554 12. RATES OF INHIBITION 



It should be noted at this point that Eqs. 12-32 limit the treatment to 

 situations in which the enzyme is saturated with substrate so that the 

 total enzyme concentration, (E^), is approximately (ES) + (EI), the con- 

 centration of free enzyme being very small. This arises from setting a = 

 (ES)/(E;) = I — i, which is true only when the enzyme reaction in the ab- 

 sence of inhibitor is proceeding at the maximal rate. This limitation of 

 the treatment was not specifically stated by Goldstein. In his experimental 

 study of cholinesterase inhibition, the substrate concentration was very 

 high, (S') being approximately 65, so that the behavior should be adequate- 

 ly represented by this treatment. However, in cases where the enzyme 

 is not saturated with substrate, and d{^S)l(U 7^ — d{'El)ldt, the situation 

 is more complex and the equations are difficult to solve directly. 



Equation 12-33 may now be integrated and the following expression 

 for the inhibition is obtained: 



(I) 



(I) + -4^ (S) 



A, 



1 — exp 





(12-34) 



This is plotted and compared with the noncompetitive rate in Fig. 12-16 

 for (S') = 5 and the marked decrease in rate may be observed. It is true 

 that Eq. 12-34 would not accurately apply to as low a specific concentra- 

 tion of substrate, but the error is not very great. The logarithmic plots 

 of these curves are shown in Fig. 12-17. The slope of the curve for com- 

 petitive inhibition is the factor multiplied by the time in the exponential 

 part of Eq. 12-34, and it may be seen that values of k-i^ and k^ can be cal- 

 culated from the slopes obtained under different conditions. The data 

 reported by Goldstein (1944) for the inhibition of dog serum cholinesterase 

 by physostigmine are shown in Fig. 12-18 and the curves have been derived 

 from Eq. 12-34 using the constants calculated by Goldstein, except that a 

 slightly lower value of K^ was chosen to obtain a closer fit to the experi- 

 mental points. As was pointed out by Goldstein, the rate of inhibition here 

 in the absence of the substrate would have been too rapid to measure but 

 in the presence of the large amounts of substrate used, the rate was slowed 

 so that it could be easily determined. Thus by this technique it is possible 

 to determine the rate constants of a reaction normally proceeding at an 

 experimentally unmeasurable rate. 



With respect to the development of competitive inhibition in such 

 systems, one may conceive of two extreme types. In one case, the sub- 

 strate rapidly dissociates from the enzyme and the limiting rate of 

 inhibition is that at which the inhibitor reacts with the free enzyme; in 

 the other case, the ES complex dissociates more slowly than the EI 

 complex is formed and thus limits the rate of inhibition. Equation 12-34 



