have not measured rates of synthesis directly, it is not unreasonable to assume that the lower mRNA abundance is manifest as a lower rate of synthesis. Thus, loss of phosphorylase is accomplished by a lower rate of synthesis and an accelerated rate of degradation. Further evidence for enhanced degradation comes from Western blot studies with a monoclonal antibody to the PLP-a transient enhancement of degradation intermediates is apparent (Cookson et al., 1992). Completely different behavior is seen in the C57BW6JdyIdy dystrophic mouse (Leyland and Beynon, 1991). This is a severe muscle wasting condition, unrelated to the less severe C57BL/10Jmd"'md" mutant that is dystrophin deficient. The mutation in the C57BL/6Jdy'dyanimal is not known. In dystrophic animals, the rate of degradation of phosphorylase is less than in normal animals (0.05/day, compared to 0.13/day) and the mRNA level is 60% of normal values. By implication, the rate of phosphorylase synthesis in these animals is severely suppressed, consistent with an overall metabolic downregulation in this condition. The failure to detect degradation intermediates such as those seen in denervation atrophy adds further credence to this view.

142


B. MUSCLE GROWTH For analysis of phosphorylase expression in muscle growth, we have focused on the chicken. In particular, we have compared animals selected for rapid growth (broiler) with animals selected for egg production (layer). In this study, we concentrated on the pectoralis muscle. During the period of the experiment, from 2 to 8 week, the pectoralis muscle increases in mass from 15 to 40 grams in the layer and from 13 to 160 grams in the broiler. The total pool of phosphorylase expands by 3-fold in the layer compared to 19-fold in the broiler. How is this dramatic expansion of the phosphorylase pool achieved in the broiler-by enhanced synthesis or by suppressed degradation? In this system, it was necessary to modify the kinetic analyses to address the issue of (a) an expanding tissue pool and (b) the potential for a time-dependent change in turnover rate throughout the growth period (Flannery et al., 1992). The results of our analyses are summarized in Fig. 3. In 3-week-old animals, when the pectoralis weight is the same in broilers and layers, the rate of phosphorylase synthesis is similar in the two strains, at approx 0.1/ day. The strains differ markedly in the rate of degradation of the enzyme: broiler, approx O.Ol/day, and layer, approx 0.OYday. As the animals grow, the two strains yield very different results. At 8 weeks in the broiler, the synthesis rate declines slightly, to 0.07/day whereas in the layer, it declines to 0.04lday. Thus, the synthesis of the enzyme is not very different in the two strains. By contrast, the rate of degradation of the enzyme declines to O.OOS/day in layer, but remains at O.Ol/day in the broiler. Thus, several factors combine to produce the different growth behaviors. In layers, the age-dependent decline in the rate of turnover of the protein and the convergence of the rates of synthesis and degradation are similar to the growth pattern seen in many species and tissues. In the broilers, rapid growth is achieved by a combination of two factors. First, even in young animals, the rate of degradation of the enzyme is low. Second, synthesis and degradation rates do not converge as quickly in the broilers, allowing for a rapid growth that soon outstrips that of the layer. It seems likely that these animals have been selected for a sustained downregulation of the degradative machinery or for an early maturationdependent decline in this machinery. Analysis of the expression of these systems in even younger animals might prove fruitful. V. VITAMIN 66 AND MCARDLE’S DISEASE

Our interest in phosphorylase turnover has led us to appreciate the importance of the phosphorylase-bound pool of vitamin B6. The behavior

143

CHAPTER 7 GLYCOGEN PHOSPHORYLASE Broiler (3weeks)

Broiler (8 weeks)

Layer (3 weeks)

Layer (8 weeks)

760

200

230

40

FIG. 3. Expression of glycogen phosphorylase in growing chicken pectoralis muscle. The total phosphorylase pool sizes are represented by the areas of the circles and are given here in milligram total followed by the concentrations (mg/g) in parentheses: broiler 3 week, 75 (5.2 2 0.4); broiler 8 week, 1410 (8.2 +- 0.2); layer 3 week, 100 (5.1 2 0.5); layer 8 week, 270 (6.6 ? 0.8). The fluxes into and out of the phosphorylase pool are presented per gram of tissue and are given in pg/day/gram wet weight tissue-this is more representative of the metabolic activity per mass of tissue, in a rapidly expanding muscle. The width of the arrows into and out of the pools are in proportion to their magnitude. The convergence of the synthesis and degradation rates in layers is apparent as is the sustained imbalance between the two processes in broilers. Unlike layers, the specific activity of the enzyme in the 8-week broiler pool is about 50% higher than the starting material, represented by a stronger shading.

of this pool has consequences for recommendations of vitamin B6 intakes (Coburn, 1990), and the inaccessibility of the pool may stress the different requirements for vitamin B6 during growth, when the phosphorylase pool is accreting, and in the adult, where vitamin B6 requirements may function primarily to sustain throughput of the fast pools. In principle, PLP could be released into the body at a maximal rate equal to that of phosphorylase degradation, but it is unlikely that phosphorylase acts as a “store” of vitamin B6 in the body. If the phosphorylase pool plays an important part in vitamin B6 kinetics, it might be anticipated that this metabolism would be disturbed in patients suffering from McMdle’s disease, a rare metabolic myopathy caused by an absence of functional muscle glycogen phosphorylase. The absence of this enzyme means that patients cannot break down their muscle glycogen reserves. Other energy sources within the muscle are rapidly depleted

144


during exercise but cannot be replenished quickly enough to allow normal muscle action. As a consequence, the clinical symptoms of the disease include muscle pain and cramps induced by exercise. A. BIOCHEMICAL HETEROGENEITY There are a number of steps in the expression of a functional protein from a gene and a disruption at any point can halt the normal expression of the protein. Consequently, there is variation between patients in their expression of phosphorylase mRNA and protein (McConchie et al., 1991). The majority of individuals do not express protein or mRNA, which is due either to a failure to transcribe the gene or due to accelerated degradation of the transcript. A smaller number of patients do express a stable transcript but it is not subsequently translated into protein and even fewer patients express protein but at low levels (approximately 1%of normal values). The majority of McArdle’s patients are characterized by a complete absence of the protein in muscle, consistent with knowledge of the mutations in McArdle’s disease. B. MOLECULAR GENETICS McArdle’s disease is a recessive condition and one normal copy of the gene seems to be adequate for normal phosphorylase activity in the muscle. In common with most molecular diseases, it is characterized by a number of different mutations (Fig. 4). The most frequently occurring of these is a single base change which generates a premature stop codon and has been

Exons

,

L281P

I

G122->TT

AF708

I

1-E

FIG. 4. Mutations in the myophosphorylase gene in McArdle’s disease. The mutations are described in the text. The most common phenotype is a complete lack of phosphorylase protein in skeletal muscle and, hence, the loss of the muscle vitamin B6 “slow” pool.

CHAPlTR 7 GLYCOGEN PHOSPHORYLASE

145

designated the R49X mutation (Bartram et al., 1993; Tsujino et al., 1993). This mutation generates a truncated peptide that is likely to be rapidly degraded but it also causes instability of the transcript-patients homozygous for the R49X mutation have no detectable mRNA (Bartram et al., 1993,1995). A rare frameshift mutation, designated the 122G --f ‘IT mutation, also causes premature termination of the protein (Bartram et a/., 1994). Three missense mutations implicated in the disease, L291P, G204S, and K542T, do not prevent translation of the protein but are postulated to have an effect on the normal function of the enzyme or its stability (Tsujino ef al., 1993, 1994). The L291P mutation may have a structural effect upon the protein whereas the G204S mutation is associated with a glycogen-binding domain and K542T affects a glucose-binding domain. In addition to the above, a mutation which deletes a single codon, AF708, may have a structural effect. Finally, a splice junction mutation at the boundary between exon 14 and intron 14 results in a 67-bp deletion in the transcript due to a frameshift that causes premature termination of the protein (Tsujino et al., 1994). These seven mutations do not explain all cases of McArdle’s disease-there are still other unknown mutations (Bartram et al., 1995). C. MCARDLE’S DISEASE AND VITAMIN B6 METABOLISM

Our interest in McArdle’s disease and vitamin B6 metabolism is stimulated by consideration of the consequences of the loss of the major, slowly metabolizing pool of vitamin B6 in the body. It is conceivable that the whole body phosphorylase-derived pool acts as a “buffer” to compensate for day-to-day variation in vitamin B6 intake. It will therefore be important to assess the rate of degradation of phosphorylase in the human, and the stable isotope method we have developed is directly applicable to this problem. Analysis of the same kinetics in McArdle’s patients will define the role of muscle phosphorylase in the compartmentalization of vitamin B6. It is conceivable that McArdle’s patients need to pay greater attention to their vitamin B6 status than normal individuals. A compromised vitamin B6 status might be of even greater significance if McArdle’s patients are more reliant on amino acid metabolism for muscle work-transaminases are PLP-dependent enzymes. Preliminary studies in our laboratory have established that McArdle’s patients show differences in vitamin B6 metabolism and that they respond quickly and dramatically to short-term changes in vitamin B6 status (Beynon er al., 1995).Whether improvement of vitamin B6 status could enhance muscle performance remains to be seen.

146


ACKNOWLEDGMENTS The work reviewed here has been supported by MRC, AFRC, and the Muscular Dystrophy Group of Great Britain and Northern Ireland. We are pleased to acknowledge many valuable discussions with, and assistance from, Steve Coburn.

REFERENCES Bartram, C.. Edwards, R. H. T., Clague, J., and Beynon, R. J. (1993). McArdle’s disease: A nonsense mutation in exon 1 of the muscle glycogen phosphorylase gene explains some but not all cases. Hum Mol. Genet. 2, 1291-1293. Bartram, C., Edwards, R. H. T., Clague, J., and Beynon, R. J. (1994). McArdle’s disease: A rare frameshift mutation in exon 1 of the muscle glycogen phosphorylase gene. Biochim. Biophys. Acra U26,341-343. Bartram. C., Edwards, R. H. T., and Beynon, R. J. (1995). McArdle’s disease: Muscle glycogen phosphorylase deficiency. Biochim. Biophys. Acia En,1-13. Beynon, R. J., Fairhunt, D., and Cookson, E. J. (1986). Turnover of skeletal muscle glycogen phosphorylase. Biomed. Biochim. Acra 45,1619-1625. Beynon, R. J., Flannery, A. V., Edwards, R. H. T., Evershed, R. P., and Leyland, D. M. (1993). Degradation of glycogen phosphorylase in normal and abnormal skeletal muscle. In “lntracellular Protein Catabolism” (J. S. Bond and A. J. Barrett, eds.), pp. 157-162. Portland Press. Beynon, R. J., Bartram, C., Hopkins, P., Toescu. V., Gibson, H., Pheonix, J., and Edwards, R. H. T. (1995). McArdle’s disease-molecular genetics and metabolic consequences of the phenotype. Muscle Nerve, Supp 3, S18-S22. Beynon, R. J., Leyland, D. M., Evershed, R. P., Edwards, R. H. T., and Coburn, S. P. (1996). Measurement of the turnover of glycogen phosphorylase by gas chromatographyhass spectrometry using stable isotope derivatives of pyridoxine (Vitamin 86). Biochem. J. 317,613-619. Cobum, S. P. (1990). Location and turnover of vitamin B6 pools and vitamin B6 requirements of humans. Ann N.Y. Acad Sci 585,76-85. Cobum, S. P., Lin, C. C., Schaltenbrand, W. E., and Mahuren, J. D. (1982). Synthesis of deuterated vitamin B6 compounds. 1. Labelled Compd Radiopharm. 19,703-716. Coburn, S . P., Lewis, D. L., Fink, W. J., Mahuren, J. D., Schaltenbrand, W. E., and Costill, D. L. (1988). Human vitamin 8-6 pools estimated through muscle biopsies. Am. J. Clin. Nutr. 48, 291 -294. Coburn, S. P., Ziegler, P. J., Costill, D. L., Mahuren, J. D., Fink, W. J., Schaltenbrand, W. E., Pauly, T. A.. Pearson, D. R., Conn, P. S., and Guilarte, T. R. (1991). Response of vitamin 8-6 content of muscle to changes in vitamin 8-6 intake in men. Am. J. Clin. NUr. 53, 1436-1442. Cookson, E. J., and Beynon, R.J. (1989). Further evaluation of cofactor as a turnover label for glycogen phosphorylase. Int. J. Biochem. 21,975-982. Cookson, E. J., Flannery, A. V., Cidlowski, J. A., and Beynon, R. J. (1992). Immunological detection of degradation intermediates of skeletal-muscleglycogen phosphorylase in vitro and in vivo. Biochem. 1. 288,291-2%. Flannery, A. V., Easterby, J. S., and Beynon, R.J. (1992). Turnover of glycogen phosphorylase in the pectoralis muscle of broiler and layer chickens. Biochem. J. 286, 915-922.


147

Johnson, L. N. (1992). Glycogen phosphorylase: Control by phosphorylation and allosteric effectors. FASEB J. 6,2274-2282, Johnson, L. N., Hu, S. H., and Barford, D. (1992).Catalyticmechanism of glycogen phosphorylase. Faraday Discuss. Chem SOC. 93,131-142. Leyland, D. M., and Beynon, R. J. (1991). The expression of glycogen phosphorylase in normal and dystrophic muscle. Bwchem. J. 278,113-117. Leyland, D. M., Turner, P. C., and Beynon, R. J. (1990). Effect of denemation on the expression of glycogen phosphorylase in mouse skeletal muscle. Biochem 1. 272,231-237. Leyland, D. M., Evershed, R. P., Edwards, R. H. T., and Beynon, R. J. (1992). Application of GUMS with selected ion monitoring to the urinalysisof 4-pyridoxicacid. 1.Chromatogr. Biomed. 581,179-185. McConchie, S . M., Coakley,J., Edwards, R. H., and Beynon, R. J. (1991). Molecular heterogeneity in McArdle’s disease. Biochim Biophys. Acta 1096,26-32. Tsujino, S., Shankse, S., and DiMauro, S. (1993). Molecular genetic heterogeneity of myophosphorylase deficiency (McArdle’s disease). N. Engl. J. Med. 329,241-245. Tsujino, S., Shankse, S., Nonaka, I., Eto, Y.,Mendell, Y.,Fenichel, G. M., and DiMauro, S. (1994). Three new mutations in patients with myophosphorylase deficiency (McArdle’s disease). Am 1.Hum. Genet. 54,44-52. Tsujino, S., Shankse, S., Martinuzzi, A., Heiman Patterson, T., and Di Mauro, S. (1995). Two novel missense mutations (E654K, U%P) in Caucasian patients with myophosphorylase deficiency. Hum. Mutat. 3,276-277.



Chapter 8 METABOLISM OF NORMAL AND MET30 TRANSTHYRETIN DENISE HANES Division of Rheumatology Department of Medicine Indiana University School of Medicine Indianapolis, Indiana 46202

LOREN A. ZECH Molecular Disease Branch National Heart, Lung, and Blood Institute National Institutes of Health Bethesda, Maryland 20892

JILL MURRELL Division of Rheumatology Department of Medicine Indiana University School of Medicine Indianapolis, Indiana 46202

MERRILL D. BENSON Division of Rheumatology Department of Medicine Indiana Universiry School of Medicine Indianapolis, Indiana 46202 and Rheumatology Section Richard L. Roudebuh- Veterans Affairs Medical Center Indianapolis, Indiana 46202

I. Introduction 11. Methods 111. Results

IV. Discussion References 149 Copyright (0 1996 by Academic Press. Inc. Ail rights of reproduction in any form reserved.

150

PART XI PROTEIN AND AMINO ACID METABOLISM

1.

INTRODUCTION

Hereditary amyloidosis is a heterogeneous disease characterized by systemic or localized deposition of fibrillar proteins which invade the extracellular spaces of organs, destroying normal tissue architecture and function. These amyloid deposits demonstrate green birefringence under polarization microscopy when stained with Congo red, and this unique characteristic is used in diagnosis.The first amyloid protein identified (Costa et al., 1978) and the one most commonly involved in hereditary amyloidosis is transthyretin (TTR).Normal TTR is a 55-kDa soluble plasma protein (20-40 mg/dl) which consists of four identical subunits held together noncovalently. TTR is synthesized in the liver and serves as a carrier for thyroxin and the retinol :retinol-binding protein complex. Over 40 amino acid mutations of TTR have been found related to hereditary amyloidosis, but the most common is the substitution of methionine for valine at position 30. This single amino acid substitution, found worldwide, gives rise to familial amyloidotic polyneuropathy Type I (FAP I). FAP is an autosomal dominant disease in which amyloid deposits show a systemic distribution in the peripheral and autonomic nervous, cardiovascular, and renal systems. Although the mutant protein is present from birth, FAP shows a delayed onset of symptoms until the third to seventh decade of life, with death usually occurring 10-15 years following onset of disease. Both individuals with FAP (Benson and Dwulet, 1983; Skinner et al., 1985; Westermark et af., 1985) and at risk nonaffected carriers of the Met30 gene (Shoji and Nakagawa, 1988) have shown lower than normal serum transthyretin levels. One explanation could be increased metabolic turnover of TTR due to the mutation; although no turnover studies have been done with the mutant proteins. The purpose of this study was to elucidate the metabolism of both normal and Met30 variant TTR in a normal and an FAP-affected individual using isotopic tracer techniques and computeraided kinetic modeling. Knowledge of the kinetics of normal and variant TTR metabolism may help in understanding amyloid fibril formation and give direction for possible prevention. 11.

METHODS

Laboratory methods and clinical procedures for collection of data were previously reported from our laboratory (Murrell, 1992). Purified native normal and Met30 variant TTR from a homozygous individual were iodinated with 1311-and IzI-labeled monochloride, respectively, and then purified by size exclusion chromatography. For the clinical experiment two human

CHAPTER 8 METABOLISM OF TRANSTHYRETIN

Normal control FAP affected

61 68

Femate Male

69.5 78.1

22 22

15 15

15 15

151

35 35

subjects, a normal control and an affected FAP individual, were simultaneously administered both isotopically labeled proteins intravenously. Blood and 24-hr urine samples were collected for 7 days. Table I shows specific patient dosing data. Plasma and urine aliquots were assayed for radioactivity. The data obtained from clinical study were analyzed through computeraided kinetic modeling. The time course of plasma decay and appearance in urine of the radiolabels was plotted for each subject and then functions were derived and a kinetic model was developed to simultaneously fit the

El FIG. 1. Kinetic model for '''I and lz5I plasma decay and urinary excretion. Compartment 1 represents plasma;compartment 7, a delay compartment,compartment 4, urine; and compartments 2 and 3, undetermined. The asterisk indicates the site of introducing tracer and the small triangles indicate the sites of sampling.

152


plasma and urine data using the NIH program SAAM, originally introduced by Berman in 1963. TTR plasma residence times were obtained from the areas under the plasma decay curves, and fractional catabolic rate calculated as the reciprocal of the residence time. Plasma volume was initially calculated as 4.5% of total body weight and then allowed to vary to give the best fitting curve. RESULTS

111.

Figure 1 shows the kinetic model and Table I1 the set of parameters which simultaneously described the plasma decay of I3'I- and I2'I-Met30TTR and the appearance of radiolabel in the urine. The radiolabels showed partial exchange with two undetermined body compartments, but a net flux back into the plasma compartment, and then disposal in urine following a 24-hr delay. Figures 2 and 3 show the modeled curves for both the normal control and FAP-affected individual, respectively. Between 80 and 90% of TABLE I1 PARAMETERS USED IN T H E MODEL AND METABOLISM VALUES

Normal subject

FAP subject

1311 '=I 1311 1251 (wild type) (Met30 variant) (wild type) (Met30 variant)

Initial conditions" (@ 10 min) K(1) IC(4) Rate coeffinents (hr-') 142.1) L ( I 2) I>( 3 ,2) L(2.3) L(0.3) f&',l) L(4.7)

Delay parameten (hr.) 1)'I (7) W 7 ) Metabolic valucs Residence time (hr) Fractional catabolic rate (hr ')

0.599 0.401

0.498 0.502

0.652 0.348

0.568 0.432

4.20E-2 5.21E-2 1.248-2 4.18E-3 0 2.32E-2 1

6.238-2 5.84E-2 1.58E-2 7.80E-3 0 2.998-2 1

6.84E-2 8.37E-2 1.98E-2 7.608-3 0 2.5OE-2 1

1.75E-1

1.85E-1 4.288-2 1.79E-2 5.99E-3 3.55E-2 1

23.8 4

23.8 4

22.7 4

22.7 4

25.77

16.64

26.04

12.40

0.039

0.060

"The model does not account for the initial rapid excretion.

0.038

0.081


153

sa

W

u)

z!a

2

10-4

TIME (HOURS)

FIG. 2. (A) Plasma lS1I and '=I decay, normal subject. (B) Cumulative fraction of dose excreted.

the radiolabels were recovered in urine within 7 days following iv administration of the labeled proteins, with as much as 50% present within the first 10 min for both subjects. TCA precipitation of urine aliquots showed over 95% of the isotope was not associated with protein (Murrell, 1992). Table I gives specific patient information and Table I1 the metabolism values derived through modeling the curves in Figs. 2 and 3. Both subjects metabolized and excreted 1251faster than 1311. 1.5-fold faster in the normal

154


2K

10-3

W

u1

i 10-4

W

z

6

o_

10-5

BF 0

a 10-6

!i

a

i8

1*0°

0.76

8

E

0.50

2

E 0.25

31 3

0.00

FIG. 3. (A) Plasma 13*1 and '"I decay, FAP subject. (B) Cumulative fraction of dose excreted.

and 2.1-fold faster in the FAP-affected individual. Residence time of I 3 l I was approximately 26 hr for both subjects, but the residence time of lz5I for the FAP-affected subject was 75% (12.4 hr) of that for the normal control (16.6 hr). IV.

DISCUSSION

The results of this study show that the 'zI-Met30-TTR is metabolized at a faster rate than normal 1311-TTRregardless of the subject's medical


155

status. This finding is in agreement with the suggestion by Hamilton et al. (1992) that the structural change in the Met30 variant of transthyretin may affect the metabolic properties of the mutant protein. In addition, several investigators have found overall serum TTR levels to be significantly lower in amyloidosis patients compared to normal controls, without distinction between normal and variant protein (Benson and Dwulet, 1983; Westermark et al., 1985; Shoji and Nakagawa, 1988). This study showed the residence time of lZI was lower but that of 13'1was equal in the FAP-affected individual compared to the normal control, indicating that only the variant protein is metabolized faster in the disease state. Thus, it is possible that both the structural changes in variant TTR and the medical status of the individual contributed to the increased rate of metabolism of Met30-TTR in the FAP affected individual. These results will contribute to our understanding of amyloid fibril formation in vivo for FAP-affected individuals. Future studies will involve more subjects, including unaffected carriers of the Met30-TTR gene. REFERENCES Benson, M. D., and Dwulet, F. E. (1983). Prealbumin and retinol binding protein Serum concentrations in the Indiana type hereditary amyloidosis. Arthrirk Rheum. 26,1493-1498. Berman, M. (1%3). A postulate to aid in model building. J. Theor. B i d . 4,229-236. Costa, P. P., Figueira, A. S., and Bravo, F. R. (1978). Amyloid fibril protein related to prealbumin in familial amyloidotic polyneuropathy. Proc. Nutl. Acud Sci U.S.A.75,44994503. Hamilton, J. A., Steinrauf, L. K., Liepnieks, J., Benson, M. D., Holmgren, G., Sandgren, O., and Steen, L. (1992). Alteration in molecular structure which results in disease: The Met30-variant of human plasma transthyretin. Biochim. Bwphys. Actu ll39,9-16. Murrell, J. R. (1992).Ph.D. Thesis, Indiana University, School of Medicine, Department of Medical and Molecular Genetics, Indianapolis. Shoji, S., and Nakagawa, S. (1988).Serum prealbumin and retinol-bindingprotein wncentrations in Japanese-type familial amyloid polyneuropathy. Eur. Neurol. 28,191-193. Skinner, M., Connors, L. H., Rubinow, A., Libbey, C., Sipe, J. D., and Cohen, A. S. (1985). Lowered prealbumin levels in patients with familial amyloid polyneuropathy (FAP) and their non-affected but at-risk relatives. Am. J. Med. Sci 289,17-21. Westermark, P., Pitkanan, P., Benson, L., Bahlquist, A., Olofsson, B. O., and Cornwell, G. G. (1985). Serum prealbumin and retinol-binding protein in the prealbumin-related senile and familial forms of systemic amyloidosis. Lab. Invest. 52,314- 318.


ADVANCES IN FOOD AND NUTRITION RESEARCH.VOL.40

Chapter 9 USE OF A FOUR PARAMETER LOGISTIC EQUATION AND PARAMETER SHARING TO EVALUATE ANIMAL RESPONSES TO GRADED LEVELS OF NITROGEN OR AMINO ACIDS M. J. GAHL,’ T. D. CRENSHAW, N. J. BENEVENGA, AND M. D. FINKE2 Departments of Nutritional Sciences and Meat and Animal Science University of Wisconsin Madison, Wisconsin 53706

I. Diminishing Returns and Dose-Response Relationships A. Rectilinear vs Curvilinear Approaches B. Four Parameter Logistic Equation-A Nonlinear Approach 11. Diminishing Returns and Protein Quality A. Relative Values Change with Respect to Curve Shape B. Protein Quality Improvement by Addition of the Limiting Amino Acid 111. Response of Rats to Each Indispensable Amino Acid References

I. DIMINISHING RETURNS AND DOSE-RESPONSE RELATIONSHIPS

A. RECTILINEAR VS CURVILINEAR APPROACHES The response of animals to graded levels of nitrogen or amino acids has been considered a linear phenomenon in the past. Currently, many of the swine growth models are still based on linear improvements in growth as increments of amino acids are added to a diet. A desirable feature of the statistical approach to defining a linear-plateau relationship is that a single nutrient level is defined as the requirement to maximize growth. A constant efficiency of nutrient use is assumed, and when the requirement is met, the

’ Present address: Farmland Industries, Inc., 3705North 139thStreet, Kansas City, KS 66109. Present address: PETsMART, lo000 North 31st Avenue, Phoenix, AZ 85051.

157 Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

158


efficiency suddenly falls to zero at the plateau (break-point or bent-stick approach). However, several investigators have shown that rats and pigs respond to increments of amino acids in a curvilinear fashion (Finke et al., 1987b; Fuller and Garthwaite, 1993; Gahl et al., 1991, 1994; Mercer et al., 1987;Robbins et al., 1979). Because of the curvilinear response to increasing nutrient intake and the asymptotic plateau (a result of using nonlinear models) defining requirements for maximum gain is difficult. Setting a point on the curve as the requirement becomes arbitrary. Even though the task of describing and interpreting curvilinear responses to nutrient inputs is more complex than the rectilinear approach, the curvilinear approach broadens the scope of information obtained in experiments conducted to compare protein qualities and responses to dietary additions of amino acids. The shape of the dose-response relationship has been an issue of debate. A theory has been proposed where a single animal’s response to graded levels of amino acid was linear and, as a result of pooling a group of these individual linear responses, a curvilinear relationship is observed (Fisher et al., 1973).The original data often used to support this theory was derived from egg production responses of laying hens. Egg output is a discrete variable while growth or protein accretion is a continuous variable; egg production may not be applicable to growth or protein accretion. Fuller and Garthwaite (1993) examined the nitrogen retention response of individual pigs to graded levels of nitrogen. A curvilinear response was observed. Implicit in the curvilinear response is the decreasing efficiency of nutrient use as the maximum gain of the animals is approached. Because of the decreased efficiency of nitrogen utilization near maximum gain, using a break-point or rectilinear estimate of the requirement would underestimate the actual requirement by 25% (Fuller and Garthwaite, 1993). The decreasing improvement in response as equal additional increments of nutrient are added to a diet is referred to as diminishing returns (Parks, 1970,1982). In addition, the concept of linear responses to graded levels of nutrient input is not consistent with the kinetics of enzymes or enzyme systems. A curved response for individual animals would be expected just as a Michaelis-Menton relationship is expected with increasing substrate concentration for an enzyme. Curved responses have been observed when lysine a-ketoglutarate reductase activity (a simple enzyme system) and lysine oxidation to CO;?(a more complicated pathway) were measured (Blemings et al., 1994). Lysine metabolism in liver homogenates may not be directly related to whole animal responses (an animal is not a big enzyme). An animal is a system of pools and fluxes, however, and there does not appear to be a set of discrete odoff switches.

CHAPTER 9 DOSE-RESPONSE RELATIONSHIPS

159

B. FOUR PARAMETER LOGISTIC EQUATION-A NONLINEAR APPROACH Diminishing returns have implications in evaluating protein quality and in determining amino acid requirements. If the diminishing returns components of response curves differ (different response curve shapes) one would expect the relative values (Hegsted er al., 1968) of proteins to differ over the range of the response. Therefore, rather than comparing proteins at equal intakes, dose-response relationships could be defined for several proteins (or protein mixtures) and relative values could be compared at specific levels of performance. Attempting to characterize enzyme kinetics by measuring the rate at a single substrate concentration would be futile. On the contrary several substrate concentrations are required to define the velocity vs substrate relationship. Therefore, several levels (usually 6 to 12) of nitrogen or amino acids would be required to define diminishing returns responses. A logistic equation has been used to describe the response of animals to graded levels of nitrogen or amino acids which takes into account the diminishing returns response (Finke et al., 1987a,b, 1989; Gahl et al., 1991, 1994). The logistic equation used previously is defined as follows (Gahl et al., 1991).

where r is gain at intake "Z" (nitrogen or weight), I equals nitrogen or amino acid intake, b is the y intercept (response to zero nutrient), R,, is the response maximum (at infinite intake), c is the parameter related to response curve shape, and k is the parameter related to response curve scale. In order to test differences among curves using this equation, parameter sharing has been used (DeLean et al., 1978;Finke et al., 1987a).The parameters for the response curves are estimated simultaneously and therefore can be forced to share common values based on biological considerations or based on statistical evaluation. For example, the response maximum for a given set of animals would be expected to be the same so R, could be forced to a common value for all curves. The parameters c and k can be shared based on a simple t test between pairs of estimates; the pair with the least significant t test would be forced to a common value in the following fit. An extra sum of squares test is used to test the effect of the constraints (parameter sharing) on the fit (Draper and Smith, 1981). The procedure is similar to performing backward elimination in multiple regression. Parameter sharing allows pooling of data across curves to estimate parameters

160


such as R,, (a region of the curve where the greatest variability occurs). Therefore more precise parameter estimates will be made. Parameter sharing is also a more powerful method of testing differences among curves compared to using the confidence limits for the parameter estimates. The NLIN procedure of SAS has been used to estimate the parameters of the logistic equation using parameter sharing to test the differences among curves (SAS, 1982).

II. DIMINISHING RETURNS AND PROTEIN QUALITY A. RELATIVE VALUES CHANGE WITH RESPECT TO

CURVE SHAPE Finke et al. (1987b) have compared the protein qualities of five nitrogen sources using the logistic equation and parameter sharing. Corn gluten meal (CGM), Mormon cricket meal (MCM), and three CGM-MCM mixtures (40% CGM-60% MCM, 50% CGM-50% MCM, and 60% CGM-40% MCM) were fed at 12 levels. The response to these nitrogen sources is shown in Fig. 1. The response to increasing nitrogen intake is curvilinear and is significantly different for each source of nitrogen. The three mixtures when fed to rats resulted in different response curve shapes but not response curve scales. The differences in protein quality that are due to curve shape is illustrated by comparing the relative values of the protein mixtures. The relative values of the proteins were compared by calculating the intake required for specific levels of performance. Using the 40 CGM60 MCM mixture as a reference, the relative values of the other proteins were calculated as the intake required for the reference protein divided by the intake required for the test protein (Proteins that required a higher intake at a specific response level had a lower relative value.) The relative values are shown graphically in Fig. 2. When rats were fed diets containing CGM as the protein source, a higher relative value was observed for mainte= 0) compared to growth (35% for R,,, = 95%). nance (48% for R,, Similar observations were made when rats were fed MCM; the relative value decreased from 111% at maintenance to 82% near maximum gain. However, when mixtures of the two protein sources were fed the relative values increased from maintenance (70%) to growth (90%). The change in relative values of proteins from maintenance to maximum gain suggests that a single estimate of protein quality at a defined intake would be misleading. Evaluating protein quality at a defined level of performance (by comparing relative intakes) would be a more accurate method.


161

4.0 3.5

3.0

2

2.5

4 2.0

1 bo

1.5

.-

1.0 0.5

0.0 -0.5

I

I

I

0

5

I

I

15 Nitrogen intake, g/2 1 d

10

20

FIG. 1. Body nitrogen gain (g) vs nitrogen intake (g) over 21 days for groups of four rats fed graded levels of corn gluten meal (CGM), Mormon cricket meal (MCM), or CGM-MCM mixtures. Lines are the best fits to the data using the logistic equation (Finke et al., 1987b). Each data point represents the mean 2 SEM of four rats.

B. PROTEIN QUALITY IMPROVEMENT BY ADDITION OF THE LIMITING AMINO ACID Addition of the limiting amino acid is one method of improving the quality of a protein. Methionine (0.4%)was added to the MCM to determine the effect of adding a limiting amino acid to a protein source. The methionine supplementation improved growth and nitrogen gain which is illustrated by the changes in both response curve scale and response curve shape; the c and k parameters were significantly different (Finke et al., 1987b). Using MCM + Met as the reference, MCM alone was always a lower quality protein and the relative value decreased as higher levels of nitrogen were fed (from 90 to 62%). The different response curve shape and scale suggest that MCM + Met when fed to rats results in a different diminishing returns response. The diminishing returns responses can be compared by examining the first derivative of the response curves (Fig. 3). The slope (dr/dZ) of the response curve is termed marginal efficiency and reflects the efficiency of use of a “small” increment of nitro-

162


60 CGM - 40 MCM CGM

I

-

95 5c

-0

FIG. 2. Relative values of corn gluten meal (CGM), Mormon cricket meal (MCM), and CGM-MCM mixtures at 0, 50, or 95% of R, (Fmke ef al., 1987b). The 40% CGM-60% MCM protein mixture was used as the control (relative value = 100%).The relative values of the other proteins and protein mixtures were calculated as the nitrogen intake required to achieve a specific nitrogen gain for the control mixture divided by the intake required for the identical response by rats fed the test proteins.

gen added to a diet. When MCM is fed to rats, diminishing returns are observed at intakes near zero (maximum efficiency of nitrogen gain was 47%) and the efficiency of nitrogen use rapidly declines and approaches zero as maximum gain is approached. However, when MCM is supplemented with methionine, the efficiency of nitrogen gain increases to a maximum of 60% at 25% of R,, and then declines as R,,, is approached. The curvilinear response and decreasing efficiency could be due to a change in the limiting amino acid as intake is increased since the pattern of amino acid requirements for maintenance is clearly different than the requirement for maintenance + growth (Benevenga et al., 1994). The change in limiting amino acid could also explain the change in relative values of proteins from maintenance to maximum gain. Amino acids could be used with different efficiencies and could evoke a different diminishing returns response when fed to rats. Ill. RESPONSE OF RATS TO EACH INDISPENSABLE AMINO ACID A series of crystalline amino acid diets were used to examine the response of rats to each indispensable amino acid when limiting (Gahl el af., 1991).

The limiting amino acid was fed at 10 levels [0-150% of the National

163


B

4.0 3.5

3 (v

0.6 0.5

a Q

0.4

9ei

3.0 2.5

W

6

2.0

0.3

.@

1.5

5

0.2

1.0 0.5

0.0

0.1

/'

MCM + .4%Met MCM

2a

0.0

-0.5 0

4

8

1216

0

4

8

1216

Nitrogen intake, g/21d FIG. 3. (A) Body nitrogen gain (g) vs nitrogen intake (g) over 21 days for groups of four rats fed graded levels of Mormon cricket meal or Mormon cricket meal with supplemental methionine.Lines are the best fits using the logistic equation (Finke ef al., 1987b). Each data point represents the mean 2 SEM of four rats. (B) Marginal efficiency (dddl) of nitrogen intake used for nitrogen gain. The marginal efficiency reflects the efficiency of utilization of an increment of nitrogen added to the diet.

Research Council (NRC), 19781 while the other amino acids were included in the diet at 35% (35-185%) above the relative level of the limiting amino acid. By using a 35% excess of the other amino acids, the amino acid of interest should remain first limiting. The 100 dietary treatments were fed for 21 days and allowed comparison of the dose-response relationships for each indispensable amino acid. The logistic equation was used to describe the response and parameter sharing was used to test differences among curves. The response curves for four of the amino acids (lysine, methionine + cystine, threonine, and tryptophan) are shown in Fig. 4. Marginal efficiencywas used to examine the efficiency of amino acid use and also the magnitude of the diminishing returns component. The efficiency for total sulfur amino acids is similar to that observed for MCM and begins to decline at very low levels of intake. Lysine, threonine, and tryptophan are utilized with a lower efficiency. However the shape of the response curves differ compared to the curve for total sulfur amino acids; there is a maximum at 35% of the requirement and then the efficiency rapidly declines and approaches zero. Diminishing returns impact on responses over at least the upper 60%of the response curve. In summary, the indispensable

164

PART I1 PROTEIN AND AMINO ACID METABOLISM 4.0 3.6 3.2

z! 2.8

s

g

2.4 2.0

Q 1.6 O 1.2 .Q 0.8 0.4 0.0 -0.4 0

0 20 40 60 80 100 120 20 40 60 80 100 120 Amino Acid Intake,% of Requirement

FIG. 4. (A) Nitrogen gain response curves generated using the parameter estimates for the logistic equation for rats fed diets limiting in one indispensable amino acid: lysine, methionine t cystine, threonine, or tryptophan (Gahl m al., 1991). The mean -t SEM for each group of four rats was plotted for each dietary amino acid concentration. (B) Marginal efficiency (dr/ d l ) of the limiting amino acid for nitrogen gain: lysine, methionine + cystine, threonine, or tryptophan. The marginal efficiency reflects the efficiency of utilization of an increment of limiting amino acid added to the diet.

amino acids are used with different efficiencies when limiting and have different diminishing returns responses. Maximum gain requirements for nitrogen or amino acid intake are more difficult to estimate because of the curvilinear nature of the dose-response relationship compared to a break-point approach which can be used to estimate a single requirement. R,,, is an asymptotic plateau and theoretically requires infinite intake. A n arbitrary decision must be made to estimate the concentration that defines maximum gain. The response curve can be used to estimate the intake required for the chosen level of performance. The required level of intake begins to rapidly increase as levels of performance above 95% R,, are attempted. Therefore, 95% R,, has been arbitrarily chosen to define the amino acid requirements of the rat using the response curves for each indispensable amino acid (Benevenga et al., 1994). The requirements for 95% R, weight and nitrogen gain are shown in Table I. As expected the requirements for nitrogen gain are 0 4 4 %higher compared to the requirements for weight gain. An assumption made in using a break-point approach is that the nutrient is used with a constant efficiency. Since the response is curvilinear, the requirement for maximum gain would be underestimated. The amino acid

165


TABLE I (g/100 g) REQUIRED

DIETARY AMINO ACID CONCENTRATION

FOR

95% R,,

FOR

G R O W H OR NITROGEN GAIN IN RATSa

Arg

His Ile Leu LYS Met + Cys Phe + Tyr Thr Trp

Val

NRC 1978

Weight gain

Nitrogen gain

Ratio nit/weightb

0.60 0.60 0.50 0.75 0.70 0.60

0.43 0.28 0.62 1.07 0.92 0.97 1.01 0.62 0.20 0.74

0.62 0.33 0.86 1.30 1.11 1.21 1.32 0.73 0.20

1.44 1.18 1.39 1.21 1.21 1.25 1.31 1.18 1.oo 1.24

0.80 0.50 0.15 0.60

0.92

‘The amino acid requirements were calculated based on the parameters of the logistic equation (Benevenga et al., 1994). The ratio was calculated as the requirement for nitrogen gain divided by the requirement for weight gain.

would also be expected to be used with the highest efficiency when limiting, Therefore, if a series of experiments were conducted to estimate requirements for amino acids, one might expect that the diet formulated based on the new requirements would not support maximum gain since the amino acids would now be colimiting and would be used with a lower efficiency. Experiments have been conducted to estimate requirements for guinea pigs using a linear approach to estimate each requirement. When a diet was formulated with the estimated requirements, an additional 30-508 of the amino acid mixture was required to obtain maximum gain (Blevins, 1983; NRC, 1995). Although the mixture estimated using the logistic curves (Benevenga et d.,1994) has not been tested, the requirements are estimated taking into account diminishing returns and would not be expected to be underestimated to the same magnitude. However, the requirements estimated from the logistic curves were obtained from rats fed diets that were first limiting in the specific amino acids. The curvilinear response of animals to graded levels of nitrogen or amino acids necessitates the description of the dose-response relationship. The relative values of proteins differ from maintenance to maximum gain if the shapes of the curves or the magnitude of the diminishing returns components differ. This implies that there is not a single “protein quality” that can be associated with a particular source of nitrogen (protein). Proteins may have a higher quality for maintenance and a lower quality for growth

166


compared to a reference protein. The change in relative values from maintenance to maximum gain may be due to a shift in the limiting amino acid. Different limiting amino acids are used with different efficiencies and have different diminishing returns components. Because of the curvilinear nature of the dose-response relationship, amino acid requirements should be estimated taking into account the decreasing efficiency associated with diminishing returns. Comparing sources and levels of proteins and amino acids taking into account diminishing returns responses has implications in swine nutrition. Diet formulation decisions should be made based on the economic value of adding increments of amino acids near maximum gain. When the linear-plateau approach is used the optimum level of nitrogen or amino acid to feed is predicted at the “break-point” which corresponds to maximum gain. However, considering diminishing returns the optimum feeding level may not be at maximum gain. The response curve and current prices could be used to predict the optimum level of inputs to feed which would maximize economic return.

