18:2/ Enzymes: Kinetics of Oxidations 343 



to always be unobservable. Under these conditions, the reaction may 

 be presented by the following nonlinear differential equations 



2 = ^(e-p-p^x - k 3 pa + k^p' - k 2 p 



2- = k 3 pa - k 5 p'a - k±p' 



(19) 

 dx 



— = -k x (e-p-p')x + k 2 p 



da . iii 



— — — k 3 pa — k 5 p a + k±a 



Except for the very end of the reaction, the "back rates," k 2 and £ 4 , 

 contribute very little to the kinetics. Considerable simplification can 

 be obtained by ignoring them; they are cross-hatched in Equation 19. 

 Even with these approximations, no exact solution in closed form exists 

 for these equations. 



Under many experimental conditions, a steady-state region is ob- 

 served. At this time, one may make quasi-static approximations. As 

 has been shown in the last chapter, these may be approximately valid a 

 considerable time after a true steady state has ceased to exist. Sym- 

 bolically, the quasi-static approximation is represented by 



dt ' dt 



Subscript one will be used as previously to indicate the values of the 

 intermediates computed by using these approximations. Solving the 

 appropriate equations in (19), it is readily apparent that 



P\ = k fh (20) 



* - (£ M (21) 



and 



'-£--*!?*'*. (22) 



For part of the reaction, the kinetics will resemble those of the hydro- 

 lases discussed earlier. However, the apparent Michaelis constant, K M , 

 will be given by the expression 



k 3 x k 3 a 

 k 5 ki 



This shows that K M depends on both x and a 



K » = i; + f (23) 



