ENZYME KINETICS
ENZYME KINETICSA Modern ApproachALEJANDRO G. MARANGONIDepartment of Food ScienceUniversity of GuelphA JOHN WILEY & SONS, INC., PUBLICATION
This book is printed on acid-free paper.Copyright 2003 by John Wiley & Sons, Inc. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.No part of this publication may be reproduced, stored in a retrieval system, or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise,except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of theappropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests tothe Publisher for permission should be addressed to the Permissions Department, John Wiley &Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail:permreq@wiley.com.Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their bestefforts in preparing this book, they make no representations or warranties with respect to theaccuracy or completeness of the contents of this book and specifically disclaim any impliedwarranties of merchantability or fitness for a particular purpose. No warranty may be created orextended by sales representatives or written sales materials. The advice and strategies containedherein may not be suitable for your situation. You should consult with a professional whereappropriate. Neither the publisher nor author shall be liable for any loss of profit or any othercommercial damages, including but not limited to special, incidental, consequential, or otherdamages.For general information on our other products and services please contact our Customer CareDepartment within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 orfax 317-572-4002.Wiley also publishes its books in a variety of electronic formats. Some content that appears inprint, however, may not be available in electronic format.For ordering and customer service, call 1-800-CALL-WILEY.Library of Congress Cataloging-in-Publication Data:Marangoni, Alejandro G., 1965Enzyme kinetics : a modern approach / Alejandro G. Marangoni.p. ; cm.ISBN 0-471-15985-9 (cloth)1.Enzyme kinetics.[DNLM: 1. Enzymes. 2. Kinetics. 3. Models, Chemical.2003]I. Title.QP601.3 .M37 2003572 .7—dc21Printed in the United States of America10 9 8 7 6 5 4 3 2 1QU 135 M311e2002014042
To Dianne, Isaac, and Joshua
CONTENTSPREFACE1 TOOLS AND TECHNIQUES OF KINETIC ANALYSISxiii11.1 Generalities / 11.2 Elementary Rate Laws / 21.2.1 Rate Equation / 21.2.2 Order of a Reaction / 31.2.3 Rate Constant / 41.2.4 Integrated Rate Equations / 41.2.4.1 Zero-Order Integrated RateEquation / 41.2.4.2 First-Order Integrated RateEquation / 51.2.4.3 Second-Order Integrated RateEquation / 71.2.4.4 Third-Order Integrated RateEquation / 81.2.4.5 Higher-Order Reactions / 91.2.4.6 Opposing Reactions / 91.2.4.7 Reaction Half-Life / 11vii
viiiCONTENTS1.31.41.51.61.2.5 Experimental Determination of ReactionOrder and Rate Constants / 121.2.5.1 Differential Method (Initial RateMethod) / 121.2.5.2 Integral Method / 13Dependence of Reaction Rates onTemperature / 141.3.1 Theoretical Considerations / 141.3.2 Energy of Activation / 18Acid–Base Chemical Catalysis / 20Theory of Reaction Rates / 23Complex Reaction Pathways / 261.6.1 Numerical Integration and Regression / 281.6.1.1 Numerical Integration / 281.6.1.2 Least-Squares Minimization(Regression Analysis) / 291.6.2 Exact Analytical Solution(Non-Steady-State Approximation) / 391.6.3 Exact Analytical Solution (Steady-StateApproximation) / 402 HOW DO ENZYMES WORK?413 CHARACTERIZATION OF ENZYME ACTIVITY443.1 Progress Curve and Determination of ReactionVelocity / 443.2 Catalysis Models: Equilibrium and SteadyState / 483.2.1 Equilibrium Model / 483.2.2 Steady-State Model / 493.2.3 Plot of v versus [S] / 503.3 General Strategy for Determination of theCatalytic Constants Km and Vmax / 523.4 Practical Example / 533.5 Determination of Enzyme Catalytic Parametersfrom the Progress Curve / 58
CONTENTS4 REVERSIBLE ENZYME INHIBITION4.14.24.34.44.5ix61Competitive Inhibition / 61Uncompetitive Inhibition / 62Linear Mixed Inhibition / 63Noncompetitive Inhibition / 64Applications / 654.5.1 Inhibition of Fumarase by Succinate / 654.5.2 Inhibition of Pancreatic CarboxypeptidaseA by β-Phenylpropionate / 674.5.3 Alternative Strategies / 695 IRREVERSIBLE ENZYME INHIBITION705.1 Simple Irreversible Inhibition / 725.2 Simple Irreversible Inhibition in the Presence ofSubstrate / 735.3 Time-Dependent Simple IrreversibleInhibition / 755.4 Time-Dependent Simple Irreversible Inhibition inthe Presence of Substrate / 765.5 Differentiation Between Time-Dependent andTime-Independent Inhibition / 786 pH DEPENDENCE OF ENZYME-CATALYZEDREACTIONS796.1 The Model / 796.2 pH Dependence of the Catalytic Parameters / 826.3 New Method of Determining pK Values ofCatalytically Relevant Functional Groups / 847 TWO-SUBSTRATE REACTIONS7.1 Random-Sequential Bi Bi Mechanism / 917.1.1 Constant [A] / 937.1.2 Constant [B] / 937.2 Ordered-Sequential Bi Bi Mechanism / 957.2.1 Constant [B] / 9590
