Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 381

10

1.0

C S

9

0.9

8

0.8

C SE

7

0.7

6

0.6

5

0.5

4

0.4

C P E

3

0.3

2

0.2

C P

1

0.1

C E0

0

0.0

0.01 0.1 1 10 100

0.01 0.1 1 10 100

k c t k c t

FIGURE. 8.27 Variations of free substrate, free product, and enzyme concentrations with time for K S ¼ 1/C E0 ;
K C ¼

0.5/C E0 . The solid lines are the predictions from Michaelis e Menten kinetics assuming the catalytic
reaction is the rate-limiting step, whereas the dashed lines are for k 1 C E0 ¼

5; K P ¼

10 k c and k 3 C E0 ¼ 10 k c , as shown earlier.

shown in Fig. 8.27 . For comparison purposes, we have also plotted the full solutions as
shown in dashed lines. One can observe that the enzyme distributions as predicted by the
Michaelis e Menten approximation are quite similar to the full solutions. Michaelis e Menten
approximation becomes suitable once the free enzyme concentration becomes nearly
constant.
Figure 8.27 shows the variations of free substrate, free product, and enzyme concentra-
tions with reaction time for a case where the catalytic reaction rate is one tenth of those of
the other two steps. The Michaelis e Menten approximation are shown as solid lines, whereas
the dashed line are full solutions. One can observe that the Michaelis e Menten approximation
agrees reasonably well with the actual system, although there is a noticeable time shift due to
the small difference among the rates of the three steps. In deriving at Michaelis e Menten
approximation, we have assumed that the rates of the other two steps are very fast. The
10-fold difference in the rate constant is still noticeable.
One can observe from Fig. 8.27 that while the variation of free substrate concentrations
for Michaelis e Menten approximation looks similar to the full solutions, there is a shift
log t (to the right) that could make the agreement closer. This is due to the fact that
the finite rates of uptake of substrate S (or enzyme complexing of S) and discharge of
P contribute to the decline of overall rate as used by Michaelis e Menten approximation.
As a result, the Michaelis e Menten approximations overpredicted the reaction rate and
thus leading to a quicker change in the bulk phase concentrations. Overall, the shape
of the curve (or how the concentrations change with time) is remarkably similar. There-
fore, if one were to correlate the experimental data, the difference between the quality of
full solutions and the quality of fit from Michaelis e Menten approximations would not be
noticeable.

8.7.3. Modified Fast Equilibrium Approximation
Comparing Figs 8.25 and 8.27 , one can observe that PSSH approximation is closer to the
full solutions. Also, PSSH approximation can be applied to a variety of systems, whereas
the rapid equilibrium approximation is more specific. Therefore, the PSSH approximations