138 4. SUBSTRATE INHIBITION AND PRODUCT INHIBITION 



Type B noncompetitive substrate inhibition (Eq. 4-11) 



1 + l/« + 1/(S') + (S')/a 

 ^^^ ""' ^ ^" 1 + (l/«) [1 + (D] + [1/(S')] [1 + d')] + (S')/a ^^"^^^ 



^^^^ ""^ ^ ^'" 1 + l/« + (D + [1/(S')] [1 + (!')] + (S')/« ^^"^^^ 



Equations 4-32 and 4-33 are special cases of 4-18. Mechanism (I) represents 

 the situation in which the inhibitor combines with the active site and 

 mechanism (II) in which it combines with an auxiKary group necessary 

 for the reaction; it is assumed that the inhibition by substrate is due to its 

 combination with this auxihary group. The expressions for the maximal 

 rate at the peak of the rate-(S) curves for the three possibilities are: 



Type A: Vo = F,„ , ^ (4-34) 



Type B (1): v, = F,„ , ^ -.^ (4-35) 



[1 +V[i +(r)]/aP 



Type B (II): v, = F,„ .- , ^ ^ -,, (4-36) 



Since a could be calculated from substrate inhibition kinetics {a was about 

 50 for xanthine oxidase), the values of K^ could be determined from each 

 of the Eqs. 4-34 to 4-36 using the measured values of Vq at the two isoxan- 

 thopterin concentrations (see Fig. 4-9). The values of K^ calculated for 

 the two isoxanthopterin concentrations should be equal and this was the 

 case only for mechanism B(I), indicating that the inhibitory substrate mol- 

 ecule combined with the auxiliary group but that the isoxanthopterin did 

 not. Combination of isoxanthopterin with both active site and auxiliary 

 group was eliminated by the linearity of plots of {v — v,) against {v — Vi)l{l). 

 A rather striking instance of competition in a substrate-inhibited system 

 is given in the study of Austin and Berry (1953) on human erythrocyte 

 acetylcholinesterase. The presence of the inhibitor 284C51 at 1.3 X 10"^ m.M 

 shifted the peak in the rate-pS curve from 2.65 to approximately 1.0 

 (a forty-five fold change in concentration) as shown in Fig. 4-15. Since 

 (So) is shifted by a factor of V 1 -f (F) (Eq. 4-23), the K, can be calcu- 

 lated to be roughly 6.4 X 10"^° M. Although the rate-pS maxima here could 

 not be accurately evaluated, and hence this value of A^j is only approxi- 



