Table Of Content

THE REGULATION OF CELLULAR SYSTEMS JOIN US ON THE INTERNET WWW: http://www.thomson.com EMAIL: [email protected] thomson. com is the on-line portal for the products, services and resources available from International Thomson Publishing (ITP). This Internet kiosk gives users immediate access to more than 34 ITP publishers and over 20,000 products. Through thomson. com Internet users can search catalogs, examine subject-specific resource centers and subscribe to electronic discussion lists. You can purchase ITP products from your local bookseller, or directly through thomson. com. Visit Chapman & Hall's Internet Resource Center for information on our new publications, links to useful sites on the World Wide Web and an opportunity to join our e-mail mailing list. Point your browser to: http://www.chaphall.com/chaphall.html or http://www.chaphall.com/chaphallllifesce.html for Life Sciences CD I p® A service of THE REGULATION OF CELLULAR SYSTEMS REINHART HEINRICH AND STEFAN SCHUSTER CHAPMAN & HALL (!) I p® International Thomson Publishing New York' Albany. Bonn' Boston' Cincinnati. Detroit· London' Madrid' Melbourne Mexico City. Pacific Grove' Paris' San Francisco' Singapore' Tokyo. Toronto' Washington Cover Design: Said Sayrafiezadeh, emDASH inc. Art Direction: Andrea Meyer Copyright © 1996 Softcover reprint of the hardcover I st edition 1996 Chapman & Hall For more information, contact: Chapman & Hall Chapman & Hall 115 Fifth Avenue 2-6 Boundary Row New York, NY 10003 London SEi 8HN England Thomas Nelson Australia Chapman & Hall GmbH \02 Dodds Street Postfach 100 263 South Melbourne, 3205 D-69442 Weinheim Victoria, Australia Germany International Thomson Editores International Thomson Publishing-Japan Campos Eliseos 385, Piso 7 Hirakawacho-cho Kyowa Building, 3F Col. Polanco 1-2-1 Hirakawacho-cho 11560 Mexico D.E Chiyoda-ku, 102 Tokyo Mexico Japan International Thomson Publishing Asia 221 Henderson Road #05-10 Henderson Building Singapore 0315 All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without the written permission of the publisher. I 2345 67 89 \0 XXX 01 00 99 97 96 95 Library of Congress Cataloging-in-Publication Data Heinrich, Reinhart, 1946- The regulation of cellular systems I by Reinhart Heinrich and Stefan Schuster. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8492-5 e-ISBN-13: 978-1-4613-1161-4 DOl: 10.\007/978-1-4613-1161-4 1. Cell metabolism-Regulation-Mathematical models. 2. Control theory. 1. Schuster, Stefan. II. Title. QH634.5.H45 1996 574.8T6---dc20 96-1594 CIP British Library Cataloguing in Publication Data available To order this or any other Chapman & Hall book, please contact International Thomson Publishing, 7625 Empire Drive, Florence, KY 41042. Phone: (606)525-6600 or 1-800-842-3636. Fax: (606) 525-7778, e-mail: [email protected]. For a complete listing of Chapman & Hall's titles, send your requests to Chapman & Hall, Dept. BC, 115 Fifth Avenue, New York, NY 10003. Die angewandte Mathematik: hat im Verlaufe der letzten Jahrhun derte eine so hohe Stufe der Ausbildung erreicht, ihre Schliisse haben einen solchen Grad von Sicherheit erlangt, daB sie unter den Wissenschaften den ersten Rang einzunehmen berechtigt ist. Sie ist der Anfang und das Ende fiir den Sternkundigen, den Techni ker, den Seemann, sie ist die feste Achse aller Naturforschung jet ziger Zeit. Nur der Biologie haben die Entdeckungen Galileis, New tons und Mariottes verhiiltnismaBig geringe Friichte getragen; fiir die Lebenserscheinungen wurden keine Formeln aufgefunden ... Throughout the last centuries, applied mathematics has attained such a high level of