Analytical solution of coupled non-linear second order differential equations in enzyme kinetics

ABSTRACT

The coupled system of non-linear second-order reaction differential equation in basic enzyme reaction is formulated and closed analytical ex-pressions for substrate and product concentra-tions are presented. Approximate analytical me-thod (He’s Homotopy perturbation method) is used to solve the coupled non-linear differential equations containing a non-linear term related to enzymatic reaction. Closed analytical expres-sions for substrate concentration, enzyme sub-strate concentration and product concentration have been derived in terms of dimensionless reaction diffusion parameters k, and us-ing perturbation method. These results are compared with simulation results and are found to be in good agreement. The obtained results are valid for the whole solution domain.

The coupled system of non-linear second-order reaction differential equation in basic enzyme reaction is formulated and closed analytical ex-pressions for substrate and product concentra-tions are presented. Approximate analytical me-thod (He’s Homotopy perturbation method) is used to solve the coupled non-linear differential equations containing a non-linear term related to enzymatic reaction. Closed analytical expres-sions for substrate concentration, enzyme sub-strate concentration and product concentration have been derived in terms of dimensionless reaction diffusion parameters k, and us-ing perturbation method. These results are compared with simulation results and are found to be in good agreement. The obtained results are valid for the whole solution domain.

Cite this paper

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

Varadharajan, G. and Rajendran, L. (2011) Analytical solution of coupled non-linear second order differential equations in enzyme kinetics.

References

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[2] Murray, J.D. (1989) Mathematical biology. Springer Verlag, Berlin.

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

[4] Roberts, D.V. (1977) Enzyme kinetics. Cambridge University Press, Cambridge.

[5] Kasserra, H.P. and Laidler, K.J. (1970) Transient-phase studies of a trypsin-catalyzed reaction. Canadian Journal of Chemistry, 48, 1793-1802. doi:10.1139/v70-298

[6] Pettersson, G. (1976) The transient-state kinetics of two-substrate enzyme systems operating by an ordered ternary-complex mechanism. European Journal of Biochemistry, 69, 273-278. doi:10.1111/j.1432-1033.1976.tb10883.x

[7] Pettersson, G. (1978) A generalized theoretical treatment of the transient-state kinetics of enzymic reaction systems far from equilibrium. Acta Chemica Scandinavica - Series B, 32, 437-446. doi:10.3891/acta.chem.scand.32b-0437

[8] Gutfreund, H. (1995) Kinetics for life sciences: Receptors, transmitters and catalysis. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511626203

[9] Fersht, A.R. (1999) Structure and mechanism in protein science: A guide to enzyme catalysis and protein folding, Freeman, New York.

[10] Silicio, F. and Peterson, M.D. (1961) Ratio errors in pseudo first order reactions. Journal of Chemical Education, 38, 576-577. doi:10.1021/ed038p576

[11] Moore, J.W. and Pearson, R.G. (1981) Kinetics and Mechanism. Wiley, New York.

[12] Corbett, J.F. (1972) Pseudo first-order kinetics. Journal of Chemical Education, 49, 663. doi:10.1021/ed049p663

[13] 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

[14] Schnell, S. and Mendoza, C. (2004) The condition for pseudo-first-order kinetics in enzymatic reaction is independent of the initial enzyme concentration. Journal of Biophysical Chemistry, 107, 165-174. doi:10.1016/j.bpc.2003.09.003

[15] Meena, A., Eswari, A. and Rajendran, L. (2010) Mathematical modelling of enzyme kinetics reaction mechanism and analytical sloutions of non-linear reaction equations. Journal of Mathematical Chemistry, 48, 179-186. doi:10.1007/s10910-009-9659-5

[16] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using pid neural networks. International Journal of Nonlinear Science and Numerical Simulation, 7, 177-182.

[17] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the homotopy perturbation method to linear and nonlinear schr?dinger equations. Zeitschrift für Naturforschung, 63, 140-144.

[18] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262.

[19] He, J.H. (2003) Homotopy perturbation method: a new nonlinear analytical Technique. Applied Mathematics and Computation, vol.135, 73-79. doi:10.1016/S0096-3003(01)00312-5

[20] He, J.H. (2003) A Simple perturbation approach to Blasius equation. Applied Mathematics and Computation, 140, 217-222. doi:10.1016/S0096-3003(02)00189-3

[21] He, J.H. (2006) Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20 (10), 1141-1199. doi:10.1142/S0217979206033796

[22] He, J.H., Wu, C.G. and Austin, F. (2010) The variational iteration method which should be followed. Nonlinear Science Letters A, 1, 1-30.

[23] He, J.H. (2003) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37-43. doi:10.1016/S0020-7462(98)00085-7

[24] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Analytical investigation of hyperbolic equations via he’s methods. American Journal of Engineering and Applied Sciences, 1 (4), 399-407.

