18: 1/ Enzymes: Kinetics of Oxidations 337 



The ratio (k 2 + k 3 a)jk l is similar to a Michaelis constant K M , except 

 that instead of being constant it depends on the value of a. 



The over-all rate V is the only rate measured for reactions of the 

 hydrolases. For catalase, it is possible to find other relationships 

 between k x and k 3 which can be measured experimentally. For example, 

 one may measure p directly and find its absolute maximum value p m . 

 [Note: />! is quasi-steady state value; p m is maximum of p x .] If a and 

 # are sufficiently large, they will not have changed appreciably from 

 their maximum values when p reaches p m . Moreover, k 2 will be 

 appreciably less than k 3 a . Accordingly, one may approximate 

 Equation 3 as 



"• * ~T^ (5) 



1 + 



ftj^O 



or as 



kiX Q 



(6) 



Either form allows one to compute k 3 /k 1 . 



The constant k 3 can be found from V by approximating Equation 4 

 at high values of x /a to give 



V = k 3 ae (7) 



Accordingly, one can compute k 1 and k 3 from measuring V and p m . 

 Moreover, one can find k 2 from the apparent Michaelis constant 

 (k 2 + k^/k^ once k x and k 3 have been measured. Thus, this system 

 allows one to find all three rate constants. 



These same constants can be found from other measurements. For 

 example, at the start of the reaction k 2 p may be neglected because p will 

 be very small. Moreover, the changes in a and x can also be neglected. 

 Then the equation for dpjdt in ( 1 ) may be rewritten as follows 



— = k x {e - p)x - k 3 pa 



This equation may be integrated directly to give, for the first part of 

 the reaction curve 



/»=AnaxO - e- (fc x*° +fc 3 a o><) (8) 



It is easy to confirm that the reaction follows a curve of this nature 

 because the half-time is constant, being given by the expression 



tp/2 . '" 2 (9) 



"•1*0 ~t~ "•3^0 



