Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 351

With the definition of

K m ¼ ½

EH ½ S

(8.76)

½ EHS

H þ

K a 1 ¼ ½

EH ½

(8.77)

EH 2 ½

½ EH

K a 2 ¼ (8.78)

E ½ H þ ½

E þ½ EH 2 ½ E 0 ¼½ EH þ½ (8.79)

and

r p ¼ k 2 ½ ES (8.80)

We can derive the following rate expression:

r max ½ S

!

r P ¼ (8.81)

H þ
K a 1

K a 2

H þ þ ½

K m 1þ þ½ S

½

or

K m ; app þ½

r max ½ S

r P ¼ (8.82)

S

where

!

H þ
K a 1

K a 2
½ H þ þ ½

K m ; app ¼ K m 1þ (8.83)

As a result of this behavior, the pH optimum of the enzyme is between pK a1 and pK a2 .
For the case of ionizing substrate, the resulting rate expression is similar to the case of
ionizing enzymes. Consider

K m

k 2

E + SH ESH E + PH

(8.84)

K a1

+

S + H

The rate expression can be developed:

r max ½ S

r P ¼ ! (8.85)

K a 1

K m 1þ þ½ S

H þ ½