Analytical expressions of the concentrations of substrate and product in enzyme inhibition process
ABSTRACT
The initial and boundary value problem in enzyme reactions mechanism for inhabitation process is discussed. Approximate analytical expressions for the concentrations of substrate and product are presented. Approximate analytical solutions of non-linear reaction equations containing non-linear terms related to enzymatic reaction mechanism are solved using Homotopy perturbation method. The relevant analytical expression for the substrate and product concentration profiles is discussed in terms of dimensionless reaction diffusion parameters α, β, γE, and γS. Numerical solution is also obtained using Matlab program. Our analytical expression compared with numerical estimation and good agreement is noted.
The initial and boundary value problem in enzyme reactions mechanism for inhabitation process is discussed. Approximate analytical expressions for the concentrations of substrate and product are presented. Approximate analytical solutions of non-linear reaction equations containing non-linear terms related to enzymatic reaction mechanism are solved using Homotopy perturbation method. The relevant analytical expression for the substrate and product concentration profiles is discussed in terms of dimensionless reaction diffusion parameters α, β, γE, and γS. Numerical solution is also obtained using Matlab program. Our analytical expression compared with numerical estimation and good agreement is noted.
Cite this paper
Muthukumar, S. and Rajendran, L. (2013) Analytical expressions of the concentrations of substrate and product in enzyme inhibition process. Natural Science, 5, 1047-1055. doi: 10.4236/ns.2013.59129.
Muthukumar, S. and Rajendran, L. (2013) Analytical expressions of the concentrations of substrate and product in enzyme inhibition process. Natural Science, 5, 1047-1055. doi: 10.4236/ns.2013.59129.
References
[1] Bisswanger, H. (2008) Enzyme kinetics. Principles and Methods. Wiley-VCH Verlag GmbH & Co., Weinheim. doi:10.1002/9783527622023 [2] Dutta, R. (2008) Fundamentals of biochemical engineering. Springer, Berlin. doi:10.1007/978-3-540-77901-8 [3] House, J.E. (2007) Principles of chemical kinetics. Elsevier Inc., Amsterdam. [4] Leskovac, V. (2004) Comprehensive enzyme kinetics. Kluwer Academic Publishers, Norwell. [5] Missen, R.W., Mims, C.A. and Saville, B.A. (1999) Introduction to chemical reaction engineering and kinetics. John Wiley & Sons, Inc., Hoboken. [6] Taylor, K.B. (2002) Enzyme kinetics and mechanisms. Kluwer Academic Publishers, Norwell. [7] Gaidamauskait, E. (2011) Computational modeling of complex reactions kinetics in biosensors. Doctoral Dissertation, Physical Sciences, Informatics (09P), Vilnius. [8] Baronas, R., Ivanauskas, F. and Kulys, J. (2009) Mathematical modeling of bio-sensors. Springer-Verlag, Berlin. [9] Berg, J.M., Tymoczko, J.L. and Stryer, L. (2002) Biochemistry. W. H. Freeman, New York. [10] Rubinow, S.I. (1975) Introduction to mathematical boilogy. Wiley, New York. [11] Murray, J.D. (1989) Mathematical biology. Springer, Berlin, 109. doi:10.1007/978-3-662-08539-4_5 [12] Segel, L.A. (1980) Mathematical models in molecular and cellular biology. Cambridge University Press, Cambridge. [13] Roberts, D.V. (1977) Enzyme kinetics. Cambridge University Press, Cambridge. [14] Rahamathunissa, G., Manisanar, P., Rajenran, L. and Venugopal, G. (2011) Modeling of nonlinear boundary value problems in enzyme-catalyzed reaction diffusion processes. Journal of Mathematical Chemistry, 49, 457474. doi:10.1007/s10910-010-9752-9 [15] Loghambal, S. and Rajendran, L. (2011) Mathematical modeling in amperometric oxidase enzyme—Membrane electrodes. Journal of Membrane Science, 373, 20-28. [16] Anitha, S., Subbiah, S., Subramaniam, S. and Rajendran, L. (2011) Analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method. Electrochimica Acta, 56, 3345-3352. doi:10.1016/j.electacta.2011.01.014 [17] Anitha, S., Subbiah, A. and Rajendran, L. (2011) Analytical expression of non-steady-state concentrations and current pertaining to compounds present in the enzyme membrane of biosensor, Journal of Physical Chemistry A, 115, 4299-4306. doi:10.1021/jp200520s [18] Uma Maheswari, M. 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 [19] Meena, A., Eswari, A. and Rajendran, L. (2011) Mathematical modeling of biosensors: Enzyme-substrate Interaction and bio-molecular interaction. In: Serra, P.A., Ed., New Perspectives in Biosensors Technology and Applications, InTech, Rijeka. doi:10.5772/19513 [20] Ghori, Q.K., Ahmed, M. and Siddiqui, A.M. (2007) Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 179-184. [21] Ozis, T. and Yildirim, A. (2007) A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 243-248. [22] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using PID neural networks. