Molecular Biology Biochemistry and Biophysics 22 Editors: A. Kleinzeller, Philadelphia· G. F. Springer, Evanston H. G. Wittmann, Berlin Advisory Editors: C. R. Cantor, New York· F. Cramer, Gattingen . F. Egami, Tokyo M. Eigen, Gattingen . F. Gros, Paris· H. Gutfreund, Bristol B. Hess, Dortmund· H. Jahrmiirker, Munich· R. W Jeanloz, Boston E. Katzir, Rehovot . B. Keil, Gif-sur-Yvette· M. Klingenberg, Munich I. M. Klotz, Evanston· F. Lynen, Martinsried/Munich W T. J. Morgan, London· K. Muhlethaler, Zurich· S. Ochoa, New York G. Palmer, Houston· I. Pecht, Rehovot . R. R. Porter, Oxford W Reichardt, Tiibingen . H. Tuppy, Vienna J. Waldenstram, M alma Herbert J. Fromm Initial Rate Enzyme Kinetics With 88 Figures Springer-Verlag Berlin· Heidelberg· New York 1975 Professor HERBERT 1. FROMM, Ph. D. Iowa State University Biochemistry and Biophysics Department Ames, IA 500lOjUSA Distributed in the British Commonwealth Market by Chapman and Hall, London ISBN-13: 978-3-642-80968-2 e-ISBN-13: 978-3-642-80966-8 001: 10.1007/978-3-642-80966-8 Library of Congress Cataloging in Publication Data. Fromm, Herbert 1. 1929-. Initial rate enzyme kinetics. (Molecular biology, biochemistry, and biophysics; v. 22). Bibliography: p. Includes indexes. i. Enzymes. 2. Chemical reaction, Rate of. I. Title. II. Series. [DNLM: 1. Enzymes-Metabolism. WI MOl95T no. 22/QU135 F932] QP601.F76. 574.1'925. 75-20206 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agree- ment with the publisher. © by Springer-Verlag Berlin Heidelberg 1975 Softcover reprint of the hardcover 1st edition 1975 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Preface Enzyme kinetics has undergone very rapid growth and development during the past fifteen years and has been well received by the biochemical community. A cursory glance at the current biochem ical literature reveals the increasing popularity of enzyme ki netics1 yet, there are very few books available to guide the enzymologist who wishes to conduct kinetic experiments. This monograph was undertaken to provide the fledgling kineticist with an outline of contemporary initial rate enzyme kinetics. A large portion of the material contained in this book is presented in a second-year, graduate-level course in biochemistry at Iowa State University. I have found that the presentation in this course has enabled students without a strong background in math ematics to undertake initial rate studies at the research bench. The monograph obviously is more comprehensive than any course could be, and should permit similar accomplishment. As the title implies, the major emphasis of this monograph is on initial rate enzyme kinetics. I considered at length the advis ability of including chapters on integrated rate equations and on the theory and application of rapid reaction kinetics, such as rapid-mixing stopped-flow, and temperature-jump kinetics. These, however, are topics that would require a good deal of space to develop if they were to be helpful to the beginner. Some deviation from initial rate kinetics was required when the topics of cooperativity and allostery were broached. A very large fraction of the research in this area of biochemistry has in volved static binding measurements, and the current literature clearly reflects this. It was necessary, therefore, to intro duce these topics within the framework of the simpler equilib rium binding models before the kinetics of allostery and coop erativity were considered. It will become quite obvious that a number of topics are omitted that might have been included inChapter IX. In an area of research such as allostery, which is in a state of flux at this writing, concepts that are not widely accepted or clearly defined are either treated superficially or not included. Books on kinetics usually cover theory and interpretation of data in the literature, but rarely present the experimental pro tocol. Chapters III and VI are devoted in large measure to instruc tion on setting up and carrying out initial rate and isotope exchange experiments. Although these sections may not provide enlightenment for the advanced student, they may serve to lower VI the energy barrier to potential experimentalists who wish to use their theoretical knowledge for practical ends. Acknowledgement Thanks are in order to the faculty members of the Department of Biochemistry and Biophysics, Iowa State University, and to my graduate students and post-doctoral fellows for their help and encouragement both before and during the writing of the book. The critical reading of sections of the manuscript by Drs. JON APPLEQUIST, JAMES ESPENSON, JAMES LUECK, DAVID METZLER, and GREGORIO WEBER is gratefully acknowledged. Special and sincere thanks to go Drs. DANIEL PURICH and DONALD SIANO whose help made writing sections of the book an educational experience for me. Finally, I wish to acknowledge the consideration and encouragement I received from my wife, KATHY, who in seeking her own liberation, did not impede my work. Ames, January 1975 HERBERT J. FROMM Contents Chapter I Nomenclature, Definitions, and Evolution of the Kinetic Mechanism •.....