REFERENCES Benevenga, N. J., Gahl, M. J., Crenshaw, T. D.. and Finke, M. D. (1994). Protein and amino acid requirements for maintenance and amino acid requirements for growth of laboratory rats. 1. Nutr. l24,451-453. Blemings, K. P., Crenshaw, T. D., Swick, R. W., and Benevenga, N. J. (1994). Lysine aketoglutarate reductase and saccharopine dehydrogenase are located only in the mitochondrial matrix in rat liver. J. Nutr. l24, 1215-1221. Hlevins, B. G. (1983). Amino acid requirements of guinea pigs. XIII. The indispensable amino acid component at levels of total nitrogen near or above the requirement. M.S. Thesis, University of Missouri, Columbia. DeLean, A., Munson, P. J., and Rodbard, D. (1978). Simultaneous analysis of families of sigmoidal curves: Application to bioassay, radioligand assay, and physiological doseresponse curves. Am. J. Physwl. 235, E97-E102. Draper, N. R., and Smith, H.(1981). “Applied Regression Analysis.” John Wiley, New York. Finke. M. D., DeFoliart, G . R., and Benevenga, N. J. (1987a). Use of simultaneous curve fitting and a four-parameter logistic model to evaluate the nutritional quality of protein sources at growth rates of rats from maintenance to maximum gain. J. Nutr. 117,1681688.

Finke. M. D., DeFoliart, G. R., and Benevenga, N. J. (1987b). Use of a four-parameter logistic model to evaluate the protein quality of mixtures of Mormon cricket meal and corn gluten meal in rats. 1. Nutr. 117, 1740-1750. Finke, M. D., DeFoliart, G. R., and Benevenga, N. J. (1989). Use of a four-parameter logistic model to evaluate the quality of the protein from three insect species when fed to rats. J. Nutr. 119,864-871. Fisher, C..Moms, T. R., and Jennings, R. C. (1973). A model for the description and prediction of the response of laying hens to amino acid intake. Br. Poult. Sci. 14,469-4&. Fuller, M.F., and Garthwaite, P. (1993). The form of response of body protein accretion to dietary amino acid supply. 1. Nutr. l23,957-%3.


167

Gahl, M. J., Finke, M. D., Crenshaw, T. D., and Benevenga, N. J. (1991). Use of a four parameter logistic equation to evaluate the response of growing rats to ten levels of each indispensable amino acid. J. Nufr. 12% 1720-1729. Gahl, M. J., Crenshaw, T. D., and Benevenga, N. J. (1994). Diminishing returns in weight, nitrogen and lysine gain of pigs fed six levels of lysine from three supplemental sources. J. Anim. Sci. 72,3177-3187. Hegsted, D. M.,Neff, R., and Worcester, J. (1968). Determination of the relative nutritive value of proteins. J. Agric. Food. Chem. 16,190-195. Mercer, L. P., Dodds, S. J., and Smith, D. L. (1987). New method for formulation of amino acid concentrations and ratios in diets of rats. J. Nufr. 117,1936-1944. National Research Council (NRC) (1978). Nutrient requirements of the laboratory rat. In “Nutrient Requirements of Laboratory Animals,” 3rd rev. ed., pp. 7-37. National Academy of Science, Washington, DC. National Research Council (1995). Nutrient requirements of the Guinea Pig. In “Nutrient Requirements of Laboratory Animals,” 4th rev. ed., pp. 103-124. National Academy of Science, Washington, DC. Parks, J. R. (1970). Growth curves and the physiology of growth. 111. Effects of dietary protein. Am. J. Physiol. 219,840-843. Parks, J. R. (1982). A Theory of Feeding and Growth of Animals.” Springer-Verlag, Heidelberg. Robbins, K. R., Norton, H. W., and Baker, D. H. (1979). Estimation of nutrient requirements from growth data. J. Nufr. 109,1710-1714. SAS (1982). “SAS User’s Guide: Statistics.” SAS Inst., Cary, NC.


Part 111 ENERGY METABOLISM



Chapter 10 TOTAL ENERGY EXPENDITURE OF FREE-LIVING HUMANS CAN BE ESTIMATED WITH THE DOUBLY LABELED WATER METHOD WILLIAM W. WONG Department of Pediatrics U S D M A R S Children's Nutrition Research Center Baylor College of Medicine Houston, Texas 77030

I. Introduction 11. Theory of the 2H2'80Method

A. Assumption of the 2H280 Method B. Equations Used in the 'HI80Method 111. Analytical Methods A. Isotopes B. Mass Spectrometric Analyses IV. Validations of the 2H2'80 Method V. Conclusion References

I. INTRODUCTION

Energy is required for muscular activity, growth, reproduction, and synthesis of metabolites such as proteins, fatty acids, nucleic acids, and steroids, which are essential to maintain basal metabolic functions as well as optimal growth and development. Numerous methods such as the food record, [13C]bicarbonate infusion, and indirect calorimetry have been used to estimate energy expenditure in humans. The food record seldom reflects the true caloric content of ethnic foods, and this procedure does not work well with children. It is also well documented that overweight individuals often underreport their food intake. The [13C]-bicarbonateinfusion method is invasive and of short duration (lo0 @day), isotope sequestration in humans results in less than 1%error in the TEE estimate. Reentry of the labeled H 2 0 and C 0 2 is most likely when premature infants are confined in incubators. However, TEE of premature infants estimated by the isotope method has been shown to be within 1%of the calorimetric estimates (Roberts et af., 1986; Jensen et af., 1992). Therefore, reentry of the labeled H 2 0 and C 0 2 in humans is not likely to affect the accuracy of the isotope method for estimation of TEE in free-living subjects.

CHAPTER 10 ENERGY EXPENDITURE! ESTIMATION

179

V. CONCLUSION

The 2H2180method has long been validated as an accurate method for estimation of TEE in small mammals and birds (Lifson and McClintock, 1966). In spite of the violation of many of the assumptions associated with the isotope method, with proper corrections, TEE estimated by the 2H2180 method has proven accurate against TEE estimated by energy balance and indirect calorimetry in infants, children, and adults. Because the isotope method is noninvasive, nonrestrictive, and does not expose the subjects to radiation, the 2H$80 method is considered the method of choice by the nutrition community for the estimation of energy requirements during infancy, growth, pregnancy, and lactation.

ACKNOWLEDGMENTS The author thanks L. Loddeke for editorial review. This project has been funded in part with federal funds from the US Department of Agriculture (USDA), Agriculture Research Service, under Cooperative Agreement No. 58-7MN1-6-100.The contents of this publication do not necessarily reflect the views or policies of the USDA, nor does mention of trade names, commercial products, or organizations imply endorsement by the US Government.

REFERENCES Black, A. E., Prentice, A. M., and Coward, W. A. (1986). Use of food quotients to predict respiratory quotients for the doubly-labelled water method of measuring energy expenditure. Hum Nutr.: Clin. Nutr. 4OC, 381-391. Coward, W. A., Prentice, A. M., Murgatroyd, P. R., Davies, H. L., Cole, T. J., Sawyer, M., Goldberg, G. R.,Halliday, D., and Macnamara, J. P. (1984). Measurement of COz and water production rates in man using ’H, ‘80-labelled HzOcomparison between calorimeter and isotope values. In “Human Energy Metabolism: Physical Activity and Energy Expenditure Measurementsin Epidemiological Research Based Upon Direct and Indirect Calorimetry” (A. J. H. van Es, ed.), pp. 126-128. EURO-NUT, The Netherlands. de V. Weir, J. B. (1949). New methods for calculating metabolic rate with special reference to protein metabolism, J. Physiol. (London) 109, 1-9. Gonfiantini, R. (1984). “Report on Advisory Group Meeting on Stable Isotope Reference Samples for Geochemical and Hydrological Investigations.” Int. At. Energy Agency, Vienna. Halliday, D., and Miller, A. (1977). Precise measurement of total body water using trace quantities of deuterium oxide. Biomed. Mass Spectrom. 4,8241. Jensen, C. L., Butte, N. F., Wong, W. W., and Moon, J. K. (1992). Determining energy expenditure in preterm infants: comparison of 2H2180method and indirect calorimetry. Am 1. Physiol. 263, R685-R692. Jones, P. J. H., Winthrop, A. L., Schoeller, D. A., Filler, R. M., Swyer, P. R., Smith, J., and Heim, T. (1988). Evaluation of doubly labeled water for measuring energy expenditure during changing nutrition. Am. J. Clin. Nutr. 41,799-804.

PART 111 ENERGY METABOLISM Klein, P. D., James, W. P. T., Wong, W. W., Irving, C. S., Murgatroyd, P. R., Cabrera, M., Dallosso. H. M., Klein, E. R., and Nichols, B. L. (1984). Calorimetric validations of the doubly-labelled water method for determination of energy expenditure in man. Hum. Nutr.: Clin. Nufr. 38C,95-106. Lifson, N., and McClintock, R. (1966). Theory of the use of turnover rates of body water for measuring energy balance. J. Theor. Biol. U,46-74. Lifson, N., Gordon, G. B., Visscher. M. B., and Nier, A. 0. (1949). The fate of utilized molecular oxygen and the source of heavy oxygen of respiratory carbon dioxide, studied with the aid of heavy oxygen. J. Biol. Chem. 180,803-811. Pflug, K. P., Schuster, K. D., Pichotka, J. P., and Forstel, H. (1979). Fractionation effects of oxygen isotopes in mammals. In “Stable Isotopes. Proceedings of the Third International Conference” (E. R. Klein and P. D. Klein, eds.), pp. 553-561. Academic Press, New York. Roberts, S. B., Coward, W. A., Schlingenseipen, K. H.. Nohria, V., and Lucas, A. 1986). Comparison of the doubly labeled water (2H280)method with indirect calorimetry and a nutrient-balance study for simultaneous determination of energy expenditure, water intake, and metabolizable energy intake in preterm infants. Am. J. Clin. Nutr. 44,315-322. Schoeller, D. A. (1988). Measurement of energy expenditure in free-living humans by using doubly labeled water. J. Nutr. 118, 1278-1289. Schoeller, D. A,, Leitch, C. A., and Brown, C. (1986). Doubly labeled water method: in vivo oxygen and hydrogen isotope fractionation. Am. J. fhysiol. 1, R1137-R1143 Stein,T. P., Hoyt, R. W., Settle, R. G., O’Toole. M., and Hiller, W. D. B. (1987). Determination of energy expenditure during heavy exercise, normal daily activity, and sleep using the doubly-labelled-water (2Hz’80) method. Am. J. Clin. Nutr. 45, 534-539. Wong, W. W., and Klein, P.D. (1986). A review of techniques for the preparation of biological samples for mass-spectrometric measurements of hydrogen-Uhydrogen-1and oxygen-18/ oxygen-16 isotope ratios. Mass Spectrom. Rev. 5,313-342. Wong, W. W., Lee, L. S., and Klein, P. D. (1987). Deuterium and oxygen-18measurements on microliter samples of urine, plasma, saliva, and human milk. Am. J. Clin. Nutr. 45,905-913. Wong, W. W.. Cochran, W. J., Klish, W. J., Smith, E. O., Lee, L. S.. and Klein, P. D. (1988). In vivo isotope-fractionation factors and the measurement of deuterium- and oxygen-18 spaces from plasma, urine, saliva, respiratory water vapor, and carbon dioxide. Am. 1. Clin Nutr. 47, 1 4 . Wong, W. W., Butte, N. F., Garza, C., and Klein, P. D. (1990). Comparison of energy expenditure estimated in healthy infants using the doubly labelled water and energy balance methods. Eur. 1. Clin. Nutr. 44, 175-184. Wong, W. W., Leggitt, J. L., Clarke, L. L., and Klein, P. D. (1991). Rapid preparation of pyrogen-free *H2I8Ofor human-nutrition studies. Am. J. Clin. Nurr. 53, 585-586. Wong, W. W., Clarke, L. L., Llaurador, M., and Mein, P. D. (1992). A new zinc product for the reduction of water in physiological fluids to hydrogen gas for 2H/’H isotope ratio measurements. Eur. J. Clin. Nutr. 46,69-71. Wong. W. W., Clarke, L. L., Llaurador, M., Ferlic. L., and Klein, P. D. (1993a). The use of isotope ratio measurecotton balls to collect infant urine samples for 2H/’H and ‘80/’60 ments. Appl. Radiar. Isot. 44(8), 1125-1 128. Wong. W. W.. Hachey, D. L., Insull, W., Opekun, A., and Kiein, P. D. (1993b). Effect of dietary cholesterol on cholesterol synthesis in breast-fed and formula-fed infants. J. Lipid Res. 34, 1403-141 1.

Part IV

METHODS FOR OBTAINING KINETIC DATA



Chapter 11 MICRODIALYSIS AND ULTRAFILTRATION ELSA M. JANLE AND PETER T.KISSINGER Biowlytical System, lnc, West Lafayene, Indiana 47906

1.

Introduction

11. Comparison of Microdialysis and Ultraliltration A. Probe Size and Physical Characteristics

B. Volume Change C. Recovery D. Sample Collection Ill. Examples of Studies Using Microdialysis and Ultrafiltration A. Ultrafiltration 8. Microdialysis IV. Summary References

I. INTRODUCTION

Microdialysis and ultrafiltration are complementary sampling techniques which have been developed for studying the interstitial space in vivo (Janle and Kissinger, 1993). Both of these techniques employ membrane probes that can be implanted in the tissue of interest, and the studies can be conducted in awake moving animals or in human subjects. Because these techniques sample low molecular weight compounds, and because they provide samples that require very little preparation prior to analysis, they hold considerable potential as tools for nutritional research. Microdialysis and ultrafiltration are separation techniques that involve moving a chemical across a semipermeable membrane. In microdialysis (Fig. lA), a fluid is pumped through the membrane capillary of a probe. The analyte crosses the membrane by diffusion. The driving force is a concentration gradient. Under ideal conditions, the perfusion fluid is isos183 Copyright 0 1996 by Academic F’ress, Inc. All rights of reproduction in any form reserved.

184

PART IV METHODS FOR OBTAINING KINETIC DATA LIQUID I

f

A MicrodialysisSampling

YUUM t

B Ultraflltration Sampling

FIG. 1. ( A ) The dialysis process results from the diffusion of molecules across the membrane wall. ( B ) In ultrafiltration. small molecules are actively pulled across the membrane.

motic with the tissue, the hydrostatic pressure is minimal, and there is no net water transfer between the perfused fluid and the animal tissue. In ultrafiltration (Fig. lB), the driving force is a pressure differential applied across a semipermeable membrane, in this case, in the form of a vacuum. The reduced pressure causes small molecules, here water and solute molecules, that cross the membrane to be pulled out of the probe ( Janle-Swain er al., 1987). Large molecules or molecules which are repelled by the membrane are prevented from crossing and are not obtained in the sample. The use of individual capillary membranes (often called dialysis fibers or hollow fibers) makes it possible to work with the small volumes needed for bioanalytical chemistry. The capillary geometry makes it easy to move microliter samples continuously and provides for a more rapid collection rate.

II.

COMPARISON OF MICRODIALYSIS AND ULTRAFILTRATION

Microdialysis and ultrafiltration are complementary techniques. They have many characteristics in common. Both methods sample the extracellular space. Both techniques allow the use of repeated sampling in small animals without withdrawing blood. This often makes it possible for animals to act as their own controls. Single animals can be studied over a longer time or may be used in crossover studies. Because so much more data can be gained from a single animal, it may be possible in many instances to

CHAPTER 11 MICRODIALYSIS AND ULTRAFILTRATION

185

reduce the number of animals needed for significant data. Both techniques make studies in awake freely moving animals possible. These methods eliminate artifacts resulting from anesthesia and withdrawal of blood (Telting-Dim et af., 1992). Both techniques provide samples that require very little preparation for analysis. Unlike blood samples, they contain neither cells nor high molecular weight proteins. Microdialysis and ultrafiltration samples are often more stable than blood samples. The analyte has been removed from enzymes that cause degradation (Lunte et al., 1991). There are also notable differences between microdialysis and ultrafiltration. In some cases, one technique may serve as well as the other. However, in other cases, one technique may be preferable. A. PROBE SIZE AND PHYSICAL CHARACTERISTICS Ultrafiltration probes (Fig. 2) consist of one or more hollow dialysis fibers connected to a single microbore, nonpermeable outflow tube. The outflow tube is connected to the source of negative pressure, either a roller pump or a Vacutainer. The probes come in various configurations with different numbers of fibers and different fiber lengths. Ultrafiltration probes tend to be larger than microdialysis probes. Typically there are one to three fibers, each 2 to 12 cm in length. The choice of a probe depends on the size of the implantation site and the desired flow rate. With an ultrafiltration probe, membrane surface area (and therefore probe size) affects the flow rate. A subcutaneously implanted probe with three 12-an fibers would yield a flow rate of 1 pl/min, which would be suitable for subcutaneous implantation in a 200-g rat or any larger animal. For subcutaneous implantation in a mouse, a probe with one or three 2-cm fibers would be appropriate.

FIG. 2. An ultrafiltration probe consists of hollow dialysis fibers attached to microbore outflow tubing (A). Negative pressure is generated either by using a peristaltic pump (B) or by attaching a needle hub and using a Vacutainer (C).

186

PART IV METHODS FOR OBTAINING KINETIC DATA

The one-fiber probe would yield a flow rate of approximately 2 plhr; the three-fiber probe would yield 5 to 8 pllhr. Microdialysis probes come in several sizes and geometries; they can be much smaller than ultrafiltration probes because the volume of sample collected does not depend on the membrane surface area (Fig. 3A). Therefore, small microdialysis probes are used when precise spatial resolution is desired. When less spatial resolution is needed and higher recoveries are desirable, longer loop-type probes can be used (Fig. 3B).

B. VOLUME CHANGE In microdialysis,when the perfusion solution is pumped through the fiber capillary, there is ideally no change in volume of the perfusate and no fluid removed from the tissue. This requires a negligible pressure gradient across the membrane. The perfusate must be isosmotic with the tissue. If either the membrane capillary or the outlet capillary is very long, the back pressure at the membrane can become sufficient for it to leak. This can be minimized by using low flow rates and membranes that have relatively low molecular weight cutoffs. In ultrafiltration, a vacuum is applied to the probe and there is a net volume loss from the tissue sampled. One must keep in mind that the loss includes both water and solutes. Removal of fluid and neurochemicals could affect the phenomena being studied. This makes the use of ultrafiltration unsuitable for some applications, such as study of brain neurochemistry.

FIG. 3. Microdialysis probes come in various geometries. Some have small membranes (A). Larger loop-type probes (B) have higher recoveries.


187

Microdialysis, however, can be carried out so that there is no volume change and a minimal amount of analyte is removed. In other instances, such as the study of metabolites or pharmacokinetics in a subcutaneous location, ultrafiltration may be the method of choice. In this case fluid is readily replenished, and the amount of the chemicals of interest being removed is small with respect to the entire pool. C. RECOVERY Recovery is the term used to quantitate the amount of analyte obtained through the probe membrane. Absolute recovery is the total amount of analyte removed from the system via the probe. Relative recovery (expressed as a percentage) is the ratio of analyte concentration in the probe effluent to that in the solution or tissue being sampled. Depending on the nature of the experiment, it may be desirable to minimize or maximize these recoveries. If, for example, the concentration of the analyte were low and the experimenter were near the lower end of the sensitivity of the assay, conditions that would maximize the relative recovery may be chosen. However, if the experimenter were studying endogenous substances with sensitive feedback mechanisms, removing significant amounts could alter the physiological parameters and yield artifactual data. In this case, absolute recovery should be minimized. In microdialysis, the recovery of analyte from the sample depends on a number of factors, including the chemistry of the analyte, temperature, perfusion rate, membrane surface area, membrane characteristics, and the nature of the sample (including its fluid volume percentage and whether it is in motion). Microdialysis is typically done at low perfusion flow rates (0.5 to 2 pl/min). As flow rate increases, relative recovery decreases, but absolute recovery increases. Relative recoveries from small membrane probes are in the 1-20% range. Substantially higher recoveries in the 5040% range can be obtained with longer loop-type probes. In ultrafiltration, analyte molecules are basically carried along with the flow of water and electrolytes. The factors determining recovery in ultrafiltration are membrane characteristics, temperature, and chemistry of the analyte. Unlike microdialysis,recovery is not dependent on flow rate, membrane surface area, or probe size. Recovery tends to be higher than for dialysis, since there is no perfusion medium to dilute the collected analyte. Ultrafiltration recovery rates are typically in the 90-100% range. This high recovery rate simplifies determination of in vivo analyte concentrations. Table I illustrates some in vitro recoveries obtainable with ultrafiltration probes.

188

PART IV METHODS FOR OBTAINING KINETIC DATA TABLE I ULTRARLTRATING RECOVERIES

Anal yte G1ucose Urea Creatinine Lactate Clutamate Aspartate Asparagine Serine GI utamine Histadine Glycine Sodium Potassium

% Recovery

98 99

102 98 96 91 83 81 81 86 91

100 102

Another factor which may indirectly affect recovery for in vivo sampling is the biocompatibility of the probe. The tissue recognizes the probe as a foreign object and may respond by forming a fibrous layer around it. In short-term studies of hours to 2 days, this is not a significant factor. For microdialysis in longer term studies, this fibrous layer would present an additional diffusion barrier and may decrease recovery. In ultrafiltration, since there is bulk flow across the membrane, recovery may not be affected by this barrier; however, the added barrier may affect the volume of fluid which is able to reach the membrane. 1. Determination of in Vivo Concentration

Determination of in vitro recovery requires only the measurement of analyte concentration in a probe sample and an unperfused sample and calculation of the percentage difference. In ultrafiltration in vivo sampling, because the entire sample comes from the tissue, the analyte concentration of the sample represents the analyte concentration of the tissue. In vivo microdialysis sampling is not as simple. One cannot use the in v i m recovery to calculate an in vivo concentration. In many cases, it is not necessary to know the exact in vivo concentration; it is sufficient to observe trends. However, if the exact concentration is desired, there are several methods of obtaining an in vivo concentration. a. Difference Method: No Net Flux. The “difference method,” developed by Lijnnroth and co-workers (1987), is based on the principle that if


189

the concentrations of analyte in both the perfusate and the tissue are equal, there will be no net transfer of analyte between membrane and tissue. Determining recovery by this method involves perfusing the probe with varying concentrations of the analyte and determining the concentration of the efflux. The difference in concentration between influx and efflux is plotted against perfusate concentration. Where the line crosses the axis (see Fig. 4), the outflow concentration is the same as the inflow concentration, and the perfusate concentration is equal to the tissue concentration. The slope indicates the relative recovery. The problem with this method is that it is time-consuming. If one is measuring a stable analyte, it works well, but if the analyte is changing, the experimenter is chasing a moving target.

b. Extrapolation to Zero Flow. The slower the flow rate of perfusion in microdialysis, the higher the recovery. If the perfusate were not flowing, there would be adequate time to reach equilibrium, and the concentration would be the same inside and outside the probe. Therefore, in the “extrapolation to zero flow” method, the pump is run at progressively slower rates and the concentration of the collected samples is determined. Concentration is plotted against flow rate and extrapolated to find zero flow rate (Jacobson ef al., 1985).

+t

SLOPE YIELDS RELATIVE RECOVERY

ZERO NET FLUX = Cln vivo)

AC (CourCin) 0

CONCENTRATION PERFUSED

Cin

FIG. 4. The difference (or “no net flux”) method of determining in vivo recovery involves perfusing the probe with varying concentrations of the analyte. The difference in concentration between influx and efflux is plotted against perfusate concentration. Where the line crosses the axis, outflow and inflow concentrations are the same, and the perfusate concentration is equal to the tissue concentration. The slope indicates relative recovery.

1%


c. Retrograde Dialysis. The “retrograde dialysis” method assumes that the diffusion of a calibrator into the tissue will be the same as the diffusion of an analyte out of the tissue (Wang et al., 1993). Therefore, if one were to put a calibrating compound in the perfusate and measure its loss, the in vivo recovery of the analyte could be estimated. If the analyte of interest were an endogenous compound, one could choose a structurally similar compound for retrograde dialysis. For an exogenous compound, one could use the compound itself to calibrate recovery. For example, if one were studying the pharmacokinetics of a drug, a preliminary study could be done in which the drug was placed in the perfusate and its loss from the perfusate was measured. This would be used as the in vivo recovery. After an appropriate wash-out time to eliminate any drug that might have been absorbed during the preliminary test, the drug could then be administered by the conventional route, and the pharmacokinetics could be studied.

D. SAMPLE COLLECTION The fact that ultrafiltration and microdialysis probes allow for repeated studies in the same animal with minimum trauma to the animal makes long-term studies possible. Microdialysis probes can be used for weeks. Ultrafiltration is especially useful for long-term studies, since ultrafiltration probes are very durable and will last for months with no decrease in recovery. In these long-term studies, there is minimal tissue reaction to the probe. A small fibrous sheath forms around the probe fibers in about a week, but then remains stable. This fibrous sheath may contribute to a decrease in flow rate during the first week in some species. Ultrafiltration probes have been used for up to 6 months in dogs (Janle et al., 1991), 55 days in mice (Janle et al., 1992), and 1 month in a human clinical study (Ash et al., 1992). In larger animals, the ultrafiltration technique permits complete freedom of movement by using the Vacutainer collection method. Equipment vests can be worn by medium-sized animals, such as dogs. With larger animals, such as horses, collection vessels can be attached to the mane or taped to the leg, depending on the site of probe insertion. Both microdialysis and ultrafiltration can be automated. The microdialysis pump can be combined with a syringe selector and computer-controlled to perfuse with different solutions on a timed schedule. The samples can be collected in a computer-controlled fraction collector at preset intervals or injected directly into an HPLC analyzer. Ultrafiltration sample collection can also be automated with the use of a pump and a fraction collector.


191

111. EXAMPLES OF STUDIES USING MICRODIALYSIS AND ULTRAFILTRATION

The literature on studies utilizing microdialysis is extensive. Ultrafiltration is a newer technique, so fewer studies have been documented. Several examples of each technique will be discussed to indicate the range of possibilities for application of these in vivo sampling techniques. A. ULTRAFILTRATION 1. Subcutaneous Glucose in Diabetics

In this study, long-fiber ultrafiltration probes were implanted subcutaneously in diabetic dogs. Since the three fibers (12 cm long each) were looped, the actual implanted length was 6 cm.An attached hub and Vacutainer were used to generate a vacuum. Each dog wore an equipment vest with a Vacutainer placed in the pocket. The implantation site was chosen such that the exit site was immediately under the equipment pocket. To obtain a sample, the Vacutainer was changed. The dogs were allowed complete freedom of movement. Glucose was monitored daily using the probe. Once a week, for as long as the probe functioned, correlation studies between blood and ultrafiltrate glucose were conducted. Glucose levels were manipulated with food and insulin. Blood and ultrafiltrate samples were taken at hourly intervals. Figure 5 shows that even after 9 weeks of implantation,

Glucose (mgldl)

500 I

400-

300200

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l OO

2

4W

6

8

W 10

Time (hours) FIG. 5. Subcutaneous ultrafiltrate glucose tracks plasma glucose in a diabetic dog, 9 weeks after probe implantation.

192


ultrafiltrate glucose tracked blood glucose very closely (Janle et al., 1992). Regression analysis on all the blood and ultrafiltrate data in this 3-month study yielded r = 0.91 and P = O.OO0 (Fig. 6). The ultrafiltrate probe has been used in a 1-month clinical trial in human diabetics to monitor glucose (Ash ef ul., 1992). Being able to monitor glucose without obtaining blood samples has opened up the possibility of doing diabetes research on small rodents which previously could be done only on larger animals. In one study (Janle et al., 1992), diabetes was induced in mice by the injection of 50 mg/kg of streptozotocin for 5 consecutive days. The mice were individually monitored for up to 55 days postinjection. In this particular study, only daily average glucose concentrations were needed to follow the progress of the disease and to quantify the metabolic derangement, so small ultrafiltrate probes were used and samples were collected once a day. It could be seen from this monitoring procedure that the development of diabetes in different animals, given the same initial dose of the drug, varied considerably (Fig. 7). Using ultrafiltrate data, the progress of the diabetes and the total exposure to hyperglycemia could be quantitated and correlated with other sequelae of the disease. 2. Subcutaneous Urea and Creatinine

Blood plasma and ultrafiltrate concentrations of urea and creatinine have been studied with subcutaneous ultrafiltrate probes in diabetic dogs. The dogs had varying degrees of kidney complications secondary to the diabetes, Ultrafiltrate Glucose (mgldl)

300-

200

.

-

m Blood Glucose (mgldl) FIG. 6. Regression analysis on all blood and ultrafiltrate glucose in a 3-month study in diabetic dogs yielded c = 0.91 and P = 0.0oO.


193

Glucose (mgldL)

6001

D

Days post injection FIG. 7. Mice treated with injections of 50 mgkg of the diabetogenic drug streptozotocin for 5 successive days. Average daily glucose levels were monitored using ultrafiltrate probes.

and urea levels ranged from 10 to 50. Animals were studied in the fasting condition and after a meal. Ultrafiltrate levels of both urea and creatinine tracked blood levels well. The correlation of blood and ultrafiltrate was 3 = 0.90 for urea and 3 = 0.97 for creatinine (Janle and Ash, 1994). 3. Distribution of Endogenous Substances in Blood, Subcutaneous Tissue, and Muscle

The horse has been used as a model to study the distribution of intravenously infused amino acids and calcium from blood to the subcutaneous and muscle interstitial fluids. Subcutaneous ultrafiltrate levels of both amino acids and calcium were higher than blood concentrations under basal conditions and after infusion. After infusion of amino acids, subcutaneous ultrafiltrate concentrations tracked the rise and fall of plasma concentrations, but ultrafiltrate concentrations were greater than plasma concentrations at all time periods. Similarly,subcutaneous calcium ultrafiltrate concentrations tracked plasma concentrations after intravenous infusion of calcium. Subcutaneous concentrations were also slightly higher than plasma concentrations. Muscle concentrations of calcium were considerably higher than either plasma or subcutaneous concentrations, both in the basal state and after infusion. Calcium concentrations remained elevated for a longer time in muscle than in plasma or in subcutaneous tissue (Sojka et al., 1995; Spehar et al., 1995). These studies demonstrate the potential of using ultrafiltration to study distribution of nutrients in different tissues.

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B. MICRODIALYSIS 1. Metabolism in Human Adipose Tissite

Microdialysis has been used to study metabolism in adipose tissue in situ. Steady-state concentrations of glycerol, glucose, pyruvate, lactate, and adenosine have been determined (Arner and Bolinder, 1991). The concentration of a metabolite is determined by two factors. One factor is the amount of metabolite delivered to the site by the vascular system; the other is the amount produced or consumed by the tissue. If production or consumption of a metabolite is small, the concentration measured is more representative of whole body concentrations. If, however, the production or consumption of the tissue is extensive, concentrations will be indicative of tissue metabolism. Steady-state glucose levels were found to be similar in blood and adipose tissue, whereas glycerol levels were higher than in blood. This reflected the local production of glycerol by fat cells. Glycerol concentrations after glucose ingestion fall to lower levels in adipose tissue than in blood and rise more after exercise. Microdialysis has also been used to demonstrate differences in metabolic activity of adipose tissue in different locations during exercise. Microdialysis can also be used to investigate differences in metabolism in different locations. For example, glycerol levels have been found to rise higher in abdominal adipose tissue than in the femoral or gluteal. Microdialysis can also be used to investigate the intricacies of hormonal control on tissue metabolism. Hagstrom-Toft and co-workers (1992) demonstrated the decrease of adipose tissue glucose during insulin infusion despite the maintenance of normoglycemia. Pyruvate and lactate concentrations increased and glycerol decreased to a greater extent in adipose tissue than in blood. Using microdialysis, beta adrenoreceptor stimulation was shown to oppose the action of insulin on glucose uptake and lipolysis but to have synergistic effects with insulin on nonoxidative glucose metabolism. 2. Effects of Ischemia on Kidney Metabolism

Ischemia may have profound effects on tissue metabolism. On the other hand, different tissues may respond differently (or to a different degree) to the same amount of ischemia. Most methods of studying metabolic changes due to ischemia require termination of the animal, and thus do not allow metabolic and survival studies in the same animal. With microdialysis, both survival and metabolic studies can be conducted. Eklund and co-workers (1991) studied the effect of ischemia on two different areas of the rat kidney-the cortex and the medulla. The ischemia


195

was induced by clamping the renal pedicle. It was found that the metabolites lactic acid, inosine, and hypoxanthine increased as a consequence of anaerobic metabolism. The increase in lactate was greater in the medulla, indicating that the medulla had a greater capacity for anaerobic metabolism than the cortex. IV. SUMMARY

Microdialysis and ultrafiltration are complementary sampling techniques that facilitate acquisition of data in awake, freely moving animals. Because the necessity of blood removal is eliminated, sampling frequency is not limited by animal size. The samples obtained by these techniques usually require no processing for analysis.

REFERENCES Amer, P., and Bolinder, J. (1991). Microdialysis of adipose tissue. J. Intern Med. 230, 381. Ash, S. R., Poulos, J. T., Ranier, J. B., Zopp, W. E., Janle, E., and Kissinger, P. T. (1992). Subcutaneous capillary filtrate collector for measurement of blood glucose. ASAIO J. 38, M416.

Eklund, T., Wahlberg, J., Ungerstedt, U., and Hillered, L. (1991). Interstitial lactate, inosine and hypoxanthine in rat kidney during normothermic ischaemia and recirculation. Actu Physiol. Scand. 143,279-286. Hagstrom-Toft, E., Amer, P., Johansson, U., Eriksson, L. S., Ungerstedt, U., and Bolinder, J. (1992). Effect of insulin on human adipose tissue metabolism in situ. Interactions with beta-adrenoceptors. Diabetologia 35,664. Jacobson, I., Sandberg, M., and Hamberger, A. (1985). Mass transfer in brain dialysis devices-a new method for the estication of extracellular amino acids concentration. J. Neurosci. Methods 15, 263. Janle, E. M., and Ash, S. R. (1994). Comparison of urea nitrogen and creathine concentrations in dog plasma and subcutaneous ultrafiltrate samples. Curr. Sep. U(4), 169. Janle, E. M., and Kissinger, P. T. (1993). Microdialysis and ultrafiltration sampling of small molecules and ions from in vivo dialysis fibers. AACC TDMR"oxic01. 14(7), 159. Janle, E. M., Swain, S. L., Hucker, E., and Ash, S. R. (1991). Long term monitoring of glucose with the capillary filtrate collector. Diabetes 40,Suppl. 1,427A. Janle, E. M., Ostroy, S., and Kissinger, P. T. (1992). Monitoring the progress of streptozotocin diabetes in the mouse with the ultrafiltrate probe. Cum Sep. 11(1), 17. Janle-Swain, E., Van Vleet, J., and Ash, S. R. (1987). Use of a capillary filtrate collector for monitoring glucose levels in diabetics. ASAZO J. 10(3), 336. LOnnroth, P., Jansson, P.-A., and Smith, U. (1987). A microdialysismethod allowingcharacterization of intercellular water space in humans. Am. J. Physiol. 253, E228. Lunte, C. E., Scott, D. O., and Kissinger,P. T. (1991). Sampling living systems using microdialysis probes. Anal. Chem. 63,773-778. Sojka, J. E., Tiedje, L. J., Spehar. A. M., Janle, E., and Kissinger, P. T. (1995). Interstitial fluid and plasma amino acid concentrations in a horse. FASEB J. 9, A747.

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Spehar, A. M.,Tiedje, L. J., Sojka, J. E., Janle, E. M.,and Kissinger, P. T. (1995). Evaluation of ultrafiltration probes for recovery of endogenous substances from subcutaneous and intramuscular spaces in horses. Proc. 13th ACUIN Forum, 1072. Lake Buena Vista, FL. Telting-Diaz, M.,Scott. D. 0.. and Lunte, C. E. ( 1 W ) . Intravenous microdialysis sampling in awake, freely moving rats. Anal. Chem 64, 806. Wane, Y., Wong, S. L., and Sawchuk, R. J. (1993). Microdialysis calibration using retrodialysis and zer-net flux: Application to a study of the distribution of zidovoudine to rabbit cerebrospinal fluid and thalamus. Pharm. Res. 10(10), 1411.


Chapter 12 MEMBRANE VESICLES PIERRE PROULX Department of Biochemistry University of Ottawa Ottawa, Ontario K I H 8M5,Canada

I. Introduction Methods A. Preparation of Intestinal Luminal Membrane Vesicles B. Preparation of Intestinal Basolateral Membrane Vesicles C. Transport Studies D. Lipid Uptake and Structural Studies Ill. Discussion References 11.

1.

INTRODUCTION

Membranes from certain types of epithelial cells can be isolated in relatively pure form using simple procedures. In the process of their isolation, they reseal to form closed vesicular structures capable of carrying out transport and other physiological functions. This is the case of the highly polarized epithelial cells of tissues such as intestine and kidney tubules, which can be used to prepare brush border and basolateral membrane vesicles. In the case of brush border membranes, because of adherence of cytoskeletal elements to the membrane, the vesicles obtained remain right side out. Such preparations offer advantages over whole-tissue or wholecell preparations since they allow measurement of processes which are specific for a single membrane type, unaltered by influences from other parts of the cell. The physical properties of these membranes can vary to some extent, depending on, for example, diet, a diseased state, or such factors as the vitamin D status of the animal. Differentiation, development, and age also influence these parameters. The lipid and fatty acid compositions as well as the fluidity of the membranes will change under these 197 Copyright 0 1996 by Academic Press. Inc. All righu of reproduction in any form reserved.

