542 12. RATES OF INHIBITION 



The rate of formation of the EI complex will be given by: 



^^ = A:i(E)(I) - k_,m) = U(^t) - (EI)] - A-_,(EI) (12-9) 



and since i = (EI)/(E^) and K^ = k_-^jk{. 



di 



—- = kAD - ik,[(l) + A-,] (12-10) 



at 



which upon integration leads to an expression showing the variation of 

 the inhibition with time: 



^ ^ —{1 - e-«=i'[(i) + s,i} (12-11) 



(I) + A^ 

 i = i^|l _ e-J:i([(i) + £,]} (12-12) 



Thus when t = 0, the inhibition is zero, but the inhibition increases with 

 time until it reaches the final equilibrium value, if. The rate of inhibition 

 at any time may be found by substituting from Eq. 12-11 into Eq. 12-10) 



di/dt = A,-,(I)e ^'t(i) + £,.] (12-13) 



Equation 12-12 may be compared to the general Eq. 12-3, which was derived 

 on the basis of no particular mechanism but only an exponential change, 

 and it is found that: 



a = k, [(I) + A',] (12-14) 



The value of a may be found by the plotting i)rocedures described in the 

 previous section and from this the rate constant can be determined. 



Equation 12-8 represents opposing reactions of the first and second or- 

 ders and is applicable ideally over the whole course of the reaction. Occa- 

 sionally, the dissociation of the complex has l^een neglected and the second- 

 order reaction alone considered: 



E + I — >EI (12-15) 



in which case it is easily shown that: 



^ = 1 _ e-fci(i)' (12-16) 



and that a plot of log (1 — i) against t will give a straight line of slope 

 — A;i(I)/2.3. Such a treatment may be satisfactory when only the initial 

 rates are measured but becomes progressively inaccurate with time; in fact, 

 Eq. 12-16 implies that the inhibition will eventually become complete 

 whatever the inhibitor concentration. 



