Fundamentals of Enzyme Kinetics

237 Pages·1979·11.571 MB·English

by  Athel Cornish-Bowden (Auth.)

To Mary Ann
Fundamentals of
Enzyme Kinetics
Athel Cornish-Bowden
Lecturer in Biochemistry,
University of Birmingham, U.K.
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First published 1979
ISBN 0 408 10617 4
© Butterworth & Co (Publishers) Ltd, 1979
British Library Cataloguing in Publication Data
Cornish-Bowden, Athel
Fundamentals of enzyme kinetics.
1. Enzymes
2. Chemical reaction, Rate of
I. Title
547'.758 QP601 79-40116
ISBN 0-408-10617-4
Typeset by Scribe Design, Gillingham, Kent
Printed and bound in England by the Camelot Press, Southampton
Preface
In this book I have set out to cover enzyme kinetics in sufficient
detail for the undergraduate student with an interest in enzymology
at the end of a degree course in biochemistry. It goes rather beyond
the essential core of enzyme kinetics that all students of biochemistry
learn, but I hope that about three-quarters of the book will be suit-
able for non-specialist teaching. The book was originally conceived
as an appropriately abridged edition of Principles of Enzyme Kinetics
(published by Butterworths, 1976), but in the event I have added
as much as I have taken away, as a result of greatly expanding the
coverage of the more practical aspects of the subject.
Most of the mathematics used in enzyme kinetics is elementary,
and the apparent difficulty of the subject owes more to the necessary
profusion of symbols and equations than to any real conceptual diffi-
culties. I have tried therefore to explain the mathematical derivations
in the book in sufficient detail to remove most of the barriers to
understanding. To test understanding, and in a few cases to carry the
theory a little beyond what is given explicitly in the text, I have
included problems at the end of each chapter, with solutions at the
end of the book.
I am grateful to the Chemistry Department of the University of
Guelph for their hospitality during the early stages of writing this
book, and for providing me with an opportunity to teach a one-
semester postgraduate course on enzyme kinetics. This experience
proved invaluable to me in the planning and writing of the book. I am
also grateful to H.B.F. Dixon, M. Gregoriou, R.H. Jackson, R.N.F.
Thorneley, C.W. Wharton and E.A. Wren, who provided many valu-
able suggestions for improvements or corrections.
Athel Cornish-Bowden
V
Principal symbols used in this book
A; a First substrate in the forward direction; its concen-
tration
A (In the sequential model) Conformation characteris-
tic of an unliganded subunit
A Constant of integration in the Arrhenius equation
a\a Ordinate intercept of general straight line; its least-
squares estimate
a ; Concentration of A at t - 0; at t = 00 (i.e. at equili-
0
brium)
A*;a* Radioactively labelled A; its concentration
B; b Second substrate in the forward direction; its
concentration
Β (In the sequential model) Conformation characteris-
tic of a subunit with ligand bound
b\b Slope of general straight line; its least-squares estimate
c (In the symmetry model) Ratio of dissociation
constants for R and Τ states
c (In the sequential model) Measure of stability of AB
interface
E; e Enzyme; its initial (or total) concentration
0
EA; [EA] Complex between Ε and A; its concentration (also
represented by single letters such asx, as defined)
EB, EP, ES, Complexes between Ε and Β, P, S, etc. Concentrations
etc. are expressed with square brackets or defined single
letters
E Arrhenius activation energy
a
e Deviation of /th observed value from calculated value
t
G Transferred group in a two-substrate two-product
reaction
h (In discussion of temperature) Planck's constant
h (In discussion of pH) Concentration of H +
h (In discussion of co-operative effects) Hill coefficient
I; / Inhibitor; its concentration
xi
xii Principal symbols
Κ Equilibrium constant
Κ Formation constant for transition state
k Rate constant
Κ Acid dissociation constant
Λ
^ ; K (In the sequential model) Subunit interaction constant
A B BB
for an AB interface; same for a BB interface
K Competitive-inhibition constant
{
K[ Uncompetitive-inhibition constant
(In multiple-substrate reactions) Inhibition constant
for A (see p. 105)
Λ
; K ; K The Michaelis constant (concentration of substrate
m m
when ν = V/2)\ its least-squares estimate; its distribu-
tion-free estimate
K, V pH-corrected value of K , V, i.e. the values K , V
m m m
would have if the enzyme were confined to its ideal
state of protonation
&m 9 Km > e t c- Michaelis constant for A, B, etc. (observed value when
inhibitors are absent and concentrations of all sub-
strates apart from the one specified are extrapolated
to infinity)
^ap ^ f/app Apparent value of K , V
P m
^exp ^ [/exp Expected value of K , V, i.e. value that would apply
m
if non-productive binding did not occur
K ; K (In the symmetry model) Dissociation constant for
R T
loss of substrate from subunit in R conformation;
same for Τ conformation
K Dissociation constant of ES
s
Κ£ Dissociation constant of EA
K Substrate-inhibition constant
si
K Dissociation constant of EX
x
Κ, K, etc. Association constant for first, second, etc. ligand in
λ 2
sequence of binding steps
k+i, k+2, etc. Rate constant for first, second, etc. steps in forward
direction
k_i, k_, etc. Rate constant for first, second, etc. steps in reverse
2
direction
k Catalytic constant (= V/e )
CSLt 0
L (In the symmetry model) Equilibrium constant
between R and Τ states in the absence of ligand
Ρ; ρ First product; its concentration
P*;p*;
p ; p^ See corresponding symbols at A
0
Ρ (In the simple theory of temperature effects)
Probability of reaction after molecular collision
pH Negative logarithm of H + concentration
Principal symbols xiii
pK. Negative logarithm of
d
Q\q \ etc. Second product; its concentration; see symbols at A
R (In the symmetry model) Conformation that binds
substrate tightly
