330 Enzyme Kinetics of Hydrolytic Reactions / 17 : 4 



As differential equations, these may be rewritten 

 % = k x '{e-p-p')x' - k 2 'p' 



& = - kx {e-p.-p') + k 2 p' 



2 = k^e-p-p^x - {k 2 + k 3 )p 



dx 



— = -k x {e-p-p')x + k 2 p 



dx dp , , 



• - v - i + ** 



Now quasi-static approximations are used; namely 



*■*<> and *:*0 



dt dt 



Michaelis constants are defined as 



k 2 + k 3 k 2 



K M = — t and A M = v 



The approximate equations can be solved to yield 



^ K' M ex 



Pl x + (K M ) ■ (x' + K' M ) 



Solving this for V, inverting and multiplying by x, one finds 



x 1 



V K 



max 



x + k m n + j 



(14) 



The Lineweaver-Burk plot of Figure 7b would then represent a series of 

 lines of constant slope, intersecting the x/V axis at points depending on 

 x'/K' M . This is illustrated in Figure 10a. 



The case of the noncompetitive inhibitor is algebraically so complex 

 that it is left to the interested reader to solve for himself. It is clear 

 from Figure 9, however, that no matter what the relative concentrations 

 of S and .S", any trace of the inhibitor S' will slow down the rate of 

 hydrolysis of S. Provided one is willing to assume that 



k" k + k k' k!" 



V = ~^~7 — " = ^ M anc * a ^ so t ^ iat 77 = ^'m = T» 



K x K x K x K x 



then one can show that 



X y=^-{x + K M )(\ +£-) (15) 



v 'max \ ^Ml 



