158 



ROBERT A. ALBERTY 



The pH dependence of the kinetic constants may be represented by equations 

 of the type 



Vs' (3) 



Vs = 



Ks = K 



1 + (U+)/KaES + KbEs/{B.+) 



1 + (H+)/A'„^ + A'6£/(H+) 



(4) 



1 + {U+)/KaES + Kf,Es/{U+) 



where Vs' and A'^-' are "pH-independent" maximum velocities and JMichaeHs 

 constants and A^ and Kb are acid dissociation constants. The plot of Vs versus 

 pH is a symmetrical bell-shaped curve and KaEs is readily calculated (Alberty 

 and Massey, 1954) from the hydrogen ion concentration at the inflection points 

 of the acidic, (H+)a , and basic, (H+)6 , branches of this curve by use of the 

 equation 



KaES = (H+)„ + (H+)6 - 4V(H)a(H)6 (5) 



The second ionization constant KtEs is then calculated from the relation 



KaES KbES = (H+)n,ax"" (6) 



where (H+),„ax is the hydrogen ion concentration at the optimum pH. 



Equations (3) and (4) are obtained from the following mechanism in which 

 the enzymatic site is considered to be a dibasic acid and the intermediate ionized 

 form of the enzyme-substrate complexes to be the catalytically-active species. 



E EF EM E 



KbE 



Ki 



bEF 



K, 



bEM 



i^bE 



F -{- EH . ' ' EHF , ' ^ EHM . ' ^ EH -\- M (75) 



^2 ^4 ^6 



■HF 



Kat 



Ka 



EF 



K. 



aEM 



Kal 



-HU 



HF EHo 



EH2F 



EH.M 



EH; HM 



where the ionization constants are all acid dissociation constants. 



In order to obtain equations (3) and (4) it is necessary to assume that the 

 steps with rate constants k-.^ and k^ are rate determining when substrate is in 

 excess and to ignore the ionization of the substrate. 



The kinetics of the forward and reverse reactions are not independent as 

 shown by Haldane (1930) for two simple mechanisms for reversible reactions. 

 The equilibrium constant for the over-all reaction given above is 



VrK^H + (H+)/A„^] (8) 



A = 



iM)eq 



(F)eq ~~ V.„K,[\ + (H+) 'A„,,] 