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different conditions and such changes may in turn affect fluidity, transport, and enzyme activities of the membranes (Proulx, 1991). Such membrane vesicles can take up lipids when incubated with lipid micelles or liposomes, a process which usually results in important compositional and fluidity changes within these membranes. Thus, the membrane vesicle can be a useful tool not only for kinetic studies of membrane functions under a particular set of conditions but also to establish structurefunction relationships in membranes under varied conditions. In effect, brush border and basolateral membrane vesicles have been widely used to study the kinetics of transport and the conditions for uptake of a great variety of substances including ions, sugars, amino acids, peptides, bile salts, fatty acids, cholesterol, other steroids, and other lipids, as well as vitamins. The influence of membrane fluidity and lipid composition on some of these transport phenomena as well as on enzyme activities have been extensively examined using such vesicle preparations. These studies have been generally very useful for the further understanding of the functioning of enzymes and transport systems in membranes (Proulx, 1991).Currently, the preparation and uses of brush border and basolateral membrane preparations will be illustrated with some underlining of recent studies involving vitamin transport and effects of vitamin D status on the properties of these membranes. II. METHODS

A. PREPARATION OF INTESTINAL LUMINAL MEMBRANE VESICLES

Brush border membranes from a number of species have been conveniently prepared by methods based on that reported by Kessler et al. (1978). In our case, female New Zealand white rabbits, weighing 4-6 lb and deprived of food for 12 hr prior to sacrificing were routinely used. The tissue from two to four animals is rinsed free of contents with ice-cold saline, everted, and further washed with saline. At this point it can be frozen and stored at -20°C or processed immediately. The epithelial cells are removed by scraping the luminal surface with a microscope slide and homogenizing the cells for 30 sec in a Waring blender using 15 vol of 0.05 M mannitol in 2 mM Tris-HCl, pH 7.4. After centrifugation for 10 min at lO,OoOg, the supernatant is kept and the pellet is resuspended in 7 vol of mannitol solution and homogenized for 1min. Following centrifugation, this supernatant is pooled with the first and filtered through a cheesecloth. Solid CaC12 is added to the filtrate to a final concentration of 10 mM and after standing

CHAPTER 12 MEMBRANE VESICLES

199

in an ice bath with occasional stirring for 20 min, the suspension is centrifuged for 10 min at 4oOOg. The pellet is discarded and the supernatant, subjected to another centrifugation (40,OOOg for 15 min). The pellet obtained is resuspended in 100 mM mannitol-10 mM Hepes adjusted to pH 7.4 with Tris (4 ml/g of original scrapings) and is homogenized with a Potter glassmeflon homogenizer by applying 10 strokes up and down at 1200 revlmin. After centrifugation of the homogenate at 40,000g for 15 min, the pellet is resuspended in mannitol/Hepes solution to yield 34 mg protein/ml. To guard against protease degradation, leupeptin (1 pg/ ml), aprotinin ( 0.12 tyrosine inhibitor unitdml), and phenylmethylsulfonyl fluoride (30 p M ) are added to all solutions used in the preparation. Vesiculation of the membranes and the absence of organelles such as mitochondria are verified by electron microscopy. The purity of the fraction can be further assessed by determining the increase in specific activity of the brush border marker enzymes, sucrase and alkaline phosphatase, and the lack of increase of specific activity for enzyme markers of other membrane or organelle fractions (Aubry et al., 1986). In the isolation procedure for brush border membranes, the use of 10mM Mg2+instead of 10 mM Ca2+for precipitating undesired organelles has been prescribed as a means of avoiding phospholipid breakdown by phospholipase A (Hauser et af., 1980). However, in our experience, the use of Ca2+does not significantly enhance phospholipid breakdown, at least when fresh tissue is used, and Mg” precipitation yields membranes more highly contaminated with basolateral membranes (Aubry et al., 1986). B. PREPARATION OF INTESTINAL BASOLATERAL MEMBRANE VESICLES

The procedure used is that of Scalera et al. (1980). Epithelial cells from one rabbit or two or three rats are isolated as indicated for the isolation of brush border membrane vesicles and suspended in 50 ml of buffer containing 250 mM sucrose, 10 mM triethanolamine hydrochloride (PH 7.6), and protease inhibitors. The suspension is homogenized for 3 min in a Waring blender and diluted 1:2 with sucrose buffer. This homogenate is centrifuged at 2SOOg for 15 min and the pellet is discarded. The supernatant is then centrifuged at 20,500g for 20 min. The resultant fluffy layer of the pellet is resuspended in 35 ml of sucrose buffer and homogenized in a glass/Teflon Potter homogenizer (20 strokes at 1200 revlmin). The membrane suspension (31.5 ml) is mixed with 3.5 ml of Percoll (final concentration 10%) by homogenizing as before with two or three strokes at 1200 rev/min. The suspension is then centrifuged at 48,OOOg for 1 hr. The Percoll gradient is fractionated from the top by pumping 60%sucrose to the

200

PART IV METHORS FOR OBTAINING KINFTlC D A T A

bottom of the centrifuge tube. After removal of the first 12 ml, basolateral membranes are found in a fraction comprising the next 3 ml. After pooling the fractions from several tubes, the basolateral membranes may be collected by centrifugation at 48,OOOg for 30 min. The membranes are obtained as a fluffy layer on top of a very solid Percoll pellet. These membranes are then be suspended in sucrose buffer. The vesiculation and purity of the fraction is assessed by electron microscopy and by determining the increase in specific activity of the basolateral marker enzyme, Na+-K+-ATPase,and the lack of enrichment of activities for marker enzymes of other organelle fractions (Hauser et af., 1980). C. TRANSPORT STUDIES isolated vesicles have been widely used for the study of transport studies. The method relies on the membrane filtration technique enabling measurements of uptake in the seconds to minutes range. For this purpose nitrocellulose filters with a pore size of 0.45 pm are very often used and apparati with a series of suction operated filtration ports are commercially available. For short intervals, the membrane suspension and the radiolabeled permeant solution can be added to separate parts of the filter, then mixed, and, after a given interval, inactivated by rapid dilution with unlabeled permeant and quick filtration, followed by washing of the filter membrane. The filter membranes can be efficiently counted directly by liquid scintillation in the presence of PCS (Amersham). Alternatively, as in the case of Ca2+uptake, the reaction can be terminated by addition of an inhibitory ion such as La3', followed by filtration and washing (Merrill et af., 1986). The normal functioning of membrane vesicles is often ascertained by testing whether they can carry out active processes such as Na' gradient-dependent transport in the case of brush border membranes. For example, when intestinal brush border membranes are incubated in the presence of labeled glucose and 150 mM NaSCN, a typical overshoot phenomenon is seen (Fig. 1). Again illustrating this point, a similar overshoot phenomenon is noticed when, for example, pyridoxine is incubated with rat kidney brush border membrane vesicles in the presence of a large initial gradient of NaSCN (Bowman et af., 1990). As the gradient becomes dissipated, the rate of permeant entry falls to equilibrium levels. To distinguish between uptake by binding to the membrane and true transport into the intravesicular space, the effect of increasing osmolarity on the process is verified. For example, adding 300 mM cellobiose to intestinal brush border membranes very markedly decreases the uptake of glucose resulting from a shrinkage of the intracellular space (Fig. 1). Similarly, for example, the equilibrium uptake of thiamine by rat small intestinal basolateral membrane vesicles

201

CHAPTER 12 MEMBRANE VESICLES 12

,

1

1.o

0

0

100

200

300

400

500

600

SECONDS FIG. 1. D-GlUCOSe uptake by intestinal brush border membranes and the effect of osmolarity. Brush border membranes (110 pg protein) were incubated for various times at 25°C with 200 pM [3H]glucose,150 mM NaSCN, 100 mM mannitol, 1 m M Hepeflris buffer, pH 7.5, in 40 p1. The vesicular uptake was terminated by a 25-fold dilution with ice-cold stop solution consisting of 100 mM mannitol, 150 mM NaCI, 0.2 mM phlorizin, and 55 MgClz in 10 mM Hepeflris buffer, pH 7.5. The uptake was measured in the presence of 300 mM ~-cellobiose (closed circles) or in its absence (open circles).

was decreased by increasing the osmolarity of the incubation medium with saccharose (Laforenza er al., 1993). Some recent studies on vitamin transport using membrane vesicles include those of vitamin B6 by rat kidney brush border membranes (Bowman et al., 1990), ascorbic acid by teleost intestinal brush border membranes (Maffia et al., 1993), biotin by human kidney brush border membranes (Baur and Baumgartner, 1992), pantothenate by human placental brush border membranes (Grassl, 1992), folate and riboflavin by rabbit intestinal brush border membranes (Said and Mohammadkhani, 1993a,b; Said et al., 1993), and thiamine by rat small intestine basolateral membranes (Laforema et al., 1993). Bile acid transport in human placental, rat ileal, and rabbit small intestinal brush border membrane vesicles (Dumaswala et al., 1993; Gong et al., 1991; Kramer et al., 1993) and the effect of vitamin D status

202


on Ca2+uptake in chick intestinal basolateral, chick and pig intestinal brush border, and rabbit renal basolateral and brush border membrane vesicles have also been recently examined (Takito et al., 1992; Kaune et al., 1992; Boutiauy et al., 1993). The use of membrane vesicles in these cases has helped elucidate mechanisms of transport, i.e, Na+-gradient dependency or other requirements, kinetic parameters, etc. Results from such studies are more easily interpreted because they are attributable to the components of a single membrane type. Composite mechanisms resulting from interacting components of tissue or whole-cell preparations are more difficult to assess. D. LIPID UPTAKE AND STRUCTURAL STUDIES The lipid composition of membrane vesicles can be studied as a function of conditions displayed in vivo in relation to lipid dietary supplementation, vitamin D status, diabetes or other diseases, age, topography, etc. Such changes are characterized by alterations in the transition temperatures and fluidity of the membrane, as measured by fluorescence anisotropy, for example, and breaks in the Arrhenius plots of transport or enzyme activity data (Proulx, 1991). Other types of studies have indicated that the lipid composition of vesicular membranes can be altered directly by interacting isolated membranes with fatty acids, cholesterol, and complex lipids containing different polar headgroups (Proulx, 1991). The lipids are added either as micelles or liposomes and the treated membranes can be isolated by centrifugation and washed free of loosely adhering lipids with buffer or buffer-containing detergent (Proulx et al., 1984). Filtration methods for such studies are to be avoided unless it can be established that the membrane filters used do not themselves markedly bind the lipid under study. Nitrocellulose filter membranes, for example, were found to be generally inappropriate for examining lipid uptake by membrane vesicles. Such studies lend themselves to establishing structure-function relationships in membranes. For example, modifying the fatty acid composition of brush border membrane lipids by in vitro uptake of unsaturated fatty acids or their methyl esters decreases the fluorescence anisotropy of diphenylhexatriene-labeled brush border membranes (i.e., increases their fluidity, Fig. 2) while increasing their rate of CaZ+uptake (Table I). In fact many investigations have dealt with attempts to establish relationships between lipid composition, fluidity, and enzyme or transport activities using membrane vesicles (Proulx, 1991). Recent studies typifying this approach are those of Ramsammy et al. (1993), using renal brush border and basolateral membranes isolated from normal and streptozotocin-induced diabetic rats, and


203

.-28

&

26 J4

K 22

c

g - .20 5 .18 .16 .14 .12

!

I

I

I

I

0

50

100

150

200

250

UPTAKE n m o l h g Prot. FIG. 2. The effect of methyloleate uptake on anisotropy of diphenylhexatriene-labeled brush border membranes. Conditions for methyloleate uptake and fluorescence anisotropy measurementswere as described (Merrill ef uL, 1987). The horizontal axis represents amounts of methyloleatetaken up. Anisotropy values are reported for measurements at 20°C (closed circles) and 37°C (open circles).

Dudeja et al. (1991), using rat small intestine brush border membranes treated with benzyl alcohol. Ill. DISCUSSION

Antipodal membranes of epithelial cells have been widely used for transport studies and recently an increasing number of investigations have dealt with mechanisms of vitamin and nutrient uptake using these preparations. With such preparations, however, rate measurements would be influenced by the presence of a substantial portion of vesicles, leaky to low molecular weight molecules. Accordingly it has been shown that actin, a marker of the cytosolic side, can be labeled by a number of reagents of molecular weights of up to 700 Da. Also studies comparing the compartmentation of radiolabeled sugars, inulin, and inulin carboxylic acid indicated that a maximum of 25% of the vesicles were sealed to small molecules (cf. references cited in Proulx, 1991). It is likely that a similar problem of leakiness exists with basolateral membrane vesicles. Nonetheless, results from many investigations based on kinetic and structural studies attest that the use of such vesicle preparations has greatly helped in elucidating mechanisms of

204

PART IV METHODS FOR OBTAINING KINETIC DATA TABLE I EFFECT OF METHYL OLEATE ON

Methyl oleate concentration (mM) 0.010 0.050 0.090 0.120 0.180 0.1806

Ca2+UFTAKE

Lipid uptake (nmoVmg protein) 14 2 68 rt 101 119 2 182 2 220 2

*

1 4 4 8 10 21

Rate of Ca2+uptake (percentage of control)' 118 rt 124 rt 133 rt 143 rt 156 2 207 2

7 8 9 7 10 29

'The control uptake was 7.5 2 1.5 nmoVmg protein per 5 min. All methyloleate values represent incubations of 1 hr with membranes except for b, in which case incubations were for 4 hr. The results represent the mean 2 SE from 6 to 20 determinations with membranes from three to five rabbits.

uptake in one or the other of the antipodal membranes and has increased our knowledge of how structural factors such as fluidity influence functional parameters. In particular, the use of intestinal brush border membranes has offered much potential for the detailed investigation of absorptive phenomena. It is likely that the absorption of lipid nutrients including fat-soluble vitamins for example, which may intercalate transiently into the bilayer and affect the physical properties of the membrane, will concurrently influence the entry of other substances. Alternatively, interactions between nutrients in the gut might be expected to mutually influence absorption of these substances. Also, there may be competition between nutrients for the same transport system. Membrane vesicles are very much suited for studying this type of problem and their use has helped resolve, for example, the mechanism whereby lecithins inhibit cholesterol uptake (Memll et al., 1987). Again, the use of isolated brush order membranes has been conducive to a better understanding of the mechanisms of interaction between lipid micelles or lipid vesicles and the intestinal luminal membrane. REFERENCES Aubry, H., Memll, A. R., and Proulx, P. (19%). A comparison of brush border membranes from rabbit small intestine prepared by procedures involving Ca2+ and Mg2+ precipitation. Biochim. Biophys. Acfu 856,610. Baur, B., and Baumgartner, R. (1992). Na+-dependent biotin transport into brush-border membrane vesicles from human kidney cortex. Ppuegers Arch. 422,499.


205

Boutiauy, I., Lajeunesse, D., and Brunette, M. G. (1993). Effect of vitamin D depletion on calcium transport by luminal and basolateral membranes of the proximal and distal nephrons. Endocrinology (Baltimore) W2,115. Bowman, B. B., McCormick, D. B., and Smith, E. R. (1990). Vitamin B6 uptake by rat kidney cells and brush-border membrane vesicles. Ann. N. Y. Acud. Sci 585, 106. Dudeja, P. K.,Wali, R. K., Hang, J. M., and Brasitus, T. A. (1991). Characterization and modulation of rat small intestine brush-border membrane transbilayer fluidity. Am. J. Physiol. 260, G586. Dumaswala, R.,Setchell, K. D. R., Moyer, M. S., and Suchy, F. J. (1993). An anion exchanger mediates bile acid transport across the placental microvillous membrane. Am. J. Physiol. 264, (31016. Zwarych, P. P., Jr., Lin, M. C., and Wilson, F. A. (1991). Effect of antiserum Gong, Y.-Z., to a 99 kDa polypeptide on the uptake of taurocholic acid by rat ileal brush border membrane vesicles. Biochem. Biophys. Res. Commun. 179,204, Grassl, S. M. (1992). Human placental brush-border membrane Na+-pantothenate cotransport. J. Biol. Chem. 267,22902. Hauser, H., Howell, K., Dawson, R. M. C., and Bowyer, D. E. (1980). Rabbit small intestinal brush border membrane preparation and lipid composition. Biochim. Biophys. Acta 602, 567. Kaune, R., Kassianoff, I., Schroder, B., and Harmeyer, J. (1992). The effects of 1,25-dihydroxy D-3 deficiency on Ca2+-transport and Ca2+-uptake into brush-border membrane vesicles from pig small intestine. Biochim. Biophys. Acfa 1109, 187. Kessler, M., Acuto, O., Storelli, C., Murer, H., Muller, M., and Semenza, G. (1978). A modified procedure for the rapid preparation of efficientlytransporting vesicles from small intestinal brush border membranes: Their use in investigating some properties of D-glucose and choline transport systems. Biochim. Biophys. Acta 506,136. Kramer, W . , Girbig, F., Gutjahr, U., Kowalewski, S., Jouvenal, K., Muller, G., Tripier, D., and Wess, G. (1993). Intestinal bile absorption: Na+-dependent bile acid transport activity in rabbit small intestine correlates with the coexpression of an integral 93-kDa and a peripheral 1CkDa bile aeid-binding membrane protein along the duodenum-ileum axis. J. Biol. Chem. 268,18035. Laforenza, U. L., Gastaldi, G., and Bindi, G. (1993). Thiamine outflow from the enterocyte: A study using basolateral membrane vesiclesfrom rat small intestine.J. Physiol. (London) 468,401. Maffia, M., Ahearn, G. A., Vilella, S., Zonno, V., and Storelli, C. (1993). Ascorbic acid transport by intestinal brush-border membrane vesicles of the teleost Anguillu anguilla. Am. J. Physiol. 264, R1248. Merrill, A. R., Proulx, P., and Szabo, A. G. (1986). Studies on calcium binding to brush border membranes from rabbit small intestine. Biochim. Biophys. Acfa 859,237. Merrill, A. R., Aubry, H., Proulx, P., and Szabo, A. G. (1987). Relation between calcium and fluidity of brush border membranes isolated from rabbit small intestine and incubated with fatty acids and methyloleate. Biochim. Biophys. Acta 896, 89. Proulx, P. (1991). Structure-function relationships in intestinal brush border membranes. Biochim. Biophys. Acfa lOn, 255. Proulx, P., Aubry, H., Brglez, I., and Williamson, D. G. (1984). The effect of phosphoglycerides on the incorporation of cholesterol into isolated brush-border membranes from rabbit small intestine. Biochim Biophys. Acta 775,341. Ramsammy, L. S., Boos, C., Josepovitz, C., and Kaloyanides, G. J. (1993). Biophysical and biochemical alterations of renal cortex membranes in diabetic rat. Biochim. Biophys. Acfa 1146, 1.

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Said, H. M.,and Mohammadkhani, R. (1993). Folate transport in intestinal brush border membrane: Involvement of essential histidine residue(s). Biochern. J. 290, 237. Said, H. M.,Mohammadkhani, R., and McCloud, E. (1993). Mechanism of transport of riboflavin in rabbit intestinal brush border membrane vesicles. Proc. SOC.Exp. Biol. Med. 202,428. Scalera, V., Storelli, C., Storelli-Joss, C., and Haase, W. (1980). A simple and fast method for the isolation of basolateral plasma membranes from rat small-intestinecells. Biochem. 1. 186,177. Takito. J., Shinki, T., Tanaka, H., and Suda, T. (1992). Mechanism of regulation of calciumpumping aaivity in chick intestine. Am J. Physiol. 262, G797.

ADVANCES IN FOOD AND NUTRITION RESEARCH,VOL. 40

Chapter 13 CULTURE OF MAMMARY TISSUE: GLUCOSE TRANSPORT PROCESSES JEFFREY D. TURNER AND ANNICK DELAQUIS Department of A n i d Science McGiH University Montreal, Quebec H9X 3V9, Canada

CHRISTIANE MALO Membrane Transport Group Department of Physiology, Faculty of Medicine University of Montreal Montreal, Quebec H3C 3J7, Canada

1. Introduction 11. Materials and Methods

A. Cells and Culture Conditions B. Experiment 1: Uptake of 2-[3H]Deoxyglucose(2-DG) and a-[14C]Methylglucose (MG) by MAC-T Cells C. Experiment 2 Uptake of 2-DG by MAC-T Cells at Different Days of Culture D. Experiment 3: Effect of Glucose Concentration in the Media on the Uptake of 2-DG by MAC-T Cells at Different Days in Culture 111. Results and Discussion IV. Conclusions References

1.

INTRODUCTION

The epithelial cells of the kidneys and the small intestine can transport glucose against a concentration gradient via a Na' - dependent transporter (Devaskar and Mueckler, 1991). In addition, in these same epithelial cells, Na+-independentglucose transporters have been identified (Harris et al., 1992). Mammary epithelial cells utilize large quantities of glucose 207 Copyright 0 1996 by Academic Press, h e . All rights of reproduction in any form reserved.

208


since the monosaccharide is the precursor of lactose, the major carbohydrate found in milk of most mammalian species. The types of glucose transporters present in mammary epithelial cells have never been determined. The objectives of the following experiments were to determine: (1) if glucose transport by mammary epithelial cells was Na+-dependent or Na+-independent, (2) changes in glucose uptake by mammary epithelial cells at different stages of growth, and (3) the effect of glucose concentration on its accumulation by mammary epithelial cells at different stages of growth. II. MATERIALS AND METHODS

A. CELLS AND CULTURE CONDITIONS An immortalized line of bovine mammary epithelial cells, the MAC-T cell line, was chosen as an experimental model. These cells form a tight monolayer in culture and have been demonstrated to differentiate when exposed to lactogenic hormones (Huynh et al., 1991). Cells were grown in Dulbecco modified Eagle medium (Gibco) containing 5 pg/ml insulin, 50 mg/liter gentamicin sulfate (Gibco), and 10% fetal calf serum (Gibco). The glucose concentration in the medium was altered for certain experiments. The medium was changed every 2 days. Cells were seeded at a rate of 1.4 X lo4 cells per 35-mrn tissue culture dish.

B. EXPERIMENT 1: UPTAKE OF 2-[3H]DEOXYGLUCOSE (2-DG) AND a-[14C]METHYLGLUCOSE (MG) BY MAC-T CELLS These two radiolabeled glucose analogues were chosen. The 2-DG is known to be accumulated in a Na+-independent manner whereas MG is accumulated in a Na+-dependent manner (Blais et aL, 1987; Malo, 1990). The uptake of 2-DG is specifically inhibited by phloretin (Pt) whereas the transport of MG is specifically inhibited by phlorizin (PZ). At Day 10 of culture the uptake of 1pCi/dish of 2-DG and MG by MAC-T cells grown in DMEM with 25 mM glucose was measured in the absence and presence of PZ (200 p M ) , Pt (200 pM),or an excess of nonradioactive substrate (50 mM).The uptake was measured at Day 10 of culture according to the following procedure. Cells were rinsed three times with 2 ml of nonradioactive transport buffer composed of 200 mg/liter CaCl2, 400 mgAiter KCl, 170 mgAiter MgS04, 125 mg/liter KH2P04, 2.38 g/liter Hepes, 6.4 gAiter NaCI, 292.2 mg/liter glutamine, and mannitol to adjust the osmolarity

CHAPTER 13 CULTURE OF MAMMARY TISSUE

209

to 300 mOsmol. Cells were then incubated 15 min at 37°C in 2 ml of transport buffer after which the buffer was changed for radioactive and inhibitor-containing buffer (same composition as described above but with radioactive substrate and inhibitors desired) (1 mudish). Cells were incubated at 37°C for 5, 10, 15, 30, 45, 60, 90, 120, 180, and 240 min. At the end of the incubation, cells were washed three times with nonradioactive transport buffer containing 200 p M PZ and 200 p M Pt. Cells were then incubated at 37°C for 2 hr in 1N NaOH (0.5 mlldish). The extract obtained was analyzed for its radioactivity and protein content using the procedure of Lowry er al. (1951). All measurements were done in duplicate.

C. EXPERIMENT 2 UPTAKE OF 2-DG BY MAC-T CELLS AT DIFFERENT DAYS OF CULTURE The uptake of 2-DG with and without Pt was measured on Days 3, 6, 9, 12, 15, 18, and 21 of culture according to the same procedure as in experiment 1 except that the uptake of 2-DG was only measured for 30 min to ensure that the uptake vs time curve was still in the linear portion. Cell number was determined at each time point. All measurements were done in triplicate. D. EXPERIMENT 3: EFFECT OF GLUCOSE CONCENTRATION IN THE MEDIA ON THE UPTAKE OF 2-DG BY MAC-T CELLS AT DIFFERENT DAYS IN CULTURE The uptake of 2-DG by MAC-T cells grown in the presence of 25,5, or 0 mM glucose was measured on Days 3, 6, and 9 of culture following the same protocol as in experiment 2. The number of cells was determined for each condition. Ill. RESULTS AND DISCUSSION

Bovine epithelial cells did not accumulate MG in a specific manner since the uptake was not altered by the presence of Pt, PZ, nor an excess of nonradioactive substrate (Fig. 1).The small accumulation of MG could be related to nonspecific uptake, diffusion, or binding to the cell surface. These results suggest that MAC-T cells grown under the present conditions do not express a Na+-dependent glucose transporter. The significant inhibition of uptake of 2-DG by an excess of nonradioactive substrate and by Pt but not by PZ demonstrated that MAC-T cells express one or several Na+-independent glucose transporters (Fig. 2). Specific uptake of 2-DG

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PART IV METHODS FOR OBTAINING KINETIC DATA 35

30 25

20

i

15

10 5 O I 5

I

1

I

I

I

I

I

I

10

15

30

45

60

90

120

180

240

Time (min.) FIG. 1. Uptake of a-methylglucose (MG)by MAC-T cells at Day 10 of culture and in the or cold a-methylglucose. presence of inhibitors phlorizin (PZ) or phloretin (a)

was also observed in rat mammary acini (Threadgold et al., 1982). These results are not surprising since mammary cells do not need to be able to transport glucose against a concentration gradient, blood concentration 16

14

12 10 8

6 4

2

0 5

10

15

30

45

60

90

120

180

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Time (min.) FIG. 2. Uptake of 2-deoxyglucose (DG) by MAC-T cells at Day 10 of culture and in the presence of inhibitors phlorizin (PZ) or phloretin (Pt)or cold 2-deoxyglucose.

CHAPTER 13 CULTURE OF MAMMARY TISSUE 10

211

pmdmg protein

8

6 4

2 0

3

6

9

12

15

18

21

Days in culture FIG. 3. Uptake of 2-deoxyglucose by MAC-T cells at different days of culture. The effect of phloretin treatment is indicated.

being higher than milk concentration: 0 . 0 5 % ~traces ~ (Larson, 1985). The uptake of 2-DG reaches a peak when expressed as pmol substratelmg protein around Day 6 of culture well before confluency is attained between Days 12 and 15 (Figs. 3 and 4). The complete inhibition 2-DG uptake in the presence of Pt throughout the 21-day period confirmed that the uptake

Day8 in culture

FIG. 4. Cell growth expressed as cell number and total cellular protein and proteidcell for MAC-T cells at different days of culture.

212

PART IV METHODS FOR OBTAINING KINETIC DATA 10

8 6 4

2 0

3

6

9

Day of culture FLG. 5. Effect of media glucose concentration on the uptake of 2-deoxyglucose by MACT cells at different days of culture. The first column is 25 mM, the second column is 5 mM, and the shaded column represents media without glucose. Bars with different letters are significantly different ( P < 0.5).

measured was specific and not due to nonspecific binding or transport or diffusion. The concommittent decrease in 2-DG uptake/mg protein and the increase in cell number or protein/dish after Day 9 can be explained by a reduced surface area per cell being in contact with the substrate as cells approach confluency or a reduced activity of the transporter as cells go from the growth state to the differentiation stage. The concentration of glucose in the medium had a significant effect on the uptake of 2-DG at Days 3 and 9 of culture (Fig. 5). The absence of glucose in the medium resulted in a significantly higher uptake of 'L-DGlmg protein than in the presence of 25 mM glucose. These results demonstrated a negative feedback of glucose concentration on the expression and/or activity of the Na+independent glucose transporter(s).

IV. CONCLUSIONS

These experiments demonstrated that monolayers of bovine mammary epithelial cells do not accumulate glucose in a Na+-dependent manner. The uptake of glucose reaches a maximum during the growth phase of the cells and diminishes as the cells reach confluency. The transport system(s) is regulated by the glucose concentration in the culture medium.

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213

REFERENCES Blais, A., Bissonnette, P., and Berteloot, A. (1987). Common charctaeristics for Na+dependent sugar transport in Caco-2 cells and human fetal colon. J. Membr. Biol. 99, 113-125. Devaskar, S. U., and Mueckler, M. M. (1991). The mammalian glucose transporters. Pedinrr. Res. 31, 1-12. Harris. D. S., Slot, J. W.,Geuze, H. J., and James, D. E. (19Sn). Polarized distribution of glucose transport isoforms in Caco-2 cells. Proc. Natl. Acad. Sci U.S.A. 89,7556-7560. Huynh, H. T., Robitaille, G., and Turner, J. D. (1991). Establishment of bovine mammary epithelial cells (MAC-T): An in vitro model for bovine lactation. Exp. Cell Res. 197, 191-199. Larson, B. L. (1985). Biosynthesis and cellular secretion in milk. I n “Lactation” (B. L. L a w n , ed.), p. 129. Iowa State Univ. Press, Ames. Lowry, 0. H., Rosebrough, N. J., Fan, A. K., and Randall, R. J. (1951). Protein measurement with the Folin phenol reagent. J. Biol. Chem. 193, 265-275. Malo, C. (1990). Separation of two distinct Na+/D-glucosecotransport systems in the human fetal jejum by means of their differential specificity for 3-0-methylglucose. Biochim. Biophys. Acra 1022,8-16. Threadgold, L. C., Coore, H. G., and Kuhn, N. J. (1982). Monosaccharide transport into lactating rat mammary acini. Biochem. J. 204,493-501. Present address: Nexia Biotechnologies, Inc., 21,111 Lakeshore Road, Ste. Anne de Bellevue, Quebec, Canada H9X 3V9.


Part V SIMULATING COMPLEX METABOLIC PROCESSES



Chapter 14 ANALYSIS OF BlOPERlODlClTY IN PHYSIOLOGICAL RESPONSES’ L. PRESTON MERCER AND DANlTA SAXON KELLEY Department of Nutrition and Food Science University of Kentucky Lexington, Kentucky 40506

I. Bioperiodicity 11. Characterization of Biological Rhythms

111. Experimental Conditions IV. Evaluation of Results A. Daily Weight Gain and Daily Food Intake Analysis B. Analysis of Rhythms V. Discussion References

I.

BlOPERlODlClTY

Bioperiodicities (rhythms) in biological phenomena have long been recognized and studied in order to gain insight into the dynamics of living organisms. Rhythms are ubiquitous phenomena which occur at all levels of biological organization and are present in subcellular units, cells, and tissues and in the organism as a whole. A rhythm has been defined as a sequence of events that repeats itself through time in the same order and at the same interval , Often, physiologic rhythms have been ignored (viewed as an epiphenomenon of homeostasis) andlor attributed to random error of measured responses. Table I compares the homeostatic view with the chronobiologic view of biology. If functional rhythmicity is present in an organism, ignoring it could impair the interpretation of physiological experiments. This work was supported by United States Department of Agriculture NRICGP Grant 9400531 and the Agricultural Experiment Station, The University of Kentucky.

217 Copyright 0 1996 by Academic Press. Inc. All tights ofreproduction in any form reserved.

218

PART V SIMULATING COMPLEX METABOLIC PROCESSES TABLE I T W O M E W S OF BIOLOGY

Homeostatic ~

~

Chronobiologic ~~

~

Regulatory mechanisms maintain constancy of “internal milieu”

Physiologic changes recur with reproducible waveform

Time ignored as source of variability

Reference time markers are essential

Steady state

Biorhythm

Bioperiodicity is an integral part of physiological responses to nutrients and diets. Recently, Halberg has stated “In the science and practice of nutrition today, the provision of a control requires the assessment of a multifrequency rhythmic structure’’ (Halberg, 1989). A mathematical model, such as the cosinor model of Halberg, can be used to establish the occurrence of rhythms and quantify the rhythm characteristics of components identified by spectral analysis. II. CHARACTERIZATION OF BIOLOGICAL RHYTHMS

According to the approach pioneered by Halberg, deterministic, biological rhythms (i.e., chronobiologic rhythms) have four measurable parameters: the mean, amplitude, acrophase, and period (Pauly, 1980). These are shown graphically in Fig. 1. ACROPHASE

PERIOD

0.0

FIG. I . A cosinor curve showing the various parameters of response.

CHAPTER 14 ANALYSIS OF BIOPERIODICITY

219

The mean of a rhythm is the average value of a continuous variable over a single cycle. When the rhythm is described by the fitting of a cosine curve, the half way point between the peaks and the troughs is known as the MESOR. Only when the data are measured equidistantly, over an integral number of cycles, will the MESOR equal the arithmetic mean. The amplitude refers to the magnitude of the response variable between its mean value and the (estimated) trough or peak. Such mathematical usage, however, is limited to rhythms which oscillate symmetrically about the mean value. The phase refers to the value of a biological variable at a fixed time. The word phasing is often used to describe the shape of a curve that depicts the relationship of a biological function to time. Acrophase is a more limited term which refers to a specified reference standard or zero time and indicates the lag in the crest of the function used to describe the rhythm. The period is the duration of one complete cycle in a rhythmic function and is equal to Ufrequency. Haus and Halberg (1980) have further categorized rhythms (by time frame) as infradian, circadian, and ultradian. Circadian rhythms are the rhythms that have been studied most extensively and have periods in the range 20-28 hr (therefore, frequencies are about 0.04 cycles per hour). There are many examples that can be cited, including rhythms in mitotic activity, metabolic processes, and susceptibility to drugs. Infradian rhythms have periods longer than 28 hr and therefore their frequencies are correspondingly lower than circadian. Some of the well known infradian rhythms are the human menstrual cycle and the annual reproductive cycle of salmon. Infradian rhythms have been identified in nutrient intake and metabolism of foodstuffs (Reinberg, 1983). A more specific type of infradian rhythm is the circasemiseptan (period approximately 3.5 day) found by Schweiger et al. (1986). Ultradian rhythms have periods shorter than 20 hr. Examples of these rhythms are the electrocardiogram, respiration, peristalsis in the intestine, etc. Rhythms may also be categorized as exogenous and endogenous (Pauly, 1980). The exogenous rhythm can be caused, driven, and/or coordinated by a force in the environment, but disappears when the driving force ceases. The endogenous rhythm has an intrinsic mechanism and its coordination lies at a cellular level, such as transcription of DNA. Rhythmicity of phospholipids, RNA, DNA, glycogen content, and mitosis has been demonstrated by Halberg et al. (1959). Endogenous rhythms have periods similar to, but statistically different from, their environmental counterparts. Those external influences (environmental factors) which are capable of entraining a rhythm are referred to as synchronizers (Minors and Waterhouse, 1981),

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PART V SIMULATING COMPLEX METABOLIC PROCESSES

and their manipulation can reset the phase of rhythms. Several environmental factors, such as light/dark cycles, sleep/wakefulness, timing of energy intake, and, presumably, qualitative dietary factors, may act simultaneously or separately on a given physiologic variable. One or the other of these external synchronizers may be dominant for the timing of the rhythm of a given function, but not for others. After a change in the synchronizer schedule, the adjustment of a rhythm to the changed environmental routine will occur with a different rate for different variables (Haus and Halberg, 1980). However, if the external synchronizer disappears, the endogenous rhythm will not disappear and will take on a characteristic called “free running.” Our goal in this manuscript is to demonstrate the protocols necessary for time-based analysis of weight gain in rats. The techniques can then be applied to other responses. 111.

EXPERIMENTAL CONDITIONS

A typical experimental design might follow one carried out in our laboratory. Ten weanling rats, (Sprague-Dawley, Indianapolis, IN) of weight range 38-60 g, were fed a nonpurified diet (Purina rodent chow No. 5001, Purina Mills, St. Louis, MO) for 2 days to acclimate after shipping. The rats were then fed a diet containing normal levels of dietary protein (25% casein). All animals were housed and fed in the animal care facility of the Division of Laboratory Animal Resources which is fully accredited by AAALAC. The rats were singly housed in suspended, wire-bottom cages and were given water (purified by reverse osmosis) and diets ad libitum. A 14: 10 light-dark cycle was maintained (light on 0600 to 2000 daily). Daily activities were carried out in the room by animal care personnel. Rats were weighed every day at the same time of day on a time integrated Sartorius balance (Brinkmann Instruments, Westbury, NY) interfaced to a computer (Dell, Austin, TX). Food intakes were calculated as disappearance from food cups with adjustments made when necessary for spillage. Since rhythm detection requires sampling involving (preferably) more than a few cycles, a long-term experiment was carried out (45 days) (Mercer et al., 1993). IV. EVALUATION OF RESULTS

A. DAILY WEIGHT GAIN AND DAILY FOOD INTAKE ANALYSIS Daily weight gain rates (dW/dt) and daily food intake rates (dFldt) for each experimental diet were calculated as


221

where W , and W,-l denote rat’s weight at times (t) and (f-1), respectively. F, and F,-] represent the food weight at times ( f ) and (t-1), respectively. We used dWldt and dFldt (rather than cumulative weight gain and food intake) to observe the daily rhythms.

B. ANALYSIS OF RHYTHMS The first step in any analysis to detect the presence of a rhythm in a response variable measured over time is to plot the data as a function of time in rectangular coordinates. This often reveals the empirical properties of a time series, including confounding features such as trend (long-term change in the mean). All time series, statistical analyses, and graphics in this study were performed using the PC-based statistics/graphics program SYSTAT (SYSTAT, Inc., 1800 Sherman Avenue, Evanston, IL 60201) and its companion time series program, MESOSAUR. Statistical analysis, such as a two-way analysis of variance may be used to test whether differences between values at various times are significant. However, statistical analyses do not provide any information about the shape, phase, amplitude, or mean level of the rhythm; they merely indicates whether the data are different from random variation. In order to quantify rhythm parameters, other mathematical techniques, such as Halberg’s cosinor model, are required. Our hypothesis assumes that the measured data follow a deterministic series model. Deterministic series are obtained when successive observations are dependent variables and any future values may be predicted from past observations (Chatfield, 1975). The cosinor method is based upon the least-squares regression of a cosine function of the form g(t) =

M

+ A - cos (of + 4) + e(t),

(1)

where g(t) is the value at time ( f ) of the regression function (more cosine and sine terms may be added if necessary to describe the response). M,A, o,and 4 denote the mean level (termed the MESOR), amplitude (half the range of oscillation), angular frequency (radians per unit time), and phase of the periodic variation, respectively. e(t) is an error term assumed to be an independent random variate with mean 0 and unknown variance u2. The angle (wt + 4) is measured in radians. Some authors refer to the frequency as the number of cycles per unit time (o/2n-). This form of

222


frequency is much easier to interpret. The period (7)of a sinusoidal cycle is equal to 2nlo or llf (unit time per cycle). A complicating factor in the analysis of growing rats is that growth is not constant but follows a trend, which must be accounted for (i.e., removed) before analysis for bioperiodicities. Figure 2 shows the data from the experiment. A definite downward trend is noted. Trends must be removed before proceeding with the analysis. Often, a simple function (growth equation) such as it logarithmic, logistic, or parabolic equation will suffice to predict the trend. Coefficients can be estimated from logarithmic, logistic, or parabolic analyses. Observed time series data are then fitted to the trend equation, residuals are recorded, and parametric analysis (for T,M,A, and 4) is carried out on the residuals of the detrended data. Trends can also be removed by “differencing” the data, i.e., replacing observed values by the difference between each value and the previous value. This approach has the advantage of not requiring an equation to “fit” the data and parametric analysis can be carried out as usual by regression. Figure 3 shows the data Erom Fig. 2 after being detrended by differencing. After detrending, continuation of the analysis requires converting the observed time-dependent data to the frequency domain. This is carried out by subjecting the data to a fast Fourier transform. Then it is possible to determine a period (if it exists) for the suspected waveform. This process 9 r

It

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FIG. 2. Observed responses (points connected by solid line). Each point is the mean of 10 rats.

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-2 0

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I

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FIG. 3. Observed responses after being detrended (points connected by solid line). Each point is the mean of 10 rats.

involves spectral analysis or, more simply, a periodogram (Fig. 4) which estimates the spectral density by plotting the square of the magnitude against frequency. This can be readily done with a computer program such as MESOSAUR. After determination of the period ( T ) , the frequency of the periodic variation (0) can be calculated using the equation w = 2dr. M,A, and (b are estimated from regression analysis after replacing w in Eq. (1) by a numerical value. The cosinor model assumes that the T(period) can be anticipated a priori based on some knowledge of the biological system being analyzed. One caveat about a period length. It could be a valid parameter or an artifact of the sampling rate; i.e., frequencies (VT)higher than 0.5 can appear at lower values due to the phenomenon of aliasing. For example, a 2-day period is close to a multiple of a l-day sampling rate. Sampling at a higher rate (i.e., weighing rats two or more times a day) would be required to determine the validity of a proposed 2-day period in rats weighed once a day. After determination of T from the periodogram, M, A, and (b can be estimated by standard nonlinear regression techniques. An equation for calculating the parameters of “daily weight gain” could take the form (based on Eq. (1)): dwg = M + A cos (2 * T / T * day + 4) where dwg is the detrended data. Using T = 3.64 k 0,089 from the periodogram of the experimental data, nonlinear regression gives M = 0.06 2 0.147, A =

224


1U

20

n 3

t

5Q

r.

10

0 0

20

10

30

PERIOD FIG. 4. Periodogram of data after fast Fourier transform. A strong peak indicates a significant period ( 7 ) of approximately 3.5 days.

0.57 -t 0.202, and 4 = -0.99 2 1.038. M is approximately zero (as expected) since the data has been detrended. To get back to the original, observed data (with its trend), M must be stated as a function of time, M = f(t), rather than carried as a constant. Therefore, the least-squares regression of a cosine function for growing animals can be represented by Eq. (l), with M replaced by a trend functions mentioned above. We then fit the exponential equation M = (7.29/(1 + exp(-5.92 + 0.14* day). This can be done by choosing appropriate equations and accepting the one giving the lowest sum of squares. Using this equation for M produces Fig. 5. V.

