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

*(English)*Zbl 1311.92072Summary: 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.

##### MSC:

92C40 | Biochemistry, molecular biology |

92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |

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\textit{K. Mallory} and \textit{R. A. Van Gorder}, J. Math. Chem. 52, No. 1, 222--230 (2014; Zbl 1311.92072)

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##### References:

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