46 



2. THE KINETICS OF ENZYME REACTIONS 



If the substrate can bind to the enzyme in the absence of the activator 

 (although the complex is not reactive) but the activator increases the affinity 

 of the enzyme for the substrate, one may write: 



A' _ EA aK, 

 E ^ EAS ^ EA + P 



K, ES aK„ 



(2-70) 



where a represents the change in substrate binding caused by the activator. 

 Assuming equilibrium concentrations of the complexes for simplicity, the 

 rate equation is found to be: 



V = V„ 



(A)(S) 



(A)(S) + a[K,K, + K,{A) + K„{S)] 



= VJ 



(S) 



(S) + K„ 



(2-71) 



where 7,„ = ^E,), VJ = ^(E,)(A)/[(A)+a K,], and K,, = K,a [(A)+ZJ/ 

 [(A)+« K,j]. If Of = 1, the rate equation simplifies to: 



F. 



(A) 



(A) + K, 



(S) 



(S) -f K, 



(2-72) 



which is similar in form to Eq. 2-61 for transfer reactions. It is clear that of- 

 ten the kinetics of activator-dependent and two-substrate reactions will 

 be very similar. However, generally « < 1 if A does facilitate the binding 

 of S; in the case of renal alkaline phosphatase, activated by Zn++, a was 

 found to be about 0.1 (Gryder et al., 1955). 



In the more general situation where the activator is not necessary for 

 the reaction but accelerates it and where the activator can either modify 

 the substrate binding or the rate of breakdown of the EAS complex relative 

 to the ES complex: 



^k 



EAS 



E + P 



^ EA + P 



(2-73) 



and V = X-(ES)+/5 ^(EAS). The rate equation can be written in a Michaelis- 

 Menten form with F„, = A;(E,) [/5(A)+a ^J/[(A)-f a ^J and K,„ = 

 a Kg [(A)+^„]/[(A)+« K^']. Several special cases can be derived from this 