DISCUSSION

Progress in the chronobiology of nutrition is impeded by the attitude “any changes around the mean physiological response should be viewed as random variability,” leading scientists to consider observed variation merely as an epiphenomenon of homeostasis. However, application of an established criterion, called the Shannon criterion, indicates the presence of an oscillation based on at least two measured points in each period (Shannon and Weaver, 1963). The results of our experiment calculated for rats either as individuals or in groups

225


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45

DAY FIG. 5. Data with predicted cosinor curve.

indicate the presence of infradian rhythms (period greater than 28 hr) in growth rates. We were initially concerned that bioperiodicity might be masked by averaging responses on nonsynchronously growing rats. This did not seem to be the case in the reported experiment as indicated by analysis on individual rats compared with groups of rats. One possible clinical application of the convergence of the disciplines of bioperiodicity and nutrition is in the area of eating disorders. Eating disorders are widespread in the United States and a clearer understanding of fundamental physiological mechanisms is essential in dealing with this problem. Single meal weight loss regimens could interact with biological rhythms of weight gain, yielding significantly different outcomes. Studies showed that when diurnally active human subjects are restricted to a single meal per day, they lose weight if the daily food (8386 kJ) was consumed as breakfast. On the other hand, when the same amount of food was consumed as an evening meal, the subject showed-on the average-either a statistically significantly smaller loss or else a weight gain (Halberg, 1989). It appears that energy consumed at one stage of a cycle is not the same metabolically as one taken at another. Bioperiodicity of physiological responses is well documented in several species (Stolz et al., 1988; Marks et aL, 1978; Kersten et al., 1980; Schumann and Haen, 1988; Li and Anderson, 1982;de Castro, 1991). Optimal nutrition thus requires not only a consider-

226


ation of what food is consumed, but of when it is consumed. The timing of food intake may be helpful in controlling body weight or, perhaps more importantly, may allow to optimally utilize a limited amount of food available at times of scarcity. In conclusion, periodicities in growth rates display infradian rhythms and these rhythms appear to be endogenous. Chronobiological analysis gives new insights into relationships between nutrition and physiological response. However, certain procedures interfere with recognition of biorhythms: use of purified rather than integrated samples, use of large samples which appear to produce synchronous activity, and the search for homeostasis rather than perturbation. Recognition of these factors will improve understanding and interpretation of many biological experiments. REFERENCES Chatfield, C. (1975). “The Analysis of Time Series: Theory and Practice,” 2nd ed. Halsted Press, New York. deCastro, J. M. (1991). Seasonal rythms of human nutrient intake and meal pattern. Physiol. Behav. So, 243-248. Halberg, F. (1989). Some aspects of the chronobiology of nutrition: more work is needed on “when to eat.” J. Nutr. 119, 333-343. Halberg, F., Halberg, E., Barnum, C. P., and Bittner, J. J. (1959). “Physiologic 24-hour Periodicity in Human Beings and Mice, the Lighting Regimen and Daily Routine. Am. Assoc. Adv. Sci., Washington, DC. Haw, E., and Halberg, F. (1980). “The Circadian Time Structure. Chronobiology.” Sijthoff KC Noordhoff, The Netherlands. Kersten, A., Strubbe, S. H..and Spiteri, N. J. (1980). Meal patterning of rats with changes in day length and food availability. Physiol. Behav. 25,953-958. Li, E. T. S., and Anderson, G. H. (1982). Self-selected meal composition, circadian rhythms and meal responses in plasma and brain tryptophan and 5-hydroxytryptamine in rats. J. Nutr. 112,2001-2010. Marks, H. G., Borns, P., Steg, N. L., Stine, S. B., Stroud, H. H., and Vates, T. S. (1978). Catch-up brain growth-demonstration by CAT scan. J. Pediat. 93,254-256. Mercer, L. P., HidvBgi, M., and Hijazi, H. (1993). Weanling rats display bioperiodicity of growth rates and food intake rates. J. Nurr. l23, 1356-1362. Minors, D. S., and Waterhouse, J. M. (1981). “Circadian Rhythm and the Human.” PSG, Inc., Boston. Pauly, J. E. (1980). “The Spectrum of the Rhythm. Chronobiology.” Sijthoff & Noordhoff, The Netherlands. Reinberg, A. (1983). “Chronobiology and Nutrition. Biological Rhythms and Medicine.” Springer-Verlag, New York. Schumann, K.,and Haen, E. (1988). Influence of food intake on the 24-hr variations of plasma iron concentration in the rabbit. Chronobiol. Int. 5,59-64. Schweiger, H., Berger, S., Kretschmer, H., Morler, H., Halberg, E., Sothern, R. B., and Halberg, F. (1986). Proc. NatL Acad Sci. U.S.A. 83,8619-8623. Shannon, C. E., and Weaver, W. (1%3). “The Mathematical Theory of Communication.” Univ. of Illinois Press, Urbana. Stolz, G., Aschoff, J. C., Born,J., and Aschoff, J. (1988). VEP, physiological and psychological circadian variations in humans. J. Neurol. US,308-313.


Chapter 15 NUTRIENT-RESPONSE: A “TOP DOWN” APPROACH TO METABOLIC CONTROL ARTHUR R. SCHULZ Department of Biochemistry and Molecular Biology Indiana University School of Medicine Indianapolis, Indiana 46202

Introduction Mathematical Treatment A. Interpretation of Nutrient-Response Curves B. Derivation of an Expression for Nutrient-Response C. A Brief Description of Metabolic Control Theory D. Biochemical Systems Theory and Nutrient-Response 111. Analysis of Data on Three Dietary Proteins IV. Conclusions References 1.

11.

I. INTRODUCTION

Analysis of nutrient-response curves may appear prosaic to some nutritionists, but these curves contain a wealth of information. Proper interpretation of nutrient-response curves has resulted in astute conclusions. For example, Max Rubner concluded in his monograph, “In spite of the varying chemistry of catabolism which apparently occurs during changes in the form of nourishment, the energy metabolism is the determining factor and focal point around which everything else revolves” (Rubner, 1902). This is a rather accurate word description of the central role played by oxidative phosphorylation in cellular metabolism. This is truly remarkable since the monograph was published approximately 25 years before the isolation and characterization of adenosine triphosphate and nearly 40 years before the identification of the process of oxidative phosphorylation. Clearly, the analysis of nutrient-response curves constitutes a worthwhile facet of mathematical modeling in nutrition. 227 Copyright 0 1996 by Academic Press,Inc. All rights of reproduction in any form reseNed.

228


II.

MATHEMATICAL TREATMENT

The first prerequisite to interpretation of nutrient-response in terms of the metabolic fate of a given nutrient is a reasonable understanding of the nutrient-response curve itself. Anyone who has looked at a nutrientresponse curve should have been convinced that the curve is nonlinear provided the curve encompasses a reasonable range of nutrient intake. 'Typical nutrient-response curves are shown in Fig. 1. At higher levels of nutrient intake, inhibition of the response may be observed in any of the foregoing curves. The equation and parameter constraints will be referred to later in the text. A. INTERPRETATION OF NUTRIENT-RESPONSE CURVES A variety of techniques have been employed to analyze these nutrientresponse curves, the worst of which is the attempt to fit these curves to a single straight line (for example, Keys er al., 1959 Hegsted et al., 1965; Said and Hegsted, 1969). Monod reported that the growth of bacterial cultures in the exponential phase of growth is a hyperbolic function of the limiting nutrient (Monod, 1949). Morgan et al. (1975) suggested that the response of higher animals to a nutrient is a rational function of nutrient intake provided that some of the parameters of the rational polynomial are arbitrarily assigned a value of zero. It has been suggested by others that the nutrient-response relationship is described better by a rational polynomial without any arbitrary assumption concerning the magnitude of any of the parameters (Schulz, 1987, 1991). There has been significant speculation on the reason why nutrientresponse curves appear to be described by rational polynomials. It has been

c n

n

FIG. 1. Possible shapes of nutrient-response curves and the parameter constraints which allow these curves to be described by a 2:2 rational polynomial.

CHAPTER 15 NUTRIENT-RESPONSE

229

suggested that the nutrient-response curve reflects the kinetic behavior of a rate-limiting enzymic reaction in the pathway by which the nutrient is metabolized (Morgan et al., 1975). On the other hand, Monod (1949) expressed the opinion that bacterial growth is too complex a process to be described by the kinetic behavior of a single enzymic reaction. In regard to this speculation, it is relevant that investigations of metabolic control at the level of individual pathways in which it is possible to obtain quantitative estimates of the control of the flux exerted by individual enzymes in the pathway indicate that, in most cases, the control of flux through a pathway is shared by more than one enzyme (see, for example, Westerhof et al., 1987).

B. DERIVATION OF AN EXPRESSION FOR NUTRIENT-RESPONSE An alternative to the empirical fitting of nutrient-response curves is the derivation of an explicit equation to describe the nutrient-response relationship (Schulz, 1992). A model for this approach lies in the stochastic approach to enzyme kinetics and biochemical cycle kinetics (Ninio, 1987; Hill, 1989; Mazur, 1991). In this treatment, the nutrient-response relationship is represented as a directed graph (digraph). The intermediate metabolites in the pathway(s) by which the nutrient is metabolized are the vertices of the digraph. It should be realized that some of the vertices may be the same metabolite found in a different pool. The allowable transitions between the metabolites are represented by the edges of the digraph. Each transition is assigned a weight which is the probability of the transition occurring. Thus, a metabolic pathway can be viewed as a series of interconnections of discrete states anyone of which might be visited during a random walk through the pathway represented by the digraph. One can calculate the probability that any specific state might be occupied during a large ensemble of random walks around the digraph. Alternately, if the transition to one or more termination states were to result in an instantaneous transition to the initial state, a similar result could be obtained by calculating the probability that a given state would be occupied at any time during a single continuous walk about the digraph (Hill, 1989). This is given as a conceptual aid and should not be construed as biochemical conversion of the termination state to the starting state. Figure 2A is an example of a very simple metabolic pathway in which the starting state is vertex 1. This state can be considered to be the dietary nutrient. If states 4 and 5 are termination states, the pathway can be represented as the closed diagram of Fig. 2B. There are two cycles in Fig. 2B one of which can be visualized as giving rise to the observed response. The expression derived for the probability of completion of either cycle is a 1:1

230


A

4e024-1.

a12n a2 1

a13n a3,

a1 2n a21

'a31

'3

a35

>5

B -

*

a13n

FIG. 2. (A) An open diagram of a metabolic pathway in which state 1 is the starting state and states 4 and 5 are termination states. (B) A closed diagram of the same metabolic pathway.

rational polynomial in nutrient intake. Figure 3 portrays a more complex pathway in which there are convergent paths each of which is dependent on nutrient intake. Figure 3B is the closed digraph if state 5 is the termina-

B

C

J2\

'ar'

4 1

\I

3

3

a

b

C

d

FIG. 3. (A) An open diagram of a metabolic pathway with parallel branches. State 1 is the starting state and state 5 is the termination state. (B) A closed diagram of the same pathway. (C)Cyclic paths of closed diagram.


231

tion state. The expression for the completion of the cycle is a 2 :2 rational function in nutrient intake (Schulz, 1992). A general equation for the nutrient-response relationship as derived from these stochastic principles is

r = -i=O

,m

2 pin’

2

1, ai 2 0, pi 2 0, where all pi f 0,

i=O

where r is the response and n is the amount of nutrient. If the measured response is a net response, then = 0. It has been shown that the foregoing general equation can describe all of the observed nutrient-response curves (Schulz, 1987). Reference to Fig. 1 shows that a 2 :2 rational function can give rise to the shapes observed in nutrient-response curves if certain constraints are placed on the parameters. This would suggest that the equation for nutrient-response is a rational polynomial, and there is no need to explain the shape of these curves in terms of a rate-limiting enzymic reaction. It is not surprising that the nutrient-response relationship is described by a rational polynomial for the rate of most biological reactions are described by this type of equation. The dimensions of the parameters in the general equation for nutrient-response are ai = response x (nutrient)’ i = 0, 1 .

pi = (nutrient)’ -

i = 0,1 . . ,1

. . m.

Thus, the numerator terms are amplitude factors and the denominator polynomial represents the total nutrient processed by the organism. This allows partitioning the nutrient processed into a number of grossly defined pools. The number of pools is determined by the number of the denominator terms. This is entirely analogous to the rational polynomial which describes the steady state rate of an enzyme-catalyzed reaction in which the denominator consists of the enzyme species and the numerator contains amplitude terms. Estimates of the parameters of the equation for the nutrient-response curve can be obtained by graphical analysis (Schulz, 1987) or by statistical analysis of the rational polynomial (Press ef af., 1992). Figure 4 shows the response of rats in terms of accumulation of body nitrogen to three different sources of dietary protein (Phillips, 1981). The sources of dietary protein were casein, peanut protein, and wheat gluten. The parameters were estimated graphically, and these estimates were “fine tuned” by simulation

232


a

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

NITROGENCONSUMED(9)

FIG. 4. Nutrient-response curves for three different proteins. The dots represent the data obtained experimentally. The solid line is the line generated for casein, the broken line represents peanut protein, and the dashed line represents wheat gluten.

(Schulz, 1987). The observed points are represented by the dots in Fig. 4. The equation which best fits the data is

The lines in Fig. 4 (solid line for casein, broken line for peanut protein, and dashed line for wheat gluten) were generated from the parameters listed in Table I. Figure 5 shows that fi, the fraction of nutrient passing through the P l n pool, increases and then decreases with increasing nitrogen intake while f2, the fraction of nutrient passing through the P2n2 pool, increases to a maximum value in an asymptotic manner. However, it should be recognized that the order of the three proteins is reversed in the two parts of Fig. 5. TABLE I EQUATION PARAMETERS'

Casein

Peanut protein ~~~

82

10.600 2.600 24.000 1.000 0.270

p3

0.005

a, a2

Po 01

a

3.700 0.900 15.ooo 1.Do0

0.060 0.002

Wheat gluten ~

2.440 0.162 13.000 1.OM) 0.008 0.001

All parameters are evaluated relative to PI.

233


-‘i

f2 -3.2 -

.lQ1 0

./ _----

- ----

.*------

I

I

I

1

I

I

4

8

12

16

22

24

,

28

i

32

n(a NJday) FIG. 5. Plots of fi and f2 vs daily dietary protein nitrogen intake. The solid line represents casein, the broken line represents peanut protein, and the dashed line represents wheat gluten.

C. A BRIEF DESCRIPTION OF METABOLIC CONTROL THEORY The object of metaboliccontrol theory is to provide a sound mathematical foundation for the quantitative estimation of the role played by individual enzymes on the control of flux through a metabolic pathway and also the control exerted by individual enzymes on the concentration of intermediate metabolites in the pathway. The general principles of metabolic control theory and biochemical systems theory can be visualized by considering the simple metabolic pathway in Fig. 6. The numbers above the arrows

234


x,

1

'M1

2

\

M2

'M,

4

1

FIG. 6. A simple metabolic pathway.

indicates the enzymes in the pathway, although both enzymes and carriers can be involved. The direction of the arrows indicates the normal flow of flux through the path, but this does not imply irreversibility of any step. The letters M i both identify and represent the concentrations of the intermediate metabolites. The substance Xo is an independent variable and lies outside of the pathway. It is important to recognize the distinction between regulation and control of a metabolic pathway. Many enzymes are subject to regulation by modifiers which may be internal and external to the pathway. Thus the activity of these enzymes may vary greatly under different physiological conditions. However, a given regulatory enzyme may or may not exert a significant effect on the control of flux through the pathway. For example. the enzymes phosphorylase and glycogen synthase are both present in liver and muscle in such high amounts that they can exert no significant role in the control of their respective pathways when they are fully active. However, when they are largely in the inactive forms they may exert an important role in the control of their pathways. It is important to recognize that the control of a pathway is a quantitative concept which must be determined by quantitative methods. A general equation for the rate of any enzymic step in the metabolic pathway is

v.I =

v, L1 - T K , 1 Mi

where M i is the substrate for enzyme j and Mi is the product of the enzymic reaction. The Michaelis constant for the substrate is K,,.,i and the K ~ Mare . product inhibition constants. The ratio of substrate to product is the kass action ratio and is represented by the symbol r . The sensitivity of the reaction rate to any pathway reactant is termed the elasticity coefficient of the enzyme to the reactant involved, and the elasticity coefficient for enzyme j with respect to substrate Mi is given by the following expression.

The elasticity can be expressed by differentiation of the general equation


235

for the rate of an enzymic reaction with respect to its substrate and multiplication by Mihi.

The elasticity coefficient consists of two terms, the first of which is thermodynamic and the second term is kinetic. The value of the thermodynamic term varies from zero when the reaction is infinitely far from equilibrium to infinity when the reaction is at equilibrium. The kinetic term varies from zero when the enzyme is saturated with substrate to a value of unity when the substrate concentration is very much less than that required to saturate the enzyme. The elasticity coefficient of the enzyme with respect to its product can be obtained in a similar manner.

The terms on the right-hand side of the foregoing expression are both negative, and this is expected because the product is an inhibitor of the enzymic reaction. It important to note that any enzyme can exhibit sensitivity to the product of the reaction even under the condition that the reaction is very far from equilibrium. The sensitivity of an enzyme to its product is often ignored in the mathematical modeling of whole animals. The reason for this oversight lies in the concept of identifiability (Jacquez and Perry, 1990;Jacquez, Chapter 19, this volume). In studies of the intact animal the effect of the product of an enzymic reaction may be difficult to identify, but that does not justify the assumption that the enzyme is insensitive to the product. Rather it emphasizes that sensitivity and identifiability are two separate concepts both of which must be recognized in mathematical modeling. If the sensitivity of the system to any reactant is not readily identifiable, then it is essential to establish means by which the sensitivity of the system to the reactant can be identified. The elasticity coefficient plays an additional essential role in metabolic control theory for it provides

236


the link between the domain of the steady-state enzyme kineticist and control of multienzyme systems. The complete development of the principles of metabolic control theory is given elsewhere (see, for example, Westerhof and van Dam, 1987; Cornish-Bowden and Cardenas, 1990) and will not be presented here. D. BIOCHEMICAL SYSTEMS THEORY AND NUTRIENT-RESPONSE The general equation for the nutrient-response relationship can be written as

This equation can be written in the factored form r=

Q>

(Ui

+ n ) (a; + n . . . (Ui + n ) - 1

0 (b; + n ) (b;- 1 + n )

. . . (bi + n)’

The foregoing equation in logarithmic form is

In r

=

a) + In (ai + n ) + In (ai - 1 + n ) + . . . + In (a1 + n) 0 - In (b; + n ) - In (b;- + n ) - . . . - In (bl + n).

In

-

‘Thus, it is not surprising that rational polynomials give rise linear segments when plotted in log-log space. In the case of many biological reactions this linearity extends over a number of order of magnitude (Savageau, 1976). This observation led Savageau to develop a power law formulation of metabolic control based on a Taylor’s series when plotted in log-log space (Savageau, 1972). It is now possible to consider Savageau’s power law, first as it applies to Fig. 6, and then as it might apply in a “top-down” approach. For any step in Fig. 6, the truncated Taylor’s series logarithmic expression is

where M i is the substrate for enzyme j and io and j~ refer to M iand vj,


237

respectively, about some operating point where the differential is evaluated at io and jo. Let In ai= In vjo - EJln Mid then In vj = In aj + EJln Mi V. = I

(y.M.Eii.

I

'

If more than one metabolite in the pathway affects vj

The concentration of an intermediate metabolite is given by

The differential equations for the pathway in Fig. 6 are

Since E: is the elasticity coefficient for an enzyme with respect to its product, it is negative because the product is inhibitory, and for convenience $ = - E:. The foregoing equations can be rearranged as

where bi = In &/ai,yo = In Xo, and yi = In Mi. The expressions for the logarithmic concentrations of the intermediate metabolites are

238


where

ID/

=

.912~23

+ E11.923+

The sensitivity of the intermediate metabolites to the nutrient is obtained by differentiation of these equations with respect to yo, and this is the sensitivity of the logarithmic concentration of the intermediate metabolites to the logarithmic concentration of the dietary nutrient. 111.

ANALYSIS OF DATA ON THREE DIETARY PROTEINS

The metabolic pools defined in the denominator of the nutrient-response equation can be viewed as intermediate metabolites. Thus, the logarithmic derivative of each metabolic pool with respect to nutrient intake provides an estimate of the sensitivity of the pool to nutrient intake. Figure 7 shows

I I I I I

I

i I

,

T I

-1

log10

"

1

i

FIG. 7. Log-log plot of f2 vs daily protein intake.


239

the log-log plot of f2, the fraction of the intake flowing through the &n2 pool vs intake for each protein. The lines at low to intermediate nutrient intake are parallel, indicating that the sensitivity of the pool to intake were identical and that the proteins differed only in their capacity for providing substrate for the P2n2pool. The data presented in Fig. 7 are summarized in Table I1 together with the sensitivity of pool P'n to intake and the sensitivity of pool P2n2to pool file.The sensitivities as indicated by the slopes of the lines are identical for each of the proteins. The sensitivity of f2 to intake is greater than that of fi, but there was no significant difference between the dietary proteins. These data suggest that the three dietary TABLE I1 SLOPES AND INTERCEPTS OF LINEAR LOG-LOG RELATIONSHIPS

Nutrient

Slope

intercept

Casein Peanut Gluten

Log response vs log intake 1.1097 2 0.0080 -0.2737 t 0.0043 1.1311 t 0.063 -0.5264 ? 0.0420 0.9702 ? 0.0046 -0.7357 t 0.0080


Log fi vs log intake -2.0115 2 0.0136 1.8066 t 0.0321 1.8104 t 0.0251 -2.4610 t 0.0163 1.8388 t 0.0185 -3.2708 t 0.0120


Log fi vs log intake -1.4392 t 0.0152 0.8315 t 0.0291 -1.2378 t 0.0137 0.8264 ? 0.0235 -1.1750 t 0.0145 0.8168 t 0.0212


Log response vs log fi 0.6128 t 0.0077 0.9596 t 0.0128 1.0085 t 0.0199 0.6233 ? 0.0098 0.5273 ? 0.0029 0.9892 2 0.0080


Log response vs log fi 1.3211 t 0.0436 1.6300 t 0.0581 1.1878 t 0.0462 1.3790 2 0.0445 0.6175 t 0.0196 1.1514 2 0.0200


Log f i vs log fl 1.11% ? 0.0631 2.2203 ? 0.0486 2.2404 2 0.0416 1.1878 t 0.0462 2.2996 2 0.0281 0.6851 t 0.0266

240


proteins differ only with respect to their ability to provide the indispensable amino acids. For example, the ratio of the analog of the intercepts of the plots of log of response vs log of intake of peanut protein to casein is 0.56 and the ratio for wheat gluten to casein is 0.34. This is consistent with the ratio of lysine in these proteins, and it is also essentially identical to the relative values of these proteins for maintenance and growth (Finke et al., 1987). The sensitivity of the response to intake, fi and f2, is not identical for all of the dietary proteins. The slopes of the log-log plots for casein and peanut protein do not differ significantly, but the sensitivity of response to wheat gluten is significantly less. This suggests that wheat gluten has an adverse effect on the final stage of protein synthesis. At the present time there is no apparent explanation for the inhibition of the later stage of protein synthesis by wheat gluten, but this analysis suggests that the differences in response to these three dietary proteins involves more differences in amino acid composition and identifies the portion of the pathway which should be the subject of further research.

IV. CONCLUSIONS

Nutrient-response curves contain a wealth of information concerning the metabolic fate of the nutrient involved. Unfortunately, this fact is often lost in the desire to obtain estimates of parameters to which one can assign high orders of statistical significance even though these parameters may have nebulous biochemical or physiological significance. The alternative described here is the mathematical analysis of the nutrient-response curve in terms of the metabolic fate of the nutrient and the integration of this analysis with the well-established principles developed in metabolic control theory and biochemical systems theory. In recent years there has been considerable progress in extending the concepts of metabolic control from simple metabolic pathways to more complex systems such as cell organelles and tissues (Brown et al., 1990; Kahn and Westerhof, 1991). This paper proposes a further extension of this trend. Finally, it is important to recognize the essential relationship between the concepts of sensitivity and identifiability in mathematical modeling in nutrition. It is too easy to overlook the fact that flux through a metabolic pathway may be sensitive to the concentration of intermediates in the pathway. If the sensitivity is not readily identifiable, conditions must be sought to render the sensitivity identifiable. However, if the sensitivity is not identifiable readily, it is incumbent on the investigator to recognize the possible effect of such sensitivity.


241

REFERENCES Brown, G. C., Hafner, R. P., and Brand, M. D. (1990). A ‘top-down’approach to the determination of control coefficients in metabolic control theory. Eur. I. Biochem. 188,321-325. Cornish-Bowden,A., and Cardenas, M. L. (1990). “Control of Metabolic Processes,” Plenum, New York. Finke, M. D., DeFoliart, G. R., and Benevenga. N. J. (1987). Use of simultaneous curve fitting and a four-parameter logistic model to evaluate the nutritional quality of protein sources at growth rates of rats from maintenance to maximum gain. J. Nurr. 117,1681-1688. Hegsted, D. M., McCandy, R. B., Myer, M. L., and Stare, F. J. (1965). Quantitative effects of dietary fat on serum cholesterol in man. Am. J. CIin. Nutr. 17,281-295. Hill, T. L. (1989). “Free Energy Transduction and Biochemical Cycle Kinetics,” pp. 39-88. Springer-Verlag, New York. Jacquez, J. A., and Perry, T. (1990). Parameter estimation: local identifiability of parameters. Am. 1.Physiol. 258, E727- E736. Kahn, D., and Westerhof, H. V. ( 1991). Control theory of regulatory cascades. 1. Theor. Biol. 153,255-285. Keys, A., Anderson, J. T., and Grande, F. (1959). Serum cholesterol in man: Diet fat and intrinsic responsiveness. Circulation 19,201-214. Mazur, A. K. (1991). A probabalistic view on steady-state enzyme kinetics. J. Theor. Biol. 148,229-242. Monod, J. (1949). The growth of bacterial cultures. Annu. Rev. Microbiol. 3,371-394. Morgan, P . H., Mercer, L. P., and Flodin, N. W. (1975). General model for nutritional responses of higher organisms. Proc. Natl. Acad. Sci. U.S.A. 72,43274331. Ninio, J. (1987). Alternative to the steady-state method: Derivation of reaction rates from first-passage times and pathway probabilities. Proc. Natl. Acad. Sci. U.S.A. 84,663-667. Phillips, R. D. (1981). Linear and nonlinear models for measuring protein nutritional quality. 1. Nutr. lll, 1058-1066. Press, W. H., Teukolsky, S. A., Vettering, W. T., and Flannery, B. P. (1992). “Numerical Recipes in C,” pp. 204-208. Cambridge Univ. Press, New York. Rubner, M. (1902). In “The Laws of Energy Consumption in Nutrition” (R. J. T. Joy, Trans.). U. S. Army Research Institute of Environmental Medicine, Washington, DC. Said, A. K., and Hegsted, D. M. (1969). Evaluation of dietary protein quality in adult rats. J. Nutr. 99, 474480. Savageau, M.A. (1972). Behavior of intact biochemical control systems. Curr. Top. Cell. Regul. 463-130. Savageau, M. A. (1976). “Biochemical Systems Analysis,” Chapter 7, pp. 118-130. AddisonWesley, Reading, MA. Schulz, A. R. (1987). Analysis of nutrient-response relationships. J. Nurr. 117, 1950-1958. Schulz, A. R. (1991). lnterpretation of nutrient-response relationships in rats. 1. Nutr. 121, 1834-1 843. Schulz, A. R. (1992). Nutrient-response: A long random walk through metabolic pathways. J. Nutr. Biochem. 3,98-103. Westerhof, H. V., and van Dam, K. (1987). “Thermodynamicsand Control of Biological FreeEnergy Transduction.” Elsevier, Amsterdam. Westerhof,H. V., Plomp, P. J. A., Groen, A. K., and Wanders, R. J. A. (1987).Thermodynamics of the control of metabolism. Cell Biophys. 11,239-267.



Chapter 16 MODELING MEMBRANE TRANSPORT RICHARD B. KING Center for Bioengineering University of Washington Seattle, Washington 98195

I. Introduction A Multiregion, Distributed Exchange Model A. Model Structure B. Model Assumptions C. Methods of Solution 111. Passive Diffusion A. Flow-Limited and Barrier-Limited Exchange B. Tracer Washout IV. Camer-Mediated Transport A. Introduction B. One-Site, Two-sided Transporter Model C. Tracer Studies D. Other Carrier Models V. Building Complex Models A. Introduction B. Differential Operators C. Organ Models D. Whole-Body Models VI. Summary References 11.

1.

INTRODUCTION

As research penetrates deeper and deeper into biological systems, we recognize increasing levels of complexity. This complexity arises from many sources, including the anatomy of the system, biochemical reactions within the tissue, feedback-control loops, and the presence of specialized transport mechanisms for specific substrates. The complexity present in the system often quickly exceeds our ability to intuitively predict its behavior, and 243 Copyright 0 1996 by Academic Press. Inc. All rights of reproduction in any form reserved.

244


artificial aids to understanding become essential. A mathematical model’ of the system that can be evaluated by a digital computer is one such aid. The development of a mathematical model begins with an hypothesis that describes the system. The hypothesis is usually expressed as a set of differential equations that can be solved by a computer program. The computer implementation of the model provides not only a description of the system, but also predicts the response of the system to changing conditions. These predictions can be compared to real-world behavior. Since it will always be the case that the model is less complex than the real system, our comparisons will most often lead us to refine the model by increasing its complexity. The principles of formulating and testing mathematical models have been described by Berman (1963). The design of a model is a compromise between attempting to describe the whole complexity of a real system and our ability to accurately deal with that complexity in the model. One approach to dealing with complexity is to begin with a set of submodels that are described as differential operators that take one or more inputs and operate on them to produce one or more outputs. These operators can then be connected together to produce more complex models. When modeling passage of substrates across membranes, the simplest system is composed of two well-stirred compartments separated by a semipermeable membrane. This simple model is easy to describe mathematically and to implement on a computer. It is also, of course, unrealistically simple. First, the assumption of complete mixing is nearly always violated. When convection is involved, transport is a distributed process. That is, a concentration gradient exists along the axis of convection (e.g., from the proximal to the distal end of the intestine, from the arterial to the venous end of a capillary). Second, substrates traverse multiple membranes when passing from the lumen of the gut to the blood and from the blood to tissue. Finally, in many instances, there are specific transport mechanisms that move substrates across the membrane. When a substrate must traverse a layer of cells to reach its target, the substrate molecules may move through gaps between the cells or they may move though the cell membrane. In the latter case, the substrate may move by pinocytosis, by “dissolving” in the membrane, or there may be a carriermediated transport mechanism for that substrate. Passage by dissolving in the membrane and passage through intercellular gaps may both be modeled as traversal of a semipermeable membrane, because the gaps present a passive diffusional barrier. Even when substrates move across membranes by passive diffusion, these membranes can have important effects on the Models used here are available from the author who can be contacted by electronic mail at [email protected].

CHAPTER 16 MODELING MEMBRANE TRANSPORT

245

dynamic response of the system since the resistance to diffusion can introduce significant delays. Carrier-mediated transport across membranes adds additional complexity to the system and, thus, to the model. For even the simplest transporter, the concentration of the transporter and its affinity for the substrate must be known before it can be modeled. Also, active transport is inherently a saturable process. Thus, to analyze the dynamics of tracer-labeled substrate, the model must account for both labeled and unlabeled substrate as the transport dynamics will depend on total substrate concentration. 11.

A MULTIREGION, DISTRIBUTED EXCHANGE MODEL

A. MODEL STRUCTURE

A model for transport and exchange of tracer between plasma, endothelial cells, interstitial fluid, mucosal cells, and the lumen of the intestine is diagrammed in Fig. 1for a single capillary-tissue unit. The model accounts for convection of tracer in the plasma (Fp)and in the lumen of the intestine (Fi). Material diffuses from region to region by conductance (PS)across the cellular membranes and through the gaps between the endothelial cells (PS,). Each of the regions is characterized by the volume of distribution

Capillary (Plasma) Endothelial Cell Interstitial Fluid Mucosal Cell Fi-

Intestinal Lumen

FIG. 1. Schematic representation of a five-region, axially distributed blood-tissue exchange model composed of plasma (p), endothelial cells (ec), interstitial fluid space (isf), mucosal cells (mc), and the intestinal lumen (i). Convection (F) takes place in both the plasma and intestinal lumen. The Vs are volumes of distribution. Bamer conductances are given by permeability-surface area products ( P S ) . Reactions or metabolic consumption within the cells are given by the clearances ( G ) .

246


(V') of the tracer being modeled. This volume may be smaller than the physical volume of the region if the tracer is excluded from some portion of the volume (e.g., exclusion from mitochondria in the cells) or larger in the presence of a mechanism that sequesters or concentrates the tracer. The model accounts for first-order consumption of tracer in the endothelial and mucosal cells (G). This model extends the four-region model described by Bassingthwaighte et al. (1989a) by including a fifth region, the intestine, with its flow. The model is axially distributed which permits the development of concentration gradients in the axial direction in each of the regions. This is accomplished by dividing the capillary-tissue unit into a number of axial segments (Nseg).The number of segments used is under the control of the modeler. The actual length of the capillary-tissue unit need not be known unless axial diffusion is included (see Model Assumptions). B. MODEL ASSUMPTIONS The explicit model assumptions are: 1. The system is linear and stationary. Flows are constant and coefficients are not concentration dependent. PSs will be nonlinear for carrier-mediated transport, but are linear for tracers present in extremely low molar concentrations (Bassingthwaighte and Goresky, 1984). 2. The transport parameters are uniform in the axial direction. Deviations from this assumption have very little effect on outflow dilution curves but the effect on intratissue concentration profiles is large (Bassingthwaighte, 1974). 3. Convection occurs only in the capillary and lumen of the intestine. 4. There is no capacitance for solute in the membranes. 5. The outer boundaries are reflecting. This means there is no exchange between neighboring capillary-tissue units. The model is inappropriate for highly diffusible tracers such as gasses and heat because of diffusional shunting (Roth and Feigl, 1981; Sharan ef al., 1989). 6. Diffusion is rapid in the radial direction in all regions. This assumption seems to be valid for small solutes in well-perfused organs such as the heart (Bassingthwaighte and Goresky, 1984) and liver (Goresky, 1963), but may be violated in some tissues. 7. There is no diffusion in the axial direction. This assumption may be violated when molecular diffusion is fast compared to convection. Bassingthwaighte et al. (1992) present methods for incorporating axial diffusion into the model.

247


C. METHODS OF SOLUTION 1. Flow Algorithm

The flow through the capillary and intestine can be modeled in a number of ways. In their distributed blood-tissue exchange models, Bassingthwaighte et al. (1989a, 1992) have used the LaGrangian, or sliding element, algorithm which is illustrated on the left in Fig. 2. This assumes plug flow with a flat velocity profile in the flowing region. This algorithm allows for fast computation and introduces no dispersion. It does, however, give the model an inherent time step, which is the time required to fill one segment of the capillary, equal to VpI(Fp- Nseg). With two flowing regions having different flows, the sliding element algorithm cannot be used. Instead a Poisson algorithm, illustrated on the right in Fig. 2, is used. This algorithm models the flowing region as a series of well-stirred tanks. At each time step, each segment of the flowing region loses some of its contents to the downstream segment and receives some of the contents of the upstream segment. If the time step is greater than

LaGrangianRow Algorithm At = V,,/(F,. N,,,)

t=o t=o+

wmT

t = At-1 -

Exchange

t...I

I

Poisson Flow Algorithm At = arbitrary Artery Vein t=O

t=o+ t = At-

I

~

I

+ff+

Convect

-

Exchange

7

1

I

t=At+

t=At+

Convect

t = 2At-

Exchange

t = 2AL

Exchange

t=2At+

Slide

t=m+

Convect

FIG. 2. Two algorithmsfor modeling flow. (Left) LaGrangiansliding fluid element algorithm. With each At the entire capillary contents slide one element downstream instantaneously.

The slide is followed by transmembrane exchanges. (Right) Poisson-stirred tanks algorithm. At each At a portion of the contents of each capillaxy segment is transferred downstream. The convection is followed by transmembrane exchanges.

248


the time required to fill one segment of the flowing region, the algorithm is repeated until the convection is completed. This algorithm has the advantage of having no inherent time step. It does, however, introduce dispersion of the input function. The dispersion is minimized when the number of axial segments is large and when the input function is dispersed rather than a spike.

2. Computation of Radial Exchanges The radial exchanges are calculated by an analytical solution to a linear set of ordinary differential equations describing the exchanges. The solutions for the concentrations in each region at time t + At are

where 0 is the concentration vector. The term eAL\I can be computed analytically using an eigenvalue and eigenvector approach. Since the model is linear and stationary, e"Afcan be calculated once when the model is initialized and applied each time the model is evaluated. The numerical methods for computing exchanges are described in detail in Bassingthwaighte et al. (1989a,b, 1992).

Ill. PASSIVE DIFFUSION

A. FLOW-LIMITED AND BARRIER-LIMITED EXCHANGE When solutes passively diffuse across barriers such as the cell membranes or the gaps between endothelial cells, the presence of the diffusion barrier may significantly alter the behavior of the solute. The magnitude of this effect depends on the relative importance of diffusion compared to convection, which may be evaluated by the ratio of barrier conductance to flow (PSIF).This is illustrated in Fig. 3, which shows the behavior of an extracellular tracer, i.e., one which only permeates the endothelial cell gaps. In Figs. 3a and 3b the gap permeability, PS,, is 100 ml g-' min-l. Figure 3a shows the outflow curves for three different plasma flows, 0.2, 1.0, and 2.0 ml g-' min..' (PSIF = 500, 100, and 50, respectively). With increasing flow, the peak of the curve shifts to the left, and mean transit time (5 = F / V ) decreases. Figure 3b shows these same curves scaled by their mean transit times. (The time axis is divided by f and the ordinate is multiplied by i, thus preserving area.) The three scaled curves superimpose because


249

a 0.6

$4

0.0

0

40

20

60

0

t, sec

2

1

3

rlf

FIG. 3. Flow-limited and barrier-limited exchange. Outflow curves for an extracellulartracer with plasma flows of 0.2 (diamonds), 1.0 (circles), and 2.0 (squares) ml g-' min-I. (a) Flowlimited exchange with PS, of 100 ml g-' min-'. (b) Curves from (a) scaled by their mean transit times.7, showing similarity scaling. (c) Barrier-limited exchange with PS, of 10 ml g-' min-'. (d) Transit time scaled curves from (c).

the transport from inflow to outflow is totally dominated flow and is not influenced, i.e., impeded, by the barrier. In contrast, Figs. 3c and 3d show the behavior of the same tracer when PS, is 10 ml g-' min-' (PWF equal to 50, 10, and 5). While the shapes of the outflow curves do not appear much different, the scaled curves no longer superimpose. Their shapes are affected by the presence of the barrier which impedes the movement of tracer out of the plasma to the extracellular fluid and its backflux into the plasma.

B. TRACER WASHOUT The rate at which solute is washed out of the capillary-tissue unit depends on the number of barriers that it must cross to reach a region in which it can be removed by convection. The washout rate is also affected by the volumes of the regions through which it is transported. Figure 4a shows the washout of tracer when the cells or the intestine are preloaded with tracer. In each case the plasma flow and all barrier conductances are 1 ml g-' min-'. (For simplicity, intestinal flow is set to zero.) Tracer preloaded in the endothelial cells washes out rapidly because

250


a

loo

, 0

b loo [

100

200

0

100

200

t, sec FIG. 4. Tracer washout after cell loading.Plasma outflow curves after: (a) loading in endothelium (circles), mucosa (squares), and intestine (diamonds) and (b) loading in the intestine with PS, of 1 (circles) and 100 (squares). Fp = 1.0 ml g-' min-', Fi = 0.0 ml g-' min-', PS, = 1.0 ml g-' min-', V, = 0.015 ml g-I, Ve. = 0.005 ml g', V'isf = 0.075 ml g-', V',,,= = 0.3 ml g-'. Vi = 0.4 ml 8.'.

the endothelial cell volume is small (0.005 ml g-l), and the tracer only has to cross one barrier to reach the plasma. The washout curve is not monoexponential because some of the tracer diffuses across the abluminal surface of the endothelial cells into the interstitial fluid, mucosal cells, and intestinal lumen before diffusing back across the barriers into the plasma. Tracer preloaded into the mucosal cells and intestinal lumen washes out much more slowly since the volumes of these regions are large (0.3 and 0.4 ml g.I, respectively) and it must traverse multiple barriers. The curve for preloading in the mucosa rises more rapidly, but at long times the washout rates are nearly identical. A large change in the conductance of the luminal mucosal cell membrane, Psi, has only a modest effect on tracer washout of tracer preloaded into the intestinal lumen, Fig. 4b. The curve rises more rapidly, but the washout rate is only slightly greater when the conductance is raised 100-fold. Figure 5 shows tracer washout when a constant infusion of tracer is made into the plasma (Fig. 5a) or intestinal lumen (Fig. 5b). Flow through the intestinal lumen is 0.05 ml g-' min-' ; membrane conductances, volumes, and plasma flow are identical to those used for Fig. 4a. The concentration of tracer in the plasma outflow rises more quickly in either case due to its much greater flow. The much higher plasma flow (20-fold greater than intestinal flow) results in nearly equal steady-state concentrations when the tracer is injected into the plasma and in a lower concentration in the plasma than in the intestine when the tracer enters through the intestine. The flow difference explains the time required for the intestinal concentration to reach steady state. Tracer in the plasma is quickly convected to


a

1.0

't =

0.5

25 1

b 0.06

-

0.04 0.02

4

0.00

0.0 0

100

200

300

0

100

m

300

t, sec FIG. 5. Tracer washout curves with constant infusion of tracer. Plasma (circles) and intestine (squares) outflow curves with infusion into: (a) plasma and (b) lumen of the intestine. Fp = 1.0 ml g-' min-', Fi = 0.05 ml g-' min-I, PSal = 1.0 ml g-I min-', Vp = 0.015 ml g-', VeC= 0.005 ml g-', V'X = 0.075 ml g-', V', = 0.3 ml g-l, V'i = 0.4 ml g-'.

the downstream end of the capillary where it can diffuse into the intestine. This "carriage effect" is much greater with plasma than with intestinal injection.

