Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 380

K P C P
1þK S C S þK P C P

C PE ¼ (8.141)

C S C P =K C
1þK S C S þK P C P

r ¼ k c C E 0 K S (8.142)

Equations (8.128) and (8.142) are quite similar when the rate constants are lumped
together. If one were to use Eqn (8.128) to correlate experimental data, one would not be
able to distinguish whether it is PSSH model or Michaelis e Menten model. The appeal of
the PSSH approach is that it may be able to approximate the reaction rate in general (i.e.
all the fluxes are considered), without imposing a particular step as the rate-limiting step
as illustrated in Fig. 8.18 . This approximation is particular useful when the beginning (reac-
tants) and end (products) are only of concern. Still, Eqn (8.142) looks simpler to use.
At time t

¼

0 in the batch reactor, the reaction is not started yet. However, the rapid equi-
librium approximated expression (8.142) requires that the uptake of substrate is already in
equilibrium. While this requirement is not of issue for reactions carried out in flow reactors
(after the transient period) or when the amount of enzyme is negligible, it becomes important
when noticeable amount of enzyme is employed in a batch reactor. For example, the concen-
tration of free substrate S can be obtained via mole balance,

K S C S 0
1þK S C S 0 þK P C P 0

C ST 0 ¼ C S 0 þC SE 0 ¼ C S 0 þ C E 0 (8.143)

The concentrations of free substrate S and product P charged into the batch reactor are C ST0
and C PT0 ¼ 0, assuming only substrate S was loaded. Since there is no P present in the initial

reaction mixture, C P0 ¼ 0. From Eqn (8.143) , we can solved for the free substrate concentra-

tion in the batch reactor as

q

ðC ST 0 C E 0 K S Þ 2 þ4K S C ST 0

C ST 0 C E 0 K S þ

C S 0 ¼ (8.144)

2

Mole balances on substrate S and product P in the reactor lead to

d C S C S C P =K C

1þK S C S þK P C P

d t ¼r ¼ k c C E 0 K S

(8.145)

d C P C S C P =K C

1þK S C S þK P C P d t ¼ r ¼ k c C E 0 K S

(8.146)

The solutions from the rapid equilibrium (Michaelis e Menten) model can be obtained by
solving Eqns (8.145) and (8.146) with initial conditions given by C S0 (Eqn 8.144 ) and C P0 ¼ 0at

¼

t 0. Subsequently, the enzyme distributions can be obtained from Eqns (8.140) and (8.141) .

One should note that the initial conditions set for Michaelis e Menten model must have taken
the substrate uptake and product discharge equilibria into consideration.
Instead of having four equations to solve, we now have only two equations to solve. The
solutions to one case consistent with the approximation to the first case discussed earlier are