[25] Loghambal, S. and Rajendran, L. (2010) Mathematical modeling of diffusion and kinetics of amperometric immobilized enzyme electrodes. Electrochimica Acta, 55, 5230-5238. doi:10.1016/j.electacta.2010.04.050

[26] Meena, A. and Rajendran, L. (2010) Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations – Homotopy perturbation approach. Journal of Electroanalytical Chemistry, 644, 50-59. doi:10.1016/j.jelechem.2010.03.027

[27] Eswari, A. and Rajendran, L. (2010) Analytical solution of steady state current an enzyme modified microcylinder electrodes. Journal of Electroanalytical Chemistry, 648, 36-46. doi:10.1016/j.jelechem.2010.07.002

[1] Rubinow, S.I. (1975) Introduction to Mathematical Biology. Wiley, New York.

[2] Murray, J.D. (1989) Mathematical biology. Springer Verlag, Berlin.

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

[4] Roberts, D.V. (1977) Enzyme kinetics. Cambridge University Press, Cambridge.

[5] Kasserra, H.P. and Laidler, K.J. (1970) Transient-phase studies of a trypsin-catalyzed reaction. Canadian Journal of Chemistry, 48, 1793-1802. doi:10.1139/v70-298

[6] Pettersson, G. (1976) The transient-state kinetics of two-substrate enzyme systems operating by an ordered ternary-complex mechanism. European Journal of Biochemistry, 69, 273-278. doi:10.1111/j.1432-1033.1976.tb10883.x

[7] Pettersson, G. (1978) A generalized theoretical treatment of the transient-state kinetics of enzymic reaction systems far from equilibrium. Acta Chemica Scandinavica - Series B, 32, 437-446. doi:10.3891/acta.chem.scand.32b-0437

[8] Gutfreund, H. (1995) Kinetics for life sciences: Receptors, transmitters and catalysis. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511626203

[9] Fersht, A.R. (1999) Structure and mechanism in protein science: A guide to enzyme catalysis and protein folding, Freeman, New York.

[10] Silicio, F. and Peterson, M.D. (1961) Ratio errors in pseudo first order reactions. Journal of Chemical Education, 38, 576-577. doi:10.1021/ed038p576

[11] Moore, J.W. and Pearson, R.G. (1981) Kinetics and Mechanism. Wiley, New York.

[12] Corbett, J.F. (1972) Pseudo first-order kinetics. Journal of Chemical Education, 49, 663. doi:10.1021/ed049p663

[13] 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

[14] Schnell, S. and Mendoza, C. (2004) The condition for pseudo-first-order kinetics in enzymatic reaction is independent of the initial enzyme concentration. Journal of Biophysical Chemistry, 107, 165-174. doi:10.1016/j.bpc.2003.09.003

[15] Meena, A., Eswari, A. and Rajendran, L. (2010) Mathematical modelling of enzyme kinetics reaction mechanism and analytical sloutions of non-linear reaction equations. Journal of Mathematical Chemistry, 48, 179-186. doi:10.1007/s10910-009-9659-5

[16] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using pid neural networks. International Journal of Nonlinear Science and Numerical Simulation, 7, 177-182.

[17] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the homotopy perturbation method to linear and nonlinear schr?dinger equations. Zeitschrift für Naturforschung, 63, 140-144.

[18] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262.

[19] He, J.H. (2003) Homotopy perturbation method: a new nonlinear analytical Technique. Applied Mathematics and Computation, vol.135, 73-79. doi:10.1016/S0096-3003(01)00312-5

[20] He, J.H. (2003) A Simple perturbation approach to Blasius equation. Applied Mathematics and Computation, 140, 217-222. doi:10.1016/S0096-3003(02)00189-3

[21] He, J.H. (2006) Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20 (10), 1141-1199. doi:10.1142/S0217979206033796

[22] He, J.H., Wu, C.G. and Austin, F. (2010) The variational iteration method which should be followed. Nonlinear Science Letters A, 1, 1-30.

[23] He, J.H. (2003) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37-43. doi:10.1016/S0020-7462(98)00085-7

[24] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Analytical investigation of hyperbolic equations via he’s methods. American Journal of Engineering and Applied Sciences, 1 (4), 399-407.

[25] Loghambal, S. and Rajendran, L. (2010) Mathematical modeling of diffusion and kinetics of amperometric immobilized enzyme electrodes. Electrochimica Acta, 55, 5230-5238. doi:10.1016/j.electacta.2010.04.050

[26] Meena, A. and Rajendran, L. (2010) Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations – Homotopy perturbation approach. Journal of Electroanalytical Chemistry, 644, 50-59. doi:10.1016/j.jelechem.2010.03.027

[27] Eswari, A. and Rajendran, L. (2010) Analytical solution of steady state current an enzyme modified microcylinder electrodes. Journal of Electroanalytical Chemistry, 648, 36-46. doi:10.1016/j.jelechem.2010.07.002