ENZYME KINETICS From diastase to multi-enzyme systems ARTHUR R. SCHULZ Indiana University School of Medicine CAMBRIDGE UNIVERSITY PRESS Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 First published 1994 Library of Congress Cataloging-in-Publication Data Schulz, Arthur R. Enzyme kinetics from diastase to multi-enzyme systems / Arthur R. Schulz p. cm. ISBN 0-521-44500-0. - ISBN 0-521-44950-2 (pbk.) 1. Enzyme kinetics. I. Title. QP601.3.S38 1994 94-28309 574.19'25-dc20 CIP A catalog record for this book is available from the British Library ISBN 0-521-44500-0 Hardback ISBN 0-521-44950-2 Paperback Transferred to digital printing 2003 Contents Preface page ix Part One: Basic steady state enzyme kinetics 1 Derivation of a rate equation 3 1.1 The role of 'diastase' in the early development of a theory 3 1.2 The basic assumptions on which derivation of an equation is based 7 1.3 The Briggs-Haldane steady state treatment of enzyme kinetic behavior 9 1.4 Estimation of steady state parameters 12 1.5 Problems for chapter 1 13 Appendix: A brief look at statistical analysis l.A.l Definition of a few statistical terms 14 1.A.2 Linear regression 15 1.A.3 Non-linear regression in enzyme kinetic analysis 18 References for chapter 1 20 2 A closer look at the basic assumptions 22 2.1 Why must the substrate concentration greatly exceed that of the enzyme? 22 2.2 What if the substrate concentration does not greatly exceed that of the enzyme? 23 2.3 Examination of the entire time-course of an enzymic reaction 25 2.4 A precise definition of steady-state velocity 27 2.5 Problems for chapter 2 28 References for chapter 2 28 3 Enzyme inhibition 30 3.1 A general model of enzyme inhibition 30 vi Contents 3.2 Quantitative estimates of steady state parameters in enzyme inhibition 34 3.3 Competitive inhibition: A limiting case of inhibition 35 3.4 Uncompetitive inhibition: A different limiting case 37 3.5 Substrate inhibition 38 3.6 Problems for chapter 3 41 Appendix: A brief discussion of polyonomial regression 42 References for chapter 3 43 4 Reversible enzyme-catalyzed reactions 44 4.1 Derivation of a rate equation by matrix inversion 44 4.2 Reformulation of the complete rate equation 50 4.3 The effect of product inhibition 52 4.4 Use of the King-Altman method to derive the rate equation 53 4.5 Problems for chapter 4 55 References for chapter 4 56 Part Two: Enzyme reaction sequence 5 Multi-reactant enzymic reactions 59 5.1 Three distinct two substrate, two product reaction sequences 59 5.2 The connection matrix method for deriving rate equations 61 5.3 Reformulation of the rate equations for multi-reactant enzymes 69 5.4 Problems for chapter 5 75 References for chapter 5 76 6 Analysis of multi-reactant enzyme kinetics 77 6.1 Analysis of the kinetic behavior of an enzymic reaction in the absence of products 77 6.2 Product inhibition as a tool in the analysis of reaction sequences 82 6.3 A reaction sequence with abortive complexes 87 6.4 Problems for chapter 6 91 References for chapter 6 91 7 Prediction of reaction sequence 92 7.1 The enzyme kineticist and the mystery novel 92 7.2 Product inhibition patterns: The clues with which the enzyme kineticist works 94 7.3 Problems for chapter 7 100 References for chapter 7 102 8 Enzyme-catalyzed isotopic exchange 103 Contents vii 8.1 Isotopic exchange in an ordered reaction sequence 104 8.2 Isotopic exchange in a random enzyme sequence 110 8.3 Problems for chapter 8 113 References for chapter 8 113 9 Kinetic isotope effect on steady state parameters 114 9.1 The basis for the kinetic isotope effect on rate constants 114 9.2 Use of the kinetic isotope effect in steady state enzyme kinetic studies 115 9.3 Problems for chapter 9 118 References for chapter 9 119 10 The effect of pH on enzyme kinetics 120 10.1 Michaelis pH functions of a simple amino acid 120 10.2 Michaelis pH functions of an enzyme 123 10.3 The effect of pH on steady state enzymic parameters 124 10.4 Effect of substrate ionization on steady state parameters 126 10.5 Problems for chapter 10 128 References for chapter 10 129 Part Three: Non-hyperbolic enzyme kinetics 11 The causes of non-hyperbolic enzyme kinetics 133 11.1 Random enzyme reaction sequences 133 11.2 The kinetic behavior of allosteric enzymes 137 11.3 Multiple enzymes catalyzing the conversion of a single substrate 138 11.4 Problems for chapter 11 138 References for chapter 11 139 12 Analysis of non-hyperbolic enzyme kinetics 140 12.1 The use of analytical geometry to analyze a substrate- saturation curve 141 12.2 Analysis of the rate equation which is a 2:2 rational function 141 12.3 Analysis of the Lineweaver-Burk plot of a 2:2 rational polynomial 143 12.4 Analysis of the Hanes plot of a 2:2 rational polynomial 149 12.5 The kinetic behavior of higher order rational polynomials 151 12.6 Problems for chapter 12 151 References for chapter 12 152 13 The effect of subunit interactions on enzyme kinetics 