Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 382

are better suited as kinetic models than the equilibrium step assumptions when true kinetic
constants are employed. However, the rate expressions from PSSH are quite similar to those
of Michaelis e Menten or equilibrium step assumptions. When utilized to correlate experi-
mental data, one would not be able to distinguish the two treatments.
To this point, we have learned now that PSSH approximation is easily visualized and
implemented to approximate reaction networks. However, the steps involved in mathemat-
ical derivation are tedious and one would almost want to solve the rates via computer
(algebraic equations or matrices). The resulting rate expressions can be simplified though
when the rate constants are properly lumped. On the other hand, the rapid equilibrium
assumption is much easier to deal with as only one rate is specified as the limiting rate, equi-
libria hold for other steps. The resulting rate expressions are simpler than those from PSSH.
Still, one needs to identify the right rate-limiting step. Comparing the PSSH and rapid equi-
librium approximations with the full solutions, one can infer that PSSH approximation is
closer to the full solution. When the rates are significantly different, i.e. there is a truly one
rate limiting step, one may not find any difference between the three solutions when the
initial moments were ignored.
To accommodate the rate difference and making rapid equilibrium approximations more
useful, we can make further approximations to recover some of the error incurred due to the
assumption of rapid equilibrium steps. This can be accomplished by

r 1 j r 2 ¼0;r 3 ¼0

r ¼ðr 1 þ r 2 þ r 3 Þ 1 z (8.147a)

k 1 C E 0
k c þ

K P
k 3 K C k 1 C E 0

1þ

r 2 j r 1 ¼0;r 3 ¼0

r ¼ðr 1 þ r 2 þ r 3 Þ 1 z

(8.147b)

k c K P

1þ

k 1 C E 0 þ k 3 K C k c

(a) (b)

10 10

C S

9 C S 9

8 8

7

7

6 6

5 5

4

4

3 3

2 2

C P

1 1 C P

0

0

0.01 0.1 1 10 100 0.01 0.1 1 10 100

k c t k c t

FIGURE. 8.28 Variations of free substrate, free product and enzyme concentrations with time for K S ¼ 1/C E0 ;
K C ¼ 5; K P ¼ 0.5/C E0 , k 1 C E0 ¼ 10 k c and k 3 C E0 ¼ 10 k c . Two cases are shown: (a) k 1 C E0 ¼ 10 k c as shown in Fig. 8.27

and (b) k 1 C E0 ¼ k c which does not qualify for single rate liming step assumption. The solid lines are the predictions

from Michaelis e Menten kinetics with the rate coefficient k c replaced by ðk c þk 1

1

C 1

E0 þK P k 1

C 1

E0 Þ 1 or the time 3

rescaled.