568 12. RATES OF INHIBITION 



not affect the rate of inactivation, i.e., k^ = ^'3 in scheme 12-48. If the total 

 decrease in enzyme activity is under consideration, the fractional inhibition 

 is given by Eq. 12-52. Since here: 



diX)ldt = h[{E,) - (X)] (12-67) 



the concentration of inactivated enzyme may be immediately written as: 



(X) = (E,)[l - e-^2«] (12-68) 



so that from the usual differential equation for f/[(EI) + (X)]/rf^ we obtain: 



dildt = A'i(I) + k, + k,K,[l - e-*^*] ^ [k,a) + k,K, + k,] i (12-69) 



which again does not integrate to a simple expression. However, the course 

 of the loss in activity is quite similar to that for the case in which the free 

 enzyme is stable, except that the activity decreases somewhat more rap- 

 idly due to the inactivation of E as well as of EI. However, if we consider 

 only the true inhibition, it may be easily shown that it develops in an 

 identical fashion to the situation where no inactivation occurs (Eq. 12-11), 

 which would be the expected result, since E and EI are inactivated at equal 

 rates. Put in another way, if experimentally both a control and an inhibited 

 run are made and the inhibition at any time calculated from the relative 

 rates, the true inhibition will be obtained. 



Finally, the situation when /12 7^ ^"3 may be discussed. Here the presence 

 of the inhibitor on the enzyme alters the rate of inactivation. If the dis- 

 sociation of EI may be neglected, the kinetics are simple because the reac- 

 tion, EI — > X, is of no consequence, both forms being inactive. From Eq. 

 12-52 and: 



mi±m. . ,,E„I, + .,E, ,12-70) 



dt 



the rate of change of i is found to be: 



dijdt = [kAl) + A^2](l - i) (12-71) 



which integrates to: 



i = 1 - e-ci-io+i-ai' (12-72) 



This is jolotted in curve 1 of Fig. 12-20 and the mcreased inhibition com- 

 pared to curve 5 may be attributed to the inactivation of E. If the reverse 

 dissociation of EI cannot be neglected and the complete scheme 12-48 

 must be treated, the true inhibition as expressed by Eq. 12-60 may be 

 obtained by the integration of: 



dijdt = k,{l) - [kMl) + A^.] + k, - k,}i + [k, - k^i' (12-73) 



