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 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. It is demonstrated that the kinetics are in influenced by the intersection of invariant slow manifolds. Based on this intersection of manifolds, we derive asymptotic solutions using both perturbation and heuristic methods using the quasi-steady state and reactant-stationary assumptions. Furthermore, we derive a particular asymptotic solution, analogous to the Schnell-Mendoza equation for Michaelis-Menten type reactions, that approximates the evolution of the observable reaction in the event that the non-observable reaction is completed much faster than 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.