IV. CARRIER-MEDIATED TRANSPORT

A. INTRODUCTION

The introduction of carrier-mediated transport requires a redefinition of the model because it represents a partial relaxation of the first model assumption. When transport is carrier-mediated, the membrane transport parameters are concentration dependent and depend on the concentration of substrate on both sides of the membrane. The effect is to modify the conductance of the membrane, a fact that we will use in redesigning the model. Carrier-mediated transport is a saturable process. The carrier has a finite concentration in the membrane. When the concentration of the substrate is high enough, all the carrier will be bound to the substrate and adding additional substrate will not result in increased transport. When a tracer is used, the carrier will exhibit competitive inhibition. Labeled substrate must compete with unlabeled substrate for available carrier. Thus, increased concentrations of unlabeled substrate may result in decreased transport of tracer. To account for this, the model must account for both tracer and mother (nontracer) substances.

252


B. ONE-SITE, TWO-SIDED TRANSPORTER MODEL 1.

Transporter Configuration

A simple one-site, two-sided transporter is diagrammed in Fig. 6. The transporter, T, can reversibly bind with the substrate, C,on either side of the membrane. The equilibrium dissociation constants, Di and Do, are the ratio of the reaction rate for the dissociation reaction to that for the association reaction (off-rate to on-rate) on the inside and outside surfaces of the membrane, respectively. The units of D are millimolar (mM, or whatever units are being used for concentration). Both the free transporter and the transporter-substrate complex, TC, can diffuse through the membrane. Po and PI are the permeabilities of the free transporter and complexed transporter, respectively. The units of P are per minute. 2.

Calculating Membrane Conductances

A concentration-dependent expression for membrane conductance can be derived by starting with Ohm's law.

Current = Driving Force Conductance. 1

(2)

In our case, current is the flux of substrate across the membrane (J),driving force is the concentration of the substrate ( C ) ,conductance is the membrane conductance ( P S ) , and Eq. (2) can be rewritten as

J PS = C'

Outside hCell Membrane -!

Inside

FIG. 6. Diagram of a one-site, two-sided transporter model.

(3)

253


The flux from the cis- to the trans-side of the membrane is

where CT is the concentration of the transporter in the tissue, mM, and

O& J = lC + DO

a i = 1 + - .Ci Di

Note that in Eq. (4) the subscripts i and o refer to the cis- and trans-sides of the membrane, respectively, rather than the inside and outside of the cell. From Eqs. (3) and (4), we can write expressions for the conductance of the outside and inside surfaces of the membrane in terms of the substrate concentrations, dissociation constants, transporter permeabilities, and concentration of transporter in the tissue.

CTPI(PO + Psi = .,[(l

+

2) + T)+ (Po

F) (1

+ :)(Po

F)] , and

+

F) + F) +

(5)

CTP,(P0+

PS,

=

Di[(l +%)(Po

(1 +?)(Po

%)] 3

+

(6)

where Psi is the conductance from the outside to the inside of the cell and PS, is the conductance from inside to outside. Figure 7 shows the conductance of a membrane with the one-site carrier with changing concentration of substrate on the outside, C,, and inside, Ci, of the membrane. As Co increases, more of the carrier is bound to the substrate, less is available to bind more substrate, and the membrane conductance decreases. Similarly, increases in Ci decrease the conductance. When Ci is 5 mM, nearly all the carrier is bound to substrate on the inside of the membrane and conductance is reduced to near zero regardless of the outside concentration. Figure 7b illustrates asymmetric transport. The dissociation constant on the outside of the membrane is three times that on the inside, and the outward conductance is three times the inward

254


-a

1.5

-*i1.0 'WJ

2

0.5

6 a, 0.0 0.0

0.5

1.o

0.0

0.5

1.o

c,, FIG. 7. Effective membrane PS in the inward (open symbols) and outward (filled symbols) directions as a function of substrate concentration on the outside of the membrane for inside concentrations (C,) of 0.02 (circles), 0.2 (diamonds), and 2.0 (squares) mM. (a) Symmetric (Do= 0,)and (b) asymmetric transport (0,= 3Di). Po = PI = 6 X sec-', 0,= mM,CT = 0.2 mhf

conductance. Changes in Po relative to PI will not produce asymmetric transport, but will alter the conductances in both directions. 3. A Blood- Tissue Exchange Model with Carrier-Mediated Transport

The model shown in Fig. 1can be modified to include the carrier-mediated transport. Figure 8 shows such a model with carriers on the membranes of

Capillary (Plasma) Endothelir Cell Interstitial Fluid Mucosal cell

V:

Intestinal Lumen

FIG. 8. Schematic representation of a five-region, axially distributed blood-tissue exchange model with carrier-mediated transport on the membranes of the mucosal cell. See the legend to Fig. 1 for an explanation of symbols.


255

the mucosal cells. This requires three changes in the implementation of the model. First, the model must account for tracer and nontracer substrate. Second, the effective conductances of the mucosal cell membranes are calculated from the substrate concentrations and the parameters of the transporter. Since the conductances are dependent on concentrations that change with time and axial position in the capillary-tissue unit, the third change is to calculate the coefficients of the updating matrix, PAtof Eq. (l),for each axial segment each time the model is evaluated. This has a impact on model performance since the calculations required to evaluate the updating matrix are costly. C. TRACER STUDIES Figure 9 shows a simulation experiment examining the behavior of the carrier-mediated transport model following a constant infusion of tracer into the lumen of the intestine. The total substrate concentration was constant at mM and the tracer concentration was mM throughout the experiment. Figure 9a shows the concentration of tracer in the outflow of the plasma and intestine. (Note the ordinate has a logarithmic scale.) The final concentration in the intestinal outflow is 0.9 X mM. Most of the tracer is passing through the intestine without being absorbed through

b

2.0

r 0

5

10

15

t, min FIG. 7. Outflow concentration (a) for plasma (circles) and intestine (squares) and average membrane conductance (b) for the intestine-mucosal cell (circles) and mucosal cell-isf (squares) exchange following constant infusion of substrate and tracer into the inflow to the intestinal lumen at a total substrate concentration of lo-* mM. See the legends to Figs. 5 and 7 for the values of the parameters for flows and volumes and of the transporter.

256


the mucosa. In this experiment, the equilibrium dissociation constant on the outside of the membrane is mM.Thus the total substrate concentration at the inflow is lo4 times the dissociation constant, causing most of the carrier to be bound on the outside surface of the membrane and not be available for transport. This is also apparent from Fig. 9b, in which the average membrane conductance is plotted as a function of time. The average is obtained from the arithmetic mean of the conductances in each axial segment; since Di equals Do, the transport is symmetric and PSO"' equals Psi".The Psi has an initial value of 1.4 ml g-' min-' but falls to a steadystate value of only 0.2. The steady-state value of PS,, is higher, 0.7 ml g-' min because it sees a much lower substrate concentration. The steady-state plasma outflow concentration, 2.6 X lo-'' mM, is much lower than that for the gut. This is explained by the low membrane permeabilities and by the much higher flow rate of the plasma (20-fold greater than that in the intestine). Note that this steady-state concentration is more than two orders of magnitude less than that in the intestine. This contrasts markedly with the relatively small difference seen when membrane conductances are not sensitive to substrate concentration, Fig. 5b. Figure 10 shows the effect on tracer extraction from the gut with constant infusion of substrate at different concentrations. Extraction measures the amount of tracer that is absorbed in a single pass through the gut. In the steady state, the extraction, E(t), is defined as

',

ts

20-

i 0

._ c . '

V

5

10.

a

W 0

0

I

1o 6

lo4

lo2

1oo

Concentration, mM FIG. 10. Extraction of tracer from the gut as a function of total concentration of the substrate. See the legend to Fig. 9 for experimental details.


257

At very low substrate concentrations, the maximum extraction is 23%.As substrate concentration increases, extraction declines to 9% at the substrate concentration of mM used in Fig. 9 and reaches a minimum of about 5% at 0.1 mM. In the real world, the extraction would continue to decline until extraction was essentially zero and all carrier was bound. The plateau at 5% is an artifact of the numerical methods used by the model to keep the solution stable.

D. OTHER CARRIER MODELS The single-site model used thus far is the simplest carrier model. Many other configurations are possible and probably exist in the body. Two examples are diagrammed in Fig. 11. The two-site carrier is a simplification of the carrier proposed for Ca-Na exchange by Wong and Bassingthwaighte (1981). A detailed discussion of different types of carriers and their dynamics is given by Stein (1986). Regardless of the carrier model used, a strategy for incorporating it into a model is to derive the equation for unidirectional flux, and then use that result to get the effective membrane conductance.

a

I- Cell Membrane -I

I- Cell Membrane -I

FIG. 11. Diagram of two carrier models. (a) Two-site carrier model. The camer ( T ) has two active sitesthat bind the same substrate (C). Only the uncomplexedand doubly complexed forms of the carrier can move across the membrane. (b) Cotransporter. Similar to (a) but the one active site of the transporter binds the first substrate (C) but the second site binds a second substrate (S).

258


V.

BUILDING COMPLEX MODELS

A. INTRODUCTION The illustrations above all use a single capillary model in an open (nonrecirculating) system. While the latter is not a limitation if the input and output concentration-time curves can be measured across the region or organ of interest, single capillary models are seldom appropriate for modeling a whole organ as this makes the assumption that the organ is internally homogeneous. Even considering the flow only, this assumption is invalid (King er al., 1985, for example), and heterogeneities also exist in other quantities such as metabolism, oxygenation, and, most likely, local membrane conductance. For many investigations, therefore, it is necessary to assemble models of whole organs or organ systems, and, for some, a model of the whole body is required.

B. DIFFERENTIAL OPERATORS One approach to the design of complex models is to build differential operators that can be assembled into a complete model in a hierarchical manner. A differential operator can be simply defined as an operator that takes one or more inputs and manipulates them in some way to produce one or more output responses continuously. (In a computer model, a differential operator becomes a function, subroutine, or procedure.) The approach is illustrated in Fig. 12 for the construction of a vascular operator that can be used to model transport through a nonexchanging

wI TI =0.95 = 2%/ rp

~,

--

2nd Order Dispersive *rator 0 1 . GI

RD = 0.48 tJ f Operator

2nd Order Dispersive

O, = 1.82 m1

I

I I

operator

C,* = 0.8

I

Pure Delay operator

i,RD

I f

CI

Cin

FIG. 12. Construction of a differential operator for vascular transport from simpler differential operaton.


259

vessel (King ef al., 1993). The operator is composed of two simpler operators, a pure delay operator and a second-order dispersive operator that provides an analytic solution to the equation

+ 2"s-dC + O2C = "2J d$

dt

where C is the output concentration, f is the input concentration, w is the natural frequency in radians per second, and l i s the dimensionless damping coefficient. Two second-order operators are connected together to make a fourth-order dispersive operator which is connected to a delay operator to form the completed vascular operator. From the parameters of the vascular operator, transit time (7) and relative dispersion (RD), the parameters of each of the simple operators can be calculated. (The differential operators are usually implemented as reentrant computer code so that only a single copy is required in a model.) C. ORGAN MODELS Differential operators can be used to build organ models. Figure 13 shows two different styles of organ models. Figure 13a shows a parallel pathway model that might be appropriate for an organ like the heart. The exchange unit in each pathway could be a distributed capillary-tissue exchange operator. The artery, arterioles, venules, and vein can be modeled by the vascular operator shown in Fig. 12. The transport parameters of each pathway can be independent to model heterogeneity in the organ. Figure 13b depicts a serial arrangement that could be appropriate for the intestine. Here the exchange units are dual flow operators like those illustrated in Fig. 8. For this model, it would be useful to make a new operator composed of an artery, vein, and exchange unit; these operators could then be linked serially to create the whole organ model. D. WHOLE-BODY MODELS

The use of differential operators facilitates the construction of wholebody models. The operators can easily be interconnected in the arrangement required for a complex closed-loop system, and the use of a hierarchical structure makes the model structure clearly visible at each level. Creating a whole-body model for a given substrate depends on having a toolkit of operators that are appropriate for the substrate. Considerable care should be taken in the design of the toolkit and in the numerical methods used. Effort spent in this design will be repaid when the compre-

260


a

.

7

I-

*

I

Arteriole

I

Venule

b CGut

in

Gut I - - - - - - -

unit

4

C

Wt

FIG. 13. Two different styles of organ models composed of simpler differential operators. (a) A parallel pathway model fed by a single artery and drained by a single vein. Each pathway has a capillary-tissue exchange unit fed by an arteriole and drained by a venule. (b) A dual flow organ with operators connected in series. The dashed line shows the basic unit of the organ composed of arterial, venous, and capillary-tissue exchange unit operators.

hensive model can be assembled with ease and produces stable, accurate results. The complexity required of the model depends, of course, on the substance being investigated. King ef al. (1993) show a simple whole-body model for an intravascular tracer that is constructed using only the vascular operator shown in Fig. 12.

VI.

SUMMARY

Many substrates cross cell membranes by processes other than passive diffusion. When the transport is carrier-mediated, e.g., facilitated diffusion, active transport, and exchange diffusion, the carrier modifies the conductance of the membrane and may either increase or decrease the flux of the substrate across the membrane. A common characteristic of all camer-


261

mediated transport is its saturability, as only a finite amount of carrier is available to bind with the substrate; even the simplest one-site carrier model exhibits saturation. Inclusion of carrier-mediated transport adds additional model parameters that describe the transporter. In addition, the model must account for both labeled (tracer) and unlabeled (mother) substrate, but this introduces no new parameters. There are many possible models for a membrane carrier. The applicability of these models must be examined for the specific substrate of interest. Many experiments aimed at measuring carrier parameters are carried out on isolated cells or cell fragments. Experiments in intact organs (either in vivo and in v i m ) are also possible. Of particular note is the “bolus sweep” method described by Rickaby et al. (1981) and Malcorps et al. (1984). The increasing sophistication of experimental procedures, data collection techniques, and computers available to investigators continues to extend the depth to which we can probe biological systems. With this increased sophistication comes increased costs in time and equipment. It behooves us then to extract the maximum amount of information from each experimental procedure. Mathematical models assist in doing so, and sophistication in model analysis should parallel that in other phases of the experiment. Increased realism brings several advantages. Simplification of a model to increase its ease of usage and speed in routine data analysis is a desirable goal, and comparing a simplified model against a more realistic model under the conditions specific to a given experiment is one way to test the simplifying assumptions. Additionally, increased model realism can bring new insight into unknown aspects of the system. All models, no matter how realistic, are always “wrong” in that they are less complex than the real system. Failure of the model to explain observed results forces us to further refine the model and teaches us something more about the system. ACKNOWLEDGMENTS This work is supported by NIH Grants RR-01243 and HL-19139.

REFERENCES Bassingthwaighte,J. B. (1974). A concurrent flow model for extraction during transcapillary passage. Circ. Res. 35,483-503. Bassingthwaighte, J. B., and Goresky, C. A. (1984). Modeling in the analysis of solute and water exchange in the microvasculature.I n “Handbook of Physiology” (E. M. Renkin and C. C. Michel, eds.), Sect. 2, Vol. IV, pp. 549-626. Am. Physiol. Soc., Bethesda, MD. Bassingthwaighte, J. B., Wang, C. Y.,and Chan, I. S. (1989a). Blood-tissue exchange via transport and transformation by endothelial cells. Circ. Res. 65, 997-1020.

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Bassingthwaighte,J. B., Chan, I. S., and Wang, C. Y.(1989b).“Blood-tissueExchange Models: BTEX30 and BTEX40(UW/BIOENG-89/1),” Report PB90-5013% FORTRAN Code; PB90-172578 Descriptive Text.” Nat. Tech. Inf. Serv., Springfield, VA. Bassingthwaighte,J. B., Chan, 1. S., and Wang, C. Y.(1992). Computationally efficient algorithms for capillary convection-permeation-diffusionmodels for blood-tissue exchange. Ann. Biomed. Eng. 24,687-125. Berman. M. (1%3). The formulation and testingof models. Ann. N.Y.Acud. Sci. 108,182-194. Goresky, C. A. (1963). A linear method for determining liver sinusoidal and extravascular volumes. Am. 1.PhysioL 204,626-640. King, R. B., Bassingthwaighte, J. B., Hales, J. R. S., and Rowell, L. B. (1985). Stability of heterogeneity of myocardial blood flow in normal awake baboons. Circ. Res. 57,285-295. King, R. B., Deussen, A., Raymond, G. R., and Bassingthwaighte, J. B. (1993). A vascular transport operator. A m J. Physiol. 265, H2196-H2208. Malcorps, C. M., Dawson, C. A., Linehan, J. H., Bronikowski, T. A., Rickaby, D. A., Herman, A . G., and Will, J. A. (1984). Lung serotonin uptake kinetics from indicator-dilution and constant-infusion methods. 1. Appl. Physiol. 57,720-730. Rickaby, D. A., Linehan, J. H., Bronikowski, T. A., and Dawson, C. A. (1981). Kinetics of serotonin uptake in the dog lung. J. Appl. Physiol. 51,405-414. Roth, A. C., and Feigl, E. 0. (1981). Diffusional shunting in the canine myocardium. Circ. Res. 48, 470-480. Sharan, M.. Jones, M. D., Jr., Koehler, R. C., Traystman, R. J., and Popel, A. S. (1989). A compartmental model for oxygen transport in brain minocirculation. Ann. Biomed. Eng. 17, 13-38. Stein, W . D. (1986). “Transport and Diffusion across Cell Membranes.” Academic Press, Orlando, FL. Wong, A. Y. K., and Bassingthwaighte, J. B. (1981). The kinetics of Ca-Na exchange in excitable tissue. Murh. Biosci. 52, 275-310.

Part VI COMPUTATIONAL ASPECTS OF MODELING



Chapter 17 ESTIMATION AND USE OF KINETIC PARAMETER DISTRIBUTIONS IN METABOLISM AND NUTRITION WILLIAM F. BELTZ Depwtmnt of Medicine University of California,San Diego La Jolla, California 92093

1. Introduction 11. Definitions and Theory A. System Model B. Experiments and the Error Model C. Likelihood D. Population Kinetic Analysis 111. Uses for Prior Parameter Distributions A. Bayesian Estimation B. Adaptive Control C. Experimental Design D. Monte Carlo Simulations IV. Applications to Metabolism and Nutrition A. Missing Values B. Sparse Data C. Covariates V. Estimation of Prior Parameter Distributions A. Naive Data Pooling B. Two-Stage Methods C. Population Kinetic Analysis VI. Identifiability Issues VII. Conclusions References

I. INTRODUCTION

In the field of pharmacokinetics, there has been much recent work on developing methods for estimating interindividual variation in kinetic model parameters, particularly in sparse data situations where there are 265 Copyright 0 1996 by Academic Press, Inc. A11 rights of reproduction in any form reserved.

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PART VI COMPUTATIONAL ASPECTS OF MODELING

sometimes fewer observations for each subject than there are parameters in the model. The underlying idea behind these methods is that fewer parameters need to be estimated if they are “population” parameters, which describe the distribution of the model parameters within a population of interest, rather than the kinetic model parameters for each of the individual studies. For example, a two-compartment model has 4 estimable parameters. If studies are performed in 100 subjects, there are 400 individual kinetic parameters to estimate. However, if a multivariate normal distribution is assumed, there are only 14 population parameters: 4 means, 4 variances, and 6 covariances. Several methods for “population kinetic analysis” have been developed. These differ primarily in the assumptions they make about the underlying distribution. Early methods assumed a multivariate normal distribution and estimated the population parameters listed in the example above. Recent work has concentrated on semiparametric and nonparametric approaches which make fewer assumptions and allow for more general distributions. These are particularly useful for detecting multimodality. Although most tracer studies in metabolism and experimental nutrition are designed to allow estimation of the individual kinetic parameters, the population methods are still useful in several situations, such as when only limited data may be obtained from a single animal or when there are missing data. The population methods discussed in this paper avoid making assumptions about missing data by simultaneously considering all the experiments in a rigorous and consistent way. For the interested reader, there are several reviews of the theory and applications to pharmacokinetics and pharmacodynamics (Steimer ef af., 1985; Sheiner and Ludden, lW), but so far, the methods have not been applied to tracer kinetic problems in metabolism and nutrition. The goal of this paper is to provide an introductory review of the theory, applications, and available software, with particular attention to how they relate to problems in metabolism and nutrition. 11.

DEFINITIONS AND THEORY

We consider the situation in which a number of similar (but not necessarily identical) experiments are performed on a number of different subjects randomly selected from a larger population of individuals. Examples include a metabolic study in which isotopically labeled lipoprotein turnover is measured in a number of patients with heart disease, a nutritional study in which vitamin turnovers are examined in some randomly selected graduate students, or an agricultural study in which calcium kinetics are studied in a few dairy cattle randomly selected from a school’s experimental herd. In

CHAPTER 17 KINETIC PARAMETER DISTRIBUTIONS

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the first example, the studied patients are assumed to be representative of all heart disease patients. In the second example, the graduate students are assumed to be representative of the public at large. In the agricultural study, the experimental cattle are assumed to be representative of dairy cattle in general. In each experiment, the goal is to make some inference about the larger population. Although it might appear that the most frequent goal of mathematical modeling is to estimate the parameters for the experimental subjects, this is usually being done as the first step toward estimating the population parameters. A. SYSTEM MODEL

The data obtained from the experiments are to be evaluated in terms of a mathematical model. Although this “system model” is only a hypothesis about how the system under investigation is thought to function, it must be mathematically well-defined so that it may be used to calculate quantitative model predictions. The system model will frequently take the form of a compartment model (Jacquez, 1985), but the ideas discussed here are applicable to other types of models as well. A very important aspect of the model is its parameters. We define the model response as q(P,t,x), where p is a vector of system parameters, t is time, x is a vector of independent variables (other than time t ) , and the details of the model are embedded in the function q. For compartmental models, the parameters are usually the intercompartmental fractional transfer coefficients and perhaps a volume of distribution. The parameters of the system model are unknown constants that may take different values for each of the experiments. Since the experimental subjects are randomly selected from the population, the model parameters are random variables and a multivariate probability distribution exists for them. We will use h( B,p,x) to denote the probability density for the system parameters, where B is a vector of “population parameters,” and x is again a vector of independent variables. The population parameters define the probability distribution h for the values of the system parameters. If h is a multivariate normal distribution, then the population parameters are the means, variances, and covariances of the distribution. We can now write the conditional probability for the values of the system parameters /3 given values for the population parameters Band the independent variable x as p(pleJ) = h(e,p,.+ As mentioned above, the system parameters are unknown and constant. One of the primary goals of modeling is to estimate the system parameters. The fact that they are unknown distinguishes them from independent variables or covariates. Note that by this definition, a variable in a model equation may be a parameter in one analysis and an independent variable

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in another. In one study for example, a volume of distribution may be estimated while in an otherwise similar study it may be known from independent measurements. The fact that the parameters must be constant will sometimes require reparameterization of a model. For example, consider a compartmental system in which the mass in the central compartment C is changing with time (nonsteady state) and the elimination rate from this compartment, R,(t), is nonlinear. Specifically, let the elimination rate follow Michaelis-Menten kinetics and be &(t) = VmaxQc(r)/(Qc(r) + &,). Now the fractional transfer coefficient is Vm,/(Qc(t) + K,) and the system parameters are V,, and K,. The distribution estimated by population analysis is often called the “prior” distribution because it is known (or assumed) before an experiment is performed. After an experiment, the new results may be incorporated to estimate a new distribution, the “posterior” distribution, which then becomes the prior for the next experiment!

B. EXPERIMENTS AND THE ERROR MODEL We now consider a series of experiments performed on the model and the observations obtained during those experiments. The statistical model is the usual one

where y,, is the ith observation for the jth experiment, q(@j,tij,xj)is the model-predicted value for yV, and eii is the residual error for this observation. The values for yij, tij, and xi are known and values for pi are to be estimated. The residual errors are random variables and we assume they are distributed according to a probability distribution of known form and quantified by perhaps unknown parameter values. This distribution comprises the “error model” and its parameters are the “error model parameters.” The usual assumption is that the errors are normally distributed with mean 0 and variance sit. In general, however, other forms may be used.

C. LIKELIHOOD

To estimate the system parameters /3, we need a quantity that measures how well our model fits the data. This is provided by the likelihood function. For experiment j , the likelihood I is defined as the probability of obtaining the actual observations, given the model, the model parameters, and the known independent variables. In this form, the likelihood is a function of the system parameters


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This likelihood for the jth experiment may be calculated by multiplying the probabilities for the individual observations

where the individual probabilities can be calculated from the error model. The model parameters may be estimated by finding the values that maximize this function. If the errors are normally distributed, maximizing this likelihood function is equivalent to minimizing the weighted least-squares function.

D. POPULATION KINETIC ANALYSIS To make sense of sparse, “routine” pharmacokinetic data, Sheiner et al. (1977) recognized that fewer parameters need to be estimated if one estimates the distribution of parameters rather than the individual parameter values for each experiment. The fundamental idea behind the population methods is that if one knows the parameter distribution h( 8), then a likelihood L for an experiment may be calculated which does not depend on actual parameter estimates for the experiment

This likelihood is calculated by weighting the parameter-dependent likelihood for each parameter value by the probability of the system parameters taking on that value. This likelihood is no longer a function of the system parameters but is a function of the population parameters. A likelihood for a several experiments (L(0))may be calculated as a product of the likelihoods for the individual experiments. The job of the various population kinetic analysis algorithms is to estimate h(P,6)by finding the 8 that maximizes L( 8). Ill. USES FOR PRIOR PARAMETER DISTRIBUTIONS

A. BAYESIAN ESTIMATION Bayesian estimation is used to estimate the model parameters, which would otherwise be unidentifiable, by taking into account the prior distri-

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bution and using this information to determine the most probable value for each parameter. The estimation is performed by finding values for system parameters for the jth experiment (pi) that maximize P( y j ( P j j j ) X h ( ~ j , O ~where j ) , y j is the vector of observations for the jth experiment and xi is the vector of independent variables for the jth experiment. In a datarich situation, P( y j ( P j j j )dominates this calculation while in a data-poor situation, h(Pj,O j j ) dominates.

B. ADAPTIVE CONTROL A frequent use of the prior parameter distribution in pharmacokinetics is for adaptive control (Schumitzky, 1986). The idea is to provide patients with individualized dosing to reach some therapeutic goal. Frequently, the goal is to maintain the blood level of a drug within some range but many other goals are possible. When a patient is treated for the first time, the appropriate kinetic parameters are unknown. The most likely parameter value for the subject may be determined from the prior distribution and the appropriate dosage estimated and applied. The “adaptive” part of the method occurs after as few as one drug level has been observed for this subject. New parameter estimates may then be obtained by Bayesian estimation and the dosage adjusted as needed. At the start, dosing is based on the best estimates from previous subjects but as therapy progresses, treatment is based more on estimates of the subject’s own parameter values.

C. EXPERIMENTAL DESIGN Experimental design is the optimization of experimental “controls” so as to maximize the likelihood of achieving a desired result. There are several goals that a researcher might wish to achieve, including minimizing the uncertainties for the estimated parameters, maximizing the ability to control specific responses of the system, and maximizing the ability to discriminate between two or more models. For any of these goals, accounting for parameter variability will provide better results or at least make the researcher aware of the range of results that might be expected. D. MONTE CARL0 SIMULATIONS

By definition, a Monte Carlo analysis is the generation of several randomly generated simulations, usually with a digital computer. There are several reasons for doing this, including evaluating statistical procedures, estimating the power of a statistical test, and verifying predictions made by other means. Clearly, to produce a set of random experiments, some


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aspect of each experiment must be drawn from a probability distribution. If an estimate for the parameter distribution within the population is available, then this can be done easily. Parameter values are selected randomly from the prior distributions for each simulated experiment. The model response is then determined and random error added to the predicted values consistent with the error model. These “observed” data may then be fit using the same or a different model, depending on the goal of the analysis, After many simulations, the distribution of results (parameter estimates, response variable values, or both) may be evaluated in light of the “known” true system. IV. APPLICATIONS TO METABOLISM AND NUTRITION

Although the mathematics is the same, kinetic modeling in metabolism and experimental nutrition differs from pharmacokinetics in at least two ways. First, while the modern methods of population kinetic analysis were developed specifically for the sparse data problem, in metabolism and nutrition we usually have adequate data for each experiment. Second, while pharmacokinetic analysis tends to be control-oriented, the ultimate goal being optimal dosing, metabolic modeling tends to be structure-oriented, the goal being to determine the true structural model. Nevertheless, as we will see, population kinetic analysis can be appropriate in some relatively common situations. A. MISSING VALUES

In the metabolic world, we are not usually dealing with the sparse data encountered by pharmacokineticists. We usually have the two-stage methods (see below) in the backs of our minds and design turnover studies that allow good identification of all parameter values for each experiment. Nevertheless, even the best planned studies sometimes go afoul. Consider the situation where the protocol calls for a final blood sample at the very end of a study. This observation might be crucial for accurately estimating the terminal slope of the plasma curve. If this sample is missed or lost, it can render the model poorly identifiable. What is frequently done in practice is that some assumption is made about the parameter values for this individual. For our example, this might be the slope of the terminal slope or the relative size of a peripheral compartment or exchange pool. The assumed value is based upon the other experiments. The strategy is therefore to model all the experiments with complete data, fix a parameter or parameters in the incomplete experiment to the mean of those obtained for the other

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experiments, and identify the remaining parameters. Although this is a common and useful procedure, there are some problems with it. First, the effect of the assumption is seldom considered, let alone examined in any detail. Second, it does not consider the uncertainty in the assumed value. Since the value of this parameter is fixed, other parameters for this experiment will often have lower coefficients of variation (fractional standard deviations) than the same parameters from experiments without the constraint. Thus values from this experiment appear to be identified with more certainty, when in fact they should be considered more suspect since they are based upon an additional assumption. Finally, if another experiment has missing data for a different part of the curve and a different parameter is to be assigned an assumed value, one can get into a loop where results from experiment A are used to fix a parameter for experiment B and results from experiment B are used to fix a parameter for experiment A. Although this procedure will probably converge, this is exactly the type of situation for which the population analysis methods were developed. In fact, they were created to specifically handle the extreme case, that in which all experiments have missing values!

B. SPARSE DATA There are good examples from the metabolic literature of studies in which the number of data observed for a single subject is limited. An example is measuring the mean residence time of low density lipoprotein in the rabbit aortic wall (Schwenke and Carew, 1989). In experiments such as these, samples of the aorta may only be obtained once, at the end of the experiment. Thus there is only one datum for each tracer used. Schwenke and Carew (1989) used two iodine tracers, administered at different times, but the compartmental model they used has four parameters. Clearly, parameter values cannot be estimated for each animal without using information from experiments in other animals.

C. COVARIATES The x in h ( P , O j ) is sometimes called a “covariate” (Wade et al., 1994) and the estimation of covariate effects is one of important goals in population kinetic analysis (Sheiner and Ludden, 1992). In pharmacokinetics, a common goal of covariate analysis is to get reliable predictors of drug clearance. The value should be obvious. If one knows how the residence time of a drug in the body depends on age, gender, body weight, renal function, and so on, then one has a good starting point from which to


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individualize dosing. In metabolism and nutrition, one might similarly be interested in how absorption rates, fractional catabolic rates, and other system parameters depend not only on age, gender, and body weight, but on dietary status, genotype, and even species. Since the covariate acts directly on the parameter distribution, the two must be estimated simultaneously. This has the additional advantage that all sources of variation are included in the estimation procedure. V.

ESTIMATION OF PRIOR PARAMETER DISTRIBUTIONS

A. NAIVE DATA POOLING In naive data pooling, observations from different subjects are pooled and treated as though they were obtained from a single experiment. This technique is fairly common but as Steimer et al. (1985) point out, the results obtained can be misleading. Averaging of data can either average out phenomena in the data that may be of interest or can suggest complexity that is really due to interindividual variation. No special software is required for naive data pooling. Any of several spreadsheets or statistical packages may be used to average the data and any of several nonlinear regression programs may be used to model the mean values. B. 'TWO-STAGE METHODS In the so-called two-stage methods, each experiment is first modeled individually to obtain parameter estimates and these estimates are then used to make inferences about the population distribution. There are two recognized two-stage methods. The standard two-stage method (STS) ignores uncertainties in the individual estimates while the global two-stage method takes them into account. 1. Standard Two-Stage Method

This is perhaps the most common method of estimating a parameter distribution, though some researchers may not be aware that this is what they are doing. It refers to modeling each experiment individually to obtain point estimates, then treating these estimates as error-free and performing standard statistical analyses on them. It does not consider that there are uncertainties in the modeling results. In particular, it doesn't allow for differences between experiments in their relative impact on the results. In the STS method, covariate effects are determined by regression and

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correlation analysis on the point estimates. No special software is required for the standard two-stage method. Any of several nonlinear regression programs may be used to model the individual experiments and any of even more statistical programs may be used to manipulate the point estimates obtained. 2.

Global Two-Stage Methods

The global two-stage method attempts to account for the uncertainty in the individual parameter estimates and considers correlations among the various model parameters. In essence, it bases the population mean on a weighted average of the individual values. The most readily available software for a global two-stage analysis is the Extended Multiple Studies Analysis (EMSA), available in the SAAMKONSAM package (Berman and Weiss, 1978). Details of EMSA are provided by Lyne et al. (1992). EMSA assumes that the distribution is normal and, as a result, EMSA will not detect multimodal distributions. As with other two-stage analyses, each experiment is first studied individually. During this stage, SAAM writes the estimated parameter values, the estimated standard deviations for the parameter values, and the parameter correlation matrix to a file. The results for the several experiments are collated into a single file which serves as input for the second step. During this stage, means and variances for the population distribution are estimated and printed. Since the parameter values and correlations for the first stage are generated by SAAM, a wide variety of compartmental and algebraic models are available for analysis. SAAM is distributed in executable binary form and is available for VAX/ VMS, Unix, PCIDOS, and Macintosh. It is available from Loren Zech at the National Institutes of Health (Building 10, Room 6B-13, National lnstitutes of Health, Bethesda, MD 20892). C. POPULATION KINETIC ANALYSIS The population analysis methods use all the available data to estimate the population. The best estimates for the parameters of an individual study are only obtained after the population distribution has been estimated by Bayesian estimation. Essentially, the various methods estimate the population parameters 0 in h(P,O,x).The methods differ primarily in the form that h ( p . 0 ~is) assumed to have. Despite the fact that all arrive at a quantitative description of h(P,@,x).the different forms have been divided into parametric, semiparametric, and nonparametric. Each of these will be described.


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1. Parametric Methods

The parametric methods assume that h(P,0;s) is multivariate normal. The population parameters (0) therefore correspond to means, variances, and covariances. The most used software package for population analysis is NONMEM (NONMEM Project Group, 1992), which stands for “nonlinear mixed effects models” and uses a parametric method. The theory behind NONMEM is described by Beal and Sheiner (1982). NONMEM is distributed as Fortran code and will therefore run on a wide variety of computers though the user must be prepared to deal with editing the code and compiling the program. Although NONMEM includes a library of the most common pharmacokinetic models, for any but the simplest of metabolic models, the user will probably have to write a subroutine to calculate the model response. NONMEM is available from the NONMEM Project Group at the University of California, San Francisco (mail stop C25.5, San Francisco, CA 94143).

2. Semiparametric Methods One limitation to the parametric approach is that any multimodal distribution will be approximated by the unimodal normal curve and an important characteristic of the population could be overlooked. Several alternative forms of h have therefore been suggested. The class of so-called semiparametric methods allows for forms of h(p,0,x) more general than the normal distribution but still places some limitations on the structure of this function. The only readily available program for semiparametric analysis appears to be NLMIX (Davidian and Gallant, 1992), which also stands for “nonlinear mixed effects models.” The method used is described as smooth nonparametric estimation (Davidian and Gallant, 1992).The class of density distributions actually allowed by the method essentially consists of a multivariate normal distribution multiplied by a polynomial, allowing multimodal, fat-tailed, and skewed densities, none of which are allowed by the parametric methods. The population parameters ( 8 ) correspond to the means, variances, and covariances of the normal distribution, as well as the coefficients of the polynomial. Like NONMEM, NLMIX is distributed as Fortran code. To use the package, the user must be prepared to do some Fortran programming, more than with NONMEM. The user must again supply a subroutine to calculate the model response, but there is no model library included with NLMIX. The source code and documentation are available by anonymous ftp from the StatLib statistical software collection at Carnegie-Mellon University. The easiest access is to make a world wide

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web (www) connection to 1ib.stat.cmu.edu and look in the “general” archive. 3. Nonparametric Methods

The nonparametric methods (Mallet et al., 1988) make no assumptions about the form of h(&0;s). The estimated probability density function takes the form of a finite number of allowable parameter vectors, the “points of support,” and a probability assigned to each point of support. Although there are an unlimited number of parameter values the estimation procedure can choose from, the final result is really a discrete multivariate probability distribution. The population parameters (0) correspond to the parameter values at the points of support and the probability assigned to each. Thus the “nonparametric” distribution is quantitative and described by a finite number of variables. After estimation, any of various smoothing techniques can be used to make the function more presentable, but calculations using the distribution are based on the unsmoothed form. Calculations can actually be faster than with other distributions because rather than integrating over a continuous function, you only need to sum over the points of support. The most readily available nonparametric software is in the USC*PACK collection of pharmacokinetic computer programs for population analysis, Bayesian estimation, and individualized dosing (Jelliffe, 1991). The rationale for the package as a whole is described in detail by Jelliffe (1986). included in the package is the NPEM2 program, which performs nonparametric population analysis on a three-compartment model using the EM algorithm (Schumitzky, 1991). The USC*PACK programs are distributed in executable binary form for PC/DOS. The programs are available from the Laboratory of Applied Pharmacokinetics at the University of Southern California (USC School of Medicine, CSC 134-B, 2250 Alcazar Street, Los Angeles, CA 90033). VI.

IDENTlFlABlLlTY ISSUES

Unfortunately, there has been minimal work on identifiability issues with respect to population kinetic analysis. The current work on identifiability of kinetic models (Jacquez, 1985; Cobelli and DiStefano, 1980; Cobelli and Saccomani, 1990) focuses on estimation for an individual experiment. With regard to population analysis and Bayesian estimation of parameters for an individual from the population, it is the population analysis step for which identifiability issues need to be considered. Once a prior distribution


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is available, Bayesian estimates can be calculated for any individual, even if there are no data from that subject. In this case, the values will be the modes of the prior and identifiability is not a problem. It is the estimation of the first prior that could be troublesome. It is also this estimation that the various procedures discussed in this paper attempt to do. Despite the lack of published theory, a couple reasonable points can be made about the identifiability of the multivariate parameter distribution. First, the system model must be identifiable for a single data-rich experiment. This means that if all the data were presumed to be from a single experiment, then the parameters for that experiment must be uniquely identifiable. Second, there must be some “information” about each parameter contained within the data set. This means that all the data together must cover the range required for an individual experiment to be identifiable. Inclusion of the covariates in the estimation step introduces some additional complications. In a recent report, Wade et ai. (1994) demonstrated with simulated data that if significant covariates were not included in the analysis, then what was in reality a monoexponential response could be interpreted as a multiexponential response. Variation between experiments was interpreted as variation within the structural model. In a sense, this is an example of using the wrong model. Nevertheless, it demonstrates some of the dangers in population analysis of questionably identifiable models and that the presence of the covariates in the population model makes the situation more complicated from an identifiability standpoint. In metabolism and nutrition, where each experiment has been designed to be complete and population analysis is used to fill in missing values and to incorporate relative uncertainties into the estimation, a good procedure would be to first examine in detail those individual studies which are the most complete. This will familiarize the user with the behavior of the model, produce initial estimates for the system parameters, provide a chance to verify that these values are reasonable, and allow the use of tools for the identifiability of individual experiments (Jacquez and Perry, 1990). After this exercise has been complete, all experiments, including those that are incomplete, can be pooled for population analysis and testing the effects of covariates. If required, the final step would be to use the estimated distributions to obtain Bayesian parameter estimates for the individual experiments. This procedure should yield the most appropriate estimates for the incomplete experiments. Finally, it must be pointed out that it is ultimately the responsibility of the user to be aware of identifiability issues. The ability of a computer program to provide parameter estimates should not always be interpreted as proving that the model is identifiable (Cobelli and Saccomani, 1990).

