RATES OF INHIBITION OF PURE ENZYMES 



553 



of E with I proceeds. It has been said that the inhibitor displaces the 

 substrate from the enzyme, which is what does occur when one considers 

 initial and final states, but this must not be interpreted as a direct displace- 

 ment, since it is unlikely than an inhibitor molecule could force a sub- 

 strate molecule from the enzyme surface in most cases. 



It might be thought that an expression for the rate of competitive 

 inhibition could be simply derived from that for noncompetitive inhibition 

 (Eq. 12-11) by multiplying K, by the factor [1 + (S)/Z,], which suffices 

 for the final inhibition, but this is not so, inasmuch as (ES) decreases 

 as (EI) increases and both f7(ES)/(/^ and d(Eil)ldt must be taken into ac- 

 count. Goldstein (1944) obtained an equation for the development of in- 

 hibition in the presence of substrate and applied it to the inhibition of 

 cholinesterase by physostigmine. This treatment will l)e followed here 

 but modified so that the results may be expressed in terms of inhibition 

 instead of activitv. There are now two simultaneous reactions: 



E + S;^ES 



(12-27) 



E +I^EI 



(12-28) 



where l\ and A-_j have been retained for the inhibitor reaction to conform 

 to the previous equations. The changes in the concentrations of the com- 

 plexes would be: 



(i(ES) 

 dt 



rf(EI) 

 dt 



= A-,(S)[(E,) - (ES) - (EI)] - A-,A^.(ES) 



(12-29) 



A^(I)r(E,) - (ES) - (EI)j - A■,A^.(EI) (12-30) 



Eliminating [(E,) — (ES) — (EI)] from these equations: 



dim) A-,(S) 



dt 



A-:(I) 



f/(El) 

 dt 



+ A-,A,(EI) 



- A-2A,,(ES) 



(12-31! 



Since i = (EI)/(E,) and a = (ES)/(E,) = 1 - i. we may write: 



and ^=^^;^ = - (E,) -^ (12-32) 



r/(EI) di 



^(ES) _ ^ di 



~dr = '^^'^-d^ 



and hence obtain an equation for dildt: 



di 

 lit 





il;k, 



iDK, + (S)g, 

 kAD + U^) 



(12-33) 



