How to derive flux control coefficients from the rate equations of classical enzyme kinetics

A recent report by Brown and Cooper demonstrated the usefulness of calculating "flux control coefficients" for each of the rate constants involved in the assumed kinetic mechanism of a single enzyme. The calculations of Brown and Cooper involved numerical differentiation. The present article substantiates this report by showing that the numerical results of Brown and Cooper can also be obtained in an explicit form. The analytical equations given establish the relationship between rigorously specified overall rate processes and "elementary rate constants," both being defined by the rate equations of classical enzyme kinetics. It is shown that analytical flux control coefficients can be obtained for all types of rate processes considered in classical enzyme kinetics, including, "initial rates," equilibrium exchange reactions, and reactions at limiting levels of substrate (and/or product) saturation. By restricting the discussion to strictly consecutive (ordered, unbranched, linear) mechanisms, the line of reasoning can be presented in a relatively simple form. The main conclusions are the following: (a) It is advantageous to carry out the analysis in terms of paired (conjugated) control coefficients. (b) Flux control analysis of "elementary rate constants" does not require any extra kinetic argument. (c) Neither the immediate aim nor the results of the presented type of analysis are directly relevant to theories of metabolic control. On the contrary, the type of control analysis considered completes classical enzyme kinetics with a new facet. (d) For illustrating its usefulness, the concept of flux control coefficients is applied to the problem of optimization of enzyme activity.