In many cases, it will not be possible to extract all four rate
constants. Thus, for example, although inspection of Eq. 10 shows
that k obs (S) should increase from k À2 when [L tot ] ( K d1 to
(k À2 + k 2 ) when [L tot ] ) K d1 , this will not always be observable.
If K d1 is low then given the nature of the typical stopped-flow
experiment, it is unlikely that it will be possible to work under
conditions where [L tot ] ( K d1 , and k obs (S) will only vary significantly with [L tot ] when k À2 ( k 2 , so that k À2 will generally be
difficult to determine. In extreme cases, K d1 may be so low that
k obs (S) may well be completely independent of [L tot ] under all
attainable experimental conditions and only the sum of the rate
constants for the isomerization step will be measurable.
If, on the other hand, the bimolecular step is the fast diffusion
controlled formation of an encounter complex with very low overall
affinity (high K d1 ~ 1 mM) then the value of K d2 would need to be
10
À3 to give a typical overall equilibrium dissociation constant of
1 μM (see Eq. 6). A typical stopped-flow experiment could then
show only a single transient process because occupancy of the
intermediate PL would always be very low. The single observed
rate might then vary linearly with [L tot ] (since deviations from
linearity would only be observed when [L tot ] approached K d1 , see
Eq. 10) with an apparent second-order rate constant of k +2 /K d1
and dissociation rate constant k À2 . When [L tot ] does approach K d1 ,
some curvature may, of course, be observed and the initial slope
can then be taken as equal to k +2 /K d1 [16]. Association rate constants measured for protein–ligand interactions are, in fact, often
significantly lower than the values predicted using theoretical calculations based on diffusion coefficients, shape, and viscosity
[2]. Scheme E with a high K d1 and a low k +2 is frequently invoked
as an explanation for the observation of these unexpectedly low
values [11, 17].
In the case of Scheme F, with the bimolecular step very much
faster than the isomerization step, the expressions for experiments
performed under the condition that L tot > (P tot + P* tot ) are [15]:
k obs F
ð Þ ¼ k 2 L tot
½
þk À2
ð11Þ
k obs S
ð Þ ¼
k À1 K d2
K d2 þ L tot
½
þ k 1
ð12Þ
Scheme F can, at least in principle, be distinguished from
Scheme E by the fact that the observed rate for the slow process
should decrease from (k À1 + k +1 ) when [L tot ] ( K d2 to k +1 when
[L tot ] ) K d2 . However, as for Scheme E, only in the most favorable
cases will it be possible to extract all four rate constants for the
reaction.
Many multistep mechanisms will consist of a series of three or
more first- and second-order reactions and it is seldom, if ever,
possible to derive analytical solutions for a kinetic analysis using
Calmodulin Target Interactions
97
constants. Thus, for example, although inspection of Eq. 10 shows
that k obs (S) should increase from k À2 when [L tot ] ( K d1 to
(k À2 + k 2 ) when [L tot ] ) K d1 , this will not always be observable.
If K d1 is low then given the nature of the typical stopped-flow
experiment, it is unlikely that it will be possible to work under
conditions where [L tot ] ( K d1 , and k obs (S) will only vary significantly with [L tot ] when k À2 ( k 2 , so that k À2 will generally be
difficult to determine. In extreme cases, K d1 may be so low that
k obs (S) may well be completely independent of [L tot ] under all
attainable experimental conditions and only the sum of the rate
constants for the isomerization step will be measurable.
If, on the other hand, the bimolecular step is the fast diffusion
controlled formation of an encounter complex with very low overall
affinity (high K d1 ~ 1 mM) then the value of K d2 would need to be
10
À3 to give a typical overall equilibrium dissociation constant of
1 μM (see Eq. 6). A typical stopped-flow experiment could then
show only a single transient process because occupancy of the
intermediate PL would always be very low. The single observed
rate might then vary linearly with [L tot ] (since deviations from
linearity would only be observed when [L tot ] approached K d1 , see
Eq. 10) with an apparent second-order rate constant of k +2 /K d1
and dissociation rate constant k À2 . When [L tot ] does approach K d1 ,
some curvature may, of course, be observed and the initial slope
can then be taken as equal to k +2 /K d1 [16]. Association rate constants measured for protein–ligand interactions are, in fact, often
significantly lower than the values predicted using theoretical calculations based on diffusion coefficients, shape, and viscosity
[2]. Scheme E with a high K d1 and a low k +2 is frequently invoked
as an explanation for the observation of these unexpectedly low
values [11, 17].
In the case of Scheme F, with the bimolecular step very much
faster than the isomerization step, the expressions for experiments
performed under the condition that L tot > (P tot + P* tot ) are [15]:
k obs F
ð Þ ¼ k 2 L tot
½
þk À2
ð11Þ
k obs S
ð Þ ¼
k À1 K d2
K d2 þ L tot
½
þ k 1
ð12Þ
Scheme F can, at least in principle, be distinguished from
Scheme E by the fact that the observed rate for the slow process
should decrease from (k À1 + k +1 ) when [L tot ] ( K d2 to k +1 when
[L tot ] ) K d2 . However, as for Scheme E, only in the most favorable
cases will it be possible to extract all four rate constants for the
reaction.
Many multistep mechanisms will consist of a series of three or
more first- and second-order reactions and it is seldom, if ever,
possible to derive analytical solutions for a kinetic analysis using
Calmodulin Target Interactions
97