384 7. INHIBITION IN MULTIENZYME SYSTEMS 



sition time is long, it may lead to a changing rate during measurement 

 and erroneous conclusions about the kinetics of the system. In the cell, 

 a prolonged transition time may make possible a marked effect on the cell 

 by the inhibition and can even secondarily prevent the re-establishment 

 of the new steady state. It is thus of some interest to look into the methods 

 for estimation of the transition time and present the factors upon which it 

 depends. 



Calculation of the transition time for inhibition in a monolinear chain 

 requires an expression for the time necessary for (B), the initial concentra- 

 tion in the uninhibited steady state, to rise to (B),, the concentration that 

 makes it possible for the steady state to be resumed in the inhibited system. 

 It is clear that d{B)ldt = i\ — v^ and becomes zero when the rates are 

 equal. The exact solution requires the integration of: 



cZ(B) ^ Fi(A) _a^-^)vm^ 



dt (A) + K, (B) + K, 



between (B) and (B),, which gives for the transition time: 



(7-70) 



(B), - (B) (1 -i)l\K, i\K, + [v, - (1 - i)F,](B), 



^ ~ v,-{\ - i)\\ [v, - (1 - i)V^Y ^ vj{, + [v, - (1 - i)V,]{B) 



(7-71) 



where Wj = (A)Fi/[(A) + K^], the rate of reaction 1. Since substitution of 

 the expression for (B), [from Eq. 7-3 with the maximal rate as (1 — i)V2 

 for noncompetitive inhibition] leads to a value of zero for the numerator 

 of the fraction after In, we see that theoretically it requires infinite time 

 to achieve a new steady state. Thus it will be convenient to calculate the 

 transition time for (B) to rise 90% of its way to the final steady state (B);. 

 This is given by: 



0.9[(B), - (B)] 2. 3(1 - i)V,K, 

 Vi - {I - OF, [vi - (1 - t)V2r 



It is interesting that the logarithmic term is unity under these specified 

 conditions and depends only on the assumed percentage rise towards (B),. 

 Although this is not the full transition time, it provides a practical measure 

 of the rate of change from one steady state to another. 



The variation of transition time with the degree of inhibition in a typ- 

 ical monolinear chain is shown in Table 7-2. As expected there is a rapid 

 rise in Jt^g as the inhibition approaches completeness. The units of J^o.g 

 will depend on those taken for v^ and v^: for example, if the rates are in 

 moles/minute, (B) and (B), must be in molar concentration and Jfg.g will be 

 expressed in minutes. The last column in Table 7-2 gives transition times 



