17:3/ Enzyme Kinetics of Hydrolytic Reactions 325 



and accordingly, one may write 



where 



T , Kr>€X 



Km = k ~^ (9) 



In this case, the intermediate p may only have the value p x for a very 

 short time. Nevertheless, it is possible that the rate of change of the 

 concentration of the intermediate complex dpjdt may be small compared 

 to k 3 p, once p has reached its maximum value p x . Should this occur, 

 then Equations 5 through 9 may be retained for most of the reaction, 

 keeping the subscript 1 on p and V, but with restriction that they apply 

 only after p has reached its maximum. This is called the quasi-static 

 approximation. 



It is an empirical fact that this last approximation is valid for the 

 reactions catalyzed by hydrolases. It is also found that k x , k 2 , and k 3 

 have such values that the time to reach p x can be ignored if x is several 

 times e. Accordingly, one may replace Equations 2 through 4, which 

 cannot be solved exactly, by the approximations 



(10) 



1 + K M /x 



V= ^f (12) 



1 + K M /x [ 4) 



.*. k 3 et = {x - x) + K M In (x /x) (13) 



Figure 6 shows the comparison of the values for x, p, and V as computed 

 numerically from Equations 2 through 4 and as plotted from Equations 

 11 through 13. Note the excellent agreement. 



Equation 12 shows that for x large compared to K M , V will have 

 the maximum value V max given by Equation 6. For x equal to K M , V 

 will be one-half of F max . This is sometimes used to find K M . In any 

 case, one may rewrite Equation 12 as 



V^V m J(l +K M /x) (12a) 



This form is particularly useful if the value of e is not known in absolute 

 concentration units. 



Equation 12a is not in a suitable form to determine graphically 

 whether the reaction obeys these kinetics. As any physics student 



