Analytical Solutions of System of Non-Linear Differential Equations in the Single-Enzyme, Single-Substrate Reaction with Non-Mechanism-Based Enzyme Inactivation

Govindhan Varadharajan, Lakshmanan Rajendran

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DOI: 10.4236/am.2011.29158                

Abstract

A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for the full range of enzyme activities has been derived. The time dependent analytical solution for substrate, enzyme-substrate complex and product concentrations are presented by solving system of non-linear differential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple analytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in terms of dimensionless reaction diffusion parameters ε, λ₁, λ₂ and λ₃ using perturbation method. The numerical solution of the problem is also reported using SCILAB software program. The analytical results are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained results are valid for the whole solution domain.

Keywords

Non-Linear Reaction Equations, Enzyme Inactivation, Homotopy Perturbation Method, Time Dependent Analytical Solution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. P. Deutscher, “Rat Liver Glutamyl Ribonucleic Acid Synthetase. I. Purification and Evidence for Separate Enzymes for Glutamic Acid and Glutamine,” Journal of Biological Chemistry, Vol. 242, No. 6, 1967, pp. 1123-1131.
[2]	J. Hernadez-Ruiz, M. B. Amao, A. N. P Hiner, F. Garcia-Canovas and M. Acosta, “Catalase-Like Activity of Horseradish Peroxidase: Relationship to Enzyme Inactivation by H2O2,” Biochemical Journal, Vol. 354, No. 2, 2001, pp. 107-114.
[3]	J. Messens, J. C. Martins, E. Brosens, K. Van Belle, D. M. Jacobs, R. Willem and L. Wyns, “Kinetics and Active Site Dynamics of Staphylococcus Aureus Reductase,” Journal of Biological Inorganic Chemistry, Vol. 7, 2002, pp. 146-156. doi:10.1007/s007750100282
[4]	S. Ho and G. S. Mittal, “High Voltage Pulsed Electrical Field for Liquid Food Pasteurization,” Food Reviews International, Vol. 16, No. 4, 2000, pp. 395-434. doi:10.1081/FRI-100102317
[5]	C. P. Yao and R. H Levy, “Inhibition-Based Metabolic Drug-Drug Interactions: Predictions from in Vitro Data,” Journal of Pharmaceutical Sciences, Vol. 91, No. 9, 2002, pp. 1923-1935. doi:10.1002/jps.10179
[6]	S. G. Waley, “Kinetics of Suicide Substrates,” Biochemical Journal, Vol. 185, 1980, pp. 771-773.
[7]	S. G. Waley, “Kinetics of Suicide Substrates. Practical Procedures for Determining Parameters,” Biochemical Journal, Vol. 227, 1985, pp. 843-849.
[8]	L. Michaelis and M. L. Menten, “Die Kinetik der Invertinwirking,” Biochemische Zeitschrift, Vol. 49, 1913, pp. 333-369.
[9]	G. E. Briggs and J. B. S. Haldane, “A Note on the Kinetics of Enzyme Action,” Biochemical Journal, Vol. 19, No. 2, 1925, pp. 338-339.
[10]	S. Schnell and S. M. Hanson, “A Test for Measuring the Effects of Enzyme Inactivation,” Biophysical Chemistry, Vol. 125, No. 2-3, 2007, pp. 269-274. doi:10.1016/j.bpc.2006.08.010
[11]	S. J. Li and Y. X. Liu, “An Improved Approach to Nonlinear Dynamical System Identification Using PID Neural Networks,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 2, 2006, pp. 177-182.
[12]	M. M. Mousa, S. F. Ragab and Z. Nturforsch, “Application of the Homotopy Perturbation Method to Linear and Nonlinear Schr?dinger Equations,” Zeitschrift für Naturforschung, Vol. 63, 2008, pp. 140-144.
[13]	J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3
[14]	J. H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5
[15]	J. H. He, “A Simple Perturbation Approach to Blasius Equation,” Applied Mathematics and Computation, Vol. 140, No. 2-3, 2003, pp. 217-222. doi:10.1016/S0096-3003(02)00189-3
[16]	J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796
[17]	J. H. He, G. C. Wu and F. Austin, “The Variational Iteration Method Which Should Be Followed,” Nonlinear Science Letters A, Vol. 1, No. 1, 2010, pp. 1-30.
[18]	J. H. He, “A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems,” International Journal of Non-Linear Mechanics, Vol. 35, No. 1, 2000, pp. 37-43. doi:10.1016/S0020-7462(98)00085-7
[19]	D. D. Ganji, M. Amini and A. Kolahdooz, “Analytical Investigation of Hyperbolic Equations via He’s Methods,” American Journal of Engineering and Applied Sciences, Vol. 1, No. 4, 2008, pp. 399-407.
[20]	S. Loghambal and L. Rajendran, “Mathematical Modeling of Diffusion and Kinetics of Amperometric Immobilized Enzyme Electrodes,” Electrochimica Acta, Vol. 55, No. 18, 2010, pp. 5230-5238. doi:10.1016/j.electacta.2010.04.050
[21]	A. Meena and L. Rajendran, “Mathematical Modeling of Amperometric and Potentiometric Biosensors and System of Non-Linear Equations—Homotopy Perturbation Approach,” Journal of Electroanalytical Chemistry, Vol. 644, No. 1, 2010, pp. 50-59. doi:10.1016/j.jelechem.2010.03.027
[22]	A. Eswari and L. Rajendran, “Analytical Solution of Steady State Current an Enzyme Modified Microcylinder Electrodes,” Journal of Electroanalytical Chemistry, Vol. 648, No. 1, 2010, pp. 36-46. doi:10.1016/j.jelechem.2010.07.002