Theory of the reactant-stationary kinetics for zymogen activation coupled to an enzyme catalyzed reaction

A theoretical analysis is performed on the nonlinear ordinary differential equations that govern the dynamics of a coupled enzyme catalyzed reaction. The reaction consists of a primary non-observable zymogen activation reaction that it is coupled to an indicator (observable) reaction, where the product of the first reaction is the enzyme of the indicator reaction. Both reactions are governed by the 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 that are analogous to the Schnell-Mendoza equation for Michaelis-Menten type reactions. These closed-form solutions approximate the evolution of the observable reaction and provide the mathematical link necessary to measure the enzyme activity of the non-observable reaction. Conditions for the validity of the asymptotic solutions are also derived and demonstrate that these asymptotic expressions are applicable under the reactant-stationary kinetics.