Cell Metabolism - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 501

Solution:
1. To derive a reasonable rate expression from the simplified pathway given in Fig. 10.15 c,
we first add some details to the reactions (stoichiometry):

k 1

2 k 1c M

M

1 þ

E M

1 $ E 2 þ

E (E10-2.1)

2 %

2

k 1

k f1

M

3 þ

E M

3 $ E

(E10-2.2)

2 %

2

k f1

k 2
k 2

k 2c M

M

2 þ

E M

2 $ E 3 þ

E (E10-2.3)

3 % 3 /

3

k 3

k 3c M

M

3 þ

E M

3 $ E 4 þ

E (E10-2.4)

4 % 4 / 4

k 3

k f2

P

1 þ

E P

1 $ E

(E10-2.5)

4 % 4

k f2

k 4c P

M (E10-2.6)

4 / 1

k 5
k 5

k 5c M

M

3 þ

E M

3 $ E 5 þ

E (E10-2.7)

5 % 5 / 5

k f3

2 þ 2 $ E

(E10-2.8)

P E

5 %

P

5

k f3

k 6c P

M (E10-2.9)

5 / 2

We have assumed that the intermediate product M 4 and desired product P 1 are essentially
the same or required no catalyst to proceed. The same holds for M 5 and P 2 . The rate of reac-
tion or rate of generation for species j is given by (Chapter 3),

N R

r j ¼ n ji r i (E10-2.10)

i ¼ 1

where r j is the reaction rate for species j, N R is the total number of reactions in the system,
n ji is the stoichiometric coefficient of species j in the reaction i,andr i istherateofreaction
for the reaction i as written. Based on the reaction system described by Eqns (E10-2.1)
through (E10-2.9) , the reaction rates can be written as

r M 1 ¼k 1 ½ M

1 ½

E

2 þk 1 ½

M E

2

(E10-2.11)

1

r M 1 E 2 ¼ k 1 ½ M

1 ½

E

2 ðk 1 þ k 1c Þ½

M E

2

(E10-2.12)

1

r M 3 E 2 ¼ k f1 ½

3 ½ 2 k f1 ½ 2

(E10-2.13)

M E M 3 E