Table Of ContentPrinciples of Enzyme Kinetics
ATHEL CORNISH-BOWDEN
M.A., D.Phil. (Oxon)
Lecturer in Biochemistry, University of Birmingham
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First published 1976
ISBN 0 408 70721 6
© Butterworth & Co 1976
Library of Congress Cataloging in Publication Data
Cornish-Bowden, Athel.
Principles of enzyme kinetics.
Includes bibliographical references and index.
1. Enzymes. I. Title. [DNLM: 1. Enzymes.
2. Kinetics. QU135 C818p]
QP601.C757 574.Γ925 75-33717
ISBN 0-408-70721-6
Typeset in 10/1 lpt Monophoto Times New Roman
Printed in England
by Butler & Tanner Ltd
London & Frome
Foreword
Kinetics is a subject of mystery and power. It is mysterious to a great many
scientists, because at some point in their careers they acquired a fear and awe
of mathematical techniques. It is powerful to many others because no other
tool in all of science has such universality. Although this potential for
mystery and power is present in many techniques, kinetics seems to be
special in its capacity for polarizing individuals.
Those who do not understand and fear mathematics tend either to have
excessive admiration for kinetics ('It's too difficult for me') or excessive con-
tempt ('Kinetics can never prove a mechanism; it can merely disprove one').
The latter statement, while true, applies to any scientific technique, so one
might ask why kinetics is a special target for criticism. The answer would
seem to lie in the twin pitfalls that (a) the explanations of kinetic procedures
are frequently confusing and imprecise and (b) the power of kinetics is
frequently over-stated and therefore leads to error.
In this book, Athel Cornish-Bowden does an admirable job in steering
between these Scylla and Charybdis of kinetics. Firstly, the core of enzyme
kinetics is explained in a simple manner which the serious biologist, who
may not be 'a mathematical type', can follow readily. Secondly, the limita-
tions of the technique and the dangers of excessive extrapolation are clearly
outlined. The reader is given a powerful weapon and warned that it can
backfire if not handled properly. This is a well disciplined book. It does not
contain everything that is known about kinetics; and that is one of its virtues.
It has distilled some of the most important areas of kinetics, treating illus-
trative sections with rigour and clarity, which should help to provide more
enthusiasm for kinetics and more recognition of its power for applications in
biology and enzymology.
DANIEL E. KOSHLAND, JR.
Preface
This book is written primarily for first-year research students in enzyme
kinetics, but I hope that it will also prove useful to more advanced research
workers and to final-year undergraduates. For the student beginning re-
search, particularly one with a first degree in chemistry or biology, it is often
difficult to find a text that goes beyond an elementary and idealized account
of enzyme kinetics, but does not assume a large amount of specialized back-
ground knowledge and understanding. There are several topics in enzyme
kinetics, such as the derivation of steady-state rate equations, the analysis
of progress curves and the statistical treatment of results, that are important
enough to be covered at an elementary level but are usually discussed
inadequately or not at all.
With a proper understanding of the principles of enzyme kinetics, the
whole subject comes within reach. It becomes more complicated, although
not more difficult, as it is developed. For this reason, it is important to
cover the elementary aspects thoroughly. If I have erred, therefore, I hope
that it is in the direction of over-explaining the simple, rather than omitting
to explain the difficult. However, I have not tried to write an exhaustive
treatise: there is little mention of three-substrate reactions, for example, not
because they are not important, but because they can be studied within the
framework developed for the study of simpler reactions. In short, they
contribute complexity rather than understanding.
Although I hope I have been consistent in important matters in this book,
I have deliberately avoided any attempt to use a slavishly consistent system
of nomenclature and symbolism. There are circumstances in which S is an
appropriate symbol for substrate, for example, and others in which A,
B . . ., are preferable; I have therefore used both. Similarly, it is an unfortu-
nate fact that one of the two principal theories of co-operativity has been
defined in terms of dissociation constants and the other in terms of associa-
tion constants, but it would be a hindrance rather than a help to the student
to re-define them in a consistent way, because this would confuse any attempt
to read the original literature.
