KINETICS OF COMPLEX ENZYME REACTION TYPES 33 



state wherein the concentrations of all complexes remain constant. Although 

 this derivation is often more laborious than the equilibrium procedure, it 

 provides constants which are of general validity since equilibrium conditions 

 are not assumed. 



First step: \\Tite out the equations for all the reactions that can occur 

 in the system and assign rate constants to each forward and backward step. 



Second step: write the equation giving the over-all rate in terms of the 

 appropriate complex or complexes, e.g. v — A:y(EX,). 



Third step: write the equation expressing the total enzyme concentration 

 in terms of all the molecular species containing the enzyme, e.g., 



(E,) = (E) ^ (EXi) ^ (EX,) + ... - (EX„) 



If the free concentrations of any other components of the system are ap- 

 preciably less than the total concentrations, due to complex formation, 

 similar equations for these components should be written. 



Fourth step: write the steady-state equation for each complex in terms of 

 the appropriate rate constants and concentrations, including all reactions 

 whereby the complex is formed or broken down, and set the over-all rate 

 equal to zero since the complex concentration is constant, e.g., 



^^^^^ = kMQ + A-_ J_,(C) + ... - k.J_,{C) - kMO - ... = 



dt 



where /(C) represents the concentrations pertinent to each reaction. 



Fifth step: these steady-state equations in conjunction with the equations 

 from step 3 provide a set of equations which may be solved for the concen- 

 tration of the rate-determining complex (EX^) and this expression can 

 be substituted in the equation from step 2. 



Sixth step: if desired, the expression for the maximal rate (F^) may be 

 determined by evaluating v at high substrate concentration and V,„ in- 

 serted into the rate equation from step 5. 



The resulting rate equation may be modified by substitution of dissocia- 

 tion constants for the corresponding ratios of rate constants or by combina- 

 tion of rate constants into single constants to facilitate interpretation. 

 Many equations derived in this manner are complex and unwieldy but oc- 

 casionally some simplification can be introduced by assuming appropriate 

 relative values for the rate constants or concentrations of the substances 

 involved, giving special cases that may be applicable under certain experi- 

 mental conditions. 