xCONTENTS7.2.2 Constant [A] / 967.2.3 Order of Substrate Binding / 977.3 Ping-Pong Bi Bi Mechanism / 987.3.1 Constant [B] / 997.3.2 Constant [A] / 997.4 Differentiation Between Mechanisms / 1008 MULTISITE AND COOPERATIVE ENZYMES1028.1 Sequential Interaction Model / 1038.1.1 Basic Postulates / 1038.1.2 Interaction Factors / 1058.1.3 Microscopic versus MacroscopicDissociation Constants / 1068.1.4 Generalization of the Model / 1078.2 Concerted Transition or Symmetry Model / 1098.3 Application / 1148.4 Reality Check / 1159 IMMOBILIZED ENZYMES1169.1 Batch Reactors / 1169.2 Plug-Flow Reactors / 1189.3 Continuous-Stirred Reactors / 11910 INTERFACIAL ENZYMES12110.1 The Model / 12210.1.1 Interfacial Binding / 12210.1.2 Interfacial Catalysis / 12310.2 Determination of Interfacial Area per UnitVolume / 12510.3 Determination of Saturation Interfacial EnzymeCoverage / 12711 TRANSIENT PHASES OF ENZYMATIC REACTIONS11.1 Rapid Reaction Techniques / 13011.2 Reaction Mechanisms / 132129
CONTENTSxi11.2.1 Early Stages of the Reaction / 13411.2.2 Late Stages of the Reaction / 13511.3 Relaxation Techniques / 13512 CHARACTERIZATION OF ENZYME STABILITY14012.1 Kinetic Treatment / 14012.1.1 The Model / 14012.1.2 Half-Life / 14212.1.3 Decimal Reduction Time / 14312.1.4 Energy of Activation / 14412.1.5 Z Value / 14512.2 Thermodynamic Treatment / 14612.3 Example / 15012.3.1 Thermodynamic Characterization ofStability / 15112.3.2 Kinetic Characterization of Stability / 15613 MECHANISM-BASED INHIBITION158Leslie J. Copp13.1 Alternate Substrate Inhibition / 15913.2 Suicide Inhibition / 16313.3 Examples / 16913.3.1 Alternative Substrate Inhibition / 16913.3.2 Suicide Inhibition / 17014 PUTTING KINETIC PRINCIPLES INTO PRACTICEKirk L. Parkin14.1 Were Initial Velocities Measured? / 17514.2 Does the Michaelis–Menten Model Fit? / 17714.3 What Does the Original [S] versus Velocity PlotLook Like? / 17914.4 Was the Appropriate [S] Range Used? / 18114.5 Is There Consistency Working Within the Contextof a Kinetic Model? / 18414.6 Conclusions / 191174
xiiCONTENTS15 USE OF ENZYME KINETIC DATA IN THE STUDYOF STRUCTURE–FUNCTION RELATIONSHIPS OFPROTEINS193Takuji Tanaka and Rickey Y. Yada15.1 Are Proteins Expressed Using Various MicrobialSystems Similar to the Native Proteins? / 19315.2 What Is the Mechanism of Conversion of aZymogen to an Active Enzyme? / 19515.3 What Role Does the Prosegment Play in theActivation and Structure–Function of the ActiveEnzyme? / 19815.4 What Role Do Specific Structures and/or ResiduesPlay in the Structure–Function of Enzymes? / 20215.5 Can Mutations be Made to Stabilize the Structureof an Enzyme to Environmental Conditions? / 20515.5.1 Charge Distribution / 20515.5.2 N-Frag Mutant / 20815.5.3 Disulfide Linkages / 21015.6 Conclusions / 21215.7 Abbreviations Used for the MutationResearch / 213BIBLIOGRAPHY217Books / 217Selection of Classic Papers / 218INDEX221
PREFACEWe live in the age of biology—the human and many other organisms’genomes have been sequenced and we are starting to understand thefunction of the metabolic machinery responsible for life on our planet.Thousands of new genes have been discovered, many of these coding forenzymes of yet unknown function. Understanding the kinetic behaviorof an enzyme provides clues to its possible physiological role. Froma biotechnological point of view, knowledge of the catalytic propertiesof an enzyme is required for the design of immobilized enzyme-basedindustrial processes. Biotransformations are of key importance to thepharmaceutical and food industries, and knowledge of the catalyticproperties of enzymes, essential. This book is about understanding theprinciples of enzyme kinetics and knowing how to use mathematicalmodels to describe the catalytic function of an enzyme. Coverage of thematerial is by no means exhaustive. There exist many books on enzymekinetics that offer thorough, in-depth treatises of the subject. This bookstresses understanding and practicality, and is not meant to replace, butrather to complement, authoritative treatises on the subject such as Segel’sEnzyme Kinetics.This book starts with a review of the tools and techniques usedin kinetic analysis, followed by a short chapter entitled “How DoEnzymes Work?”, embodying the philosophy of the book. Characterizationof enzyme activity; reversible and irreversible inhibition; pH effects onenzyme activity; multisubstrate, immobilized, interfacial, and allostericenzyme kinetics; transient phases of enzymatic reactions; and enzymexiii
