324 Enzyme Kinetics of Hydrolytic Reactions /I7 : 3 



general acceptance. In the absence of additional data, it is the simplest 

 explanation of the observed rates of reactions catalyzed by hydrolases. 



The simultaneous differential Equations 2-4 are nonlinear. In 

 general, an analytical solution does not exist in closed form, subject to 

 arbitrary initial conditions. The equations can be solved by numerical 

 computation, by analog computation, or by making suitable approxima- 

 tions. The last method is so successful for these equations that it is the 

 only one presented here. 



In order to understand the approximations, consider the behavior 

 of p and x for various values of the initial substrate concentration x 

 while holding e constant. In all cases, p must be zero at the beginning 

 of the reaction [t = 0) and will return to zero at the end of the reaction. 

 At the beginning of the reaction dp/dt, the rate of change of p given by 

 Equation 2, must be positive; at the end it will be negative. Some 

 place in between, there will be a maximum value of/?, designated by 

 p x , at which time dp/dt will vanish. At this time, one may rewrite 

 Equation 2 as 



dp 

 It 



= = k x (e - p x )x - (k 2 + k 3 )p 1 (5) 



p=pi 



It seems reasonable that as x is made increasingly large, the maximum 

 value p x will be reached increasingly rapidly, and also that p x will 

 approach the total enzyme concentration e. At very large values of 

 x , it likewise would seem that the value of p x will remain close to that 

 of e for a comparatively long time. When this is true, the velocity V 

 will have a maximum value V x given by 



**-J 



hPi — k 3 e (6) 



p = pi 



To show this, one may combine Equations 1 and 2 to give 



dx dp 



~ dt = i + k ^ 



and use the relationships 



= and p\—^e for large x 



dp 

 It 



p=pi 



If one solves Equation 5 for p x in the case where x Q is not as large as 

 considered above, one finds 



*-rnc ■ (7) 



