FATE OF INHIBITORS IN THE ANIMAL 415 



The derivation of the equations above may be briefly indicated. Equa- 

 tions 8-6 and 8-7 may be written in operational form: 



{D + k,)ao) = hih) 



{D + k,+ k,){h) = kAlo) 

 and solved to give: 



[D' + {2k, + k,)D + k,k,]{li) = 



from which the values of u and /? can be determined for the general solution: 



(IJ = C,e' + C,e'^' 

 (I.) = Cge"' + ^e^* 



The constants C^ ... C4 may be related by differentiating these expressions 

 and substituting back in Eq. 8-6 or 8-7, and may finally be evaluated 

 by imposing the conditions when ^ = 0, where (IJ = (Io)o and (I,) = 0. 

 The behavior of this system may be illustrated for the special case 

 in which the rate of entrance is equal to the rate of disappearance of the in- 

 hibitor (k^ — k^. Equations 8-8 and 8-9 are plotted- in Fig. 8-3, from 

 which it may be seen that as (Iq) falls, (I,) rises in the cell, reaches a max- 

 imum and then decreases, as expected. The inhibition exerted on a hypo- 

 thetical enzyme in the cell is also shown in the figure, assuming the inhi- 

 bition to be rapidly reversible. In most cases when inhibitors are adminis- 

 tered to animals, the enzyme inhibitions in the various tissues will rise 

 and fall in some such manner. The deviations observed when k^ does not 

 equal k^ are shown in Fig. 8-4. The maximal (I,) reached depends on the 

 relative values of k-^ and k<£, the greater k^^ is relative to k.^, the higher will 

 be (Xi)jnax- The maximal level of (I,) and the time required to reach this may 

 be obtained by differentiation of Eq. 8-9 and setting it equal to zero; 

 for the special case when k-^ = k^. 



(I,)™„, = 0.275(I,)o (8-12) 



Ux = 0.861 /^^ 



where A- = k^ = k^. 



It has been assumed that disappearance of the inhibitor in the cell is 

 first-order. If the inactivation is enzymic, it may be necessary to modify 

 the treatment by introduction of the appropriate rate expressions in Eq. 

 8-6 and 8-7. The difference in rates of disappearance for the two types of 

 reaction is shown in Fig. 8-5; the enzymic reaction is initially slower and 

 eventually faster than the first-order reaction, when the over-all rates 

 of the two are comparable. However, the modification of the above treat- 



