KINETICS OF COMPLEX ENZYME REACTION TYPES 35 



(IV) Case 4. The over-all rate depends primarily on the enzyme reaction 

 but the rate of formation of S' is near enough to A;2(ES') to exert an effect. 

 A steady-state treatment putting f7(S')/f?^ = cl{ES)ldt = gives the follow- 

 ing general equation: 



F,„(S) 



(S) +(1/A-J(F« -V) +K'K„ 



(2-28) 



where K' = ^_o/^'o ^^^cl K„, = (Jc_i + J^2)l^v This equation is similar, but 

 not identical, to that of Lineweaver and Burk (1934) who did not include 

 the back reaction from S' -> S. They showed how the constants may be 

 graphically evaluated by varying 7„, in certain ways. 



When diffusion of substrate occurs through a membrane into a spherical 

 or cylindrical region, additional complications arise due to the relations 

 between volume and membrane area, as well as to possible lack of homo- 

 geneity in the system, and the rate equations are quite complex (Blum, 

 1956). Blum and Jenden (1957) have treated the situation where the con- 

 centration of substrate decreases with increasing distance from the boun- 

 dary membrane due to utilization of the substrate by an homogeneously 

 distributed enzyme. It is interesting that inhibitors competing with the 

 substrate in such a system would inhibit to different degrees in various 

 regions, the over-all inhibition being greater than expected from the exter- 

 nal concentration of substrate and the kinetics of the isolated enzyme. 



The Concentration of Enzyme Is Greater Than That of the Substrate 



This situation is rare in studies on isolated enzymes but intracellularly 

 may be predominant over systems operating according to classic kinetics. 

 For the present only the extreme case where (S,) is so low that (E) = (E^) 

 will be considered; a more general treatment will be given in the next chap- 

 ter. One may now put (S^) = (S) + (ES) and the resulting rate equation is: 



where K,„ = {l:_i-^k2)lki. This expression is seen to be identical in form 

 to Eq. 2-11 except that the rate is now linear with (S,) and hyperbolic with 

 respect to (E^); ^■2(E,) is not equated with F,„ since under the conditions 

 assumed the rate will not become maximal upon increasing (S^). The recip- 

 rocal equation: 



1 1 Km 



V k,(&,) ^A:,(E,)(S,) 



shows that if Ijv is plotted against 1/(S^), a straight line that would pass 

 througlj the origin would be obtained but from which no constants may be 