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 177-182. doi:10.1515/IJNSNS.2006.7.2.177 [23] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the Homotopy perturbation method to linear and nonlinear Schrodinger equations. Zeitschrift für Naturforschung, 63, 140-144. [24] Margret PonRani, V. and Rajendran, L. (2010) Analytical expression of non steady-state concentration profiles at planar electrode for the CE mechanism. Natural Science, 2, 1318-1325. doi:10.4236/ns.2010.211160 [25] 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 [26] Venugopal, K., Eswari, A. and Rajendran, L. (2011) Mathematical model for steady state current at ppo-modified micro-cylinder biosensors. Journal of Biomedical Science and Engineering, 4, 631-641. doi:10.4236/jbise.2011.49079 [27] Muthukumar, S. and Rajendran, L. (2012) Concentration of species in two species oscillator using Homotophy peturbation method. Global Journal of Theoretical and Applied Mathematics Sciences, 2, 99-108. [28] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. [29] He, J.H. (2003) Homotopy perturbation method: A new non-linear analytical technique. Applied Mathematics and Computation, 135, 73-79. doi:org/10.1016/S0096-3003(01)00312-5 [30] 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
[1] Bisswanger, H. (2008) Enzyme kinetics. Principles and Methods. Wiley-VCH Verlag GmbH & Co., Weinheim. doi:10.1002/9783527622023 [2] Dutta, R. (2008) Fundamentals of biochemical engineering. Springer, Berlin. doi:10.1007/978-3-540-77901-8 [3] House, J.E. (2007) Principles of chemical kinetics. Elsevier Inc., Amsterdam. [4] Leskovac, V. (2004) Comprehensive enzyme kinetics. Kluwer Academic Publishers, Norwell. [5] Missen, R.W., Mims, C.A. and Saville, B.A. (1999) Introduction to chemical reaction engineering and kinetics. John Wiley & Sons, Inc., Hoboken. [6] Taylor, K.B. (2002) Enzyme kinetics and mechanisms. Kluwer Academic Publishers, Norwell. [7] Gaidamauskait, E. (2011) Computational modeling of complex reactions kinetics in biosensors. Doctoral Dissertation, Physical Sciences, Informatics (09P), Vilnius. [8] Baronas, R., Ivanauskas, F. and Kulys, J. (2009) Mathematical modeling of bio-sensors. Springer-Verlag, Berlin. [9] Berg, J.M., Tymoczko, J.L. and Stryer, L. (2002) Biochemistry. W. H. Freeman, New York. [10] Rubinow, S.I. (1975) Introduction to mathematical boilogy. Wiley, New York. [11] Murray, J.D. (1989) Mathematical biology. Springer, Berlin, 109. doi:10.1007/978-3-662-08539-4_5 [12] Segel, L.A. (1980) Mathematical models in molecular and cellular biology. Cambridge University Press, Cambridge. [13] Roberts, D.V. (1977) Enzyme kinetics. Cambridge University Press, Cambridge. [14] Rahamathunissa, G., Manisanar, P., Rajenran, L. and Venugopal, G. (2011) Modeling of nonlinear boundary value problems in enzyme-catalyzed reaction diffusion processes. Journal of Mathematical Chemistry, 49, 457474. doi:10.1007/s10910-010-9752-9 [15] Loghambal, S. and Rajendran, L. (2011) Mathematical modeling in amperometric oxidase enzyme—Membrane electrodes. Journal of Membrane Science, 373, 20-28. [16] Anitha, S., Subbiah, S., Subramaniam, S. and Rajendran, L. (2011) Analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method. Electrochimica Acta, 56, 3345-3352. doi:10.1016/j.electacta.2011.01.014 [17] Anitha, S., Subbiah, A. and Rajendran, L. (2011) Analytical expression of non-steady-state concentrations and current pertaining to compounds present in the enzyme membrane of biosensor, Journal of Physical Chemistry A, 115, 4299-4306. doi:10.1021/jp200520s [18] Uma Maheswari, M. 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 [19] Meena, A., Eswari, A. and Rajendran, L. (2011) Mathematical modeling of biosensors: Enzyme-substrate Interaction and bio-molecular interaction. In: Serra, P.A., Ed., New Perspectives in Biosensors Technology and Applications, InTech, Rijeka. doi:10.5772/19513 [20] Ghori, Q.K., Ahmed, M. and Siddiqui, A.M. (2007) Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 179-184. [21] Ozis, T. and Yildirim, A. (2007) A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 243-248. [22] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using PID neural networks. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 177-182. doi:10.1515/IJNSNS.2006.7.2.177 [23] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the Homotopy perturbation method to linear and nonlinear Schrodinger equations. Zeitschrift für Naturforschung, 63, 140-144. [24] Margret PonRani, V. and Rajendran, L. (2010) Analytical expression of non steady-state concentration profiles at planar electrode for the CE mechanism. Natural Science, 2, 1318-1325. doi:10.4236/ns.2010.211160 [25] 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 [26] Venugopal, K., Eswari, A. and Rajendran, L. (2011) Mathematical model for steady state current at ppo-modified micro-cylinder biosensors. Journal of Biomedical Science and Engineering, 4, 631-641. doi:10.4236/jbise.2011.49079 [27] Muthukumar, S. and Rajendran, L. (2012) Concentration of species in two species oscillator using Homotophy peturbation method. Global Journal of Theoretical and Applied Mathematics Sciences, 2, 99-108. [28] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. [29] He, J.H. (2003) Homotopy perturbation method: A new non-linear analytical technique. Applied Mathematics and Computation, 135, 73-79. doi:org/10.1016/S0096-3003(01)00312-5 [30] 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