The emphasis throughout this book is on understanding enzyme kinetics,
and not on information about specific enzymes. It is in no sense a catalogue
of the properties of enzymes. Not only are there already several books that
fulfil that role admirably, but there also seems to be a real need for a book
that discusses the principles of enzyme kinetics at an intermediate level. I
hope that this book will help to fill that need.
Many mathematically inclined books begin with a descriptive introduc-
tion, designed, presumably, to lull the reader into a false sense of security.
This book does not follow that format, because I believe that it should be
made clear at the outset that enzyme kinetics is not a subject for anyone who
is frightened of simple algebra or simple calculus. Chapter 1 is a resume of
chemical kinetics, much of which should be familiar to the reader, but it is
included in order to establish the knowledge that will be assumed in later
chapters. It also includes a brief discussion of dimensional analysis, which
I believe to be by far the most powerful simple method for detecting algebraic
errors.
Chapters 2, 4, 5 and 6 cover the essential characteristics of steady-state
kinetics as taught in innumerable biochemistry courses, and require little
special discussion here. I have deviated slightly from common practice by
treating V\K as a parameter in its own right, at least as important as K ,
m m
because many aspects of enzyme kinetics are far simpler to understand and
classify in terms of Fand V\K rather than Fand K . Some may feel that the
m m
section in Chapter 6 on temperature dependence is rather short. This is
because I feel that very few of the large number of studies on the temperature
dependence of enzyme activity are of any value. It is rare for conditions to be
sufficiently favourable that a temperature study can usefully be carried out.
Chapter 3 sets out to explain as simply as possible the most useful methods
for deriving steady-state rate equations: the student of complex mechanisms
soon discovers that the method taught in the context of very simple mechan-
isms is virtually useless because of the hopelessly complicated algebra that it
engenders. Although the King-Altman method has been outlined in several
textbooks, its principle is to be found only, as far as I know, in the original,
difficult, paper. However, I believe that anyone who often uses such an
important method ought to have some understanding of its theoretical basis,
and so I have tried to explain this as simply as possible. The chapter also
includes some important developments from the King-Altman method that
have been made in recent years.
The study of co-operativity (Chapter 7) has developed apart from the
mainstream of enzyme kinetics, and it has often been neglected in textbooks.
It has developed with its own conventions, such as the more common use of
association constants than dissociation constants. In this chapter particu-
larly, and to some extent throughout the book, the temptation to invent new
symbols and terminology has been strong, but I have not consciously suc-
cumbed to it anywhere, except for the use of h for the Hill coefficient. This
exception seemed justified by the very strong objection to n, which is grossly
misleading, and the typographically cumbersome nature of n (particularly
H
when used as an exponent).
Chapter 8 concerns an aspect of enzyme kinetics that has been almost
completely ignored by biochemists for 60 years, for reasons that have lost
much of their original force. I believe that it is time for integrated rate
equations to regain the respectability that they lost with the classic work of
Michaelis and Menten.
Chapter 9 is an introduction to the study of fast reactions, but it does not
pretend to be a comprehensive account, which would require a separate
book. Instead, I have tried to cover those aspects of fast reactions that ought
to be familiar to anyone who is working mainly on studies in the steady state,
but who feels that the need for transient-state studies may arise occasionally.
Chapter 10 is an introduction to the statistical aspects of enzyme kinetics.
Many biochemists apparently believe that it is unnecessary to understand
this topic, but they deceive themselves. The continued widespread use of the
Lineweaver-Burk plot is evidence of the laziness of the majority who cannot
be bothered to discover the most basic information about data analysis.
I am grateful to Dr. J. R. Knowles and to Dr. D. E. Koshland for stimulat-
ing and developing my interest in enzyme kinetics, and to several colleagues,
particularly Dr. R. Eisenthal, Mr. A. C. Storer, Dr. C. W. Wharton and
Dr. E. A. Wren, for many helpful comments on the first draft of this book.