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VII.

CONCLUSIONS

The population estimation methods developed by pharmacokineticists to handle sparse data have potential use in the relatively data-rich studies encountered in metabolism and experimental nutrition. They provide a consistent and logical method to combine information from several experiments, accounting for interindividual variation in a theoretically sound manner. Anyone working in kinetic analysis should be aware of these tools, their advantages, and their limitations.

REFERENCES Beal, S. L., and Sheiner, L. B. (1982). Estimating population kinetics. CRC Crit. Rev. Biomed. Eng. 9, 195-222. Berman, M., and Weiss, M. F. (1978). “SAAM Manual,” DHEW Publ. No. (NIH) 78-180 U.S. Govt. Printing Office,Washington, DC. Cobelli, C., and DiStefano, J. J., 111 (1980). Parameter and structural identifiability concepts: A critical review and analysis. Am. J. Physiol. 239, R7-R24. Cobelii, C., and Saccomani, M. P. (1990). Unappreciation of a priori identifiability in software packages causes ambiguities in numerical estimates. Am. J . Physiol. 258, E1058-El059. Davidian, M., and Gallant, A. R.(1992). Smooth nonparametric maximum likelihood estimation for population pharmacokinetics, with application to quinidine. J. Pharmacokinet. Biopharm. 20,529-556. Jacquez, J. A. (1985). “Compartmental Analysis in Biology and Medicine.” Univ. of Michigan Press, Ann Arbor. Jacquez, J. A., and Perry, T. (1990). Parameter estimation: Local identifiability of parameters. Am. J. Physiol. 258, E l U - E l 3 6 Jelliffe, R. W. (1986). Clinical applications of pharmacokinetics and control theory: Planning, monitoring, and adjusting dosage regimens of aminoglycosides, lidocaine, digitoxin, and digoxin. In “Topics in Clinical Pharmacology and Therapeutics” (R. F. Maronde, ed.), pp. 26-82. Springer-Verlag, New York. Jelliffe, R. W. (1991). The USC*PACK PC programs for population pharmacokinetic modeling, modeling of large kinetiddynamic systems, and adaptive control of dosage regimens. I n “Fifteenth Annual Symposium on Computer Applications in Medical Care” (P. D. Clayton, ed.), pp. 922-924. McGraw-Hill, New York. Lyne, A., Boston, R., Pettigrew, K., and Zech, L. (1992). EMSA: A SAAM service for the estimation of population parameters based on model fits to identically replicated experiments. Comput. Methods Programs Biomed. 38,117-151. Mallet, A., MentrC, F., Steimer, J.-L., and Lokiec. F. (1988). Nonparametric maximum likelihood estimation for population pharmacokinetics, with application to cyclosporine. 1. Pharmacokinet. Bwpharm. 1 6 3 1 1-327. NONMEM Project Group (1992). “NONMEM Users Guides.” University of California, San Francisco. Schumitzky, A. (1986). Stochastic control of pharmacokinetic systems. In “Topics in Clinical Pharmacology and Therapeutics” (R. F. Maronde, ed.), pp. 13-25, Springer-Verlag, New York.


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Schumitzky, A. (1991). Nonparametric EM algorithmsfor estimatingprior distributions.Appl. Math. Comput. 45,143-157. Schwenke, D. C., and Carew, T. E. (1989). Initiation of atherosclerotic lesions in cholesterolfed rabbits. 11. Selective retention of LDL vs. selective increases in LDL permeability in susceptible sites of arteries. Arteriosclerosis (Dallas) 9,908-918. Sheiner, L. B., and Ludden, T. M. (1992). Population pharmacokineticddynamics.Annu. Rev. PharmacoL Toxicol. 32,185-209. Sheiner, L. B., Rosenberg, B., and Marathe, V. V. (1977). Estimation of population characteristics of pharmacokinetic parameters from routine clinical data. J. Pharmacokinet. Biop h a m . 5,445-479. Steimer, J.-L., Mallet, A., and Mentrd, F. (1985). Estimating interindividual pharmacokinetic variability. In “Variability in Drug Therapy: Description, Estimation, and Control” (M. Rowland, L. B. Sheiner, and J.-L. Steimer, eds.), pp. 65-111, Raven Press, New York. Wade, J. R., Beal, S. L., and Sambol, N. C. (1994). Interaction between structural, statistical, and covariate models in population pharmacokinetics analysis. J. Pharmcokinet. Biopharm 22,165-177.



Chapter 18 ESSENTIAL NUMERICAL SUPPORTS FOR KINETIC MODELING SOFTWARE: LINEAR INTEGRATORS R. C. BOSTON AND T. MCNABB Clinical Studies, NBC, School of Veterinary Medicine University of Pennsylvania Kennett Square, Pennsylvania 19348

P. C. GREIF AND L. A. ZECH Laboratory of Mathematical Biology National Institute of Health Bethesda, Maryland 20892

I. Introduction 11. Linear Systems of Differential Equations

Ill. Modeling Software and Linear Integrators A. The Chu Berman Constant Coefficient Differential Equation Solver B. Evaluation of the Linear Integrators IV. Conclusion References

I.

INTRODUCTION

Computers and computer software have had a major impact on the capacity of investigators to probe systems. The power and sophistication of computers have meant that, on the one hand, very complex tasks, such as the population driven analysis of multiple experimental studies (Lyne et af., 1992, Berman and Weiss, 1978) have become possible, and, on the other hand, the effort needed to accomplish tasks, e.g., the manipulation of models and their structural organizations, has been drastically reduced. In developing and refining computer software for a particular (or target) community, care needs to be taken to ensure that the gamut of needs of the community are met. For the modeling community, whose goals seem 281 Copyright 0 1996 by Academic Press, Inc. Ail rights of reproduction in any form reSeNed.

282


primarily linked to understanding and explaining aspects of systems (domains of investigation), this essentially amounts to the provision of a framework to expedite model specification and appraisal. Here specification embraces the incorporation of knowledge, conjecture, information, and data into a schema suitable for computer-supported probing and appraisal involves the evaluation of the conjecture against the information base. We can identify four classes of support included in modeling software fabricating this framework, viz., 0 0

Modeling constructs Processing support Manipulation facilities Visual guides.

Modeling constructs are the elements by which models are constructed, and as such they need to have features which render them easily conceptualized as representing entities from the (or any reasonable) domain of investigation. In addition they are of course underpinned by intuitively obvious mathematical structures and require a straight forward set of rules to which they are susceptible in terms of their manipulation. The key is that their functional and abstract senses must be consistent, intuitive, and yet flexible. Examples of (diverse) modeling constructs can be drawn from the SAAM repertoire (see Table I). Processing support involves, for example, invocation of the machinery underpinning the modeling constructs. Models, as far as the user is concerned, are primarily graphic and lexical entities manipulated in a style which logically advances the investigation. However, to advance the investigation implies access to translation machinery to translate these superficial structures to codified mathematical forms, access to solving machinery to provide requested model predictions, and access to other types of processing machinery as well. The collective responsibility of processing support is to

TABLE I SAAM’s

MODELING CONSTRUCT WU\MPLE FROM

‘

L

~

(OR ~MODELING ~

~

CONSFRUCT LIBRARY)

Construct

Purpose

Conceptual mapping

L(i,j) M(i)

Identifies rate, mechanism, and transfer sense Steady-state pool size Differential equation solution Resetting function

Uptake mechanism Tissue space Perturbation response Experimental perturbation

F(i) QO( j )

~

~

CHAPTER 18 LINEAR INTEGRATORS

283

facilitate the integrated analysis of models. The Consam (Boston etal., 1981) modeling software provides the following (examples of) processing support. 0 0 0

0

-

Model Model solving Parameter refinement Batch processing

DECK SOLVe ITERate SAAM

command command command command

The manipulation facilities of a modeling environment comprise the collection of resources that expedite user interaction with the model. Here model entry, model modification, and model retention and retrieval are examples of what we have in mind. In addition, provision of support to compare the utility and viability of alternate structural organizations of a model is needed. From the user interface perspective this aspect of modeling software development is extremely challenging on the one hand, easy and efficient access to model manipulation needs to be provided, and, on the other hand, a layer of protection is required to ensure that each modeling step the user takes is not in conflict with either the software rules or past modeling steps. Finally, critical cues, as visual guides, need to be provided to enable the modeler to appraise progress. For example, the adequacy of data to facilitate identification of a particular parameter or the flexibility of a model to mimic temporal patterns manifest in the data may be issues at the back of the investigators mind as the modeling episode advances. Modeling software must anticipate these and address them in terms that the user is comfortable with as opposed to terms which a scientific programmer might respond to. In this note we discuss one of the major processing support issues arising in conjunction with the development of modeling software, solving linear systems of differential equations. We start by defining linear systems and describe how they arise in the investigation of biological systems and then move on to relate how they are solved. A key concern of this article is how efficient are the various approaches available for solving linear systems.

II. LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

A system of ordinary (as opposed to partial) differential equations is linear if the coefficients of the included derivatives are constant, viz., N

a$' =h(t ).

i=l

284


If N is equal to 1 the system is first order and if h ( t ) = 0, the system is homogeneous. This article will deal with first-order linear homogeneous systems sometimes referred to as initial value problems, viz., Y = LY,

(2)

where

and

The mathematical description of biological processes are rarely representable in terms of systems of the form Eq. (2). The reasons for this are implicit in the form of Eq. (2). First Y in Eq. (2) is self-regulating, yet a biological process would have its own (if any) in-built demand for Y. Second, Y in Eq. (2) transports itself, yet most biological processes have highly refined carriers to meet their needs in conjunction with metabolic deficiencies (see Fig. 1).

1

Resorption Signal (+)

FIG. 1 . A typical biological system including metabolic feedback signals.


285

The question then arises why study first order linear systems (of the form of Eq. (2)) and facilitate their incorporation into modeling software when their relevance seems at best questionable. To answer this we need to note the following. 0 The most useful characterization of a biological system may not impose a full dynamic representation of the system but rather a detailed account of the system in a particular metabolic state. For example the characterization of a disease may be most usefully (for example, from the diagnostic perspective) represented in terms of a detailed exposure of its salient aspects at a particular metabolic state of the host 0 The response of a nonlinear system at steady state (see above) to a small perturbation is linear, i.e., describable as a set of first order linear differential equations (of the form Eq. (2)). Furthermore, kinetic models represented by such a set of linear differential equations contain many helpful pieces of information concerning the host nonlinear system, viz., the number of exchanging (metabolic) pools, the rate of exchange among the (metabolic) pools, and the size of the (metabolic) pools. If the nonlinear system represented as

Y‘

=

h(Y) + U

(5)

is perturbed by amount Y*, then the time profile of Y* will be given by y‘* = h‘ (Y)Y*,

(6)

where h(.) is some (nonlinear) function, h ’ ( ) is the Jacobian of h and I/ is a steady input to Y. In Fig. 2A we show that the response of a nonlinear system (of the form Eq. (5)) in steady state to a small perturbation is linear (with parameter predictable from Eq. (6)). In Fig. 2B we demonstrate the effect of non steady state on the response. In Fig. 2C we see the effect of an excessively large perturbation on the response. Further, in Fig. 2D we see the effect of the degree of nonlinearity of the system on the perturbation response. For small systems involving three or fewer components (or exchanging pools in the instance of kinetic models) the solution to Eq. (2) can be mapped analytically from its exponential form

Y = Aexp-”‘ to its “kinetic” model form

(7)

286


1

.

-

I

-

1

.

-

-

-

1

-

.

-

1

Theoretfcd mode( Y' = -2,OY

*: --. -..

*.

.--.-..--..

-.-'%.-.-.

--.. --... simulation

-....-...

... -... a-.

1

I

I

I

5.2

5.4

5.6

I

5.8

I

j

6.0

TIME FIG. 2. ( A ) The response of a nonlinear system at steady state to a small perturbation. The theoretical and simulated responses are presented. (B) The difference in the responses of a nonlinear system to a small perturbation when it is at steady state versus when it is not at steady state. (C) The response of a nonlinear system at steady state to a large perturbation. (D) The effect of the degree of nonlinearity on the response of a nonlinear system at steady state to a small perturbation. Predicted plateaus and slopes are presented in parentheses.


-

B

1

I

I

287

'

I

I Y'= -0.01 Y 2 Y(O)=O

-

1

UF=lO

$ C

8. rn

a" 8 L

'0;

-.-C C

0

Z

1

I

0

2

.

.

I

.

4

.

.

l

.

.

.

I

6

8

6

8

TIME

TIME FIG. 2. Continued

288


TIME

%C

:0:1.00; 2 .-s lu 4I2 0.10-

Nonilnear resporrse due

c.

n"

0.01#

...--.

I

I . .

.

- 1

. . . . . .I

.

*

...I

289


.................................................................

f

:.'

P=3

P-5

P=9

(0.465)

(0.631)

(0.774)

0.11

-

Y' = -1O.OY YO) 0 W-1

0.0

1

2

P

.

.

.

1

.

.

4

TIME

TIME FIG. 2. Continued

.

1

.

6

.

I

8

290


where, of course, some, or possibly all, of the Lij in Eq. (2) may be negative. It will nevertheless be found most efficient to solve Eq. (2) numerically for the required solution points (time points for which solutions are required) using numerical integrators commonly supported in modeling software. 111.

MODELING SOFIWARE AND LINEAR INTEGRATORS

Since we are dealing with first order linear homogeneous systems the computational complexity associated with solving these systems can be considered in terms of the L matrix, though the number of solutions and the integration domain will be secondary issues as well. There are from our perspective four classes of problems of particular interest in the linear homogeneous set: stiff problems (in which the diagonal elements of the coefficient matrix vary over a large range), problems with oscillating solutions, problems whose coeffkient matrix is diagonally dominant (in which the diagonal elements of the coefficient matrix are all considerably larger that the surrounding elements), and other problems. Modeling software not only needs to provide a battery of numerical integrators to address these four classes of problems but also should have the potential to guide the user in terms of matching the integrator used for a problem with the problem characteristics. For our comparative evaluation of SAAM’s Chu Berman (Chu and Berman, 1974) integrator (also referred to as model code 10 in SAAM and CONSAM) we will be considering as alternate integrators the Runge Kutta 412 integrator (Press er al., 1987), the Butirsch Stoer integrator (Press ef al., 1987), and Petzold’s DASSL integrator (Petzold, 1983). Our selection of this group of integrators was driven by the following considerations. Runge Kutta (RK) is currently available in the SAAM software, is a tried and proven procedure, and is known to perform at least competitively for nonstiff systems. Bulirsch Stoer (BS) is gaining considerable appeal as a standard, or default, integrator in all but stiff systems with excellent accuracy reports (Acton, 1970, Kahaner et al., 1989). Petzold’s DASSL (Petzold) reflects the state-of-the-art in dealing numerically with stiff differential algebraic systems and certainly warrants competitive evaluation against the foregoing. Table I1 shows the computer language, compiler, and arithmetic precision used in conjunction with generation of executable versions of these integrators. A. THE CHU BERMAN CONSTANT COEFFICIENT DIFFERENTIAL EQUATION SOLVER The Chu Berman constant coefficient differential equation solver is a single step integrator which operates on the same principle as the accuracy


291

TABLE I1 COMPILERS USEDa

Integrator

Language

Compiler

SAAM 31'

Fortran C C Fortran C Fortran

Watcom Watcom Watcom Watcom Borland Watcom

CBCCDS' R K ~ Petzold' XCJ BSg

Precision

Size

SP DP DP DP

32 64 64

Long double DP

80 64

64

Details of compilers used to generate machine images of the code for the four integrators. SP denotes single precision, DP denotes double precision. Size is the number of binary digits used for numerical coding. All compilation and processing was performed on a Gateway 2000 486DX2-50computer. The Chu-Berman numerical integrator in the code of SAAM31. CBCCDS denotes a code C-coded version of the Chu Berman constant coefficientdifferential solver. RK denotes the variable step size fourth order Runga Kutta integrator. Petzold denotes Petzold's DASSL. XC denotes values obtained from analytical solutions. 8 BS denotes Bulirsch Stoer integrator.

'

controlled version of the Runge Kutta integrator (see also Rice, 1983). Figure 3A presents an overview of its step-by-step processing flow and the critical boxes (numbered processing steps in Fig. 3A) are exploded in Figs. 3B and 3C to expose the necessary detail. The goal of the integrator is to provide acceptably accurate solutions to the differential equations at the user nomin;;ted (solution) points (Fig. 4A) with the greatest possible computational efficiency. The process starts with the determination of an integration step (box 2 Fig. 3B) (which may fall short of the required solution point, Fig. 4B) and the estimation of a differential equation solution coinciding with this step (box 3 Fig. 3B). To derive the solution the integrator uses a deconvolution technique in estimating the contribution of the linear term in the Taylor series about the dominant diagonal of the coefficient matrix. We see that the diagonal terms in fact translate to component exponentials in the first stage of deriving the solution. Next the error of the solution is estimated using the quadratic term in the Taylor series expansion (box 4 Fig. 3C). If the error is unacceptably large, the step is halved and the entire process is repeated for this new step (box 5 Fig. 3C and Fig. 4C). If the error is tolerable the step is accepted and a new step proposed.

292


w h more ~ sdutions needed

Else

Initialize integration

w not sotution point calculated Use Do

Catadate integration point estimat?

-

Advance integration point I

[-A&'

Advance solution points

FIG. 3. (A) An overview of the operation of the Chu-Berman constant coefficient solver (CBCCS). (B) Exploded details of boxes 1.2, and 3 of A showing the determination of the minimum step size, the determination of the actual step size, and the determination of the first de solution estimate for the integration point. (C) Exploded details of boxes 4,5, and 6 of A showing the solution acceptance process, the step reduction process, and the step expansion process.

The integrator uses a number of intelligent features to improve its performance. 0 If a proposed integration step is within 30% of the last step, the step taken is one of the size of the last step (Fig. 4D). 0 If the accuracy warrants it, the step is expanded, thus potentially reducing the number of steps per solution point (box 6 Fig. 3C). 0 If an integration step would take us to within 30% of a solution point, a step size necessary to hit the solution point is taken (Fig. 4E). 0 Decayed components of stiff systems cease to influence step size calculations once their magnitude falls below some acceptably small value.


B

293

i Minimum Step

MinStep =

StepFactor

my+/) 2

SfepSize(h)= StepFactor

step S k

I

3 Solutlon Estimate

QI

PI

=

=

exp(a,h)- 1 a,

exp(a,h)- 1-a,h a:

FIG. 3. Continued

B. EVALUATION OF THE LINEAR INTEGRATORS 1.

Test Problem Selection

To examine integrator performance 10 test problems were selected according to the following criteria. Used in the literature specifically for testing integrators. Comprised problems for which each integrator was potentially well suited as well as problems for which it was likely that they were not well suited. 0

0

294

PART VI COMPUTATIONAL ASPECTS OF MODELING c

4 -A

If X,

= M/X{

Crtterlon

H}

< TolembleRelativeError

Then accept step 6, = a,Ax; I

AX; = X : - X ; - h i ; x; = x )

5

- h'P, + (2s,h'a, Else If h > MinStep h Thenhe2 Else h = MinStep

step Reduction

6 !Step Expension

Gain Control

hch(

TolerableRelati~Emr

xm

FIG. 3. Continued

0 Of the form likely to be encountered in practical kinetic modeling episodes. 0 Linear or reducible to a linear form. 0 Analytic solution available or derivable. 0 Modifiable to exhaustively test the integrators to avoid test machine features as factors influencing the results.

Aspects of the test problems are summarized in Table 111. To enable accurate assessment of the speed of the integrators four combinations of solution requests were specified, a t series, a e series, an m series, and a replicated t series. The t series was a typical irregular set of around 10solution point requests (in fact each test set included roughly this number of solution requests), similar to a simple modeling run, the e series was an exponentially expanding set of solution points similar to what may be present in a data set relating to a tracer experiment, and the m series was


..

295

lntegratlonSteps IdentMed by Inlegrat01

to achleve desired acculclcy

FIG. 4. (A) Solution points negotiated during an integration run. (B) Integration steps negotiated in conjunction with determination of solutions. The first solution point requires four integration steps, the second solution point requires two and the third and fourth solution points require only one integration step each. (C) A demonstration of the step halving process. The initial integration point has an unacceptably large error and so the step must be halved. (D) Improved processing speed is achieved by wherever possible using the calculations performed for the “last” step. (E) When an integration step will advance the solution to within (or beyond) 30% of a required solution point the actual step taken is one that will assure a solution point “hit.”

2%


L

J

Calculated StepActualstepCalcukted Step Actual step calculated step Actual step

Step she of last IntegruHcm step

.I

FIG. 4. Continued

a uniformly expanding set of points similar to what would be created in conjunction with SAAM’s data generation facility. Finally, to allow large enough runs to remove the influence of the computer clock on our speed tests we replicated the t series runs for up to 10,OOO repeats. In Table IV we summarize the solution points associated with each of the test sets

297


E L \ \

\ \

Step tolerance Calculated step size

* 30%

Actual step size

//

//

Integration point j

Solution point i

FIG. 4. Continued

2.

The Speed of the Integrators

In Tables VA and VB we summarize the computation time take for each integrator with each problem set, Here we note the following. For problems involving 100 solution requests the CBCCDS was consistently faster than the other integrators except for the Petzold integrator when applied to test t5. 0 For the heaviest test involving some 10,000 solution requests the superiority of the Petzold integrator in dealing with very stiff systems emerged. 0 For its preferred domain of operation the CBCCDS was as fast as Petzold’s integrator, 13 times faster that the RK integrator and 5 times faster than the BS integrator. 0 For small problem sets (e and rn series) representative of small kinetic modeling runs CBCCDS, RK, and BS had substantial speed advantages over the Petzold scheme. CBCCDS was slightly superior to the other two. 0

TABLE 111 DtrAI1.S OF THfi TEST PROBLEMS

ID

Source

t2 13

Lapidus and Seinfeld (1971, p. 84. I ) Lapidus and Seinfeld (1971, p. 84. 11) Lapidus and Seinfeld (1971, p. 84. 111)

t5

Lapidus and Seinfeld (1971, p. 84. V I )

t13

Chemical kinetics

11

Math yl'

=

Y2I

=

Y3, =

(14

Test 05, problem 14

120

Braun (1986, p. 379)

121

Braun (1 986, p. 381 )

- Y1

y yl' = y2 + yl yz' = 0.0 y I ' = -49.9~2 - 0 . 1 ~ ~ yz' = -5oy2 y3' = 70y2 - 1 2 0 ~ ~ YI' = Yz - YI ,v1' =

Yl

- 2yz + ys

y2 - y3

Analytic Vl(O) = 1.0 Y1(0)= 1.0 y (0) = 0.0 v2(0) = 1.0 YI(O) = 2.0 y2(0) = 1.0 y3(0) = 2.0 y,(O) = 2.0

yI(x) = e-' yl(r) = e' y1(x) = e' - 1 y2(x) = 0.0 yl(x) = e-"'" + e-5k y*(x) = e-50x y&) = e-* + e-l2Ox

yz(0) = 0.0

y(x)=1+-+2 y2(x) = I - e-3r

y3(0) = 1.0 y3(x) = 1

Yl'

= -Yl

y,(O) = 1.0 y (0) = -1.0 y (0) = 0.0 yz(0) = 0.0 y3(0) = 1.0 y (0) = 0.0 yz(0) = 0.0

t30

Lapidus and Seinfeld (1971, p. 84. V)

171

Chu and Berman (1974)

e-3x

Y3' =

-6Yl - 3Y3

y3(0) = 1.0

yl'

yz

y1(0) = 1.0

=

yz(0) = 1.0 Y2' = -yi yl' = -200~1- 1991~2+ 2000~3 199~4 yl(0) = 10.0 yz(0) = 1.0 Yz' = -Y2 y3' = 0.0 y3(O) = 1.0 y4(0) = 0.0 Y4' = Y2 - Y4

e-x

2

e-3x e-' +-2 2

yl(x) = e-' y&) = -e-' y1(x) = 0.0 y2(x) = 2 c Xsin x cos x y&) = 2e-' cos x2 - cX

4 18 18 y2(x) = - x - - + -e-7x I 49 49 12 x + 40 y3(x) = - -+-22-7~ 7 49 49 yI(x) = sin x + cos x y2(x) = cos x - sin x yl(x) = 10 + 10e-2m - xe-x - IOe-' yz(x) = e-x y&) = 1.0 y4(x) = x c X


299

TABLE IV SOLUIlON POINTS USED FOR EACH TEST PROBLEM AND SERIES

Problem

Number of compartments evaluated

Normal points (t series) 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0 0.01, 0.05, 0.1, 0.5, 1.0, 2.0,4.0, 6.0, 8.0, 10.0 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0 0.001. 0.005, 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 20.0, 40.0 0.001, 0.005, 0.01, 0.028, 0.05, 0.1, 0.2, 0.5, 0.7, 1.0, 2.0, 10.0 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0, 2.0, 3.0, 5.0, 7.0 0.001, 0.005. 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0,2.5 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0,2.5 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0.2.5 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0

tl

n t3 t5

fl 113 t14

a0 121 130

Exponential solution (e series) All

All

Number of compartments evaluated All

All

1, 2,4, 8, 16, 32, 64

Incremental solution points ( m series) 10,20,30,40,50,60,70,80,!30,100

3. Integrator Precision

In Table VI we summarize the accuracy of each integrator for each test problem. Here accuracy is defined to be the lowest number of digits of agreement between the exact solution and the particular integrator’s results for each of the component responses and considering each solution point. From Table W we note the following. 0 For all tests the CBCCDS integrator gave at least the accuracy its settings suggested. 0 For its preferred problem set the CBCCDS integrator was convincingly more accurate that the others. 0 The Petzold and BS integrators each failed on a number of problems. 0 The RK integrator performed with reasonable consistency on all problems.

300

PART V1 COMPUTATIONAL ASPECTS OF MODELING

TABLE VA PROCESSINGTIME IN SECONDS FOR EACH INTEGRATOR AND TEST PROBLEM, USING THE

e

A N D ITl SERIES OF SOLUTION POINTS

Problem ID.

CBCCS CMatcom

CBCCS 100 ClWatcom

RK UWatcom

Petzold F/Watcom

BS FNatcom

SAAM 31 FlWatcom

0.22 0.16 0.22 0.17 0.49 0.22 0.33 0.66 0.22 0.33

0.05 0.05 0.05 1.92 2.86 0.06 0.11 0.22 0.22 0.06

0.109 0.059 0.059 0.109 11.758 0.168 0.109 1.148 0.281 0.820

0.33 0.38 0.72 0.1 1 0.55 0.16 0.50 1.04 0.17 0.88

0.05 0.1 1 0.11 2.96 5.05 0.11 0.11 0.28 0.33 0.11

1.100 1.369 1.480 1.590 4.117 2.359 1.590 2.359 2.199 2.371

el

0

0.05

0.06

a

0.05

e3

0 0.1 1

0.1 1 0.05

0.06 0.11 0.17 1.5 0

e5 el7 ct3 el3 e20 e21 e30

0.05 0 0.06 0.17 0.06 0.05

8.6 4.7 0.88 0.1 1 1.4 2.6 6.1

0.05 0.38 0.06 0.16

CBCCS CMatcom ml

0.06

m2

0.0

m3

0.0 0.11 0.1 1

m5

in77

tnl4

0.0 0.0

m20

0.16

tn2 1 m30

0.06 0.1 1

m13

0.28 0.22 0.44 20.00 11.00 2.9 0.33 33.00 9.2 23.00

0.05 0.11 0.16 0.27 0.06 0.06 0.22 0.11 0.22 2.3

0 Even for tests for which the CBCCDS integrator was unsuited it returned solutions of acceptable accuracy.

IV. CONCLUSION

From this work we may draw the following conclusions. 0 With the exception of two instances with problem t20, the CBCCDS integrator delivered accuracy of better than 1%irrespective of the nature of the linear problem. 0 Systems of linear differential equations for which the stiffness is greater than 1000or those for which the solution involves periodic functions present the greatest problems for the CBCCDS integrator.

TABLE VB PROCESSING TIME IN SECONDS FOR EACH INTEGRATOR A N D TEST PROBLEM, FOR MULTIPLE RUNS OF THE f-SERIES SOLUTION POINTS

CBCCS Integrator: Repeat: 11 12 13 t5 177 113 114 120 f21 130

10

100

1000

10000

10

0 0 0 0.88 0.22 0.05 0

0.11 0.05 0.16 8.4 1.8 0.28 0.17 0.33 0.5 0.93

0.83 0.77 1.4

7.9 7.7 13 840 180 27 11 30 48 da

0.44 0.39 0.5 0.33 0.49 0.44 0.17 0.43 0.55 0.6

0.05 0.06 0.11

84

18 2.7 1.2 3 4.8 9.2

BS

RK

Petzold 100

1000

loo00

1000

loo00

lo00

loo00

0.44

0.93 0.88 1.32 I .43 1.65 1.32 0.82 1.2 1.21 da

5.33 5.38 8.18 10.82 11.21 8.79 6.37 8.24 8.07

11 11 19 130 250 18 6.6 21 22 21

110 110 190 1300 2500 180 66 210 220 da

3.9 3.57 6.32 1173.7 491.09 12.36 4.01 9.28 54.70 5.88

38.78 36.03 61.9 11737.25 4910.34 123.58 39.93 92.61 547 da

0.38 0.6 4.9 0.66

0.55 0.22 0.49 0.55 0.66

da

TABLE VI INTEGRATOR ACCURACP

CBCCS 1

tl

n t3 6 n7 t13 114 220 R1 t30

16115 16/16 16114

I14 1116 914 16116 16116 I14 1013

2

16116 16116 16115 114 16116 1012 8/4

913

3

1614 16116 914 12/2 913

RK 4

1

1617 1sn i6n 1517

1113

ion

1318 16 n 16116 916 i6n

2

16/16 1616 16110 1118 16/7 1617 9fl 1 6fl

BS

Petzold

3

1616 16116 1318 1611 1019

4

16/9

1 1219 1219 1311 911 1211 1318 1219 16/16 1016 1319

2

3

4

16116

8/0 11/8

1518 1219 1317

ion

1219

910 16/16 1318 1319 1118

ion

1

2

1519 1619 1517 9fl 14n 16/8 16/9 16116 310 1619

16116 1210 1418 la8 16/9 1418 310 16/9

3

4

1211 16116 16/8

1418

1319 310

“ The accuracy of each integrator for each test problem. Here accuracy is defined as the worst number of digits of agreement or any solution within each test problem run when comparing the particular integrator results with the “exact” results. Each compartment accuracy is indicated by its ordinal assignment of digits. The slash ( I ) separates compartment precision.


303

0 The CBCCDS integrator was the fastest of all integrators for problems of small to medium size. 0 Although a consistent performer the slowness of the Runge Kutta integrator rule it out in favor of the Petzold integrator in those areas where an alternative to the CBCCDS integrator is needed. 0 The Bulirsch Stoer integrator performed with consistent accuracy but again was too slow to compete with Petzold’s scheme as an alternative to the CBCCDS integrator. 0 The CBCCDS integrator is justifiably placed in SAAM as the default integrator.

REFERENCES Acton, F. S. (1970). “Numerical Methods that Work. Harper & Row, New York. Berman, M., and Weiss, M.F. (1978). “SAAM Manual,” DHEW Publ. No. (NIH) 78-180. U.S. Govt. Printing Office Washington, DC. Boston, R. C., Greif, P. C., and Berman, M. (1981). Conversational SAAM: An interactive program for the kinetic analysis of biological systems. Comput. Methods Program Biomed. l3,lll-119. Braun, M. (1986). “Differential Equations and Their Applications.” Springer-Verlag, New York. chu, S. C., and Berman, M. (1974). An exponential method for the solution of systems of ordinary differential equations. Commun. ACM 17,699-702. Frost, A. A., and Peason, R. G. (l%l).“Kinetics and Mechanism.” Wiley, New York. Kahaner, D., Moler, C., and Nash, S. (1989). “Numerical Methods and Software.” Prentice Hall, Englewood Cliffs, NJ. Lapidus, L., and Seinfeld, J. H. (1971). “Numerical Solution of Ordinary Differential Equations.” Academic Press, New York. Lyne, A., Boston, R., Pettigrew, K., and Zech, L. (1992). EMSA A SAAM service for the estimation of population parameters based on model fits to identically replicated experiments. Comput. Methods Programs Bwmed. 38,117-151. Petzold, L. (1983). A description of DASSL A differentiavalgebraicsystem solver. In “Scientific Computing” (R. Stepleman et 01, eds.), pp. 65-68. lMACS/North-Holland Printing Co., Amsterdam. Press, W. H., Flannery, B. P., Teukolsky, S. A,, and Vettling, W. T. (1987). “Numerical Recipes: The Art of Scientific Computing.” Cambridge Univ. Press, New York. Rice, J. R. (1983). “Numerical Methods, Software, and Analysis: IMSL Reference Edition.” McGraw Hill, New York.


Chapter 19 IDENT1FlABILITY JOHN A. JACQUEZ Departments of Physiology and Biostatistics The University of Michigan Ann Arbor, Michigan 48109

1. Introduction 11. Examples

111.

1v. V.

v1. VII.

VIII.

A. Initial Velocity in Michaelis-Menten Kinetics B. A Two-Compartment Model Classification of Parameters A. Observational Parameters B. Basic Parameters Parameter Identifiability and Estimation Identiliability: Definitions A. Local Identifiability B. Global Identifiability C. Structural Identifiability D. Model Identifiability E. Conditional ldentifiability F. Interval ldentifiability and Quasi-identiliability Methods of Checking Identifiability A. Methods for Linear Systems with Constant Coefficients B. Methods for Nonlinear Systems Local ldentifiability at a Point A. Least Squares B. Local ldentifiability C. Correlations between Identifiable Parameters Conclusion References

I. INTRODUCTION In 1956, Mones Berman and Robert Schoenfeld published a paper that was concerned with the information content of measurements on some 305 Copyright 8 1996 by Academic Press. Inc. All rights of reproduction in any form reserved.

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compartments of a multicompartment system. They pointed out that in general an n-compartment linear system with constant coefficients has n2 rate constants in the coefficient matrix; the entries in the coefficient matrix are invariants, i.e., fixed, but unknown. Assuming the eigenvalues are distinct and there are no unusual circumstances, measurement of the time course of one compartment gives only 2n - 1parameters that are functions of the invariants. For each additional compartment one obtains only n 1 more such parameters. In theory we need measurements on all compartments to obtain enough information to determine all n2 invariants. From the connectivities of our models, some of the rate coefficients are zero so there may not actually be n2 invariants. Even so the same issue is present: if we measure only a subset of the compartments how d o we know there is enough information in the measurements to uniquely define all of the rate coefficients? Suppose there are 2 transfer coefficients that are zero and that we can estimate P compound parameters (parameters that are functions of the basic rate parameters); then there are n2 - 2 - P degrees of freedom in the data. If P = n2 - 2, one has P equations, generally nonlinear, in P unknowns and it may be possible to solve for the values of the basic parameters. However, there is no guarantee that P nonlinear equations in P unknowns has a unique solution. If n2 - 2 - P > 0 the basic rate parameters are determinable as functions of n2 - 2 - P free variables. Note that some of the invariants (the basic rate parameters) may be determinable; in that case, if d are determinable, the remaining n2 - 2 - d are functions of n2 - 2 - P - d free variables. Using similarity transformations and imposing constraints such as positivity of the nonzero invariants Berman and Schoenfeld showed that the values of the free variables are constrained to fall in subspaces of the space of the free variables. In their paper, Berman and Schoenfeld (1956) presented an example of a three-compartment model and an experiment in which only compartment 1 and the excretion from compartment 1 could be measured. With the measurements available there were two degrees of freedom left in the measurements. Figure 2 in their paper shows the subspaces in the two free variables in which all permissable models must fall. That work comes very close to what is now called the identifiability problem; however, they did not separate out the effects of measurement error. They summarized their work in a subsequent paper (1958) on the information content of data in which they did distinguish between the two sources of uncertainty. “The uncertainty in these values arises from the fact that the collected data may not be sufficient to define the system completely and that the collected data have associated fluctuations.”

CHAPTER 19 IDENTIFIABILITY

307

In the meanwhile, work on this problem was being done in statistics (Koopmans and Reiersol, 1950), econometrics (Fisher, 1959), and systems engineering (Astrom and Eykhoff, 1971) where identifiability was defined as a distinct problem. It was not until 1970 that a paper on identifiability of a compartmental system appeared. That was the paper of Bellman and Astrom ("70) which is now cited as the first paper to address the identifiability problem for compartmental systems. Since that time, a growing body of literature on identifiability for compartmental models and other types of models of biological systems has come to dominate the literature on identifiability. For compartmental systems, although the problem has not been solved in full generality, a large amount of theory is available and we have a clear picture of its extent and difficulties. For an introduction, see the review of Cobelli and DiStefano (1980) and Chap. 14 of my book (Jacquez, 1985). For a simplified introduction see Jacquez (1987). More advanced papers by Cobelli, DiStefano, Godfrey, and others can be found in the collection edited by Eric Walter (1987). A catalog of identifiability properties of three-compartment models is given by Norton (1982). In conclusion, a crucial step in the development of our ideas was the recognition that the two problems: Are the observations obtained sufficient to specify the invariants of the model?, and, What are the effects of the errors of measurement?, are separable and should be treated in that order. The first problem is that of the identifiability of the parameters, the second concerns the propagation of errors through the calculations and their effects on the estimates, the estimation problem. 11.

EXAMPLES

To introduce the idea of identifiability, we start with two very simple problems, one from enzyme kinetics and one from compartmental modeling. In the next section, we classify the parameters; section IV then gives a general statement of the identifiability problem. A. INITIAL VELOCITY IN MICHAELIS-MENTEN KINETICS

Consider a one-substrate, one-product enzyme reaction, as shown in Fig. 1. Suppose one can do the experiment of measuring the initial velocity of kl

k3

k2

k4

E + S - E S a E + P FIG. 1.

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the formation of product, P, at a series of substrate concentrations, S. If the rate of formation of the intermediate complex, ES, is rapid in relation to the rate of formation of product, P, it is well known that the initial velocity shows saturation kinetics in the substrate concentration, as given by Eq. (1) for the initial forward velocity, vr.

Here, [S] is standard chemical notation for concentration of S. VMfand Kmf are parameters that are functions of the basic kinetic parameters, k l , k2, k3, and of the total enzyme concentration, Eo, as given by

Both VMf and K,nf can be estimated from the initial velocity experiment, but notice that the basic kinetic parameters may not be determinable. If we know Eo, k3 is determined by Eq. (2); it is identifiable. However, unique solutions for k , and k2 cannot be obtained and k4 does not even appear in Eq. (2) and (3). Now suppose that one can d o the same experiment for the backwards initial velocity of formation of substrate at different concentrations of product. Under the same assumptions, the backwards initial velocity is given by

The analogs of Eqs. (2) and (3) are now

For this experiment, if we know Eo, k2 is identifiable, but although k3 and k4 influence Kmb, they are not identifiable. Not only is kl not identifiable, it does not even appear in Eqs. ( 5 ) and (6).


309

It is clear that only if one knows Eo and can do both of the above experiments, can one obtain estimates of all four basic kinetic parameters. 3. A TWO-COMPARTMENT MODEL

Consider the simple two compartment system shown in Fig. 2. We suppose the inflow to compartment 1 is constant and the system is at a steady state. The experiment consists of injecting an impulse of D units of tracer into compartment 1and measuring the tracer concentration in compartment 1. The solution for total tracer in compartment 1 is given by

The observation function or response function (the function that describes what is observed without the errors of observation) is

where V1 is the volume of distribution in compartment 1. Notice that the actual observations are samples at different times of the observation function with some experimental error added. To check identifiability, we only need to examine the observation function. From y, one can determine the two compound parameters

4 = -D Vl

FIG. 2.