perfection and such a degree of certainty in its conclusions that it is entitled to take the first place among the Sciences. Mathematics is the be-all and end-all for the astronomer, the engineer, and the seaman; it is the solid basis of all natural sciences today. Only for biology, the yield of the discoveries of Galilei, Newton and Mariotte has been comparatively small; no formulae have been found for the phenomena of life ... ROBERT MAYER, Die organische Bewegung in ihrem Zusammen hange mit dem Stoffwechsel (Organic motion in its relation to me tabolism), Heilbronn, 1845 Contents Dedication vi Preface xiii Symbols xvii 1. INTRODUCTION 1 2. FUNDAMENTALS OF BIOCHEMICAL MODELING 9 2.1. Balance Equations 10 2.2. Rate Laws 14 2.2.1. Generalized Mass-Action Kinetics 14 2.2.2. Various Enzyme-Kinetic Rate Laws 17 2.2.3. Thermodynamic Flow-Force Relationships 28 2.2.4. Power-Law Approximation 36 2.3. Steady States of Biochemical Networks 38 2.3.1. General Considerations 38 2.3.2. Stable and Unstable Steady States 40 2.3.3. Multiple Steady States 46 2.4. Metabolic Oscillations 52 2.4.1. Background 52 vii viii Contents 2.4.2. Mathematical Conditions for Oscillations 53 2.4.3. Glycolytic Oscillations 55 2.4.4. Models of Intracellular Calcium Oscillations 59 2.4.5. A Simple Three-Variable Model with Only 66 Monomolecular and Bimolecular Reactions 2.4.6. Possible Physiological Significance of Oscillations 71 3. STOICHIOMETRIC ANALYSIS 75 3.1. Conservation Relations 78 3.1.1. Linear Dependencies Between the Rows of the 78 Stoichiometry Matrix 3.1.2. Non-negative Conservation Relations 80 3.1.3. Conserved Moieties 87 3.2. Admissible Steady-State Vectors and the Null-Space 91 3.2.1. Linear Dependencies Between the Columns of the 91 Stoichiometry Matrix 3.2.2. Block-Diagonalization of the Null-Space Matrix 92 3.2.3. Non-negative Flux Vectors 94 3.2.4. Elementary Flux Modes 95 3.3. Thermodynamic Aspects 102 3.3.1. A Generalized Wegscheider Condition 102 3.3.2. Strictly Detailed Balanced Subnetworks 109 3.3.3. Onsager's Reciprocity Relations for Coupled Enzyme 110 Reactions 4. TIME HIERARCHY IN METABOLISM 112 4.1. Time Constants of Metabolic Processes 112 4.2. The Quasi-Steady-State Approximation 116 4.3. The Rapid-Equilibrium Approximation 123 4.4. Modal Analysis 134 5. METABOLIC CONTROL ANALYSIS 138 5.1. Basic Definitions 139 5.2. A Systematic Approach 141 Contents ix 5.3. Theorems of Metabolic Control Analysis 148 5.3.1. Summation Theorems 148 5.3.2. Connectivity Theorems 150 5.3.3. Calculation of Control Coefficients Using the 151 Theorems 5.3.4. Geometrical Interpretation 153 5.4. Control Analysis of Various Systems 155 5.4.1. General Remarks 155 5.4.2. Elasticity Coefficients for Specific Rate Laws 156 5.4.3. Control Coefficients for Simple Hypothetical 160 Pathways 5.4.3.1. Unbranched Chains 160 5.4.3.2. A Branched System 169 5.4.4. Control of Erythrocyte Energy Metabolism 170 5.4.4.1. The Reaction System 170 5.4.4.2. Basic Model 171 5.4.4.3. Interplay of ATP Production and ATP 177 Consumption 5.4.4.4. Glycolytic Energy Metabolism and Osmotic 188 States 5.4.5. A Simple Model of Oxidative Phosphorylation 192 5.4.6. A Three-Step Model of Serine Biosynthesis 196 5.5. Time-Dependent Control Coefficients 198 5.6. Are Control Coefficients Always Parameter Independent? 