•.•••.........•... A. Nomenclature.............................. 3 B. Evolution of Initial Rate Kinetics........ 6 References. . . . . . . . • . . . . . • . . • . . . . . . . . . . . . . . 20 Chapter II Derivation of Initial Velocity Rate Equations 22 A. Definitions and Derivations .......•....... 22 1. Steady-State........................... 22 2. Initial Velocity ...•.....•..•.......... 29 3. The Maximal Velocity and Michaelis Constant. . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . 30 4. Reverse Reaction Parameters and Rate Constants. . . . . . . . • . • • . . . . . . . . . . . . . . . . . . 31 B. The Equilibrium Assumption................ 31 C. Derivation of Complex Steady-State Rate Equations................................. 33 D. Derivation of the Rate Equation Using the Rapid Equilibrium Assumption.............. 36 1. The Random Bi Bi Mechani sm. . . . . . . . . . . . . 36 2. The Ordered Bi Bi Mechanism............ 37 E. Derivation of Initial Rate Equations Using a Combination of Equilibrium and Steady- State Assumptions......................... 38 F. Derivation of Steady-State Rate Equations by Using the Digital Computer ............. 40 References. . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Chapter III Experimental Protocol and Plotting of Kinetic Data................................. 41 A. General Considerations.................... 41 B. Analysis of Radioactive Substrates and De- termination of Radiopurity................ 45 C. pH Effects................................ 46 D. Substrate Concentration................... 47 E. Studies of Forward and Reverse Reactions.. 49 F. Studies of Nucleotide Dependent Enzymic Re- actions. . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 G. The Kinetic Assay ................•..•....•. 56 VIII 1. The Continuous Assay •••..........•.•... 56 2. The Stop-Time Assay •.••..•••.•.....•..• 60 H. Plotting Methods. .• . . . • . . . • . . •. . .• . • . . . • . . 62 1. Graphical Procedures...................... 69 J. The Point of Convergence of Sequential Double Reciprocal Plots as a Criterion of Kinetic Mechanism......................... 73 K. Protocol and Data Plotting for Three Sub- strate Systems............................ 75 L. Graphical Methods for Differentiating be tween Steady-State and Equilibrium Ordered Bi Bi Mechanisms.......................... 81 References................................ 81 Chapter IV Use of Competitive Substrate Analogs and Alternative Substrates for Studying Kinetic Mechani sms. . . . • • • • . . . . . • . . . . . . . . . . . . . • . . • . . • . 83 A. Competitive Inhibition.................... 83 B. Partial Competitive Inhibition ........••.• 86 C. Noncompetitive Inhibition ................. 88 D. Uncompetitive Inhibition .................. 91 E. Nonlinear Enzyme Inhibition ....•.••..•.... 92 F. The Use of Substrate Analogs for Studying Kinetic Mechanisms........................ 94 1. Bireactant Enzymic Systems ..•.......... 94 2. Terreactant Systems.................... 101 3. Kinetic Studies of AdenylosuccinateSyn- thetase Using Dead End Inhibitors ...... 108 G. Cleland's Rules for Dead End Inhibition ... 110 H. The Stereochemical Nature of Enzyme and Substrate Interaction ................•.... 112 1. Kinetics of Enzyme Specificity ..........•. 115 J. The Kinetics of Transition State Analogs .. 116 References ......•..•.....•.•.............. 119 Chapter V Product, Substrate, and Alternative Substrate Inhibition. • . . . • . • • • . . . • . . . • . . . . • . . . . . . . . . . .. 121 A. Product Inhibition Experiments ......•.•..• 121 1. Experimental ProtocoL................. 121 2. One Substrate Systems •................. 122 3. Two Substrate Systems .................. 125 4. Abortive Ternary Complex Formation •.... 127 5. Calculation of Rate Constants from Product Inhibition Experiments ......... 138 6. Noncompetitive Product Effects ••••.••.• 139 B. Substrate Inhibition •••••••••.•••.•.•••••. 144 1. A Simple Model for Substrate Inhibition. 144 2. Two Substrate Systems ••••••••.