GENERALIZED KINETICS OF REVERSIBLE INHIBITION 57 



and combined with the equilibrium expressions 3-3 to 3-6 the rate equation 

 is obtained: 



r =V ^^)^"-^^'- + ^^I)] (3-9) 



"' (S)(I) + a{K,{^) + KAD + K,K,] ^ ^ 



where r, is the rate in the presence of inhibitor and F„; = A'(E,). This 

 equation may be rewTitten in the Michaehs form: r, = 7„/ (S)/[(S)+^„/] 

 where 7,,/ = F„J« Z, + AI)]/[« ^- + (I)] and K,,' ^ K,ia K, + amj 

 [a J^(+/?(I)], so that the kinetics with respect to substrate follow the clas- 

 sic Michaelis-Menten theory although the constants now depend on the in- 

 hibitor affinity and concentration. For inhibitors it is generally true that 

 u > \ and /? < 1, although this is not strictly necessary since the rate 

 may be decreased even if « < 1 providing (3 is sufficiently low. 



The inhibition produced may be expressed in various ways; we shall uni- 

 formly use the fractional inhibition i, which may be defined as follows: 



1 = 1 - a = 1 - (i\lv) (3-10) 



where a is the fractional activity. Substituting expressions for v and t\ 

 from Eqs. 2-7 and 3-9 into Eq. 3-10 gives: 



(I)[(S)(1 - ^) ^ KAa - iS)] ^^^^^ 



(I)[(S) +aK,] + ^,[a(S) + aK,] 



The various types of inhibition are characterized by different values 

 of a and /?. Five characteristic types of inhibition may be considered on 

 this basis. 



(1) Completely conipetitice inhibition {a = oo): the inhibitor completely pre- 

 vents the combination of substrate with the enzyme. 



(I) 



(I) + ^,[1 + (^)IK,] 



(3-13) 



(2) Partially competitive inhibition (oo > a > 1 and fi = 1): the inhibitor 

 only partially hinders binding of the substrate and does not affect the rate of 

 breakdown of the substrate complex. 



V, = F„ ^^ (3-14) 



"^ (S) + K,{[aa) + aKlliil) + «Z,]} 



(^)("-l) (3-15) 



(I) [a + (8)IK,] + ccK, [1 + {^)jK,'\ 



