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A transformed time-dependent Michaelis-Menten enzymatic reaction model and its asymptotic stability. (English) Zbl 1311.92072
Summary: 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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[1] Michaelis, L; Menten, ML, Die kinetik der invertinwirkung, Biochem. Z., 49, 333-369, (1913)
[2] Frenzen, CL; Maini, PK, Enzyme kinetics for two-step enzymic reaction with comparable initial enzyme-substrate ratios, J. Math. Biol., 26, 689-703, (1988) · Zbl 0714.92006
[3] Schnell, S; Mendoza, C, Closed form solution for time-dependent enzyme kinetics, J. Theor. Biol., 187, 207-212, (1997)
[4] Tzafriri, AR, Michaelis-Menten kinetics at high enzyme concentrations, Bull. Math. Biol., 65, 1111-1129, (2003) · Zbl 1334.92185
[5] Tzafriri, AR; Edelman, ER, The total quasi-steady state approximation is valid for reversible enzyme kinetics, J. Theor. Biol., 226, 303-313, (2004)
[6] Golicnik, M, Explicit reformulations of time-dependent solution for a Michaelis-Menten enzyme reaction model, Anal. Biochem., 406, 94-96, (2010)
[7] Golicnik, M, Explicit analytic approximations for time-dependent solutions of the generalized integrated Michaelis-Menten equation, Anal. Biochem., 411, 303-305, (2011)
[8] Abu-Reesh, IM, Optimal design of continuously stirred membrane reactors in series using Michaelis-Menten kinetics with competitive product inhibition: theoretical analysis, Desalination, 180, 119-132, (2005)
[9] Uma Maheswari, M; Rajendran, L, Analytical solution of non-linear enzyme reaction equations arising in mathematical chemistry, J. Math. Chem., 49, 1713-1726, (2011) · Zbl 1303.92154
[10] Vogt, D, On approximate analytical solutions of differential equations in enzyme kinetics using homotopy perturbation method, J. Math. Chem., 51, 826-842, (2013) · Zbl 1402.92225
[11] Liang, S; Jeffrey, DJ, Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Commun. Nonlinear Sci. Numer. Simul., 14, 4057-4064, (2009) · Zbl 1221.65281
[12] Turkyilmazoglu, M, Some issues on HPM and HAM methods: a convergence scheme, Math. Comput. Model., 53, 1929-1936, (2011) · Zbl 1219.65083
[13] S.S. Motsa, S. Shateyi, Y. Khan, A new piecewise spectral homotopy analysis of the Michaelis-Menten enzymatic reactions model. Nume. Algorithms (in press) (2013) · Zbl 1296.65099
[14] A.L. Lehninger, D.L. Nelson, M.M. Cox, Lehninger Principles of Biochemistry (W.H. Freeman, New York, 2005)
[15] J.P. LaSalle, S. Lefschetz, Stability by Lyapunov’s Second Method with Applications. Academic, New York (1961) · Zbl 1334.92185
[16] Degn, H; Olsen, LF; Perram, JW, Bistability, oscillation, and chaos in an enzyme reaction, Ann. N.Y. Acad. Sci., 316, 623-637, (1979)
[17] E. Fehlberg, Low-Order Classical Runge-Kutta Formulas with Step Size Control and Their Application to Some Heat Transfer Problems. NASA Technical Report 315 (1969)
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