xivPREFACEstability are covered in turn. In each chapter, models are developedfrom first principles, assumptions stated and discussed clearly, andapplications shown.The treatment of enzyme kinetics in this book is radically differentfrom the traditional way in which this topic is usually covered. In thisbook, I have tried to stress the understanding of how models are arrivedat, what their limitations are, and how they can be used in a practicalfashion to analyze enzyme kinetic data. With the advent of computers,linear transformations of models have become unnecessary—this bookdoes away with linear transformations of enzyme kinetic models, stressingthe use of nonlinear regression techniques. Linear transformations are notrequired to carry out analysis of enzyme kinetic data. In this book, Idevelop new ways of analyzing kinetic data, particularly in the study ofpH effects on catalytic activity and multisubstrate enzymes. Since a largeproportion of traditional enzyme kinetics used to deal with linearizationof data, removing these has both decreased the amount of informationthat must be acquired and allowed for the development of a deeperunderstanding of the models used. This, in turn, will increase the efficacyof their use.The book is relatively short and concise, yet complete. Time is today’smost precious commodity. This book was written with this fact in mind;thus, the coverage strives to be both complete and thorough, yet conciseand to the point.ALEJANDRO MARANGONIGuelph, September, 2001
ENZYME KINETICS
CHAPTER 1TOOLS AND TECHNIQUES OFKINETIC ANALYSIS1.1 GENERALITIESChemists are concerned with the laws of chemical interactions. The theories that have been expounded to explain such interactions are basedlargely on experimental results. Two main approaches have been used toexplain chemical reactivity: thermodynamic and kinetic. In thermodynamics, conclusions are reached on the basis of changes in energy and entropythat accompany a particular chemical change in a system. From the magnitude and sign of the free-energy change of a reaction, it is possible topredict the direction in which a chemical change will take place. Thermodynamic quantities do not, however, provide any information on the rateor mechanism of a chemical reaction. Theoretical analysis of the kinetics,or time course, of processes can provide valuable information concerningthe underlying mechanisms responsible for these processes. For this purpose it is necessary to construct a mathematical model that embodies thehypothesized mechanisms. Whether or not the solutions of the resultingequations are consistent with the experimental data will either prove ordisprove the hypothesis.Consider the simple reaction A B C. The law of mass action statesthat the rate at which the reactant A is converted to product C is proportional to the number of molecules of A available to participate inthe chemical reaction. Doubling the concentration of either A or B willdouble the number of collisions between molecules that lead to productformation.1
2TOOLS AND TECHNIQUES OF KINETIC ANALYSISThe stoichiometry of a reaction is the simplest ratio of the number ofreactant molecules to the number of product molecules. It should not bemistaken for the mechanism of the reaction. For example, three moleculesof hydrogen react with one molecule of nitrogen to form ammonia: N2 3H2 2NH3 .The molecularity of a reaction is the number of reactant molecules participating in a simple reaction consisting of a single elementary step. Reactions can be unimolecular, bimolecular, and trimolecular. Unimolecularreactions can include isomerizations (A B) and decompositions (A B C). Bimolecular reactions include association (A B AB; 2A A2 ) and exchange reactions (A B C D or 2A C D). The lesscommon termolecular