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

摘要

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 themodel 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.