(9)

310


Thus, knowing D, V 1is uniquely determined (identifiable). However, kol and k21 are not uniquely determined, only the sum kol + k21 is determined, and ko2 has no effect on the observations. Ill. CLASSIFICATION OF PARAMETERS

Both of the above examples illustrate a number of important points about the basic parameters of the system (Berman’s invariants) and the parameters that are determinable by a particular experiment. 1. The first point is that one has to clearly distinguish between the basic parameters and the parameters that are determinable by an experiment. I have called the latter observational parameters and denote them by the symbol i = 1, . . . .Notice that in each of the above experiments, the observational parameters are functions of the basic kinetic parameters. 2. A basic kinetic parameter may or may not influence the observations in a particular experiment. If the observational parameters are not functions of a particular basic parameter, the basic parameter can be changed without affecting the observations. Such a parameter is insensible in the experiment and is called an insensible parameter. If a basic parameter does influence the observations in an experiment, it is sensible by that experiment. However, a sensible parameter may or may not be uniquely determined (identifiable) by the experiment. In each of the above examples, there were sensible parameters that were identifiable and others that were not identifiable. 3. Basic parameters may also be introduced by an experiment. We have talked so far as though the basic parameter set consisted of the invariants of the system and that the observational parameters are functions of the those invariants. That is very often the case and it is easy to think of the problem in those terms. However, it is not quite that simple. The design of an experiment may also introduce basic parameters. A good example is the impulsive input (dose) D in example 2 above. If that is not known and is to be estimated, it appears as a basic parameter in Eq. (8). We conclude this section with a summary of the classification of the parameters. A. OBSERVATIONAL PARAMETERS The observational parameters are determined by the experimental design and are functions of a basic parameter set.


311

B. BASIC PARAMETERS The basic parameters are the system invariants (kinetic parameters of the system) plus possibly some parameters introduced by the experimental design. For a given experiment, they may be: (i) insensible, i.e., do not influence the observations, or (ii) sensible, i.e., influence the observations in the experiment. In that case, they may be: (a) identifiable or (b) nonidentifiable. IV. PARAMETER IDENTlFiABlLlTY AND ESTIMATION

With that background, let us proceed to a more formal statement of the identifiability problem and distinguish between the identifiability problem and the estimation problem. We do experiments on systems in the real world but analyze the results on models of experiments done on models of the systems. We have in mind a model of the systemwhich incorporates current hypotheses of its structure, rate laws, and values of some of the parameters. An experiment involves adding inputs and making measurements (outputs). So we are concerned with models of the experiments. Let x be the vector of state variables of the model. The inputs in the experiment are often described as the product of a matrix B and a vector of possible inputs, u.The inputs are combinations of the components of the vector u, i.e., Bu. For given initial conditions and input to the model, the time course of change in the vector of state variables is usually given by a set of differential equations.

where 8 is a vector of basic parameters and TQ gives the initial conditions. To fully specify an experiment, we also have to give the observations. The observations are usually specified by giving the vector function of the state variables that describes what is measured.

We call y the observation function; y = G (I, 4) is also called the response function. In engineering it is common to refer to input-output experiments, the response function being the output, in the information sense. With compartmental systems that could be confused with the material outputs or outflows from the system so I will try to use only the terms response

312


function or observation function. Remember that the experiment could also introduce some basic variables, which are not explicitly shown in Eq. (12). The actual observations, zir are samples of the observation function at different times with added experimental errors of measurement.

zi

y(r;, 4)

+ Ei

i

=

1, .

. . , n,

where E , is the vector of measurement errors at sample time ti. If the model is a compartment model with constant fractional transfer coefficients. the equation corresponding to (11) is

In Eq. (14), K is the matrix of transfer coefficients. The components of K are basic or structural parameters of the model. If the observations are linear combinations of the compartments, the observation function is given by Eq. (15), in which C is the observation matrix. y = cq. Basic parameters could be introduced by the experimental design by way of the initial conditions, 90, the inputs Bu, and the observational matrix C. Now we can restate the identifiability problem. Equation (12) determines the observational parameters (4;); the Cpi are by definition the compound parameters that are uniquely determined by the observation functions. For the given model and experiment, are the basic parameters, the 0, uniquely determined by the 4i?That is the identifiability problem. Notice that data collection is not needed to solve that problem. We only have to know the model of the experiment. Then we calculate the observation functions and check to see if the 4 are uniquely determined. Given that the parameters of interest are identifiable, one can then proceed to collect data and estimate the 6, However, there is no point to doing the experiment if the parameters of interest are not identifiable! One final point: the term identification is sometimes used to mean the estimation of parameters from the data and in some papers one may find both terms, identifiability and identification. Do not be confused. Identifiability has the meaning we have just given. Identification refers to the actual estimation of the parameter values.


V.

313

IDENTIFIABILITY: DEFINITIONS

Now that it is clear that identifiability is concerned with the question of uniqueness of solutions for the basic parameters from the observation function of a given experiment, we have to introduce the various types of identifiability that have been defined in the literature. A. LOCAL IDENTIFIABILITY If the observation function for an experiment determines a finite number of values for a parameter, the parameter is locally identifiable. Auxiliary information may be needed to decide which one of the values is the appropriate one for the physiological system you are working on. This includes cases of symmetry in models in which two or more parameters play equivalent roles, so their values can be interchanged. B. GLOBAL IDENTIFIABILITY If a parameter is localiy identifiable but the observation function determines exactly one solution in the entire parameter space, that parameter is globally identifiable for that experiment. Thus, global identifiability is a subcategory of local identifiability. The term unique identifiability is equivalent to global identifiability.

C. STRUCTURAL IDENTIFIABILITY A property of a parameter is structural if it holds for almost all values of the parameter, i.e., almost everywhere in parameter space. The qualification, “almost everywhere” means that the property may not hold on a special subset of measure zero. Thus a parameter could be globally identifiable almost everywhere but only locally identifiable for a few special values. Structural global or local identifiability are generic properties that are not dependent on the values of the parameters, in the almost everywhere sense (Walter, 1982). D. MODEL IDENTIFIABILITY If all of the parameters of a model are globally identifiable, the model is globally identifiable for that experiment. If all of the parameters are identifiable but at least one is not globally identifiable, the model is only locally identifiable.

314

PART VI COMPUTATIONAL ASPECIS OF MODELING

E. CONDITIONAL IDENTIFIABILITY If for a model and experiment a parameter is not identifiable but setting the values of one or more other parameters makes it identifiable, the parameter is identifiable conditioned on the parameters that are preset. Note that by “setting a parameter” we mean we assign a value to it and then treat it as known, i.e. remove it from the parameter set. F. INTERVAL IDENTIFIABILITY AND QU ASI-IDENTIFIABILITY Following up on the work of Berman and Schoenfeld (1956, 1958), DiStefan0 (1983) has used the term interval identifiability to describe the restriction of a parameter to a subspace by the constraints of a problem. It is possible for such an interval to be small enough for a parameter to be “identifiable for practical purposes” and DiStefano calls that quasi-identifiability. VI. METHODS OF CHECKING IDENTlFlABlLlTY

For very simple problems, such as the two examples in section 11, one can often determine identifiability by inspection of the observation function. However, as soon as the models get more complex, that is no longer possible. A number of methods are available for checking identifiability. Here, I want to give brief descriptions of them. For more details see the books by Carson er al. (1983), Godfrey (1983), Jacquez (1985), and Walter (1982). The methods differ for linear and nonlinear systems; I present them in that order. First, let us make clear the distinction between linear and nonlinear systems and linear and nonlinear parameters. For a linear system, the rates of change of the state variables are given by linear differential equations. Such systems have the superposition or input linearity property. By that I mean, the response to a sum of two inputs equals the sum of the responses to the individual inputs. In contrast, the rates of change of the state variables of nonlinear systems are given by nonlinear differential equations, and superposition does not hold. When applied to the parameters of a system, the terms linear and nonlinear have entirely different definitions; they then refer to the way the parameters appear in the solutions for the state variables or the observation functions. Suppose x is a state variable and the solution of the differential equation is of the form x =c

+ Of(r).

(16)


315

The parameter 8 is a linear parameter because it appears linearly in the solution. For the solution of a one compartment system with two parameters, the initial value and the excretion coefficient, the solution is q = qoe-".

(17)

The initial value qo is a linear parameter and A is a nonlinear parameter. Even for linear systems, many of the parameters appear nonlinearly in the solutions. A. METHODS FOR LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS

Some simple topological properties of the connection diagram should be checked first. They provide necessary but not sufficient conditions for identifiability. (i) Znput and output connectability. There must be a path from some input to each of the compartments of the model and there must be a path from each compartment to some observation site. (ii) Condition on number of parameters. The number of unknown parameters must not exceed a number which depends on the topology of the system; see Carson et al. (1983) for the method of calculation. For checking parameter identifiability, three methods have received most attention. 1. The Similarity Transformation Method

Consider a system for which K has been subjected to a similarity transformation to give a system with a coefficient matrix P-lKP, where P is nonsingular. Recall that under a similarity transformation, the eigenvalues do not change. Impose on P-'KP all the structural constraints on K and require that the response function of the system with matrix P-'KP be the same as that of the system with matrix K. If the only P that satisfies those requirements is the identity matrix, all parameters are globally identifiable. If a P # I satisfies the requirements, one can work out which parameters are not identifiable and which are.

2. The Modal Matrix Method The matrix whose columns are the eigenvectors is the modal matrix. In this approach, one looks at the response function to see if the eigenvalues

316


and the components of the modal matrix are identifiable; both are of course functions of the basic parameters. This method has not been developed as fully as a formal method as the next one and seems to offer no advantages. 3.

The Laplace Transform or Transfer Function Method

This method is simple in theory and is the most widely used, although it becomes quite cumbersome with large models. First we note that if a linear model is identifiable with some input in an experiment, it is identifiable from impulsive inputs into the same compartments. That allows one to use impulsive inputs in checking identifiability even if the actual input in the experiment is not an impulse. Take Laplace transforms of the system differential equations and solve the resulting algebraic equations for the transforms of the state variables. Then write the Laplace transform for the observation function (response function). That will be of the form

The coefficients, tpi,, are the observational parameters and are functions of the basic parameters. That gives a set of nonlinear algebraic equations in the basic parameters. The hard part is to determine which of the basic parameters are uniquely determined by this set of nonlinear equations. To illustrate its use, let us apply this method to example B of section 11. The equations for tracer flow are

The Laplace transforms are

Thus.


317

We do not really need Q2for this problem. The transform of the observation function is

That gives us

B. METHODS FOR NONLINEAR SYSTEMS

Although there is a large literature on identifiability for linear systems with constant coefficients, less has been done on nonlinear systems. Two general properties should be remembered. Whereas for linear systems one can substitute impulsive inputs for the experimental inputs for the analysis of identifiability, one cannot do that for nonlinear systems. One must analyze the input-output experiment for the actual inputs used. That is a drawback. On the other hand, experience shows that frequently the introduction of nonlinearities makes a formerly nonidentifiable model identifiable for a given input-output experiment. Two methods are available. 1.

Taylor Series

A method used widely depends on expanding the observation function (response function) in a Taylor’s series around t = 0’ (Pohjanpalo, 1978). The coefficients of the expansion are functions of the basic parameters and are the observational parameters. Although there are an infinite number of coefficients, only a finite number are independent. As one adds coefficients from terms of higher and higher order, eventually one reaches coefficients that are no longer independent of the preceeding ones. One problem is that it is not always obvious when that point is reached.

2. Similarity Transformation The method of similarity transformations has been extended to nonlinear systems (Vajda et al., 1989). I have had no experience with this method but point out that the similarity transformation method for linear systems is often much more work than the Laplace transform method.

318


VII. LOCAL IDENTlFlABlLlTY AT A POINT

A large part of the literature on identifiability is concerned with checking global and/or local identifiability of models. That is a problem of great interest from the viewpoint of basic theory but can also be very difficult because basically it is the problem of solving simultaneous nonlinear algebraic equations. A less difficult problem but one that gives considerable useful information is that of testing local iaentifiability at a given point in parameter space. In physiology we are concerned with testing hypotheses and focus on those parameters that have to be estimated to test a hypothesis. Furthermore we often have auxilliary information and results from other experiments that provide initial estimates of the parameter values. Even if our estimates are rough, tests of local identifiability at a number of points provide sufficient information for practical applications. In fact, for compartmental models with constant fractional transfer coefficients,we are assured that if a parameter is locally identifiable at one value it is locally identifiable for almost all values of the parameter (Eisenfeld, 1986; Walter, 1982), so we obtain structural local identifiability. It is natural to develop the theory in terms of the two levels of parameters, the basic parameters, 6, and the observational parameters, 4" which are identifiable functions of the basic parameters. For problems of low dirnensionality it is easy to generate the Cpi explicitly as functions of the 6, and check identifiability on the functional relations, Cbi = fi(&. . . . ,@,). For problems of even moderate magnitude the algebraic work involved in finding the +iand solving the equations may become limiting. An important finding is that if one has initial estimates of the basic parameters one can determine local identifiability numerically at the initial estimates directly without having to generate the observational parameters as explicit functions of the basic parameters. That is the approach used in the IDENT programs which use the method of least squares (Jacquez and Perry, 1990; Perry, 1991). It is important to realize that the method works for linear and nonlinear systems, compartmental or noncompartmental. Furthermore, for linear systems it gives structural local identifiability. A. LEAST SQUARES Let there bep basic parameters and assume we have estimates, elo, . . . , ePo.For the 6, set at these estimates, calculate a set of values of the observation function at ri points in time for ri > p. Then linearize the observation function in the parameters ek around the estimates which now play the


319

role of known values. What follows can be repeated for a number of values for the parameters or it can be done sequentially as one obtains more estimates of the values of the parameters

The superscript O means the term is to be evaluated at the known values (the estimates) of the parameters. Notice that the ej are not measurement errors, they are truncation errors in the expansion. Form the squared deviations around the initial estimates and s u m over the set of calculated values of the response function, as in

Notice that y j - CY = 0. Next, find the least squares estimate of Aek. Let A 8 = (A&, . . . , A8,) be the least squares estimates of A 0 and g be the sensitivity matrix

g=

. .

. .

The normal equations for the estimates A6, are given by

If the determinant of g'g is nonzero, except possibly on a subspace of the parameter space of lower dimensions, the model is locally identifiable; if the system is a linear system, it is structurally locally identifiable.

B. LOCAL IDENTIFIABILITY 1. Insensible Parameters

Examine the columns of g. For the insensible parameters the corresponding columns have only zero entries.

320


2. Sensible Parameters Suppose we delete the columns of g corresponding to the insensible parameters to form the matrix g, and also delete the insensible parameters from the parameter vector. Then the normal equations corresponding to the sensible parameters are given by

gPT&AdS = 0. For the identifiable parameters, 8, we should obtain for solutions, Adk = 0. For the nonidentifiable parameters there should not be such unique solutions. To check that, use row reduction to row echelon form, with pivoting on maximum elements. When completed, the equations should have the following form. (i) Rows with one nonzero entry on the diagonal. The column indexes of the diagonal elements of these rows give the identifiable parameters. (ii) Rows with nonzero entries on the diagonal and elsewhere. The column indexes of the nonzero elements of these rows give the nonidentifiable parameters. (iii) Rows with all zero entries. These are evidence of redundancy in the equations. C. CORRELATIONS BETWEEN IDENTIFIABLE PARAMETERS

The final sLep is to find the painvise correlations for the identifiable parameters. Even if identifiable it may be difficult to estimate two parameters separately in the presence of measurement error if they are highly correlated. To check that, we reduce the matrix gTg6by eliminating rows and columns corresponding to the nonidentifiable parameters to obtain glTgl.Since glTglis the matrix corresponding to the identifiable parameters, it is nonsingular so we can invert it to obtain (gFg,)-I. The correlation matrix is obtained by dividing the i,j element of (glTg,)-*by the square root of the product of the ith and jth diagonal elements.

VIII.

CONCLUSION

In experiments in the biological sciences one generally cannot sample all state variables (compartments). In that case, some parameters of the system may not be uniquely determined (identifiable) by the observations of the experiment. In order to check identifiability, one need only examine


321

the model of the experiment; there is no need to do the experiment. Check identifiability before committing resources to doing the experiment. REFERENCES Astrom, K. J., and Eykhoff, P. (1971). System identification-a survey. Automatica 7,123-162. BeUman, R., and Astrom, K. J. (1970). On structural identifiability. Math. Biosci. 7,329-339. Berman, M., and Schoenfeld, R. L. (1956). Invariants in experimental data on linear kinetics and the formulation of models. J. Appl. Phys. 27,1361-1370. Berman, M., and Schoenfeld, R. L. (1958). Information content of tracer data with respect to steady-state systems. I n Symposium on Information Theory in Biology,” pp. 181-186. Pergamon, New York. Carson,E. R., Cobelli, C., and Finkelstein, L. (1983). “The Mathematical Modeling of Metabolic and Endocrine Systems.” Wiley, New York. Cobelli, C., and DiStefano, J. J., I11 (1980). Parameter and structural identifiability concepts and ambiguities: A critical review and analysis. Am. J. Physiol. 239, R7-R24. DiStefano, J. J., 111 (1983). Complete parameter bounds and quasiidentifiability conditions for a class of unidentifiable linear systems. Math. Biosci. 65, 51-68. Eisenfeld,J. (1986). A simple solution to the compartmental structural-identifiabilityproblem. Math. Biosci. 7’9,209-220. Fisher, F. M. (1959). Generalization of the rank and order conditions for identifiability. Econometrics 27, 431-477. Godfrey, K. (1983). “Compartmental Models and Their Application.” Academic Press, New York. Jacquez, J. A. (1985). “Compartmental Analysis in Biology and Medicine.” 2nd ed. Univ. of Michigan Press, Ann Arbor. Jacquez, J. A. (1987). Identifiability: The first step in parameter estimation. Fed Proc., Fed. Am. SOC.Exp. Bwl. 46,2411-2480. Jacquez, 3. A., and Perry, T. (1990). Parameter estimation: Local identifiability of parameters. Am. J. Physiol. zS8, E727-E736. Koopmans, T. C., and Reiersol, 0. (1950). Identification of structural characteristics. Ann. Math. Slat. 21, 165-181. Norton, J. P. (1982). An investigation of the sources of nonuniqueness in deterministicidentifiability. Math. Biosci. 60, 89-108. Perry, T. (1991). “IDENT 11. Identifiability for Compartmental Models. RFKA Documentation for IDENT.” University of Michigan, Ann Arbor. Pohjanpalo, H. (1978). System identifiability based on the power series expansion of the solution. Math. Biosci 41,21-33. Vajda, S., Godfrey, K. R., and Rabitz, H. (1989). Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci 93,217-248. Walter, E. (1982). “Identifiability of State Space Models,” Lect. Notes Biomath. No. 46. Springer-Verlag,New York Walter, E.,ed. (1987). “Identifiability of Parametric Models.” Pergamon, Oxford.



Chapter 20 DYNAMIC SYSTEMS AND NEURAL NETWORKS: MODELING IN PHYSIOLOGY AND MEDICINE SAMIR I. SAYEGH Physics Department Purdue Universiw Ft. Wayne, Indiana 46805

I. Introduction II. Linear Systems Modeling 111. Nonlinear Systems, Chaos, and Fractional Dimensions IV. Geometric Interpretation of Models and Dynamical Systems V. Neural Networks VI. Conclusion References

I.

INTRODUCTION

This paper deals with a general introduction to dynamic systems as well as a brief introduction to neural networks. In particular, it is intended for those involved in modeling physiological and biochemical systems where the questions of appropriate dimension and existence of a model are important considerations. Other complementary approaches to modeling should be considered depending on the problem at hand (Collins, 1992; Webber and Zbilut, 1994; van Rossum et a[., 1989). One of the main goals of the following paragraphs is to emphasize the importance for scientists involved in modeling to understand the implications and limitations of dimensionality, nonlinearity, and the related nature of the problem they are attempting to solve. The choice of a modeling approach is strongly determined by such considerations. One of the great advantages of neural networks and other techniques of multivariate analysis is that they are applicable to phenomena with a limited level of definition (Benigni and Giuliani, 1994). Advantages of dimensionality reduction in323 Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any fonn reserved.

324


clude “escaping chaos” and allowing for the qualitative analysis of the system at hand. II. LINEAR SYSTEMS MODELING

Whether one is dealing with a system by using dynamic systems or neural network approaches, the level of difficulty involved is quite high for nonlinear systems and relatively trivial for linear systems. The most general linear system can be represented by the equation d d d t = x or d d d t = a.Here .r can represent any vector of variables that depends on the independent variable which in general stands for time. Such a linear system is limited in its capability to represent complex phenomena and exhibits only a couple of behaviors. The first kind of behavior is exponential decay. Indeed a function that is its own derivative is clearly an exponential function. With a negative exponent, one gets exponential decay. With a complex exponent or a pure imaginary exponent, one gets an oscillatory behavior which typically translates into a limit cycle. So if we talk in terms of attractors, namely, a structure that will attract the motion in the sense that the evolution of a system will end up being such a portion of a phase space, then the only attractors that are associated with linear systems will be point attractors, i.e., fix points, and limit cycles, i.e., periodic behavior. Of course, one could also get a positive exponentially developing solution, but that would require an infinite amount of energy that is not assumed to be available to the systems we usually model. Therefore, in a sense, linear systems can be considered to be trivial. From a computational point of view all one needs LO do is to compute an exponential. That could be an exponential of a constant multiplied by the time as it develops or a sine or cosine function. More generally, if the dimension of the system is higher than one, one is dealing with a constant which is a matrix and in order to exponentiate a matrix, the standard approach is to diagonalize the matrix and exponentiate the eigenvalues, again a standard procedure that is built into a number of commercial software packages. The motion can then be essentially reconstructed by these techniques. Though these techniques can display some subtleties, one can still in a general sense classify these as very simple techniques in comparison to the complexity that one encounters with nonlinear systems. Ill. NONLINEAR SYSTEMS, CHAOS, AND FRACTIONAL DIMENSIONS As soon as one deals with nonlinear systems, one encounters a richness that is absent from linear systems. One of the richer aspects of nonlinear

CHAPTER 20 DYNAMIC SYSTEMS AND NEURAL NETWORKS

325

systems has been called chaos and one of the characteristics of chaos is the sensitive dependence on initial conditions. What could be discussed at this point is a definition of chaos and some development of this definition, particularly in relationship to the notion of the dimension of the system. The main goal of dimensional considerations should be the characterization of the system at hand, the identification of the most appropriate modeling approach, and attempts at the reduction of the dimensionality. There are at least two reasons to try and reduce the dimensionality of a system. One reason is to avoid chaos or to show that the underlying system is nonchaotic, which quite often is a very important result to establish. The other reason is to be able to perform a qualitative analysis of the system and to gain insight into the workings of the dynamics. The working definition of chaos is essentially a sensitive dependence on initial conditions, which means that slightly different initial conditions lead to very different final conditions or very different states after a finite time of evolution of a system. In order to understand the geometric interpretation of dimensionality considerations, one has to look at the concept of the phase space in which only the dependent variables are represented. Figure 1 shows such a construct. Points in phase space represent the state of the system at a given time. As time evolves and the state of the system changes, the point traces a curve. As Fig. 1 shows again, this curve is not allowed to intersect itself if the system is to be considered deterministic. The reason is that if the system is deterministic, the complete specification of the state of the system given by a point should predict all future evolution of the system. At an intersection point, we would clearly have two different evolutions of the system that are diverging from the same point, that is, the same initial state is leading to different futures, in violation of the assumption of determinism. Therefore, one cannot assume any intersection of the curve tracing the evolution of the system. Now since chaos has been defined as “the exponential separation of trajectories leading to sensitive dependence on initial conditions,” and if one is in the presence of a system with finite

No Determinism FIG. 1.

326


energy, or, better yet, a dissipative system that is losing energy, as the system loses energy trajectories have to become restricted to a limited region of phase space. The evolution in time cannot lead to explosive behavior and has to remain finite or in a finite boundary. The curves have to return therefore to the region in phase space from which they originated. However, as they return, they are not allowed to intersect, in order to preserve determinism. This leads to the contradiction between a lowdimensional system, i.e., a two-dimensional system, and the existence of chaos. The conclusion, therefore, is that a two-dimensional system cannot display chaotic behavior, i.e., there is not enough room in two dimensions to allow for the exponential separation of the trajectories while preserving the constraint of no intersection that preserves determinism (Fig. 2). One can see how in a higher dimensional system, if one is allowed a little more than two dimensions, one could preserve both chaotic behavior and determinism. That is illustrated in Fig. 3 where the trajectory is returning to the general region of interest where it originated. Instead of intersecting, it crosses over and it crosses over because it has been given a certain amount of extra “space” in the form of increased dimensionality. In a three dimensional system that is dissipative, i.e., that loses energy, the dimensionality of the system collapses. That is, the potential dimension of the motion is going to be less than three because the system is losing energy. However, if the system is a chaotic one, the eventual motion cannot be on a two-dimensional manifold, for the reasons explained above. The conclusion therefore, is that, the eventual structures on which the trajectory ends up evolving has dimension less than three but more than two (Fig.

No Chaos in 2D

FIG. 2.


327

OKin>2d FIG. 3.

4). In a three-dimensional differential equation the system is going to collapse on a structure that is strictly less than three in dimension, but strictly more than two.Such a strange observation leads us immediately to the concept of fractional dimensionality and the attractor associated with such a fractional dimensionality is called a strange attractor, due to its unusual nature. The fractional dimensionality is also described as a “fractal.” A fractal object such as a “C curve” may have some unusual properties. The properties are that it has a fractal dimension but this fractal dimension is not a fraction. In the case of the C curve it is equal to two. The reason this object is still a fractal relates to a definition of fractal dimension. First, one defines two concepts of dimensions: the topological dimension, which corresponds to our usual concept of a dimension, and a so-called HausdorffBesicovic dimension. If for a given object the two dimensions defined are different, the object is said to have a fractal dimension. In the case of the

3

3Var.Chaos

u-u 1D

OD

2 D < d < 3 D FRACML FIG. 4.

328

PART V1 COMPUTATIONAL ASPECTS OF MODELING

C curve, the topological dimension is equal to one and the HausdorffBesicovic dimension is equal to two and that is why it is considered fractal, despite the fact that neither of these dimensions is itself fractional. In most cases, however, a fractal has a fractional dimension. IV. GEOMETRIC iNTERPRETATiONOF MODELS AND DYNAMICAL SYSTEMS

The following discussion will concentrate on a geometric definition of differential equations and on actual reduction of a dimensionality. As Fig. 5 shows, a differential equation can be considered to be a collection of arrows. As Fig. 6 shows, the solution to the differential equation consists of a collection of trajectories which are everywhere tangent to the collection of arrows. Figures 7 and 8 show another example of a differential equation, with a collection of arrows, in phase space, and their solution, the trajectories, that are tangent to the arrows. In Fig. 9, the two differential equations are displayed simultaneously. The intent of that figure is to show that these two particular differential equations have a very special property, namely, that they are orthogonal, in the sense that the arrows of one differential equation take the arrows of the other differential equation into each other. That is, it maps arrows of one equation into arrows of the other. As a consequence, if one were to draw a solution to one of the differential equations, then the arrows of the other differential equation would take

How to reduce d?

- 1 A Di€fczentialEquation(U) FIG. 5.


329

A Solution FIG. 6.

this solution into another solution. This is illustrated in Fig. 10. Having this property, if one could display just one solution to a differential equation, the arrows of the other equation would generate all other solutions. One can intuitively see that such a construct is going to collapseheduce the dimension of a differential equation by one. It turns out that the mathematical condition for such a situation to arise is rather simple and is expressed by the commutator of the arrows corresponding to the two differential

FIG. I.

330


FIG. 8.

equations being zero. The computation of the commutator is very inexpensive and can be performed by hand or by any symbolic manipulation software such as Reduce, Maple, or Mathematica. There is also a slightly more general formulation where the commutator can be simply equal to a multiple of one of the differential equations. It is worthwhile to note that the commutator condition expressed, which might be unfamiliar to some

FIG. 9.


331

FIG. 10.

readers, encompasses most techniques of integration of elementary differential equations. The importance of such reduction techniques in lowdimensional models is that they can make the difference between the need to deal with chaotic system and the possibility of avoiding them altogether (Sayegh and Jones, 1986). We now examine another aspect of dimensionality reduction, namely, the one that provides for a qualitative analysis of a differential equation. A very good example, physiologically of great interest, is that of the Hodgkin-Huxley equations. Basically the Hodgkin-Huxley equations give rise to behavior of the voltage across a membrane as a function of time. So one has an axon and the voltage across the axon is a function of time. One particular time behavior is that of an action potential development and subsequent evolution of that action potential as a function of time. The equations that were written by Hodgkin and Huxley can basically be understood in the following simplified or modified description. Namely, one variable will represent voltage, and several other variables will describe the conductances of the ion channels across which sodium and potassium ions flow. These conductances themselves are functions of voltage and time and need therefore be defined in terms of a differential equation. While historically things were described a little differently, all in all, Hodgkin and Huxley had one equation for the membrane potential and three more for describing conductance. They had a total of four differential equations. Four differential equations is a manageable number and one that might benefit from reduction because two is very close to four. Indeed, two prevents chaos and allows two dimensional qualitative analysis of the differential equation. The reduction to two dimensions is precisely what Nagumo et al. (1962) and Fitzhugh (1961) have done. They have done a

332


two-dimension analysis of a Hodgkin and Huxley equation. So, essentially by eliminating the phase dynamic and by lumping two of the variables together, Fitzhugh and Nagumo were able to write a two-dimensional system that is represented by

dv - _ - v(a dr

dw- bv

dr

-

v)(v - 1) -w

- yw.

Figure 11 illustrates the dv/dr = 0 and the dw/dt = 0 curves which partition the two-dimensional phase space @to several regions, each region having a certain sign for dv/dt and dw/dt. For example, the lower right quadrant below the sine-like dv/dt = 0 curve would have dv/dt > 0 and dw/dt > 0. One can start with a zero potential through the injection of a current raise the potential to a certain value. Through simple qualitative analysis that takes into account only the signs one can trace the evolution of the system under different excitation conditions. Through such an analysis, one first establishes the existence of a threshold; i.e., if the system is not pushed hard enough, there is no action potential that results and the system settles back to the point and to the voltage that represents the resting condition. However, if the system is pushed hard, current is injected in such a way as to exceed a certain threshold and an action potential results. This can be traced qualitatively and it does display all the known characteristics of an action potential as represented in Fig. 12. The technique of dimensionality reduction is particularly powerful but obviously has some limitations. One obvious limitation is in cases where

I

FIG. 11.


333

Action Potential

FIG. 12.

the number of variables is very large. An example which we have been interested in recently is that of modeling a portion of the hippocampus in the place cell phenomenon as well as the onset of epilepsy (Traub and Miles, 1991; Jaboori et al., 1995). An assembly of 10,OOO neurons each is represented by about 20 compartments and each compartment represented by about five variables. Thus one has on the order of a million variables to deal with, and the reduction of the system by one variable is useless. Other techniques, perhaps from statistical physics, should be brought to bear on such high-dimensional problems. It is not unusual, however, that realistic good models relating to physiology have a lower dimension that can be possibly amenable to the techniques presented above. V. NEURAL NETWORKS

The remaining is a discussion of some historical and introductory aspects of neural networks. Figure 13 represents a brain as one of the most naive and minimal models of a learning system consisting of two neurons and one synaptic weight. Figure 14is an extension showing the same two neurons and a synaptic weight. The underlying principle of a weight implementing a mapping between input x and output y through a simple multiplication of the input by a weight w. The problem of mapping or learning as represented is the problem of finding the perfect w such that y = wx where x and y are known. Clearly one can find w through simple division, and the problem is solved. The above problem of learning can be generalized in at least three different directions. The first direction for generalization simply involves

334


its model FIG. 13.

an increase in the number of input and output lines in order to represent a more complex mapping of patterns or vectors to be learned. The second direction is that of an increase in the number of patterns where a large number of patterns becomes allowed, possibly largely exceeding the number of weights or parameters in the learning system. In the case of linear problems, it turns out that the solution to these generalized problems is still essentially given by a “generalized division” or matrix inversion. The above two generalizations, however, are restricted to solving a class of relatively simple problems. The third direction of generalization is that of a more complex network that is needed to handle more complex problems. The network is more complex both in the sense that it has at least an additional layer of nodes and weights (a so-called hidden layer) and the fact that it needs to introduce a degree of nonlinear processing in order to allow for the learning of the more complex tasks. The standard procedure that is then used to find the unknown weights is that of gradient descent, i.e., a smooth continuous motion in the direction X

Y

W

1 y=wx w =y/x

Solved FIG. 14.


335

that minimizes an error function. Another way of seeing the search procedure that is needed is simply to think in terms of feedback. If one is to start from the arbitrary value of the weight w, produce an output using that weight w, and then change the w via feedback in proportion to the error produced, one does get the same procedure as the one dictated by gradient descent. Notice one must also change w in proportion to the input in order to give credit to different inputs, i.e., if an input has caused a large error, one would like to correct proportionally to that input line. Now the feedback equation can be turned into a differential equation where the change in weight is turned into dw and time is introduced in the form of dr. When one looks at this equation, one sees that the derivative of w is now proportional to w itself plus some extra constant term. As indicated in the beginning of this article, these equations have simple exponential behavior. In this particular case, the initial weight w exponentially can evolve toward the best solution w*. w* can be evaluated and turns out to be the same as the minimum of the error function as previously discussed. The history of neural networks, at least in its popular version, has its angels and demons. One of them is Marvin Minsky, referred to as the devil. Minsky and Papert wrote a book entitled Perceptrons (1969) in which they showed, among other things, some of the limitations of the neural networks that were popular at the time. This was interpreted by some as a very restrictive limit on all kinds of neural networks and it was interpreted by some as being the end of an era of funding for the then popular neural networks in order to make room for the more classical A1 approaches promoted by Minsky and others. In 1982, John Hopfield published a paper that dealt with associative memory that had a compelling analogy to a system of electron spins and could therefore be treated in a reasonably rigorous fashion. It is believed by some that the combination of Hopfield’s reputation and the timing of the publication of the paper has given a new impetus to the field of neural networks that was reinforced by the error backpropagation algorithm (Rumelhart and McClelland, 1986) or, perhaps, the reintroduction of an algorithm that was discovered in the early 1970s but remained unnoticed (Werbos, 1974). Here one wants to think of neural networks as universal mapping devices taking any input sequence to any output sequence of patterns. The true angel was Kolmogorov, (1963) who has proved a theorem in the theory of function of several variables that translate to the universality of a class of neural networks. The neural networks that are universal, however, need to be nonlinear. Linear networks do suffer from the limitations that were outlined by Minsky and others and briefly discussed above. It is therefore necessary to apply nonlinear networks to hard problems. The disadvantage of such an approach is the appearance of local minima

336


that have plagued search techniques in the past (Fig. 15). In other words as one uses more powerful techniques allowing one to handle tougher problems, one is no longer guaranteed that the solutions found will be optimal. The following summarize the advantages of neural networks. (1) Any method can be implemented, no matter how complex, provided one has a complex enough network to perform such a task. A complex

enough network usually means a network with at least three layers of nodes and nonlinear units and an undefined number of hidden units in the intermediate layer between input and output. (2) Neural networks can generalize from the input-output pattern that have been shown. That is, through training, not only do they fit the given input and output, they fit them in such a way that new, previously unseen patterns can be predicted correctly in a large number of cases. An underlying physiological model need not be present. (3) This implies a certain robustness of neural network to perturbation of the patterns that were originally shown as well as a robustness to perturbation of the network itself in the form of destruction or modification or some of the nodes. (4) Neural networks are universal and friendly. The universality aspects usually mean the same as the implementation of arbitrary mapping as in (1) while the friendliness stems from a generalized architecture that is generally provided nowadays in a large number of commercially available software packages which come with a very similar interface. The uninitiated user can quickly adapt to such systems and switch comfortably between different systems. (5) One of the great advantages of neural networks is that the problem is formulated directly on its architecture, i.e., one can solve the problem

But for non trivial NN non linear

FIG. 15.


337

and formulate the corresponding parallel architecture simultaneously. This is to be contrasted with algorithmic approaches where the problem is first solved and, as the problem is solved, a fast, probably parallel architecture is sought to efficiently implement the solution. VI.

CONCLUSION

Once again one can simply conclude that linear systems, although they enjoy a mathematical apparatus of great simplicity and power, have their limitations as to the richness of representation and the limited class of problems they can address. It is only when one deals with nonlinear problems and nonlinear techniques that the wealth of realistic physiological and physical phenomena can be tackled. The use of statistical techniques should be reserved for problems with a very large number of variables. However, for the more common physiologically realistic situations a combination of dynamical system techniques and neural network processing can yield powerful, useful, and elegant results.

REFERENCES Benigni, R., and Giuliani, A. (1994). Quantitative modeling and biology: The multivariate approach. Am 1. Physiol. 266(5, Pt.2). R1697-Rl704. Collins, J. C. (1992). Resources for getting started in modeling. 1.Nurr. 122, Suppl. 3,695-700. Fitzhugh, R. (1%1) Impulses and physiologicalstates in theoretical models of nerve membrane. Biophys. 1. 1,445-466. Hopfield, J . J. (1982). Neural networks and physical systems with emergent collective computational properties. Proc. Natl. Acad. Sci. U.S.A. 79. Jaboori, S., Sampat, P., and Sayegh, S. (1995). Analyzing the hippocampal place cell phenomenon by modeling the visual pathway. In “The Neurobiology of Computation: Proceedings of the Third Annual Computation and Neural Systems Conference” (J. Bower, ed.). Kluwer Academic Publishers, New York. Kolmogorov, A. N. (1963). On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR 144,679-681. Ame. Math. SOC.(Engl. Transl.) 28,55-59. Minsky, M., and Papert, S. (1969). “Perceptrons.” MIT Press, Cambridge, MA. Nagumo, J. S., Arimoto, S., and Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proc IRE 50,2061-2071. Rumelhart, D. E., and McClelland, J. L. (1986). “Parallel Distributed Processing.” MIT Press, Cambridge, MA. Sayegh, S. I., and Jones, G. L. (1986). Symmetries of differential equations. J. Phys. A: Math. Cen 19,1793-1800. Traub, R. D., and Miles, R. (1991). “Neuronal Networks of the Hippocampus.” Cambridge Univ. Press, New York. van Rossum J. M.,de Bie J. E., van Lingen, G.,and Teeuwen, H. W. (1989). Pharmacokinetics from a dynamical systems point of view. J Pharmacokinet. Biophurm 17(3), 365-392.

338


Webber C.L., Jr., and Zbilut, J . P. (1994). Dynamical assessment of physiological systems and states using recurrence plot strategies. 1. Appl. Physiof. 76(2), 965-973. Werbos, P. (1974). Beyond regression: New tools for prediction and analysis in the behavioral sciences. Ph.D. Thesis, Harvard University Committee on Applied Mathematics, Cambridge, MA.

ADVANCES IN FOOD AND NUTRITION RESEARCH, VOL.40

Chapter 21 GRAPH THEORETICAL METHODS FOR PHYSIOLOGICALLY BASED MODELING HONG ZHANG Department of Mathematical Sciences Indian University - Purdue University Fort Wayne, Indiana 4805

ZHEN ZHANG Department of Biometry and Epidemiology Medical University of South Carolina Charleston, South Carolina 29425

I. Introduction 11. The Graph Model

111. Computer Implementation IV. Analysis of Models

References

I. INTRODUCTION

We propose a new mathematical approach to the physiologically based biological modeling (such as pharmacokinetic or nutritional system modeling) using a mathematical concept called graphs. A graph is defined as an object which consists of a set of vertices, a set of edges, and an incidence function which describes the incidence relation between the vertices and edges. Graph theory is a well-developed branch of mathematics and has been successfully applied to various practical problems such as electrical circuit analysis and computer network design. It has also been used in modeling biological systems ( Jacquez, 1985). In many applications, the graph theoretical method is proven to be a natural and effective technique for system analyses and designs. 339 Copyright 0 1996 by Academic Ress, Inc. All rights of reproduction in any form reserved.

340


The graph for a physiologically based model is based on the Aow diagram of the model. The vertices of the graph correspond to the junctions in the flow diagram. Each body region (or compartment) is represented by an edge in the graph. Two edges are adjacent if the corresponding body regions are linked in the flow diagram. The graph approach is especially useful for implementing automatic modeling systems. A computer program is written to illustrate the application. The program provides a convenient graphic interface for users to set up the graph and enter the parameters. Based on the graph model the program automatically generates the equations and solves the system. The graph method is also useful for theoretical analysis of the models. We will present some theoretical results obtained by using the graph model and applying results from graph theory. In particular, we give a method to determine a minimal set of flow rates in a model and we derive a necessary and sufficient condition for certain systems to be minimal.

11.

THE GRAPH MODEL

The graph for a model is based on the flow diagram of the model. The vertices of the graph correspond to the junctions in the flow diagram. Each tissue region (or compartment) is represented by an edge in the graph. Two edges are adjacent if the corresponding body regions are linked in the flow diagram. The edges of the graph will be oriented. It is natural to orient the graph according to the directions of flows. An example of a physiologically based model is shown in Fig. 1. The corresponding directed graph for the model is given in Fig. 2.

Fat FIG. 1. A model.

I

CHAPTER 21 PHYSIOLOGICALLY BASED MODELING

341

FIG. 2. Graph for model in Fig. 1.