reactions can also take place (A B C P).The task of a kineticist is to predict the rate of any reaction under agiven set of experimental conditions. At best, a mechanism is proposedthat is in qualitative and quantitative agreement with the known experimental kinetic measurements. The criteria used to propose a mechanismare (1) consistency with experimental results, (2) energetic feasibility,(3) microscopic reversibility, and (4) consistency with analogous reactions. For example, an exothermic, or least endothermic, step is mostlikely to be an important step in the reaction. Microscopic reversibilityrefers to the fact that for an elementary reaction, the reverse reactionmust proceed in the opposite direction by exactly the same route. Consequently, it is not possible to include in a reaction mechanism any stepthat could not take place if the reaction were reversed.1.2 ELEMENTARY RATE LAWS1.2.1 Rate EquationThe rate equation is a quantitative expression of the change in concentration of reactant or product molecules in time. For example, consider thereaction A 3B 2C. The rate of this reaction could be expressed asthe disappearance of reactant, or the formation of product:rate 1 d[B]1 d[C]d[A] dt3 dt2 dt(1.1)Experimentally, one also finds that the rate of a reaction is proportionalto the amount of reactant present, raised to an exponent n:rate [A]n(1.2)
ELEMENTARY RATE LAWS3where n is the order of the reaction. Thus, the rate equation for thisreaction can be expressed as d[A] kr [A]ndt(1.3)where kr is the rate constant of the reaction.As stated implicitly above, the rate of a reaction can be obtained fromthe slope of the concentration–time curve for disappearance of reactant(s) or appearance of product(s). Typical reactant concentration–timecurves for zero-, first-, second-, and third-order reactions are shown inFig. 1.1(a). The dependence of the rates of these reactions on reactantconcentration is shown in Fig. 1.1(b).ReactantConcentration10080n 0n 160n 24020n 300510Time(a )15202.0Velocity1.5n 3n 01.00.5n 1n 20.0012 3 4 5 6 7 8Reactant Concentration9 10(b )Figure 1.1. (a) Changes in reactant concentration as a function of time for zero-, first-,second-, and third-order reactions. (b) Changes in reaction velocity as a function of reactant concentration for zero-, first-, second-, and third-order reactions.
4TOOLS AND TECHNIQUES OF KINETIC ANALYSIS1.2.2 Order of a ReactionIf the rate of a reaction is independent of a particular reactant concentration, the reaction is considered to be zero order with respect to theconcentration of that reactant (n 0). If the rate of a reaction is directlyproportional to a particular reactant concentration, the reaction is considered to be first-order with respect to the concentration of that reactant(n 1). If the rate of a reaction is proportional to the square of a particularreactant concentration, the reaction is considered to be second-order withrespect to the concentration of that reactant (n 2). In general, for anyreaction A B C · · · P, the rate equation can be generalized asrate kr [A]a [B]b [C]c · · ·(1.4)where the exponents a, b, c correspond, respectively, to the order of thereaction with respect to reactants A, B, and C.1.2.3 Rate ConstantThe rate constant (kr ) of a reaction is a concentration-independent measure of the velocity of a reaction. For a first-order reaction, kr has unitsof (time) 1 ; for a second-order reaction, kr has units of (concentration) 1(time) 1 . In general, the rate constant of an nth-order reaction has unitsof (concentration) (n 1) (time) 1 .1.2.4 Integrated Rate EquationsBy integration of the rate equations, it is possible to obtain expressions thatdescribe changes in the concentration of reactants or products as a functionof time. As described below, integrated rate equations are extremely usefulin the experimental determination of rate constants and reaction order.1.2.4.1 Zero-Order Integrated Rate EquationThe reactant concentration–time curve for a typical zero-order reaction,A products, is shown in Fig. 1.1(a). The rate equation for a zero-orderreaction can be expressed asd[A] kr [A]0dt(1.5)Since [A]0 1, integration of Eq. (1.5) for the boundary conditions A A0 at t 0 and A At at time t, At td[A] krdt(1.6)A00
