How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 563

4

P 1

M 3

3

M 1 M 2

S

2

1

5

M 4

P 2

6

FIGURE 11.3 A simple metabolic pathway producing two products P 1 and P 2 from a single limiting substrate S.

k 4 c E 3 þ P 1

M 3 $ E 3 / (11.15f)

k 5

M 2 $ E 2 þ E 5 % M 2 $ E 5 þ E 2 (11.15g)

k 5

k 5 c M 4 $ E 5

M 2 $ E 5 / (11.15h)

k 6 c P 2 þ

M 4 $ E 5 / E 5 (11.15i)

M 4 $ E 5 % v E j E j þ E 5 ; j ¼ 1 ; 2 ; 3 ; 5 (11.15j)

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

X

N R

r j ¼ v ji r i (11.16)

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, and r i is the rate of reaction for the
reaction i as written. Based on the reaction system described by Eqns (11.15a) through
(11.15i) , the reaction rates can be written as

r S ¼k 1 S½ E 1 þk 1 ½S E 1 (11.17a)

r SE 1 ¼ k 1 S½ E 1 ðk 1 þ k 1c Þ½ SE 1 (11.17b)

r M 1 E 1 ¼ k 1c ½ SE 1 k 2 ½ E 2 ½ M 1 E 1 þk 2 ½ M 1 E 2 ½ E 1 (11.17c)

r M 1 E 2 ¼ k 2 ½ M 1 E 1 ½ E 2 ðk 2 ½ E 1 þk 2c Þ½ M 1 E 2 (11.17d)

r M 2 E 2 ¼ k 2c ½ M 1 E 2 k 3 ½ M 2 E 2 ½ E 3 þk 3 ½ M 2 E 3 ½ E 2 k 5 ½ M 2 E 2 ½ E 5 þk 5 ½ M 2 E 5 ½ E 2 (11.17e)

r M 2 E 3 ¼ k 3 ½ M 2 E 2 ½ E 3 ðk 3 ½ E 2 þk 3c Þ½ M 2 E 3 (11.17f)

r M 2 E 5 ¼ k 5 ½ M 2 E 2 ½ E 5 ðk 5 ½ E 2 þk 5c Þ½ M 2 E 5 (11.17g)

r M 3 E 3 ¼ k 3c ½ M 2 E 3 k 4c ½ M 3 E 3 (11.17h)