EA and EP. In practice, enzymes have different confor-
mations at the various stages of the catalytic cycle. Thus,
they have open forms where the reactants are free to
bind and dissociate, and closed forms in which catalysis
takes place. The chemistry may also take place in more
than one step, so that intermediates are present. Figure 4
shows a profile where a conformation change converts
the open EA form to a closed, precatalytic EA
p
form.
This undergoes a catalytic reaction to give an EI
intermediate complex, which then is further converted
to the closed EP
p
form. A final conformation change
gives the open EP form, from which P can dissociate.
Rate-Limiting Step for V/K
The two independent kinetic constants for an enzymatic
reaction are the maximum velocity, V, and the ratio of V
and the Michaelis constant, V/K. There is a separate V/K
for each substrate. Each of these parameters varies
separately with pH, ionic strength, temperature, and the
concentrations of other substrates, products, or inhibi-
tors. The Michaelis constant, K, while it measures the
apparent dissociation constant of the substrate in
the steady state, is not an independent constant, but
just the ratio of V and V/K. Thus when one speaks of
“rate-limiting steps” one must specify whether one is
referring to V or V/K.
The rate-limiting step for V/K is the one with the
highest barrier in the free-energy profile. V/K involves
the combination of enzyme and substrate and reaction
through the first irreversible step, which normally is
release of the first product. Later steps that involve
further conformation changes in the enzyme and release
of other products do not affect V/K. Thus the slow
release of NADH from dehydrogenases, which often
limits the maximum velocities of these enzymes, has no
effect on V/K. As a result, the V/K value often is limited
more by the chemistry of the reaction than V,and
isotope effects on the chemistry are more fully expressed
(see entry on kinetic isotope effects). Except for some
slow mutants, full rate-limitation by the chemical
reaction is unusual, however, and the conformation
changes that precede and follow the chemical
reaction are usually partly rate-limiting. The equation
for an isotope effect on V/K is:
x
ðV=KÞ¼ð
x
k þ c
f
þ
x
K
eq
c
r
Þ
ð1 þ c
f
þ c
r
Þ½4
where x defines the isotope effect (D, T, 13, 15, 18 for
deuterium, tritium,
13
C,
15
N,
18
O) and the leading
superscript indicates the ratio of the parameters for light
and heavy isotopes.
x
k is the intrinsic isotope effect on
the chemical step, and
x
K
eq
the equilibrium isotope
effect on the reaction. The constants c
f
and c
r
are
forward and reverse commitments to catalysis and are
the ratio of the rate constant for the isotope sensitive
step to the net rate constant for release from the enzyme
of either the substrate (for c
f
) or the first product (for c
r
).
For the simple mechanism
E þ A O
k
1
k
2
EA O
k
3
k
4
EA
p
O
k
5
k
6
EPQ
p
O
k
7
k
8
EPQ
!
k
9
EQ þ P !
k
11
E þ P þ Q
½5
c
f
¼ðk
5
=k
4
Þð1 þ k
3
=k
2
Þ c
r
¼ðk
6
=k
7
Þð1 þ k
8
=k
9
Þ½6
The commitments thus consist of partition ratios of
intermediates and these ratios correspond to the
differences in barrier heights in a free-energy profile. In
the profile shown in Figure 4, c
f
and c
r
will be small if the
isotope effect measured involves reaction of EI to EP
p
because the barrier between EI and EP
p
is the highest
one. But if the isotope effect measured is on reaction of
EA
p
to EI, c
f
will be small, but c
r
will be large. Thus one
will see largely the equilibrium isotope effect, as the EA
p
to EI step approaches equilibrium.
In the definition of c
f
in eqn. [5], k
5
/k
4
is the internal
part of the commitment (c
f-in
) and k
3
k
5
/(k
2
k
4
) is the
external part (c
f-ex
). A sticky substrate is one where the
net rate constant for reaction of the initial collision
complex to give products is faster than the rate constant
for dissociation (k
2
). This ratio is called the stickiness
ratio (S
r
) and is given by S
r
¼ c
f-ex
=ð1 þ c
f-in
þ c
r
Þ in
terms of the parameters of eqn. [5].
Rate-Limiting Steps for V
The rate-limiting step for the maximum velocity depends
on the definition of rate limiting. One definition is the
“slowest” step, or the one with the highest individual
barrier in the forward direction. Another definition is the
“least-conductive” step, which is the one that sees the
highest total barrier to reach an irreversible step. A third
definition is the “most-sensitive” step. This is the one
FIGURE 4 Free-energy profile showing conformation changes in the
EA and EP complexes, as well as an intermediate between the two
activated complexes. The levels of (E þ A) and (E þ P) are not shown.
SUBSTRATE BINDING, CATALYSIS, AND PRODUCT RELEASE 125