BioPHYSICAL chemistry

Sequential first-order reactions

Another possible reaction is one that consists of sequential processes.
In terms of the gumballs, there are two dispensers but an anxious
child must wait until the gumballs pass from the first to the second
and then out of the second before receiving the gumball (Figure 7.5).
Such a reaction involving two sequential first-order steps can be
written as beginning in state A, and progressing through state B,
followed by state C:

(7.15)

Since each of the two steps is reversible, there are two forward rate
constants, identified as kf 1 and kf 2 , and two backward rate constants,
identified as kb 1 and kb 2 for the first and second steps, respectively.
Rate equations can be written for each species. The change in the
concentration of A is caused by both loss to state B with the rate
constant kf 1 and gain due to the reverse reaction from B with the
rate constant kb 1. The change in C arises from the step of B going to
C with the rate constant kf 2 or the decay of C back to B with the rate
constant kb 2. The most intricate changes occur for state B as the amount
can increase due to the forward step of A to B or the backward step
of C to B, whereas the amount of B can decrease due to both the
reverse step of B to A and the forward step of B to C:

(7.16)

These relationships can be solved to yield the time dependence of each
species in terms of the four rate constants. Although this reaction can be
solved explicitly, more complex reactions are best solved as numerical solu-
tions using a computer program. An understanding of complex reactions
can be achieved if certain assumptions are utilized. For example, if the
reverse rates are all much smaller than the forward rates, the process is
essentially irreversible. If one of the forward rates is much smaller than
the other, than the overall rate will be determined by the slowest rate
that is often termed the rate-limiting rate. Another example is the steady-
state approximation in which the intermediate state is assumed to not

dC

d

BC

t

=−kkfb 22 [] []

dB

d

AB BC

t

=−−+kkkkfbfb 1122 [ ] [] [] []

dA

d

AB

t

=−kkfb 11 []+ []

ABC↔↔

k

k

k

k

b

f

b

f

1

1

2

2

CHAPTER 7 KINETICS AND ENZYMES 139

Figure 7.5The sequential

transfer of gumballs.