Analytical solution of coupled nonlinear rate equations. I. Michaelis-Menten kinetics

An approximate analytic solution to the Michaelis-Menten kinetic equations is developed which is free of the traditional steady-state hypothesis and is of a simpler form than previous perturbation approaches, so that the relaxation process can be traced in detail. The method involves coordinate transformation of the rate equations and subsequent solution of an integral equation which displays evolution of the system in time as the unfolding of a memory function whose value at any instant is contingent upon the past history of the system. A memory function of the type described here is not present in linear kinetics, so that it can serve to define the distinctly nonlinear features of a kinetic process. The scheme can be enlarged for development of approximate analytical solutions to nonlinear rate equations descriptive of open systems which can evolve into limit cycles or show chaotic behavior.