R Gas constant
R Ratio of X concentrations for 90% and 10% saturation
x
S; s; etc. Substrate; its concentration; see symbols at A
SS Sum of squares of deviations
Τ (In the symmetry model) Conformation that binds
substrate weakly
Τ Absolute temperature (in kelvins)
t time
A
V\ V\ V* Maximum velocity, i.e. value of ν at saturation; its
least-squares estimate; its distribution-free estimate
γ- r/app. i/exp See unter K
m
ν Rate of reaction (at t = 0 if not otherwise specified)
v Initial rate (subscript omitted when unambiguous)
0
v* Rate of transfer of radioisotopic label
ν Calculated rate
Vi; Ρ Value of V in forward direction; in reverse direction
X Ligand of unspecified nature (e.g. substrate, product,
activator, inhibitor)
X* Transition state
x, y Concentrations of intermediates (always defined
when used)
Y Fractional saturation
Ζ Collision frequency
AG* Free energy of activation
AH0 Standard enthalpy of reaction
AH* Enthalpy of activation
AS* Entropy of activation
r Relaxation time
Chapter 1
Basic principles of chemical kinetics
1.1 Order of a 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 molecules that are altered in the reaction. Thus, a
reaction A -* Ρ is unimolecular (sometimes called monomolecular),
a reaction A + Β -> Ρ is bimolecular, and a reaction A + Β + C Ρ
is trimolecular (or termolecular). The order is a description of the
kinetics of the reaction and defines how many concentration terms
must be multiplied together to get an expression for the rate of reac-
tion. 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 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 usually the same as the molecular-
ity. However, many reactions consist of sequences of unimolecular
and bimolecular steps, and the molecularity of the complete reaction
need not be the same as its order. Indeed, for a complex reaction it
is often not meaningful to define an order, as the rate often cannot
be expressed as a product of concentration terms. As we shall see in
later chapters, this is almost universal in enzyme kinetics, and even
the simplest enzyme-catalysed reactions do not have simple orders.
In spite of this, the concept of order is important in the understand-
ing of enzyme kinetics, because the individual steps in enzyme-
catalysed reactions nearly always do have simple orders, usually being
first or second order. The binding of a substrate molecule to an
enzyme molecule is a typical example of a second-order bimolecular
reaction in enzyme kinetics; whereas conversion of an enzyme-
substrate complex into products or into another intermediate is a
typical example of a first-order unimolecular reaction.
For a first-order reaction A P, the velocity ν can be expressed as
v = af = ~ar = ka = k(a° ~p) ( M)
in which a and ρ are the concentrations at any time t of A and Ρ
l
2 Basic principles of chemical kinetics
respectively, and k is a first-order rate constant. At the start of
reaction t = 0 and a = a. By the stoicheiometry of the reaction,
0
every molecule of A that is consumed is converted into a molecule
of P, and so the concentrations of A and Ρ are related by the equation
a + ρ = a. Equation 1.1 can readily be integrated by separating the
0
two variables ρ and i.e. by bringing all terms in ρ on to the left-
hand side and all terms in t on to the right-hand side:
Therefore
-In (a — p) = kt + a
0
in which α is a constant of integration. This can be evaluated by
allowing for the fact that there is no product at the start of the reac-
tion, i.e. ρ = 0 when t = 0. Then, a = -In (a ), and so
0
In [(a - p)/a] = -kt
0 0
Taking exponentials of both sides we have
(a - p)/a = exp (-kt)
0 0
which can be rearranged to give
ρ = a [1 - exp (-kt)] (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 integration must always be included and calculated when
kinetic equations are integrated; they are rarely found to be zero.
The commonest type of bimolecular reaction is one of the form
A + Β ^ Ρ + Q, in which two different types of molecule, A and B,
react to give products. In this case the rate is likely to be given by a
second-order expression of the form
ν = apjàt = kab = k(a — p)(b — p) (1.3)
0 0
in which k is now a second-order rate constant. (Notice that conven-
tional symbolism does not, unfortunately, indicate the order of a
rate constant.) Again, integration is readily achieved by separating
the two variables ρ and t:
For readers with limited mathematical experience, the simplest and
Order of a reaction 3
most reliable method for integrating the left-hand side of this equation
is to look it up in standard tables of integrals. It may also be done by
multiplying both sides of the equation by (b — a) and separating
0 0
the left-hand side into two simple integrals:
Hence
-In (a - p) + In (b - p) = (b - a )kt + a
0 0 0 0
Putting ρ = 0 when t = 0, we find a = In (b/a ), and so
0 0
ao(b - ρ)
In B 0B . = (b-a)kt
b(a - P)_ 0 0
Q 0
or
αΦ - ρ)
0 0 exp [φ -a)kt] (1.4)
b(a - ρ) 0 0
0 0
The following special case of this result is of interest: if a is very
0
small compared with b, then ρ must also be very small compared
0
with b at all times, because ρ can never exceed a on account of the
0 0
stoicheiometry of the reaction. So (b — a ) and (b — p) can both
0 0 0
be written with good accuracy as b and equation 1.4 simplifies to
0
ρ = a [1 - exp (-kbt)]
0 0
This 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 kb is a pseudo-first-order rate constant. The
0
situation occurs naturally 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 evalua-
tion of the rate constant, as I shall discuss in Section 1.5.
Trimolecular reactions, such asA+B + C-*P+...,do not
usually consist of a single trimolecular step, and consequently they
are not usually third order. Instead the reaction is likely to consist
of two or more elementary steps, such as
A+ X
and
X + C -* Ρ
If one step in such a reaction is much slower than the others, the rate