It has gained much from their advice, and has lost numerous errors. Doubt-
less some remain, as all books that contain many equations contain errors,
and I shall greatly appreciate it if they can be brought to my attention.
ATHEL CORNISH-BOWDEN
1
Basic Principles of Chemical Kinetics
1.1 Order of reaction
A chemical reaction can be classified either according to its molecularity or
according to its order. The molecularity is defined by the number of mole-
cules that are altered in the reaction. Thus, a reaction A -> products is
unimolecular or monomolecular, a reaction A + B -> products or 2A -> pro-
ducts is bimolecular, and a reaction A + B + C -> products is trimolecular or
termolecular. (The illogical variations in prefixes is a consequence of the un-
fortunate propensity of scientists for inventing new words from what they
imagine to be classical roots without first determining what the roots mean.
In this book, I shall use the first of each of the above alternatives, albeit with
misgivings.) The order is a description of the number of concentration terms
multiplied together in the rate equation. Hence, in a first-order reaction, the
rate is proportional to one concentration, in a second-order reaction it is
proportional to two concentrations or to the square of one concentration,
and so on.
For a simple reaction that consists of a single step, or for each step in a
complex reaction, the order is generally the same as the molecularity. How-
ever, many reactions consist of a sequence of unimolecular and bimolecular
steps, and the molecularity of the complete reaction need not be the same as
its order. Reactions of molecularity greater than 2 are common, but reactions
of order greater than 2 are very rare. It should also be noted that neither the
molecularity nor the order of a reverse reaction need be the same as the
corresponding molecularity or order of the forward reaction. This is an
important consideration for metabolic reactions, which are often reversible
and can be made to proceed in either direction by adjusting the concentra-
tions of reactants.
For a first-order reaction A -► P, the velocity, v, can be expressed as
follows:
v = ^ = ka = k(a -p) (1.1)
0
1
BASIC PRINCIPLES OF CHEMICAL KINETICS
where a and p are the concentrations of A and P, respectively, and are
related by the equation a + p = a , t is time and k is a,first-order rate constant.
0
This equation can readily be integrated as follows:
kdt
JJ aaoo~~PP JJ
Therefore,
— ln(a — p) = fei + a
0
where a is a constant of integration, which can be evaluated by defining the
time scale so that a = a , p = 0 when t = 0. Then, a = —ln(a ), and so
0 0
ln[(fl -p)/fl<>] = ~kt
0
which can be rearranged to give
p = a [l-exp(-/ci)] (1.2)
0
It is important to note that the constant of integration, a, was included in
this derivation, evaluated and found to be non-zero. Constants of integra-
tion must always be included and calculated when kinetic equations are
integrated; they are very rarely found to be zero.
A simple bimolecular reaction, 2A -► P, is likely to be second order, with
rate v given by
= dp/dt = ka2 = k(a -2p)2 (1.3)
v 0
where k is now a second-order rate constant. (Notice that conventional sym-
bolism does not, unfortunately, indicate the order of a rate constant.)
Then,
kdt
JJ ((aa--22pp)r2 JJ
00
Therefore,
= /ci + a
2(a -2p)
0
and, putting p = 0 when t = 0, we have a = l/2a , so that
0
Reactions of this type are not unknown, but they are rare, and bimolecular
reactions are much more commonly of the type A + B -* P, in which the two
reacting molecules are different:
v = dp/dt = kab = k(a -p)(b -p) (1.5)
0 0
In this instance, we have
Jdp/[(fl -p)(ft -p)] = J/cdi
0 0
which can be integrated to give
2
ORDER OF REACTION
1 \η(^ϊ =* + «
b -aoJ V*o-P,
0
and putting p = 0 when ί = 0 and rearranging, we obtain
a (b -p)^
0 0
In = (b -a )kt
0 0
PO(<*O-P\
or
flo(fcp-p)
= exp[(b -a )ki] (1.6)
0 0
The following special case of this result is of interest: if a P b , then p
0 0
must be insignificant compared with a at every stage in the reaction, and so
0
(a — p) can be written simply as a . In this case, equation 1.6 simplifies to
0 0
p = b [\-exp(-ka t)~\
0 0
which is of exactly the same form as equation 1.2, the equation for a first -
order reaction. This type of reaction is known as a pseudo-first-order reaction,
and ka is a pseudo-first-order rate constant. The situation arises most often
0
when one of the reactants is the solvent, as in most hydrolysis reactions, but
it is also advantageous to set up pseudo-first-order conditions deliberately,
in order to simplify evaluation of the rate constant, as we shall discuss in
Section 1.5.