204 5.6.1. Posing the Problem 204 5.6.2. A System Without Conserved Moieties 204 5.6.3. A System with a Conserved Moiety 206 5.6.4. A System Including Dynamic Channeling 210 5.7. Normalized Versus Non-normalized Coefficients 212 5.8. Analysis in Terms of Variables Other Than 216 Concentrations and Fluxes 5.8.1. General Analysis 216 5.8.2. Concentration Ratios and Free-Energy Differences as 221 State Variables x Contents 5.8.3. Entropy Production as a Response Variable 222 5.8.4. Control of Transient Times 223 5.8.5. Control of Oscillations 230 5.9. A Second-Order Approach 233 5.10. Metabolic Regulation from the Viewpoint of Control 240 Analysis 5.1O.l. Coresponse Coefficients 240 5.10.2. Fluctuations of Internal Variables Versus Parameter 243 Perturbations 5.10.3. Internal Response Coefficients 245 5.10.4. Rephrasing the Basic Equations of Metabolic Control 247 Analysis in Terms of Coresponse Coefficients and Internal Response Coefficients 5.11. Control Within and Between Subsystems 249 5.12. Modular Approach 256 5.12.1. Overall Elasticities 256 5.12.2. Overall Control Coefficients 261 5.13. Flux Control Insusceptibility 267 5.14. Control Exerted by Elementary Steps in Enzyme Catalysis 273 5.15. Control Analysis of Metabolic Channeling 278 5.16. Comparison of Metabolic Control Analysis and Power- 283 Law Formalism 5.17. Computational Aspects 288 6. APPLICATION OF OPTIMIZATION METHODS AND THE 292 INTERRELATION WITH EVOLUTION 6.1. Optimization of the Catalytic Properties of Single 295 Enzymes 6.l.l. Basic Assumptions 295 6.1.2. Optimal Values of Elementary Rate Constants 298 6.1.3. Optimal Michaelis Constants 307 6.2. Optimization of Multienzyme Systems 309 6.2.l. Maximization of Steady-State Flux 309 Contents xi 6.2.2. Influence of Osmotic Constraints and Minimization 316 of Intermediate Concentrations 6.2.3. Minimization of Transient Times 322 6.3. Optimal Stoichiometries 325 6.3.1. Optimal Properties of the Pentose Phosphate Pathway 326 6.3.2. Optimal Location of ATP-Consuming and ATP- 327 Producing Reactions in Glycolysis 6.3.3. Concluding Remarks 340 References 343 Index 367

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THE REGULATION OF CELLULAR SYSTEMS JOIN US ON THE INTERNET WWW: http://www.thomson.com EMAIL: [email protected] thomson. com is the on-line portal for the products, services and resources available from International Thomson Publishing (ITP). This Internet kiosk gives users immediate access to more than 34 ITP publishers and over 20,000 products. Through thomson. com Internet users can search catalogs, examine subject-specific resource centers and subscribe to electronic discussion lists. You can purchase ITP products from your local bookseller, or directly through thomson. com. Visit Chapman & Hall's Internet Resource Center for information on our new publications, links to useful sites on the World Wide Web and an opportunity to join our e-mail mailing list. Point your browser to: http://www.chaphall.com/chaphall.html or http://www.chaphall.com/chaphallllifesce.html for Life Sciences CD I p® A service of THE REGULATION OF CELLULAR SYSTEMS REINHART HEINRICH AND STEFAN SCHUSTER CHAPMAN & HALL (!) I p® International Thomson Publishing New York' Albany. Bonn' Boston' Cincinnati. Detroit· London' Madrid' Melbourne Mexico City. Pacific Grove' Paris' San Francisco' Singapore' Tokyo. Toronto' Washington Cover Design: Said Sayrafiezadeh, emDASH inc. Art Direction: Andrea Meyer Copyright © 1996 Softcover reprint of the hardcover I st edition 1996 Chapman & Hall For more information, contact: Chapman & Hall Chapman & Hall 115 Fifth Avenue 2-6 Boundary Row New York, NY 10003 London SEi 8HN England Thomas Nelson Australia Chapman & Hall GmbH \02 Dodds Street Postfach 100 263 South Melbourne, 3205 D-69442 Weinheim Victoria, Australia Germany International Thomson Editores International Thomson Publishing-Japan Campos Eliseos 385, Piso 7 Hirakawacho-cho Kyowa Building, 3F Col. Polanco 1-2-1 Hirakawacho-cho 11560 Mexico D.E Chiyoda-ku, 102 Tokyo Mexico Japan International Thomson Publishing Asia 221 Henderson Road #05-10 Henderson Building Singapore 0315 All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without the written permission of the publisher. I 2345 67 89 \0 XXX 01 00 99 97 96 95 Library of Congress Cataloging-in-Publication Data Heinrich, Reinhart, 1946- The regulation of cellular systems I by Reinhart Heinrich and Stefan Schuster. