Iterative approximate solutions of kinetic equations for reversible enzyme reactions

Author(s)
Sarbaz H. A. Khoshnaw

ABSTRACT

We study kinetic models of reversible enzyme reactions and compare two techniques for analytic approximate solutions of the model. Analytic approximate solutions of non-linear reaction equations for reversible enzyme reactions are calculated using the Homotopy Perturbation Method (HPM) and the Simple Iteration Method (SIM). The results of the approximations are similar. The Matlab programs are included in appendices.

Cite this paper

Khoshnaw, S. (2013) Iterative approximate solutions of kinetic equations for reversible enzyme reactions.*Natural Science*, **5**, 740-755. doi: 10.4236/ns.2013.56091.

Khoshnaw, S. (2013) Iterative approximate solutions of kinetic equations for reversible enzyme reactions.

References

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[2] Schnell, S. and Maini, P.K. (2000) Enzyme kinetics at high enzyme concentration. Bulletin of Mathematical Biology, 62, 483-499. doi:10.1006/bulm.1999.0163

[3] Varadharajan, G. and Rajendran, L. (2011) Analytical solutions of system of non-linear differential equations in the single-enzyme, single-substrate reaction with non mechanism-based enzyme inactivation. Applied Mathematics, 2, 1140-1147. doi:10.4236/am.2011.29158

[4] Varadharajan, G. and Rajendran, L. (2011) Analytical solution of coupled non-linear second order reaction differential equations in enzyme kinetics. Natural Science, 3, 459-465. doi:10.4236/ns.2011.36063

[5] Li, B., Shen, Y. and Li, B. (2008) Quasi-steady state laws in enzyme kinetics. The Journal of Physical Chemistry A, 112, 2311-2321. doi:10.1021/jp077597q

[6] Murray, J.D. (1989) Mathematical biology. Springer, Berlin. doi:10.1007/978-3-662-08539-4_5

[7] Rubinow, S.I. (1975) Introduction to mathematical boilogy. Wiley, New York.

[8] Segel, L.A. (1980) Mathematical models in molecular and cellular biology. Cambridge University Press, Cam bridge.

[9] Hanson, S.M. and Schnell, S. (2008) Reactant stationary approximation in enzyme kinetics. The Journal of Physical Chemistry A, 112, 8654-8658. doi:10.1021/jp8026226

[10] Briggs, G.E. and Haldane, J.B.S. (1925) A note on the kinetics of enzyme action. Biochemical Journal, 19, 338-339.

[11] Gorban, A.N. and Shahzad, M. (2011) The michaelis-menten-stueckelberg theorem. Entropy, 13, 966-1019.

[12] Meena, A., Eswari, A. and Rajendran, L. (2010) Mathematical modeling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations. Journal of Mathematical Chemistry, 48, 179186. doi:10.1007/s10910-009-9659-5

[13] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. doi:10.1016/S0045-7825(99)00018-3

[14] Dawkins, P. (2007) Differential equations. http://tutorial.math.lamar.edu/terms.aspx

[15] Gorban, A.N., Radulescu, O. and Zinvyev, A.Y. (2010) Asymptotology of chemical reaction networks. Chemical Engineering Science, 65, 2310-2324. doi:10.1016/j.ces.2009.09.005

[16] Kargi, F. (2009) Generalized rate equation for single substrate enzyme catalyzed reactions. Biochemical and Biophysical Research Communications, 382, 157-159.

[17] Flach, E.H. and Schnell, S. (2010) Stability of open path ways. Mathematical Biosciences, 228, 147-152. doi:10.1016/j.mbs.2010.09.002

[18] Pedersen, M.G., Bersani, A.M., Bersani, E. and Cortese, G. (2008) The total quasi steady-state approximation for complex enzyme reactions. Mathematics and Computers in Simulation, 79, 10101019. doi:10.1016/j.matcom.2008.02.009

[19] Goeke, Ch.S., Walcher, S. and Zerz, E. (2012) Computing quasi-steady state reductions. Journal of Mathematical Chemistry, 50, 14951513. doi:10.1007/s10910-012-9985-x

[1] Maheswari, M.U. and Rajendran, L. (2011) Analytical solution of non-linear enzyme reaction equations arising in mathematical chemistry. Journal of Mathematical Chemistry, 49, 1713-1726. doi:10.1007/s10910-011-9853-0

[2] Schnell, S. and Maini, P.K. (2000) Enzyme kinetics at high enzyme concentration. Bulletin of Mathematical Biology, 62, 483-499. doi:10.1006/bulm.1999.0163

[3] Varadharajan, G. and Rajendran, L. (2011) Analytical solutions of system of non-linear differential equations in the single-enzyme, single-substrate reaction with non mechanism-based enzyme inactivation. Applied Mathematics, 2, 1140-1147. doi:10.4236/am.2011.29158

[4] Varadharajan, G. and Rajendran, L. (2011) Analytical solution of coupled non-linear second order reaction differential equations in enzyme kinetics. Natural Science, 3, 459-465. doi:10.4236/ns.2011.36063

[5] Li, B., Shen, Y. and Li, B. (2008) Quasi-steady state laws in enzyme kinetics. The Journal of Physical Chemistry A, 112, 2311-2321. doi:10.1021/jp077597q

[6] Murray, J.D. (1989) Mathematical biology. Springer, Berlin. doi:10.1007/978-3-662-08539-4_5

[7] Rubinow, S.I. (1975) Introduction to mathematical boilogy. Wiley, New York.

[8] Segel, L.A. (1980) Mathematical models in molecular and cellular biology. Cambridge University Press, Cam bridge.

[9] Hanson, S.M. and Schnell, S. (2008) Reactant stationary approximation in enzyme kinetics. The Journal of Physical Chemistry A, 112, 8654-8658. doi:10.1021/jp8026226

[10] Briggs, G.E. and Haldane, J.B.S. (1925) A note on the kinetics of enzyme action. Biochemical Journal, 19, 338-339.

[11] Gorban, A.N. and Shahzad, M. (2011) The michaelis-menten-stueckelberg theorem. Entropy, 13, 966-1019.

[12] Meena, A., Eswari, A. and Rajendran, L. (2010) Mathematical modeling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations. Journal of Mathematical Chemistry, 48, 179186. doi:10.1007/s10910-009-9659-5

[13] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. doi:10.1016/S0045-7825(99)00018-3

[14] Dawkins, P. (2007) Differential equations. http://tutorial.math.lamar.edu/terms.aspx

[15] Gorban, A.N., Radulescu, O. and Zinvyev, A.Y. (2010) Asymptotology of chemical reaction networks. Chemical Engineering Science, 65, 2310-2324. doi:10.1016/j.ces.2009.09.005

[16] Kargi, F. (2009) Generalized rate equation for single substrate enzyme catalyzed reactions. Biochemical and Biophysical Research Communications, 382, 157-159.

[17] Flach, E.H. and Schnell, S. (2010) Stability of open path ways. Mathematical Biosciences, 228, 147-152. doi:10.1016/j.mbs.2010.09.002

[18] Pedersen, M.G., Bersani, A.M., Bersani, E. and Cortese, G. (2008) The total quasi steady-state approximation for complex enzyme reactions. Mathematics and Computers in Simulation, 79, 10101019. doi:10.1016/j.matcom.2008.02.009

[19] Goeke, Ch.S., Walcher, S. and Zerz, E. (2012) Computing quasi-steady state reductions. Journal of Mathematical Chemistry, 50, 14951513. doi:10.1007/s10910-012-9985-x