Trimolecular reactions, such as A + B + C -► P, do not usually consist of a
single trimolecular step, and consequently they are not usually third order.
Instead, the reaction usually consists of two or more elementary steps, such as
A + B-+X
X + C -P
If one step in such a reaction is much slower than the others, the rate of the
complete reaction is equal to the rate of the slow step, which is accordingly
known as the rate-determining (or rate-limiting) step. If there is no clearly
defined rate-determining step, the rate equation is likely to be complex and
to have no constant order. Some trimolecular reactions do display third -
order kinetics, with v = kabc, where k is now a third-order rate constant, but
it is not necessary to assume a three-body collision (which is inherently very
unlikely) in order to account for this. Instead, we can assume a two-step
mechanism, as above but with the first step rapidly reversible, so that the
concentration of X is given by x = Kab, where K is an equilibrium constant.
The rate of the reaction is determined by that of the slow second step:
v = k'xc = k'Kabc
where k! is the rate constant for the second step. Hence the observed third -
order rate constant is actually the product of a second-order rate constant
and an equilibrium constant.
Some reactions are observed to be zero order, i.e. the rate appears to be
constant, independent of the concentration of reactant. If a reaction is zero
order with respect to only one reactant, this may simply mean that the re-
3
BASIC PRINCIPLES OF CHEMICAL KINETICS
actant enters the reaction after the rate-determining step. However, some
reactions are zero order overall, i.e. independent of all reactant concentra-
tions. Such reactions are invariably catalysed reactions and occur if every
reactant is present in such large excess that the full potential of the catalyst
is realized. Examples of this behaviour will be seen when enzyme catalysis
is discussed.
1.2 Determination of the order of a reaction
The simplest means of determining the order of a reaction is to determine the
rate at different concentrations of the reactants. Then a plot of log(rate)
against log(concentration) gives a straight line with a slope equal to the
order. If all of the reactant concentrations are altered in a constant ratio, the
slope of the line is the overall order. It is usually useful to know the order with
respect to each reactant, however, which can be found by altering the con-
centration of each reactant separately, keeping the other concentrations
constant. Then the slope of the line will be equal to the order with respect to
the variable reactant. For example, if the rate is second order in A and first
order in B,
dp/dt = ka2b
then
log dp/dt = log /c + 2 log fl + log b
Hence a plot of log dp/dt against log a (with b held constant) will have a
slope of 2, and a plot of log dp/dt against log b (with a held constant) will
have a slope of 1. Both plots should give the same intercept of log k on the
log v axis, so they provide a useful check on one another. It is important to
realize that if the rates are determined from the slopes of the progress curve
(i.e. a plot of concentration against time), the concentrations of all of the
reactants will change. Therefore, if valid results are to be obtained, either the
initial concentrations of all the reactants must be in stoichiometric ratio, in
which event the overall order will be found, or (more usually) the 'constant'
reactants must be in large excess at the start of the reaction, so that the
changes in their concentrations are insignificant. If neither of these alterna-
tives is possible or convenient, the rates must be obtained from a set of
measurements of the slope at zero time, i.e. of initial rates. This method is
usually preferable for kinetic measurements of enzyme-catalysed reactions,
because the progress curves of enzyme-catalysed reactions often do not
rigorously obey the simple rate equations for extended periods of time. In
practice, the progress curve for an enzyme-catalysed reaction often requires
a more complicated equation than the integrated form of the rate equation
derived for the initial rate.
1.3 Dimensions of rate constants
Dimensional analysis is a technique that deserves to be used much more
4
