Theory of the reactant-stationary kinetics for a coupled enzyme assay

A theoretical analysis is performed on the nonlinear ordinary differential equations that govern the dynamics of a coupled auxiliary enzyme catalyzed reaction. The assay consists of a non-observable reaction and an indicator (observable) reaction, where the product of the first reaction is the enzyme for the second. Both reactions are governed by the single substrate, single enzyme Michaelis-Menten reaction mechanism. Using singular perturbation methods, we derive asymptotic solutions that are valid under the quasi-steady-state and reactant-stationary assumptions. In particular, we obtain closed form solutions, analogous to the Schnell-Mendoza equation for Michaelis-Menten type reactions, that approximate the evolution of the observable reaction. Conditions for the validity of the asymptotic solutions are also rigorously derived showing that these asymptotic expressions are applicable under the reactant-stationary kinetics.