Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 341

Compulsory Order Reactions
Pseudosteady-State Hypothesis
In compulsory order, steady-state systems, one substrate must be added prior to the addi-
tion of the second substrate. The reaction pathway can thus be visualized as:

k R

k Q

EAB EPQ EP + Q

k -R

k P

k B k -B

(8.19)

E + P

B
+

k A

E + A EA

k -A

The substrates must be added in the order A and then B, and products are released in
compulsory order as well. We can write the equation below for the total amount of enzyme
in the system as

½ 0 ¼½ þ½ þ½ þ½ þ½ (8.20)

E E EA EAB EPQ EP

The analysis proceeds by assuming that all ES complexes are at pseudosteady state and that
the rate of formation of P is given by

r P ¼ k P ½ EP (8.21)

Expressions for each of the ES and enzyme e product complexes can now be written:

0 ¼ r EA ¼ k A ½ E ½ A k A ½ EA k B ½ EA ½ B þk B ½ EAB (8.22)

0 ¼ r EAB ¼ k B ½ EA ½ B k B ½ EAB k R ½ EAB þk R ½ EPQ (8.23)

0 ¼ r EPQ ¼ k R ½ EAB k R ½ EPQ k Q ½ EPQ (8.24)

0 ¼ r EP ¼ k Q ½ EPQ k P ½ EP (8.25)

Using these equations, we can eliminate the intermediates starting from Eqn (8.25) to obtain

k Q
k P ½

½ EP ¼ EPQ (8.26)

k R

½ EPQ ¼

k R þk Q ½

EAB (8.27)

k B ½ B k B ½ B

½ EAB ¼ ½ EA ¼ ½ EA (8.28)

k R k R
k R þ k Q

k R k Q
k R þk Q

k B þ k R k B þ

k A ½ A k A ½ A

½ EA ¼ ½ E ¼ ½ E (8.29)

k B k B ½ B k B k R k Q

k B ðk R þ k Q Þþk R k Q ½

k A þ k B ½ B k A þ B

k R k Q
k R þ k Q

k B þ