ENZYME RATE AND SUBSTRATE CONCENTRATION 15 



derived in the following direct manner. The reaction sequence may be 



written as: 



hi ^ 



E+S;z±ES^E+P (2-1) 



where E represents the enzyme, S the substrate, ES the enzyme-substrate 

 complex, P the product or products, and Jc^, k_i, and ^'2 are rate constants. 

 The rate is assumed to be proportional to the concentration of the complex: 



^' = -^ = ^-^(ES) (2-2) 



at 



The equilibrium concentration of the complex may be obtained from the 

 expression for the dissociation: 



A;_i (E)(S) 



and the conservation equation relating the forms of the enzyme existing 

 during the reaction: 



(E,) = (E) + (ES) (2-4) 



where (E^) and (E) represent the total and free enzyme concentrations 

 respectively. Therefore: 



and substituting this value in Eq. 2-2: 



A-.(E,)(S) 



(S) +K 



(2-6) 



Since the maximal rate at high enzyme-saturating substrate concentration 

 in equal to ^^^(E^), Eq. 2-6 may be rewritten in the usual form: 



where F„; is the maximal rate under the specified experimental conditions. 

 This equation gives the expected hyperbolic curve when v is plotted 

 against (S) and a sigmoid curve when a logarithmic scale for (S) is used 

 (Fig. 2-2). The constant K is commonly termed the " Michaelis constant " 

 and, under the present assumptions, is the dissociation constant for the ES 

 complex and an inverse measure of the affinity of the enzyme for the sub- 

 strate. Vy„ and K for a particular enzyme reaction may be determined 