ELEMENTARY RATE LAWS5100[At]80slope kr60402000102030t405060Figure 1.2. Changes in reactant concentration as a function of time for a zero-orderreaction used in the determination of the reaction rate constant (kr ).yields the integrated rate equation for a zero-order reaction:[At ] [A0 ] kr t(1.7)where [At ] is the concentration of reactant A at time t and [A0 ] is theinitial concentration of reactant A at t 0. For a zero-order reaction, aplot of [At ] versus time yields a straight line with negative slope kr(Fig. 1.2).1.2.4.2 First-Order Integrated Rate EquationThe reactant concentration–time curve for a typical first-order reaction,A products, is shown in Fig. 1.1(a). The rate equation for a first-orderreaction can be expressed asd[A] kr [A]dt(1.8)Integration of Eq. (1.8) for the boundary conditions A A0 at t 0 andA At at time t, At td[A] krdt(1.9)A0 [A]0yields the integrated rate equation for a first-order reaction:lnor[At ] kr t[A0 ][At ] [A0 ] e kr t(1.10)(1.11)
6TOOLS AND TECHNIQUES OF KINETIC ANALYSISFor a first-order reaction, a plot of ln([At ]/[A0 ]) versus time yields astraight line with negative slope kr (Fig. 1.3).A special application of the first-order integrated rate equation is in thedetermination of decimal reduction times, or D values, the time requiredfor a one-log10 reduction in the concentration of reacting species (i.e.,a 90% reduction in the concentration of reactant). Decimal reductiontimes are determined from the slope of log10 ([At ]/[A0 ]) versus time plots(Fig. 1.4). The modified integrated first-order integrated rate equation canbe expressed as[At ]tlog10 (1.12)[A0 ]Dor[At ] [A0 ] · 10 (t/D)(1.13)0ln [At]/[Ao] 1 2slope kr 3 4 5 60102030t405060Figure 1.3. Semilogarithmic plot of changes in reactant concentration as a function oftime for a first-order reaction used in determination of the reaction rate constant (kr ).0D 10tlog10[At]/[Ao] 1 2 3 4 5D 601020
principles of enzyme kinetics and knowing how to use mathematical models to describe the catalytic function of an enzyme. Coverage of the material is by no means exhaustive. There exist many books on enzyme kinetics that offer thorough, in-depth treatises of the subject. This book stresses understanding and practicality, and is not meant to .
Question #1: What is the effect of enzyme concentration on enzyme activity? 1. Set up 3 fresh cups of 1% H 2 O 2 that are 4 cm deep. 2. Begin with the enzyme solution. Make a dilution of the enzyme so that you have 3 strengths of enzyme: one at 200% enzyme strength ( 4 mL solution), one at
5 Kinetics of Single-Substrate Enzyme Reactions 109 5.1 The Time Course of Enzymatic Reactions / 109 5.2 Effects of Substrate Concentration on Velocity / 111 5.3 The Rapid Equilibrium Model of Enzyme Kinetics / 113 5.4 The Steady State Model of Enzyme Kinetics / 115 5.5 The Significance of k and K / 120 5.6 Experimental Measurement of k and K .
(2) Investigate the enzyme-bound iron content of catalase by using a modified ferrozine method. (3) Perform a protein assay to determine the enzyme concentration of a given sample. (4) Study the effects of reaction environment (temperature, pH) on the rate of the enzyme-catalyzed reaction. Enzyme kinetics is governed by a series of equations.
existed on earth. Enzymology: How to monitor enzyme catalysed reactions. How does an enzyme catalyse reactions? Enzyme classification. Mechanism of enzyme action. Discuss at least two examples. Enzyme kinetics and kinetic 3 7. Information Transfer Purpose: The molecular basis of coding and decoding genetic information is universal.
It is always best to check the enzyme activity in advance. In the ICT support there is a datalogging sheet on monitoring an enzyme-catalysed reaction. The Core Practical requires investigation of enzyme and substrate concentration. Having completed the practical investigating enzyme conc
enzyme activity. A brewer may use exogenous enzyme supplementation if there is concern that endogenous enzyme levels will not be sufficient. Barley variety, pre-harvest sprouting and methods of malting, kilning and mashing may all influence endogenous enzyme levels. Exogenous enzyme mixtures can correct issues like stuck mashes and low extract .
B) Simple kinetics C) Advanced kinetics. Optional temperature control is possible. This method is espe-cially well suited for preliminary experiments, i.e. determina-tion of the kinetic of a reaction (speed, range of linearity). b) “Simple kinetics”: as in a); additionally, units and conver-
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