Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 342

Back substitution of [EA] to [EAB], [EPQ], and [EP], we express all the concentrations of
complexes to [E], [A], and [B]. Applying the enzyme balance, Eqn (8.20) , we then obtain.

½ E 0

<
: 1þ

=

½ ¼

E

k A ½ A k B k R

k R þ k Q

k P þ k Q
k P

1þ ½ B 1þ

k B k R k Q
k B ðk R þk Q Þþk R k Q ½

k R k Q
k R þk Q

;

k A þ B k B þ

(8.30)

The overall rate expression is then

½ E 0 ½ A ½ B

r P ¼

k P þ k Q þ k R þ k Q
k R k Q

k A ðk B k R þk B k Q þ k R k Q Þ

þ ½

k A þ

B k B k R þk B k Q þk R k Q

½ A þ ½ A ½ B

k A k B k R k Q k B k R k Q

(8.31)
Under conditions where both substrates are present in excess, Eqn (8.31) can be reduced to

0
k P þ k Q þ k R þk Q
k R k Q

½ E

r P ¼ ¼ r max (8.32)

If only one of the substrates is present in excess, we retain the terms containing the other
substrate and obtain the reaction rates as functions of the limiting substrates:

½ E 0 ½ A

r P ¼ (8.33a)

k P þ k Q þ k R þ k Q
k R k Q

k A þ

½ A

½ E 0 ½ B

r P ¼ (8.33b)

k P þ k Q þ k R þ k Q
k R k Q

k B k R þ k B k Q þ k R k Q
k B k R k Q

þ ½ B

We recover the single (limiting)-substrate rate expression as discussed in the previous section.

Fast Equilibrium Step Analysis
By assuming that the ES complexes are all in equilibrium, the analysis is considerably
simplified. Let

k A
k A ; K B ¼

k B
k B ;

k R
k R

K A ¼ K R ¼ (8.34)

where K's are dissociation constants for the complexes.