How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 564

r M 4 E 5 ¼ k 5c ½ M 2 E 5 k 6c ½ M 4 E 5 (11.17i)

r P 1 ¼ k 4c ½ M 3 E 3 (11.17j)

r P 2 ¼ k 6c ½ M 4 E 5 (11.17k)

Assume that P 2 is the intracellular products. Since E 1 ,E 2 ,E 3 ,andE 5 are all part of the
cellular material, we further simplify Eqn (11.15j) and incorporate it into the enzyme
balances lead to:

½ E 1 T ¼½ E 1 0 þ c 1 P 2 ¼½ E 1 þ½ SE 1 þ½ M 1 E 1 (11.18a)

½ E 2 T ¼½ E 2 0 þ c 2 P 2 ¼½ E 2 þ½ M 1 E 2 þ½ M 2 E 2 (11.18b)

½ E 3 T ¼½ E 3 0 þ c 3 P 2 ¼½ E 3 þ½ M 2 E 3 þ½ M 3 E 3 (11.18c)

½ E 5 T ¼½ E 5 0 þ c 5 P 2 ¼½ E 5 þ½ M 2 E 5 þ½ M 4 E 5 (11.18d)

Mass balances in a batch reactor with constant medium volume lead to

dC j
dt ¼ r j

(11.19)

¼

0, C j ¼

¼

At time t S 0 . The total enzyme concentrations at time 0 are each

assumed to be 2% of the total substrate S 0 , except [E 3 ] T0 ¼

0 for all j except S

0.04 S 0 . Equation (11.19) can be inte-
grated with OdexLims and the solution is shown in Fig. 11.4 .In Fig. 11.4 , we have assumed that
the increase in the amount of each enzyme is 5% of the intercellular product P 2 or c j ¼ 0.05,

except E 3 is increased by 10% of P 2 or c 3 ¼ 0.1. One can observe that the substrate

1.0

S

0.8

P 1

0.6

0.4 S 0 - S - P 1

P 2

0.2

0.0

0.0 0.5 1.0 1.5 2.0

k 1 E 0 t

FIGURE 11.4 A comparison of the full solution and approximate solution (single-step Michaelis e Menten
equation) for the reaction network showing in Fig. 11.3 . The solid lines are based on the full solution, while the
dotted lines are based on the approximated model. The kinetic constants are k j S 0 ¼

k jc ¼ 0.01 k j S 0 ; k j ¼ k 1 , for all j

except j ¼ 1. Also, k 1 ¼ 0.01 k 1 S 0 .