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8492-5 e-ISBN-13: 978-1-4613-1161-4 DOl: 10.\007/978-1-4613-1161-4 1. Cell metabolism-Regulation-Mathematical models. 2. Control theory. 1. Schuster, Stefan. II. Title. QH634.5.H45 1996 574.8T6---dc20 96-1594 CIP British Library Cataloguing in Publication Data available To order this or any other Chapman & Hall book, please contact International Thomson Publishing, 7625 Empire Drive, Florence, KY 41042. Phone: (606)525-6600 or 1-800-842-3636. Fax: (606) 525-7778, e-mail: [email protected]. For a complete listing of Chapman & Hall's titles, send your requests to Chapman & Hall, Dept. BC, 115 Fifth Avenue, New York, NY 10003. Die angewandte Mathematik: hat im Verlaufe der letzten Jahrhun derte eine so hohe Stufe der Ausbildung erreicht, ihre Schliisse haben einen solchen Grad von Sicherheit erlangt, daB sie unter den Wissenschaften den ersten Rang einzunehmen berechtigt ist. Sie ist der Anfang und das Ende fiir den Sternkundigen, den Techni ker, den Seemann, sie ist die feste Achse aller Naturforschung jet ziger Zeit. Nur der Biologie haben die Entdeckungen Galileis, New tons und Mariottes verhiiltnismaBig geringe Friichte getragen; fiir die Lebenserscheinungen wurden keine Formeln aufgefunden ... Throughout the last centuries, applied mathematics has attained such a high level of perfection and such a degree of certainty in its conclusions that it is entitled to take the first place among the Sciences. Mathematics is the be-all and end-all for the astronomer, the engineer, and the seaman; it is the solid basis of all natural sciences today. Only for biology, the yield of the discoveries of Galilei, Newton and Mariotte has been comparatively small; no formulae have been found for the phenomena of life ... ROBERT MAYER, Die organische Bewegung in ihrem Zusammen hange mit dem Stoffwechsel (Organic motion in its relation to me tabolism), Heilbronn, 1845 Contents Dedication vi Preface xiii Symbols xvii 1. INTRODUCTION 1 2. FUNDAMENTALS OF BIOCHEMICAL MODELING 9 2.1. Balance Equations 10 2.2. Rate Laws 14 2.2.1. Generalized Mass-Action Kinetics 14 2.2.2. Various Enzyme-Kinetic Rate Laws 17 2.2.3. Thermodynamic Flow-Force Relationships 28 2.2.4. Power-Law Approximation 36 2.3. Steady States of Biochemical Networks 38 2.3.1. General Considerations 38 2.3.2. Stable and Unstable Steady States 40 2.3.3. Multiple Steady States 46 2.4. Metabolic Oscillations 52 2.4.1. Background 52 vii viii Contents 2.4.2. Mathematical Conditions for Oscillations 53 2.4.3. Glycolytic Oscillations 55 2.4.4. Models of Intracellular Calcium Oscillations 59 2.4.5. A Simple Three-Variable Model with Only 66 Monomolecular and Bimolecular Reactions 2.4.6. Possible Physiological Significance of Oscillations 71 3. STOICHIOMETRIC ANALYSIS 75 3.1. Conservation Relations 78 3.1.1. Linear Dependencies Between the Rows of the 78 Stoichiometry Matrix 3.1.2. Non-negative Conservation Relations 80 3.1.3. Conserved Moieties 87 3.2. Admissible Steady-State Vectors and the Null-Space 91 3.2.1. Linear Dependencies Between the Columns of the 91 Stoichiometry Matrix 3.2.2. Block-Diagonalization of the Null-Space Matrix 92 