A transformed time-dependent Michaelis–Menten enzymatic reaction model and its asymptotic stability

Abstract

The dynamic form of the Michaelis–Menten enzymatic reaction equations provide a time-dependent model in which a substrate $S$ reacts with an enzyme $E$ to form a complex $C$ which in turn is converted into a product $P$ and the enzyme $E$. In the present paper, we show that this system of four nonlinear equations can be reduced to a single nonlinear differential equation, which is simpler to solve numerically than the system of four equations. Applying the Lyapunov stability theory, we prove that the non-zero equilibrium for this equation is globally asymptotically stable, and hence that the non-zero steady-state solution for the full Michaelis–Menten enzymatic reaction model is globally asymptotically stable for all values of the model parameters. As such, the steady-state solutions considered in the literature are stable. We finally discuss properties of the numerical solutions to the dynamic Michaelis–Menten enzymatic reaction model, and show that at small and large time scales the solutions may be approximated analytically.