153 13.1 The effect of subunit interactions on rate constants 153 viii Contents 13.2 The effect of sequential subunit interactions 155 13.3 The effect of partially concerted subunit interactions 158 13.4 The effect of fully concerted subunit interactions 160 13.5 The effect of exclusive allosteric subunit interactions 162 13.6 Problems for chapter 13 164 References for chapter 13 165 Part Four: Control of multi-enzyme systems 14 Control of linear multi-enzyme systems 169 14.1 Definition of the parameters of metabolic control 170 14.2 Application of the sensitivity theory to the control of linear multi-enzyme systems 173 14.3 The relationship between sensitivity theory and metabolic control theory 179 14.4 The effect of feedback and feed forward loops on the control of a linear pathway 183 14.5 The quantitative estimation of control coefficients 185 14.6 Problems for chapter 14 186 Appendix 186 References for chapter 14 187 15 Control of branched multi-enzyme systems 189 15.1 Application of the sensitivity theory to branched pathways 189 15.2 Flux control in substrate cycles 194 15.3 Problems for chapter 15 197 Appendix 198 References for chapter 15 198 16 Biochemical systems theory 199 16.1 Power law formulation of control of a linear multi- enzyme system 199 16.2 Power law formulation of control of linear multi-enzyme pathways when enzyme activities are variable 205 16.3 Power law formulation for branched pathways 207 16.4 The future of metabolic control in biology 209 16.5 Problems for chapter 16 210 Appendix 210 References for chapter 16 211 Part Five: Solutions to problems Author index 241 Subject index 244 Preface One goal of this textbook is to provide the reader with an orderly development of steady state enzyme kinetics from the early formulations, through analysis of the reaction sequence of multi-reactant enzymes, to the analysis of non-hyperbolic enzyme kinetics, and finally to the control of multi-enzyme systems. The material included in this book has formed the basis of lectures on enzyme kinetics which have been given as a portion of a course on enzyme chemistry. It is hoped that it will be useful not only to the reader who is enrolled in a formal course in enzyme kinetics or enzyme chemistry, but also to readers who wish to familiarize themselves with enzyme kinetics in a self-study program, and also to the readers who wish to review the principles of steady state enzyme kinetics. The book contains numerous equations, but neither the equations nor the derivations of the equations constitute the primary objective. Rather, it is crucial that the information contained in an equation be correlated correctly with the kinetic behavior of the enzyme. Hence, it is the kinetic behavior of the enzyme which mandates the structure of the rate equation. The task which is presented to the enzyme kineticist is to visualize the enzyme model which is consistent with the rate equation. The reader will note that there are few references to individual enzymes in this textbook. A deliberate objective has been to present the fundamentals of enzyme kinetics in general terms rather than in terms of specific enzymes. The basis for this approach is the conviction that an objective investigation of the kinetic behavior of an enzyme-catalyzed reaction should be pursued in a manner which is cognizant of basic principles rather than an attempt to 'fit' the data obtained with one enzyme to the behavior of some other enzyme. The need to impose a realistic limit on the size of this textbook has IX x Preface led to the omission of some important materials, for example, the derivation of rate equations for enzyme-catalyzed reactions based on stochastic principles [J. Ninio, Proc. Natl. Acad. Sci. USA 84:663 (1987); A. K. Mazur, J. Theor. Biol. 148: 229 (1991)]. Likewise, the structural approach to metabolic control theory developed by Reader and Mazat [C. Reder, J. Theor. Biol. 135: 175 (1988)] is not included. The omission of these and other important topics from this textbook reflects only the limitation of space. It is hoped that this textbook will provide sufficient background to motivate the reader to study the foregoing papers as well as other valuable publications. I am indebted to Dr. William F. Bosron, Dr. Robert Eisenthal, Dr. David M. Giobson and Dr. Robert A. Harris for their willingness to read the manuscript of this book. I deeply appreciate their comments. I also wish to thank Dr. Robin C. Smith of Cambridge University Press for his helpful suggestions during the preparation of the manuscript. Finally, I acknowl- edge those who have contributed so much to the inspiration and comple- tion of this book, namely, my parents, who nurtured and guided me in my early life, my wife, Marian, who has loved and encouraged me, the teachers who taught and challenged me and the students who questioned and stimulated me. Part One Basic steady state enzyme kinetics