enzyme reactions that proceed in two steps, the parameter KMis given
by the ratio of the rates (eqn 7.44).
(7.50)
When the enzyme operates under the condition that kf 2 <<k 1 b, then the
equation reduces to:
(7.51)
In this case, KMis a measure of the affinity of the enzyme for its substrate.
However, for many enzymes this limit of the rates does not hold and the
interpretation of KMis not straightforward, especially when the enzyme
undergoes a multistep process.
To describe the enzyme activity, a more general rate constant, kcat, is
used to describe the limiting rate under saturating conditions. When the
mechanism is complex, this parameter represents a function of several
rate constants. However, for a simple two-step reaction, the new para-
meter can be equated to the forward rate of the second step (kcat=k 2 ).
In this case the Michaelis–Menten equation (eqn 7.45) can be written in
terms of kcat:
(7.52)
The parameter kcatis afirst-order rate constant and is called theturno 9 er
numberas it is equivalent to the number of substrate molecules that are
converted to product within a certain time when the enzyme is saturated
with substrate. Together with the KMvalue, the kcatprovides a measure
of the kinetic efficiency of the enzyme (Table 7.3). When the substrate
concentration is small compared to KM, this relationship can be simplified:
(7.53)
The initial velocity is now dependent upon the concentrations of the two
reactants: the total enzyme and substrate. This relationship is a second-
order rate equation with the ratio kcat/KMbeing a second-order rate
constant. One way to compare the efficiencies of different enzymes or
the turnover of different substrates by one enzyme is to make use of the
ratio kcat/KM, termed the specificity constant. The upper limit of this ratio
is the rate at which the enzyme and substrate can diffuse together in an
V
k
K
k
0 = +[]≈
cat total
M
[E ][S] cat total
S
[E ][[S]
E ][S]
M
cat
M
K total
k
K
= [
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
V
k
KS
f k
0
2 []
[]
[
=
+
=
Etotal][S E][
M
cat total SS
M S
]
K +[]
K
k
k
f
f
M=
2
1
K
kk
k
fb
f
M=
21 +
1
156 PARTI THERMODYNAMICS AND KINETICS
