Page 293 - Introduction to chemical reaction engineering and kinetics

274 Chapter 10: Biochemical Reactions: Enzyme Kinetics

SOLUTION
(4 Only steps (11, (3, and (5) of the general scheme above are involved in competitive
inhibition. We apply the SSH to the complexes ES and EI to obtain the rate law:

r,, = k,c,c, - k-,c,, - k,.lCES = 0 (1)
t-l-21 = k2cEcI - k-,c,, = 0 (2)
cEo = CE + CES + CEI (3)
(In the corresponding material balance for I, it is usually assumed that cI >> cnr, because
cn itself is usually very low; cr is retained in the final expression for the rate law.)

rP = krlCES (4)

From (l),

cE = ck-l + krl)cES _ KmCES - (5)
-
k , % CS
where K,,, is the Michaelis constant, equation 10.2-17.
From (2),

k2cEcI _ KmCESCI
cEI = k-, (6)
K2%
(using (5) to eliminate cn), where K2 = k-,/k,, the dissociation constant for EI.
From (3) (5), and (6), on elimination of cn and cnr,

CEO
cES = (7)
%+l+E
Substituting (7) in (4) we obtain the rate law:

i-p = krl cEo VOUlXCS (10.4-8)
= cs + K,(l + cIIK2)

The effect of inhibition is to decrease rp relative to rp given by the Michaelis-Menten
equation 10.2-9 (c, = 0). The extent of inhibition is a function of cr.
(b) To show the effect of inhibition on the Michaelis parameters V,,, and Km, we compare
equation 10.4-8 with the (uninhibited form of the) Michaelis-Menten equation, 10.2-9.
Vmax is the same, but if we write 10.4-8 in the form of 10.2-9 as

rp = VlTU?XCS (10.4-9)
cs + Kn,app

the apparent value Km,app is given by
(10.4-10) 1
K ww = K,(l + cIIK2)

and Kmapp > Km.