section 6.2
Catalysis
89
equals its rate of breakdown, and steady-state kinetics
then applies.
2. Only initial velocities are measured. Hence, during
the period of measurement, [S] is not significantly
depleted as [P] increases.
3.
[ES] is constant during the period of measurement,
which means that (a) enough time has elapsed since
mixing E and S for [ES] to build up; (b) not enough
time has passed for the rate of formation of ES
to decrease owing to substrate depletion; and
(c)
k2 < k -
1
, so that [ES] can build up. If
k2
> k-i,
ES breaks down as rapidly as it is formed, and a
steady-state can never be established.
These expressions can both be solved for
k2:
k2
v
[ES]
and
k2 =
Vmax
ÏË d '
Therefore,
v
_ Vmax or
V
_ [ES]
[ES] “ [ET] OI ^max
[ET]'
Substituting the latter equation for [ES]/[ET] in Equa-
tion (6.3), we obtain
[S] —
~ ~ ~
-f [S ]).
^ max
Rate of formation of ES
d[
ES]
dt
= k
i [E] [S]
=
([E T] - [ES]) [ S ] .
Rearranging and solving for v,
[S]V max
K m
+ [S]
(6.4)
Rate of breakdown of ES
d[ES]
dt
= k_! [ES] +
k2
[ES]
= (*_j +
k2)
[ES].
Under steady-state conditions, the rate of formation of
ES equals its rate of breakdown, so that
kx
([Et] - [ES]) [S] = (*_,+
k2)
[ES].
(6.1)
Rearranging the above Equation yields
([E T] - [ES]) [S] = * _ i
+ k 2
[ES]
kx
(6.2)
Note that
Km
(the
Michaelis constant)
is
not
an equilib-
rium constant but a ratio of rate constants.
Equation (6.2) can be rearranged in the following ways:
[S][ET] - [S][ES] =
Km[
ES]
or
[S][ET] = [ES](/fm + [S])
or
[S]
=
(Km
+ [S]).
(6.3)
[E T]
It is usually difficult to measure [ES] in a reaction mixture.
Consequently, Equation (6.3) is not useful experimentally.
On the other hand, the velocity (v) and the maximum
velocity
(Vmax)
are readily determined by a variety of
methods.
As indicated in Figure 6-4, Vmax is the limiting value that
v approaches as [S] -* oo. In that situation, all enzyme
molecules have substrate bound to them, and so [E ] = 0
and [ET] = [ES], Thus, v = k
2
[ES], and Vmax = k
2
[ET].
Equation (6.4) is known as the
Michaelis-Menten equa-
tion,
and the following points should be noted.
1.
Km
is a constant for a particular enzyme and
substrate and is independent of enzyme and
substrate concentrations.
2
. Vmax depends on enzyme concentration, and at
saturating substrate concentration, it is independent
of substrate concentration.
3.
Km
and Vmax may be influenced by pH, temperature,
and other factors.
4. A plot of v versus [S] fits a rectangular hyperbolic
function (Figure 6-4).
5. If an enzyme binds more than one substrate, the
Km
values for the various substrates can be used as a
relative measure of the affinity of the enzyme for
each substrate (the smaller the value of
Km,
the
higher the affinity of the enzyme for that substrate).
6
. In a metabolic pathway,
Km
values for enzymes that
catalyze the sequential reactions may indicate the
rate-limiting step for the pathway (the highest
Km
corresponds roughly to the slowest step).
7. When
k
i
k2,
[ES] is assumed to be at
equilibrium with [E] and [S], ES is dissociating
more often to yield E and S than to yield product.
Under this condition,
the dissociation constant of the
enzyme-substrate complex
for this equilibrium
(ES ^ E + S) is
But
[E] [S] = U
4
[ES]
kx
k-\
+
k2