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A new piecewise spectral homotopy analysisof the Michaelis-Menten enzymatic reactions model. (English) Zbl 1296.65099
Summary: In this paper we report on a novel method for solving systems of nonlinear differential equations which is an extension of the spectral homotopy analysis method (SHAM). The proposed method extends the application of the SHAM to initial value problems that model the Michaelis-Menten enzymatic reaction equation. Results from the proposed method are compared with Runge-Kutta routines as a measure of accuracy and efficiency.

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
Software:
ISHAM; Matlab
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References:
[1] Michaelis, L; Menten, ML, Die kinetic der invertinwirkung, Biochem. Z., 49, 333-369, (1913)
[2] Henri, V, The orie ge ne rale de l’action de quelques diastases, C. R. Hebd. Acad. Sci., 135, 916-919, (1902)
[3] Briggs, GE; Haldane, JBS, A note on the kinetics of enzyme action, Biochem. J., 19, 338-339, (1925)
[4] Goldstein, A, The mechanism of enzyme-inhibitor-substrate reactions, J. Gen. Physiol, 27, 529-580, (1944)
[5] 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
[6] Murray, J.D.: Mathematical Biology. Springer-Verlag, Berlin Heidelberg (1989) · Zbl 0682.92001
[7] Schnell, S; Mendoza, C, Closed form solution for time-dependent enzyme kinetics, J. Theor. Biol., 187, 207-212, (1997)
[8] Schnell, S; Mendoza, C, The condition for pseudo-first order kinetics in enzymatic reactions is independent of the initial enzyme concentration, Biophys. Chem., 107, 165-174, (2004)
[9] Schnell, S; Mendoza, C, Enzyme kinetics of multiple alternative substrates, J. Math. Chem., 27, 155-170, (2004) · Zbl 0993.92019
[10] Schnell, S; Maini, PK, Enzyme kinetics at high enzyme concentrations, Bull. Math. Biol., 62, 483-499, (2000) · Zbl 1323.92099
[11] Tzafriri, AR; Edelman, ER, The total quasi-steady state approximation is valid for reversible enzyme kinetics, J. Theor. Biol., 226, 303-313, (2004)
[12] Tzafriri, AR, Michaelis-Menten kinetics at high enzyme concentrations, Bull. Math. Biol., 65, 111-1129, (2003) · Zbl 1334.92185
[13] 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)
[14] Golicnik, M, Explicit reformulations of time-dependent solution for a Michaelis-Menten enzyme reaction model, Anal. Biochem., 406, 94-96, (2010)
[15] Golicnik, M, Evaluation of enzyme kinetic parameters using explicit analytic approximations to the Michaelis-Menten equation, Biochem. Eng. J., 53, 234-238, (2011)
[16] Kumar, A; Josic, K, Reduced models of networks, J. Theor. Appl. Mech., 278, 87-106, (2011) · Zbl 1307.92107
[17] Bajzer, Z., Strehler, E.E.: About and beyond the Henri-Michaelis-Menten rate equation for single-substrate enzyme kinetics. Biochem. Biophys. Res. Commun. (2012). doi:10.1016/j.bbrc.2011.12.051
[18] Kargi, F, Generalized rate equation for single-substrate enzyme catalyzed reactions, Biochem. Bio. Res. Comm, 282, 57-159, (2009)
[19] Shanmugarajan, A; Alwarappan, S; Somasundaram, S; Lakahmanan, R, Analytical solution of amperometric enzymatic reactions based on homotopy perturbation method, Electrochica Acta., 56, 3345-3352, (2011)
[20] Sevukaperumal, S; Eswari, A; Rajendran, L, Analytical expression pertaining to concetration of substrate and effectiveness factor for immobilized enzymes with reversible Michaelis Menten kinetics, Int. J. Comput. Appl. (0975-8887), 33, 46-53, (2011)
[21] Joy, RA; Meena, A; Loghambai, S; Rajendran, L, A two-parameter mathematical model for immobilized enzymes and homotopy analysis method, Nat. Sci., 3, 556-565, (2011)
[22] Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University (1992) · Zbl 1334.92185
[23] Motsa, SS; Sibanda, P; Shateyi, S, A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Commun. Nonlinear Sci. Numer. Simul., 15, 2293-2302, (2010) · Zbl 1222.65090
[24] Motsa, SS; Sibanda, P; Awad, FG; Shateyi, S, A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem, Comput. Fluids, 39, 1219-1225, (2010) · Zbl 1242.76363
[25] Makukula, Z.G., Sibanda, P., Motsa, S.S.: A note on the solution of the von Kármń equations using series and Chebyshev spectral methods. Boundary Value Problems, Volume 2010, Article ID 471793, 17 pages. doi:10.1155/2010/471793 · Zbl 1207.35248
[26] Makukula, Z., Sibanda, P., Motsa, S.S.: A Novel Numerical Technique for Two-dimensional Laminar Flow Between Two Moving Porous Walls, Mathematical Problems in Engineering, vol. 2010, Article ID 528956, p. 15 (2010). doi:10.1155/2010/528956 · Zbl 1195.76387
[27] Motsa, SS; Marewo, GT; Sibanda, P; Shateyi, S, An improved spectral homotopy analysis method for solving boundary layer problems, Bound. Value Prob., 2011, 3, (2011) · Zbl 1270.35023
[28] Motsa, S.S., Shateyi, S.: A New Approach for the Solution of Three-Dimensional Magnetohydrodynamic Rotating Flow over a Shrinking Sheet, Mathematical Problems in Engineering, vol. 2010, Article ID 586340, p. 15 (2010). doi:10.1155/2010/586340 · Zbl 1202.76157
[29] Sibanda, P; Motsa, SS; Makukula, Z, A spectral-homotopy analysis method for heat transfer flow of a third grade fluid between parallel plates, Int. J. Numer. Methods Heat Fluid Flow, 22, 4-23, (2012) · Zbl 1356.80073
[30] Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press (2003) · Zbl 0714.92006
[31] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin (1988) · Zbl 0658.76001
[32] Trefethen, L.N.: Spectral Methods in MATLAB, SIAM (2000) · Zbl 0953.68643
[33] Laidler, K.J.: Theory of the transient phase in kinetics, with special reference to enzyme systems. Can. J. Chem. 33, 1614-1624
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