3.2.3. Non-negative Flux Vectors 94 3.2.4. Elementary Flux Modes 95 3.3. Thermodynamic Aspects 102 3.3.1. A Generalized Wegscheider Condition 102 3.3.2. Strictly Detailed Balanced Subnetworks 109 3.3.3. Onsager's Reciprocity Relations for Coupled Enzyme 110 Reactions 4. TIME HIERARCHY IN METABOLISM 112 4.1. Time Constants of Metabolic Processes 112 4.2. The Quasi-Steady-State Approximation 116 4.3. The Rapid-Equilibrium Approximation 123 4.4. Modal Analysis 134 5. METABOLIC CONTROL ANALYSIS 138 5.1. Basic Definitions 139 5.2. A Systematic Approach 141 Contents ix 5.3. Theorems of Metabolic Control Analysis 148 5.3.1. Summation Theorems 148 5.3.2. Connectivity Theorems 150 5.3.3. Calculation of Control Coefficients Using the 151 Theorems 5.3.4. Geometrical Interpretation 153 5.4. Control Analysis of Various Systems 155 5.4.1. General Remarks 155 5.4.2. Elasticity Coefficients for Specific Rate Laws 156 5.4.3. Control Coefficients for Simple Hypothetical 160 Pathways 5.4.3.1. Unbranched Chains 160 5.4.3.2. A Branched System 169 5.4.4. Control of Erythrocyte Energy Metabolism 170 5.4.4.1. The Reaction System 170 5.4.4.2. Basic Model 171 5.4.4.3. Interplay of ATP Production and ATP 177 Consumption 5.4.4.4. Glycolytic Energy Metabolism and Osmotic 188 States 5.4.5. A Simple Model of Oxidative Phosphorylation 192 5.4.6. A Three-Step Model of Serine Biosynthesis 196 5.5. Time-Dependent Control Coefficients 198 5.6. Are Control Coefficients Always Parameter Independent? 204 5.6.1. Posing the Problem 204 5.6.2. A System Without Conserved Moieties 204 5.6.3. A System with a Conserved Moiety 206 5.6.4. A System Including Dynamic Channeling 210 5.7. Normalized Versus Non-normalized Coefficients 212 5.8. Analysis in Terms of Variables Other Than 216 Concentrations and Fluxes 5.8.1. General Analysis 216 5.8.2. Concentration Ratios and Free-Energy Differences as 221 State Variables x Contents 5.8.3. Entropy Production as a Response Variable 222 5.8.4. Control of Transient Times 223 5.8.5. Control of Oscillations 230 5.9. A Second-Order Approach 233 5.10. Metabolic Regulation from the Viewpoint of Control 240 Analysis 5.1O.l. Coresponse Coefficients 240 5.10.2. Fluctuations of Internal Variables Versus Parameter 243 Perturbations 5.10.3. Internal Response Coefficients 245 5.10.4. Rephrasing the Basic Equations of Metabolic Control 247 Analysis in Terms of Coresponse Coefficients and Internal Response Coefficients 5.11. Control Within and Between Subsystems 249 5.12. Modular Approach 256 5.12.1. Overall Elasticities 256 5.12.2. Overall Control Coefficients 261 5.13. Flux Control Insusceptibility 267 5.14. Control Exerted by Elementary Steps in Enzyme Catalysis 273 5.15. Control Analysis of Metabolic Channeling 278 5.16. Comparison of Metabolic Control Analysis and Power- 283 Law Formalism 5.17. Computational Aspects 288 6. APPLICATION OF OPTIMIZATION METHODS AND THE 292 INTERRELATION WITH EVOLUTION 6.1. Optimization of the Catalytic Properties of Single 295 Enzymes 6.l.l. Basic Assumptions 295 6.1.2. Optimal Values of Elementary Rate Constants 298 6.1.3. Optimal Michaelis Constants 307 6.2. Optimization of Multienzyme Systems 309 6.2.l. Maximization of Steady-State Flux 309 Contents xi 6.2.2. Influence of Osmotic Constraints and Minimization 316 of Intermediate Concentrations 6.2.3. Minimization of Transient Times 322 6.3. Optimal Stoichiometries 325 6.3.1. Optimal Properties of the Pentose Phosphate Pathway 326 6.3.2. Optimal Location of ATP-Consuming and ATP- 327 Producing Reactions in Glycolysis 6.3.3. Concluding Remarks 340 References 343 Index 367

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