••••••••• 146 IX C. Alternative Substrate Inhibition .•.•...... 152 1. Alternative Substrates Acting as Inhi- bi tors Only............................ 152 2. Bireactant Systems .......•.•...•......• 153 3. Terreactant Systems.................... 156 D. Alternative Product Inhibition ••..•...•.•. 158 E. Multisite Ping Pong Mechanisms .....•..••.. 158 F. Enzymes with Identical Substrate-Product Pairs. • . . . . . • . . . • . . • . . . . . . . • . . . • . . . • . • . . •. 159 References •.....•....................•.... 160 Chapter VI I sotope Exchange............................. 1 61 A. Abortive Complex Formation ..........•.•... 163 B. Derivation of Rate Equations .•....•....... 165 1. The Equilibrium Case: Ping Pong Bi Bi .. 165 2. The Steady-State Case: Ordered Bi Bi (Theorell-Chance) ..•................... 167 3. Random Bi Bi........................... 1 72 4. Theorell-Chance Mechanism ........•.•... 172 C. Substrate Synergism ....••....•.•.....•.... 172 D. Calculation of Kinetic Parameters .....•.•. 174 1. The Ping Pong Bi Bi Meehanism .......... 174 2. The Random Bi Bi Mechanism (Rapid Equilibrium) ............•.•.......•.... 175 E. Exper imental ProtocoL.................... 175 F. Isotope-Trapping •...•...•.•.•..•.......•.. 183 References. . . . . . • . . . . . . . . . . . . . . . . . . . . . . . •. 184 Chapter VII Isomerization Mechanisms and the p and Haldane Relationships ......•..•...................... 186 A. The ~ Relationships ....................... 186 B. The Haldane Relationships ..•••..•......•.. 189 1. Ordered Bi Bi.......................... 190 2. Rapid Equilibrium Random Bi Bi ......... 190 3. Steady-State Random Bi Bi ......•.•..... 191 C. Isomerization Mechanisms ..........•....... 193 References ..........................•..... 199 Chapter VIII The Effect of Temperature and pH on Enzyme Activity .•..••..•......•.•................... 201 A. Effect of pH on Enzyme Kinetics ........•.• 201 1. pH Functions........................... 202 2. The Effect of pH on Unireactant Models. 204 3. Evaluation of Ionization Constants ..... 208 4. Bisubstrate Systems .......•..•..•.•.•.. 213 5. Cooperative Proton Binding ......•.•...• 214 6. Identification of Amino Acid Residues from Studies of pH Kinetics ...•........ 215 x 7. Some Limitations in the Study of pH Kinetics. • . . . • . . . . . • • • . • • . • . • • . . . . . . . .. 21 7 8. Choosing a Buffer for Kinetic Experi- ments ..•...••••.....•.•.••...." . . . . . . . .. 220 9. The pH Kinetics of the Fumarase Reaction •......•.•..•.•.•••..•......•.. 221 B. The Effect of Temperature on Enzyme Catal- yzed Reactions •..•..••.•••.•..........•••. 224 1. Collision Theory and the Arrhenius Equation ........................•...... 225 2. Transition-State Theory ...•....•...•... 227 3. Significance of Activation Enthalpy and Activation Entropy •.••••......•....•... 230 4. Application of Transition-State Theory to the a-Chymotrypsin Reaction ..•....•. 233 References .•....•...•.••...•....•...•..... 234 Chapter IX Cooperativity and Allostery •.......•....•.••• 236 A. Cooperativity ..•........•..•......•....•.. 237 1. The Hill Equation ........•......•..•... 237 2. The Adair Equation..................... 241 3. The Scatchard Plot ••.••••...•...••..... 244 B. Molecular Models .•..•....•..•......•....•. 244 1. The Monod Model .•........•....••..••.•. 245 2. The Adair-Koshland Model ..•.......•.... 248 3. Subunit-Subunit Polymerization •..•..... 250 4. Protein Isomerization .•.••.........••.. 252 C. Kinetic Models .......••.••.••..........•.• 254 1. Kinetic Models Involving Subunit-Sub- unit Interaction ......•.•.......•.•.... 254 2. Kinetic Models Involving Alternative Path~ays of Substrate Addition and Enzyme Isomerization .•.........•....... 256 D. Allostery ...•.........•.•.••.............• 266 1. Nonsigmoidal Systems................... 267 2. The Monod ModeL....................... 269 3. The Adair-Koshland Model .........••.•.. 273 4. Enzyme Isomerization Mechanisms .•...•.. 274 5. Kinetic Models....... . . • • . • . . . • . . • • • . •• 275 E. Product Effects •.•.••..•••.•.•.•.•••••.••. 276 References .•...•.•.•.••..•.•••••.••..•..•. 278 Appendix I Rate Equations, Determinants, and Haldane Ex pressions for Some Common Kinetic Mechanisms. 281 Appendix II A Computer Program for Deriving Enzyme Rate Equations. • • . • • . . • • • . . • • • • . • • • . . . . . . • • . . . • . .. 295 Appendix III Plotting and Statistical Analysis of Kinetic Data Using the OMNITAB Program .....•.••..•.•. 307 Subj ect Index............................................. 31 3