On a nonelementary progress curve equation and its application in enzyme kinetics

The analytical equation describing progress curves of an enzyme catalyzed reaction acting upon the Michaelis-Menten mechanism has been known for the case in which only the free enzyme incurs a loss of its activity, either spontaneously or as a result of an irreversible inhibitor action. The solution of differential equations which defines the rates of enzyme inactivation and substrate utilization is expressed by a nonelementary function in equation of an implicit type that precludes direct calculation of the extent of reaction at any time. Previously, the implicit equations have been rearranged to the alternative formulas and solved by the Newton-Raphson method, but this procedure may fail when used upon the presented equation. For this reason the other root-finding numerical method was applied, and the enzyme kinetic parameters of such numerically solved implicit equation for the reaction mechanism of irreversibly inhibited acetylcholinesterase were fitted to the experimental data by a nonlinear regression computer program.