This method of constructing the graph may seem unusual because it associates tissue regions to edges rather than vertices of the graph. However, this method will provide a more natural and mathematically convenient description of the topological structure of a physiologically based model. For conventional (not physiologically based) compartment models, representing the compartments with vertices is certainly the natural approach, since the links between compartments are usually specified separately. Physiologically based models, on the other hand, often involve more complicated mutually dependent connections among the tissue regions. For example, three or more different compartments may be connected at a common point and clearly their flow rates will be related. In this case using the vertex representation will require an additional procedure of separating the connections to several single connections between two compartments because an edge of a graph has only two end points and it cannot represent a connection among three or more compartments. The edge representation will be much more flexible because there is no restriction on the number of edges at a vertex and virtually any type of reasonable interconnections among the tissue regions can be naturally described by this method. This method will give a topological representation that is closer to the original model and that is mathematically easier to manipulate, especially when the structure of the model is complex. This type of graph model is also commonly used in electrical engineering for modeling complex electrical circuits (Deo, 1974). The graph provides a simple description of the topological structure of the model. For simplicity, we assume that the model is flow-limited and that the tissue total concentration is combined with the equilibrium blood total concentration. Because we are mainly concerned with the modeling methodology and the generalization of the method to more complicated models is straightforward, the discussion of this simplified case is sufficient.

342


The mathematical equations for the model can be derived from the oriented graph of the model, the parameters of the tissue regions, and a few simple general rules. The rules are fixed for all models of a certain general category. They are not specific to a particular model and they apply to all tissue regions of the model identically. Consequently this graph model provides a simple and convenient way for computer implementations. A computer modeling program will implement the fixed set of rules. Users only need to enter the graph and the parameters for their model. The program will be able to automatically perform all the necessary calculations for any given graph and the associated parameters. There is no need to specify any special rule for any individual region. All characteristics of a tissue region are completely determined by its parameters and its relation with other tissue regions which is given by the graph. The rules for the models we considered here are given as follows. 1. At each vertex, the sum of flow rates Qi directed away from the vertex is equal to the sum of flow rates directed into the vertex. 2. At each vertex, the sum of mass transfer rates away from the vertex is equal to the sum of mass transfer rates into the vertex. 3. For an edge started at a vertex, the mass transfer rate directed into the edge is proportional to the flow rate on the edge. 4. For an edge terminated at a vertex, the mass transfer rate directed away from the edge is QiCi/R1where Ri is the tissuehlood partition coefficient.

We will use the incidence matrix to represent a graph. Let G be a directed graph with m vertices and n edges. The incidence matrix M = [Mv] of G is a m X n matrix defined as

M..=

I

1, -1, 0,

if vi is the initial vertex of ej if vi is the terminal vertex of ej otherwise.

We will also need to separate the positive and negative parts of the incidence matrix. Let M be the incidence matrix of a directed graph. The positive and negative parts of M, denoted by M, and M-,are defined as

I

1, if Mij = 1 1, i f M V = -1 and Mii- = 0, otherwise 0, otherwise.


343

Clearly we have M = M + - M-. The graph for the model shown in Fig. 1is given in Fig. 2. Its incidence matrix, positive and negative parts are

M , t l -1 - 1 1

-1 1

-1 1

I

1 0 0 0

M + = [o 1 1 11 0 1 1 1 M- = [l 0 0 01. We will derive the equations for a model in matrix form. The parameters for tissue regions in a model are denoted by the following matrices.

V = diag( V1, . . . ,V,) the diagonal matrix of volumes of tissue regions. Q = diag(Q2, . . . , QJ the diagonal matrix of blood flow rates. R = diag(R1, . . . , Re) the diagonal matrix of partition coefficients. C = (Cl, . . . , QTthe vector of concentrations in the tissue regions. the vector of rates of injections. In = (Znl, . . . , the vector of rates of elimination. Ex = (Exl, . . . , We need an matrix operation that cannot be conveniently expressed as conventional matrix operations. This operation, denoted by AD, acts on a square matrix A and produces a diagonal matrix with row sums of the square matrix at the diagonal entries. Let

A=

then

[

a11

AD=

+ a12 + . . . + a1, 0 0

0

a2,

+ a22 + . . . + 0

... 0 ... 0 ... . . . an1+a,2+.

..+a,

1.

344


For example, if

then,

Given the rules, incidence matrix of the graph, and parameters defined above, we may derive the differential equations for the model directly using only matrix operations.

Theorem 1 The system of differential equation for the model with the graph and parameters defined above is given by V

dC dt = (FM,=M-

-

f ) Q R - ' C + In - Ex,

where

Prooj At an edge e; of the graph, which corresponds to an tissue region in the model, the rate of change of mass satisfies the mass balance equation

dC; = ri + Inj - si " 5

EX^,

where r; and s; denote the mass transfer rates directed into and away from the edge respectively. Clearly s; can be determined by Rule 4.

To determine ri we apply Rule 2 and Rule 3. Let the initial vertex of e;


345

be vi. Then the jth row of M - indicates the edges directed to vi. The sum of all mass rates directed into the vertex is given by n

where (M-)jk denotes the entry (j, k) of M-. The algebraic sum of the mass rates at a vertex is 0 by Rule 2. Hence the above total sum is redistributed among the edges started at this vertex. The amount of mass transfer rate directed into an edge is proportional to the flow rate by Rule 3. Therefore, the mass transfer rate into the given edge is

Combining the equations into matrix form, we obtain the system equation in the theorem. Example. Consider the model shown in Fig. 1 (cf. Bischoff, 1987).

1

0

0

0

0

0

QdQi

By theorem 1, we obtain the system equation V1 * dCi/dt V2 * dCddt

[: :::$] [ =

-Qi/Ri Q2/R1 QdRi Q3iR1

QdR2 -Q2/R2

0 0

QdR3 0 -Q3/R3

0

If the above matrix equations are expanded in component forms, we will obtain the usual differential equations for the model. In our graph modeling technique for physiologically based systems, parallel edges can be handled naturally since we use the incidence matrices for representation. However, loops (edges with same initial and terminal

346


vertices) are not allowed. There is a special case that two junctions are directly connected. This may cause a loop in the graph if the junctions are identified as one vertex. We can resolve the problem by introducing a special region with volume 0 represented by an edge that corresponds to the connection between the two vertices. The above procedure can still be applied to produce the correct equations for the system. The only difference is that the equation for the special edge does not have the derivative term since the volume is 0. Hence the resulting system matrix equation is an algebraic-differential equation. This type of equations have been studied extensively and several numerical methods are available (Gear, 1971;Zwillinger 1989). This special case is illustrated by the following example. Example. Consider the system in Fig. 3 (6. Gibaldi and Pemer, 1982). The corresponding graph is shown in Fig. 4. Note that q is the special edge used to avoid a loop. It corresponds to a region with volume 0. The incidence matrix is

*

.t

Muscle


e, FIG. 4. Graph for model in Fig. 3.

M =

[

1 -1 0 0 -1 -1 -1 0 1 - 1 - 1 0 0 0 - 1 0 1 1 1 1 1

By theorem 1 the system equation can be written as

-1 0 1

-1 1

347

348


Since V3 = 0 and R3 = R , = 1, from the third equation we have C3 = C1. Substituting the variable C3 using this algebraic equation, we obtain the usual equation for tissue region 1.

V,dCZ = dt R,

+

& R4

4

Q2 - -cz. R2

Ill. COMPUTER IMPLEMENTATION This graph theoretical approach to the models has several advantages. It provides a systematical way to obtain the mathematical equations foq the models. This is especially useful for computer implementation and simulation of the model. To illustrate this application, we implemented a computer program for constructing and simulating the models. The program runs under Microsoft Windows with an easy-to-use interface. Users establish a graph model by clicking the appropriate tool buttons and drawing the vertices and edges with a mouse. Parameters for a compartment are entered through a dialogue box. The model can be conveniently edited by using operations such as move, cut, copy, and paste. Once the graph model with the associated parameters is established, the program will calculate the equations automatically. It will also solve the system numerically and plot the results of simulation. The program achieves the generality and simplicity by applying theorem 1. It is conceptually straightforward to implement. The program maintains an incidence matrix of the model based on user inputs. Each edge object contains data structures for various parameters associated with the region. With the incidence matrix and the parameters on edges, the system equation is calculated through direct matrix operations according to theorem 1. Numerical methods are used to solve the resulting system equation. The program can be used as a tool for teaching and research. It is available from the first author at the e-mail address [email protected]. IV. ANALYSIS OF MODELS

The graph method provides a convenient mathematical tool for the analysis of the models. Here we present some results on the models obtained through the properties of their graphs. Clearly the flow rates Qi are not always independent because of Rule 1. A natural problem is to find a minimal set of flow rates that completely determines all Row rates in the model.


349

Theorem 2 A minimal set of flow rates that completely determines all flow rates contains e - n + 1values. Proof: Let q = (Qb Q2,

. . . ,Q,)?

Because of Rule 1,the vector q sat-

isfies

M 4 = 0. Hence the vector q is in the cycle space of the graph. By a result in graph theory, the row space of the incidence matrix is the bond space of the graph and the nullspace of the incidence matrix (the orthogonal complement of the row space) is the cycle space of the graph. Since the graph we consider here is always connected, the cycle space of the graph has dimension e n + 1 (cf.Bondy and Murty, 1976). Consequently, among the e flow rates, only e - n + 1 of them are linear independent. n - 1 values can be expressed as linear combinations of the basis vectors. There are also techniques in graph theory to determine a basis for the cycle space, which yields a set of independent values of flow rates. One simple method involves a spanning tree of the graph. A spanning tree of a graph is a connected subgraph with the same vertex set as the original graph and with no cycles. Let T be a spanning tree of the graph. Then all values on edges not in T form an independent set which generates all values in the graph. For example, in Fig. 1, the flow rates Q,, Q3. and Q4 form an independent set. The flow rate Q2 can be expressed as a linear combination of Q,, Q3, and Q4. For the model given in Fig. 3 and Fig. 4, e2 and e3 form a spanning tree of the graph. Hence the flow rates Q2 and Q3 can be expressed as linear combinations of other rates. Another application of the graph model is to derive equivalence for certain parts of the model. In analysis and simulation of the model, it is often desirable to combine several components to an equivalent single component. The overall reduction of a model to a minimal system will be of great value not only for computations but also for theoretical study of essential characteristics of the system and comparison to the traditional compartment modeling. The graph model will facilitate the development of such equivalence transformations. We will derive a necessary and sufficient condition for minimality of a special type of models. The models are similar to the one illustrated in Fig. 1. It contains one central region in one direction and several other regions connected to the central region in the other direction. Theorem 3 A linear system defined above is equivalent to a system with fewer states if and only if there exist two edges satisfying

350


-

Qi

ViRi

Qi

VjRi

In order to prove the theorem, we need some concepts and results from linear system theory. For convenience, we state the results here (see Kailath, 1980, for details). A linear system is defined by

du _ - AX + By dt

=

cx.

A linear system is said to be controllable if the system can be taken to any desired state x by controlling the input function. A linear system is said to be observable if the states x ( t ) can be determined from the observation of the output function. The following is a useful result for testing controllability and observability.

Lemma 1 (Popov-Belevitch-Hautus tests) 1. A linear system is noncontrollable if and only if there exists a nonzero row vector q such that qA = Aq and q B = 0. 2. A linear system is nonobservable if and only if there exists a nonzero column vector p such that A p = Ap and Cp = 0. The following result establishes the connection between the minimality of a system and its controllability and observability. Lemma 2 A linear system is minimal if and only if it is both observable and controllable.

Proof of theorem 3. If the condition is satisfied, we may combined the two edges into one with the following new parameters.

v. = vj + vj Q* = Qi + Qj

C*= (ViCi + VjCj)/(Vi + Vj) R* = (VjRj + VjRj)/(Vj + Vj). It is straightforward to verify that the new system is equivalent to the original system. Conversely if the system is not minimal, then it is noncontrollable or nonobservable. By Popov-Belevitch-Hautus tests, one of the two cases i s true.


351

Case 1: qV-’ (FM+TM- - Z)QR-l = A q and qb = 0 41 = 0 Q2

m=O + . . . + - qQm

-92

V2R2 VmRm Qi = A q , i = 2, . . . ,m. --qi ViRi

Case 2: V-’ (FM+TM-

- Z)QR-’p

=

Ap and cp = 0

PI = 0 pQ2

VlR2

2

Qm + . . . + pV1Rm m=0

Either case will lead to the condition in the theorem. Example. For the system in Fig. 1, let Q3 = 2.0, V3 = 1.0, R3 = 0.2, Q4 = 3.2, V4 = 0.8, and R4 = 0.4. Then the system is not minimal since we have

According to theorem 3, compartments 3 and 4 can be combined to form a new compartment with parameters V* = V3 f V4 = 1.8 Q* = V , f V4 = 5.2 C* = (V3C3 + VdC4)I(V3 3- V4) = 0.56C3 + 0.44 C4 R* = (V3R3 + V4R4)I(V3 + V4) = 0.29.

The new system has only three tissue regions (or compartments), but it is equivalent to the original system. From the observations of compartments C1 and C2, the new system is indistinguishable from the original system. The new combined concentration C8 behaves as a weighted average of original values C3 and C4.

REFERENCES Bischoff, K. B. (1987). Physiologicallybased pharmacokineticmodeling,I n “Pharmacokinetics in Risk Assessment-Drinking Water and Health,” Vol. 8, pp. 36-61. National Academy Press, Washington DC.

352


Bondy, J. A., and Murty, U. S. R. (1976). “Graph Theory with Applications.” North-Holland Publ., New York. Deo, N. (1974). “Graph Theory with Applications to Engineering and Computer Science.” Prentice-Hall, Englewood Cliffs, NJ. Gear, C. W. (1971). Simultaneous numerical solution of differential-algebraicequations. lEEE T r m . Circuit Theory CT-18(1), 89-95. Cibaldi, M., and Pemer. D. (1982). “Pharmacokinetics,” 2nd ed., Dekker, New York. Jacquez, J. A. (1985). “Compartmental Analysis in Biology and Medicine,” 2nd ed. Univ. of Michigan Press, Ann Arbor. Kailath, T. (1980). ”Linear Systems.” Prentice-Hall, Englewood Cliffs, NJ. Zwillinger, D. (1989). “Handbook of Differential Equations.” Academic Press, San Diego, CA.

INDEX

A Absorption, see also Membrane vesicle p-carotene, 32,41,58-60,69-70,71 folate, 82 population kinetics, 273 Accumulation, folate, 101-104 para-Acetamidobenzoylglutamate, 86,91 Action potential, 331-332 Adaptive control, 270 Adipose tissue p-carotene, 49, 61 microdialysis, 194 Adrenals, vitamin A, 9, 11 Alcoholism, 84-85, 91 Amino acids dose-response, 157-160,162-166 protein quality improvement, 161-162 substitution, 150 para-Aminobenzoylglutamate, 86, 91 Amyloidosis, 150-155 Amyloid protein, 150; see also Transthyretin metabolism Analyte recovery, 187-190 Antioxidant status, 58 apABG, see paraAcetamidobenzoylglutamate APE, see Atom percentage excess Area under concentration-time curve, 50-52,60,70 Area under moment curve, 50-52 Atom percentage excess, 67,68, 71-72 Attractor, strange, 327 AUC, see Area under concentration-time curve AUMC, see Area under moment curve

B Basolateral membrane, 198-200 Bayesian estimation, 269-270 Biochemical systems theory, 233,236-238 Biological rhythm, see Bioperiodicity Bioperiodicity, 217-226 categories, 219-220 characterization, 218-220 food intake, weight gain, 220-226 human subject, 225 Brain neural network model, 333-334 vitamin B6, 112 Break-point approach, see Linear system; Rectilinear approach Brush border membrane, 197-199,200-204 BS integrator, see Bulirsch Stoer integrator Bulirsch Stoer integrator, 290, 297,299-303

C Calorimetry indirect, 171, 172,178-179 room, 171, 172 Cancer, 57 Capillary membrane, 184, see also Membrane transport; Membrane vesicle Carbon dioxide, energy expenditure, 173-175,177-178 Carbon-13 tracer, 64-75, 171-172 Cardiovascular disease, 57-58 &Carotene metabolism, 25-54,55-79; see also Retinol; Retinol-binding protein; Retinyl ester; Vitamin A metabolism absorption, 32,41,58-60, 69-70.71 353

354

INDEX

&Carotene metabolism (continued) adipose tissue, 49, 61 animal model limitations, 61-62 antioxidant status, 58 background information, 26-29,56 bioconversion, 59-60 biological effects, 57-58 cancer, 57 cardiovascular disease, 57-58 catabolic rate. 61 chylomicrons, 32, 37,41,62-63, 69, 71 compartmental modeling, 26-54 behavior predicted, 43-50 construction, 32-35 empirical descriptions, 45, SO-52 final version, 41,43 iteration, 35-43 materials, methods, constraints, 29-32 statistical evaluation, 40-43 unobservable behavior, 4 - 5 0 dietary factors, 59,60 distribution, 69-70 enterocyte, 36-37.38-39 gastrointestinal delay, 36-37 human subject, 29-32,62-67 immune function, 58 organs. 33,49 liver, 38-40,46-48,60,69,71 tracer methods, 26.30-31,42-43,63-75 carbon-l3,64-75 dosage, 65 GC-C-IRMS analysis, 66-68,71,73-74 GC-C-IRMS data, 68-70 nontracer versus, 62-64 plasma analysis, 65-66 utility, 64.70-75 transport, 60-61 vitamin A conversion, 49-50, 59 Carotenoid, 59-61; see also @-Carotene metabolism Camer-mediated transport, 251-257, 261-262 Casein, 231-232,238-240 Catabolic rate 8-carotene, 61 folate, 88-89, 90 vitamin A, 7, 13 CBCCDS integrator, 297,299-303 cDNA, see Complementary DNA Chaos, 324-326

Cholesterol, 57-58; see also Lipoprotein Chromatography GC-C-IRMS, 66-70, 71,73-74,177 HPLC, 29,60,66,72,74 Chronobiologic rhythm, 218 see also Bioperiodicity Chronobiology, 224-226; see also Bioperiodicity Chu Berman integrator, 290-292 Chylomicrons 8-carotene, 32, 37, 41, 62-63, 69, 71 vitamin A, 4, 16 Circadian rhythm, 219; see nlso Bioperiodicity 8-C metabolism, see 8-Carotene metabolism Commutator. differential equation, 329-330 Compartment, definition, 13 Compartmental modeling &carotene, 26-54 dynamic systems, 333 folate, 100-101 graph theory, 340-341,348-351 identifiability, 306, 307, 309-310, 312 membrane transport, 244 parameters, 34.40-43 population kinetics, 267 transthyretin, 152-154 vitamin A, 9-21 Complementary DNA, 96, 99, 101; see also Genetic transfection Computational modeling, see Dynamic system; Graph theory; Identifiability problem; Kinetic parameter distribution; Linear integrator; Neural network Concentration-time curve, 50-52, 60, 70 Conditional identifiability, 314 Conductance, membrane, 248-249, 252-257,331-332 Connectivity, modeling, 27, 32, 43 identifiability, 306, 315 CONSAM software, 6, 9, 21,32, 274, 290 Corn gluten meal, 160 Cosinor method, 221 Covariate, population kinetics, 272-273 Creatinine, ultrafiltration, 192-193 Curvilinear approach, see Nonlinear system CVD.see Cardiovascular disease

INDEX Cytosolic folate, 99, 100-101 Cytotoxicity, 104

D DASSL integrator, 290 Deuterium, 116,173-174 DG uptake, see Glucose, transport Diabetes, 191-192 Dialysis micro-, 183-190,194-195 retrograde, 190 Dialysis fiber, 184, 185-186 Difference method, microdialysis, 188-189 Differential equation background information, 283-290 commutator, 329-330 dynamic system, 328-333 graph model, 344-348 linear integrators, 283-292 Differential operator, 258-259 Diffusion, passive, 248-251 Dimensionality, 325-333 fractional, 326-328 Hausdorff-Besicovic,328 reduction, 328-333 Diminishing return, 157-166 Directed graph, 229-230,340-341 Distribution, see also Kinetic parameter distribution posterior, 268 prior, 268,269-271,273-276 DNA, see Complementary DNA; Genetic transfection Dose-response, amino acids, 157-160, 162-166 Doubly labeled water method, 171-180 assumptions, 173-174 equations, 174-175 isotopes, 175-176 mass spectrometry, 176-178 validations, 178-179 Down syndrome, 113 Dynamic system, 323-338 differential equations, 328-333 geometric interpretation, 328-333 linear modeling, 324 neural networks, 333-337 nonlinear modeling, 324-328

355 E

Eating disorder, 225 EHT, see Extrahepatic tissue Elasticity coefficient, 234-236 Endogenous rhythm, 219,226; see also Bioperiodicity Endothelial cell, 245,246, 248, 250 Energy metabolism, 171-180,227; see also Nutrient-response curve Enrichment ratio method, 45 Enterocyte, 36-37,38-39 Enzyme, metabolic control theory, 233-236; see also Substrate Epilepsy, 333 Epithelial tissue cell membrane, 197-204 mammary cells, 207-212 vitamin A, 20 Equilibration, vitamin B6 metabolism, 126 Error measurement, 307 Error model, 268 Erythrocyte folate, 87-88 vitamin B6, 111 Escherichia coli, 97-101, 105 Estimation likelihood, 268-269 parameter, 311-312 population kinetics, 266-277 prior, 273-276 Exogenous rhythm, 219; see also Bioperiodicity Extrahepatic tissue, see also names of specific organs 8-carotene, 32,49, 61 vitamin A, 9, 11, 17-21 Extrapolation, zero flow, 189 Eyes, vitamin A, 9, 11,19

F Familial amyloidotic polyneuropathy, 150-155 FAP, see Familial amyloidotic polyneuropathy Feedback, neural network, 335 Flow algorithm, 247-248 Flow diagram, 340 Flow rate, 342-345, 348-351

356

INDEX

Folate metabolism, 81-93.95-106 absorption. 82 alcoholism, &1-85,91 background information, 81-83, 95-97 catabolism, 85-90 cytosolic, 99, 100-101 dietary factors, 82, 89 erythrocytes, 87-88 Escherichia coii, 97-101, 105 folypolyglutamate chain length, 97-100 folypolyglutamatesynthetase, 97-1 01 accumulation. 101-104 chain length, 97-100 companmentat ion, 100-1 01 cytotoxdty, 104 rttectlverIe~,98-100 retention, 97-98 genetic transfection, 95.96-101, 105 human subject, 86-90 intestines, 82, 201 membrane vesicles, 201 mitochondrial, 99, 100-101 pregnancy, 86, 91 RDAs, 83 stable-isotopic studies, 86-90 tracer methods, 86-90 turnover, 83-85, 89-90 Folic acid, 82, 83.87-88; see also Folate metabolism Folylpolyglutamate synthetase, 97-101 Food intake, bioperiodicity, 220-226 Food rerord. 171 Forcing function method, 19-21 Fractal, 327-328 Fractional catabolic rate @-carotene.61 folate, 88-89, 90 population kinetics, 273 transthyretin, 152 vitamin A, 7, 13 Fractional dimension, 326-328 Fractional standard deviation, 29, 41 Fractional transfer coefficient, 31, 40-43, 267-268,312 FSD, see Fractional standard deviation FTC, see Fractional transfer coefficient

G Gas chromatography, IRMS, f&-70,71, 73-74, 171

Gastric cancer,57 Gastrointestinal delay, 36-37 GC-C-IRMS, see Gas chromatography, IRMS Genetic transfection, 95, 96-101, 105 GI delay, see Gastrointestinal delay Global identifiability,313 Glucose transport, 207-213 ultrafiltration, 191-192 Glycine, 98-99 Glycogenolysis, 136 Glycogen phosphorylase metabolism, 135-147; see also Vitamin B6 metabolism cofactor labeling, 137-140 role, 135-136 growth, muscle, 142 labeling methods, 137-140 radiolabeling, 137- 138 stable isotope, 138-140 McArdle’s disease, 142-145 messenger M A , 141,144, 145 model systems, 140-142 pyridoxal phosphate, 135-136 turnover, 124, 137-140, 142 vitamin B6, 115-116, 118,121 wasting, muscle, 140-141 Graded levels, nutrient, 157-166 Graph, definition, 339 Graph theory, 339-352 computer implementation, 348 model analysis, 348-351 model building, 340-348

H Hausdorff-Besicovicdimension, 328 HDL, see Lipoprotein Heart, membrane transport, 259 High-performance liquid chromatography, 29,60,66,72,74 Hippocampus, modeling, 333 Hodgkin-Huxley equations, 331-332 Hollow fiber, 184 HPLC, see High-performance liquid chromatography Human subject bioperiodicity, 225

357

INDEX p-carotene metabolism, 29-32.62-67 energy metabolism, 176, 178 folate metabolism, 86-90 total energy expenditure, 171-180 transthyretin metabolism, 150-154 vitamin B6 metabolism, 115,121, 145

IRMS, see Isotope ratio mass spectrometry Irreversible utilization, 12 Ischemia, 194-195 isotope ratio mass spectrometry, GC, 66-70,71,73-74, 177 Isotope tracer, stable, see Stable isotope tracer

I identifiability conditional, 314 global, 313 interval, 314 lo~al,313,318-320 quasi, 314 structural, 313 Identifiability problem, 305-321 background information, 305-310 checking methods, 314-317 linear systems, 315-317 nonlinear systems, 317 classification of identifiability, 313-314 classification of parameters, 310-311 correlations, 320 estimation problem, compared to, 311-312 nutrient-response analysis, 235,240 population kinetics, 276-277 single point, 318-320 identifiable parameter, 311 IDENT software, 318 IDL, see Lipoprotein Immune function, 58 Incidence matrix, graph theory, 342-347 Indirect calorimetry, 171, 172,178-179 Infradian rhythm, 219,225,226; see ako Bioperiodicity Initial value problem, 284,325; see also Differential equation Insensible parameter, 311,319 Integrator, linear, see Linear integrator intestines &carotene, 49 folate, 82, 201 membrane transport, 245-251,255-257, 259 membrane vesicle, 198-199 vitamin A, 9, 11, 19 vitamin B6.109-110,115,121 Iodination, 8

J Jejunum, 82,115; see also intestines

K Kidneys &carotene, 33,49 membrane vesicle, 197, 200,201,202 microdialysis, 194-195 vitamin A, 4, 9, 11,20-21 vitamin B6, 201 Kinetic parameter distribution, 265-279 adaptive control, 270 applications, 271-273 Bayesian estimation, 269-270 covariates, 272-273 identifiability, 276-277 missing values, 271-272 Monte Carlo simulations, 270-271 prior, estimation, 273-276 naive data pooling, 273 nonparametric, 276 parametric, 275 semiparametric, 275-276 two-stage methods, 273-274 prior, uses, 268,269-271 sparse data, 265-266,272 theory, 266-269 error model, 268 likelihood, 268-269 parameter estimation, 269 system model, 267-268

L Lactation, 112-113 LaGrangian flow, 247 Laplace transform, identifiability, 316-3 17 LDL, see Lipoprotein Learning problem, 333-335

INDEX

358 Least squares bioperiodicity, 221-224 identifiability, 318 Leukoplakia, oral, 57 Ligand binding, 122-127 Likelihood estimation, 268-269 Linear integrator, 281-303 background information, 281-283 Bulirsch Stoer, 290,297,299-303 CBCCDS, 297,299-303 L7u Berman, 290-292 DASSL, 290 differential equations, 283-292 evaluation, 293-300 Petzold, 297,299-303 Runge Kutta, 290,291,297,299-303 Linear system, see also Rectilinear approach graph model, 349-351 identifiability. 315-317 limitations, 324, 335, 337 membrane transport model, 246 Lipid uptake, 197, 198, 202-203 Lipoprotein HDL, 38-39,57 IDL, 69 LDL, 38-39,58,62,69 VLDL, 38-39,69 Liver 8-carotene, 38-40,46-48,60,69, 71 vitamin A, 4, 21 compartmental modeling, 14-18 empirical compartmental analysis, 12-14 whole-body models, 9-11 vitamin B6, 108, 110-112,115, 121 Local identifiability, 313 Logistic equation, nutrient response, 159-160 Lungs cancer, 57 &carotene. 49 vitamin A, 9. 1 1

M Mammary tissue culture, 207-213 Manipulation facilities, modeling, 282, 283 Maple software, 330

Mass spectrometry &carotene, 66-70.71, 73-74 total energy expenditure, 176-178 Mathematical modeling, importance, 26, 244,261 Mathematica software, 330 Matrices graph theory, 342-347 identifiability,315-316 McArdle’s disease, 142-145 Mean residence time, see Residence time Membrane conductance, see Membrane transport, conductance Membrane probe, 183-184, 185-186 Membrane transport, 243-262 background information, 243-245 carrier-mediated, 251-257,261-262 saturability, 251, 261 single-site, 252-255 tracer studies, 255-257 conductance, 248-249,252-257,331-332 differential operators, 258-259 multiregion model, 245-248 assumptions, 246 solution methods, 247-248 organ models, 259 passive diffusion, 248-251 potential, 331-332 tracer washout rate, 249-250 whole-body models, 259-260 Membrane vesicle, 197-206 brush border, 197-199,200-204 intestinal basolateral, 198-200 intestinal luminal, 198-199 lipid uptake, 197, 198,202-203 preparation, 199-200 structure, 202-203 transport, 200-202,203,204 MESOSAUR software, 221,223 Messenger RNA, 141, 144, 145 Metabolic control theory, 233-236 Metabolic pathway, 107-110, 227,233-234, 240 Methionine FAP disease, 150-155 supplementation, 161-162 Met30 transthyretin, 150-155 MG uptake, see Glucose, transport Michaetis constant, 234

359

INDEX Michaelis-Menten kinetics enzyme substrate, 158 initial velocity, 307-309 population kinetics, 268 vitamin B6, 117 Microdialysis, 183-190,194-195 Microgravity, 121 Missing values, 271-272 Mitochondrial folate, 99,100-101 Modal matrix, identifiability, 315-316 Modefing, see &o Compartmental modeling; Parameter; Software connectivity, 27, 32,43, 306, 315 constructs, 282 dynamic systems, 323-338 Modeling, computational aspects, see Dynamic system; Graph theory; Identifiability problem; Kinetic parameter distribution; Linear integrator; Neural network Moment curve, 50-52 Monte Carlo simulation, 270-271 Mormon cricket meal, 160-163 mRNA, see Messenger RNA Mucosal cell, 245,246,250 Muscle,see also Glycogen phosphorylase metabolism growth, 142 ultrafiltration, 193 vitamin 86 metabolism, 115-116, 120-121 wasting, 140-141

N Naive data pooling, 273 Neural network, 333-337 Nitrogen dose-response, 157-160,164-166 protein quality, 160-162 NLMIX software, 275 Nonidentifiable parameter, 311 Nonlinear system advantages, 158,317,337 chaos, 324-326 dimensionality, 325-328 identifiability, 317 linear differential equations, 285-289 neural networks, 335-336

nutrient response analyses, 157-166, 228-229 NONMEM software, 275 NPEM2 software, 276 Nucleic acid synthesis, 81 Nutrient-response curve, 157-166,227-241 biochemical systems theory, 233,236-238 data analysis. 238-240 equation derivation, 229-232 graded level intake, 157-166 metabolic control theory, 233-236 perturbation response, 285-289 rational polynomials, 228-229, 231

0 Oral leukoplakia, 57 Oxygen-18, energy metabolism, 173-178

P pABG, see para-Aminobenzoylglutamate Parameter, see also Kinetic parameter distribution classification,310-311 definition, 34 estimation, 311-312 evaluation, 40-43 identifiable, 311 insensible, 311,319 nonidentifiable, 311 sensible, 311, 320 Parameter sharing, 159-160 Parenchymal cell, 15-17,40 Passive diffusion, 248-251 PC, see Parenchymal cell Peanut protein, 231-232,238-240 Perturbation response, 285-289 Petzold integrator, 297,299-303 Pharmacokinetics graph theory, 339 population analysis, 265-278 Phloretin, 208-209,211 Phlorizin, 208-209 Phosphorylase,see Glycogen phosphorylase metabolism Physiologic rhythm, 217; see alro Bioperiodicity PLP, see F'yridoxal phosphate Poisson flow, 247

360

INDEX

Polyglutamate, 97, 99 Polyglutamylation, 91, 103 Popov-Belevitch-Hautustests, 350-351 Population kinetics, see Kinetic parameter distribution Posterior distribution, 268 Potential, action, 331-332 Pregnancy, 86,91, 112-113 Prior distribution, 268.269-271.273-276 Probe, sampling, 183-184, 185-186.188, 190 Processing support, modeling, 282-283 Protein, see also Glycogen phosphorylase metabolism; Retinol-binding protein; Transthyretin metabolism amyloid, 150 binding, 117, 122-127 nutrient-response, 231-232,238-240 peanut, 231-232,238-240 quality. 160-162, 165-166 recombinant, 8 Pt, see Phloretin Pterin, 86.91 Pteroylglutamate, 81-82,97,9!2, 103; see also Folate metabolism Pteroylglutamic acid, 82; see also Folic acid Pyridoxal phosphate, 121-122, 123, 135; see also Vitamin B6 metabolism as phosphorylax cofactor, 136-140, 142- 14s Pyridoxamine metabolism, 121-122, 123; see also Vitamin 86 metabolism Pyridoxic acid, 112,116,117,120-125, 127 Pyridoxine metabolism, 121-122, 123; see &o Vitamin B6 metabolism PZ.see Phlorizin

Q Quasi-identifiability,314

by vitamin B6, 137-138 for vitamin 86, 114-116 RBP, see Retinol-binding protein Recombinant protein, 8 Recommended daily allowance, folate, 83 Recovery, analyte, 187-190 Rectilinear approach, 157-158; see also Linear system Reduce software, 330 Residence time &carotene, 44-46.61 folate, 88-89, 90 transthyretin, 152-155 Respiratory quotient, 175 Response curve, see Nutrient-response curve Retinoid, 38-39,44, 47-49 Retinol 8-carotene, 30, 31, 33, 66 empirical data, 50-52 final model, 43-47 intermediate models, 39 vitamin A, 3, 9, 11, 14, 15-16 pool size, 21-22 turnover, 12-13, 14, 19.21 Retinol-binding protein 8-carotene, 32-33, 38,46 vitamin A, 4, 8, 9, 11, 14, 16-17 kidneys, 21 Retinyl ester @-carotene,32, 59, 66, 69 intermediate models, 38-40 turnover, 47 vitamin A, 4. 11, 16 Retrograde dialysis, 190 Rhythm, biological, see Bioperiodicity RK integrator, see Runge Kutta integrator RNA, see Messenger RNA Room calorimetry, 171,172 Runge Kutta integrator, 290, 291, 297, 299-303

R Radical exchange, computation, 248 Radioactive tracer, 64;see also Stable isotope tracer glycogen phosphorylase, 137-138 membrane vesicles, 200,203 t ransthyreti n, 150- 152

S SAAM software, 6,9,31,61, 126-127,274 linear integrator, 290, 296, 303 modeling construct, 282 Sampling probe, 183-184, 185-186,188,190 SAS software, 50, 160

361

INDEX Saturability, membrane transport, 251, 261 SC, see Stellate cell Sensible parameter, 311,320 Sensitivity analysis, 7, 234-235, 238-240 Separation technique, see Microdialysis; Ultrafiltration Shannon criterion, 224 Similarity transformation, identifiability, 315,317 Simulation, see also Bioperiodicity; Membrane transport; Nutrientresponse curve graph model, 348 population kinetics, 270-271 Skin, 9,49 Software, 281-283; see also Linear integrator CONSAM, 6,9,21,32,274,290 IDENT, 318 Maple, 330 Mathematica, 330 MESOSAUR, 221,223 NLMIX, 275 NONMEM, 275 NPEM2, 276 Reduce, 330 SAAM, see SAAM software SAS, 50, 160 SYSTAT, 221 USC*PACL, 276 Sojourn time p-carotene, 44-46,49,51-52, 61 vitamin A, 12-13 Sparse data situation, 265-266,272; see also Kinetic parameter distribution Spectrometry, mass, see Mass spectrometry Stable isotope tracer, see also Radioactive tracer p-carotene, 64,72,74 folate, 86-90 glycogen phosphorylase, 138-140 total energy expenditure, 175-176 by vitamin B6.138-140 for vitamin B6.116 Standard deviation, fractional, 29, 41 Stellate cell, 16-17.40 Strange attractor, 327 Structural identitiability,313

Subcutaneous analysis, ultramtration, 191-193 Substrate, 158,244-245,260-261; see also Membrane transport Supplementation, methionine, 161-162 Synchronizer, bioperiodicity, 219-220 SYSTAT software, 221

T TEE, see Total energy expenditure Testes, vitamin A, 9, 11 Timing, food intake, 225-226 Total energy expenditure, 171-180 Tracer, see also Radioactive tracer; Stable isotope tracer carbon-l3,64-75,171-172 extraction, 256-257 modeled, 245-251,255-257 Tracer method p-carotene, 26,30-31,42-43.63-75 energy expenditure, 175-176 folate, 86-90 glycogen phosphorylase, 137-140 membrane transport, 249-250,255-257 membrane vesicles, 200,203 transthyretin metabolism, 150-152 vitamin A, 5, 7-8 by vitamin B6, 137-140 for vitamin B6, 114-116 Tracer washout rate, 249-250 Transfection, genetic, 95,96-101, 105 Transfer coefficient, fractional, 31, 40-43, 267-268.312 Transfer function, identifiability, 316-317 Transthyretin metabolism, 4, 13-14, 149-1 55 human subject, 150-154 MET30,150-155 modeling, 151-154 tracer methods, 150-152 Triglyceride, 62-63 TTR, see Transthyretin metabolism Turnover folate, 83-85, 89-90 glycogen phosphorylase, 124,137-140, 142 vitamin A, 12-13, 14, 19, 21

INDEX Turnover (continued) vitamin 86, 114-127 water, 173 Two-stage methods, population analysis, 273-274

U Ultradian rhythm, 219; see also Bioperiodicity LJltrafiltration, 183-193, 195 Urea, ultrafiltration, 192-193 USC*PACL software. 276

V Vacutainer, 185, 190 Visual guide, modeling, 282,283 Vitamin A metabolism, 3-24; see also pCarotene metabolism; Retinol; Retinol-binding protein; Retinyl ester background information, 3-5 catabolic rate, 7, 13 chylomicrons, 4, 16 cornpartmental analysis, 9-21 empirical, 11-14 model, 14-21 disposal rate, 4-5,7, 12, 21 -22 experimental design, 5-7 organs, 4.9, 11, 19-21; see also Liver tracer, 5 , 7-8 turnover, 12-13, 14, 19,21 whole-body models, 8-1 1 Vitamin R6 metabolism, 107-132; see also Glycogen phosphorylase metabolism brain, 112 clearance values, 110-1 11 compartmental modeling, 114-116 conservation effect, 117, 120-121, 127 dietary factors, 117 doses, 117-118, 119-120 equilibration, 126 erythrocytes, 111 flushing effect, 122, 127 glycogen phosphorylase, 115-116, 118, 121 human subject, 115, 121, 145 intestines, 109-110, 115, 121

kidneys, 201 kinetics, 113-127, 137-140 growth requirements, 116-117 tracers, radioactive, 114-116 tracers, stable isotope, 116 lactation, 112-113 liver, 108, 110-111, 112, 115, 121 McArdle’s disease, 142-145 membrane vesicles, 201 metabolic pathways, 107-110 model refinement, 117-127 dosing protocols, 119-120 fasting, 120-121 form comparisons, 121-122, 123 microgravity, 121 physiological observations, 118-119 protein-binding, 122- 127 muscle, 115-116, 120-121 oral versus intravenous intake, 115, 127 plasma, 110-111 pregnancy, 112-113 protein binding, 117, 122-127 protein intake, 117 tissue distribution, 112 turnover, 114-127 Vitamin D status, 197, 198, 201-202 Vitamin transport, membrane vesicle, 201-202,203,204 VLDL, see Lipoprotein Volume change, tissue sample, 186-187

W Water, doubly labeled, see Doubly labeled water method Water turnover, 173 Weight gain, bioperiodicity, 220-226 Wheat gluten, 231-232,238-240 Whole-body model folate, 83 membrane transport, 259-260 vitamin A, 8-11 vitamin B6, 116-117 World Wide Web, semiparametric tools, 275

2 Zero flow, extrapolation, 189

Mathematical Modeling in Experimental Nutrition: Vitamins, Proteins, Methods

Advances in Food and Nutrition Research Volume 40 Mathematical Modeling in Experimental Nutrition: Vitamins, Proteins, Methods

Vitamins in animal and human nutrition

Molecular Modeling of Proteins

Experimental Methods in Biology

Molecular Modeling of Proteins (Methods in Molecular Biology)

Proteins (Methods in Molecular Biology)

Sports nutrition: fats and proteins

Sports Nutrition: Fats and Proteins

G Proteins (Methods in Neurosciences)

Mathematical Modelling in Animal Nutrition

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