Applied and Computational Mathematics

Submit a Manuscript

Publishing with us to make your research visible to the widest possible audience.

Propose a Special Issue

Building a community of authors and readers to discuss the latest research and develop new ideas.

Mathematical Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that Follow the Michaelis – Menten Mechanism

In this paper, mathematical models of immobilized enzyme system that follow the Michaelis-Menten mechanism for both reversible and irreversible reactions are discussed. This model is based on the diffusion equations containing the non-linear term related to Michaelis-Menten kinetics. An approximate analytical technique employing the modified Adomian decomposition method is used to solve the non-linear reaction diffusion equation in immobilized enzyme system. The concentration profile of the substrate is derived in terms of all parameters. A simple expression of the substrate concentration is obtained as a function of the Thiele modulus and the Michaelis constant. The numerical solutions are compared with our analytical solutions for slab, cylinder and spherical pellet shapes. Satisfactory agreement for all values of the Thiele modulus and the Michaelis constant is noted. Graphical results and tabulated data are presented and discussed quantitatively to illustrate the solution.

Mathematical Modeling, Nonlinear Differential Equations, Modified Adomian Decomposition Method, Michaelis-Menten Kinetics, Immobilized Enzyme

Krishnan Lakshmi Narayanan, Velmurugan Meena, Lakshman Rajendran, Jianqiang Gao, Subbiah Parathasarathy Subbiah. (2017). Mathematical Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that Follow the Michaelis – Menten Mechanism. Applied and Computational Mathematics, 6(3), 143-160.

Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. R. Aris, The Mathematical Theory Of Diffusion And Reaction In Permeable Catalysts, Clarendon, Oxford. 1975.
2. P. Cheviollotte, Relation between the reaction cytochrome oxidase-Oxygen and oxygen uptake in cells in vivo-The role of diffusion, J. Theoret. Biol. 39: 277-295, (1973).
3. M. Moo-Young and T. Kobayashi, Effectiveness factors for immobilized-enzyme reactions, Can&. J. Chem. Eng. 50: 162- 167 (1972).
4. S. A. Mireshghi, A. Kheirolomoom and F. khorasheh, Application of an optimization algotithem for estimation of substrate mass transfer parameters for immobilized enzyme reactions, Scientia Iranica, 8(2001)189-196.
5. Benaiges, M. D., Sola, C., and de Mas, C.: Intrinsic kinetic constants of an immobilized glucose isomerase. J. Chem. Technol. Biotechnol., 36, 480-486 (1986).
6. Shiraishi, F. ’Substrate concentration dependence of the apparent maximum reaction rate and Michaelis-Menten constant in immobilized enzyme reactions’. Int. Chem. Eng., 32, 140-147 (1992).
7. Shiraishi, F., Hasegawa, T., Kasai, S., Makishita, N., and Miyakawa, H.: Characteristics of apparent kinetic parameters in a packed bed immobilized enzyme reactor. Chem. Eng. Sci., 51, 2847-2852 (1996).
8. Lortie, R. and Andre, G.: On the use of apparent kinetic parameters for enzyme bearing particles with internal mass transfer limitations. Chem. Eng. Sci., 45, 1133-l 136 (1990).
9. Hemrik Pedersen, EnmoAdema, K. Venkatasubramanian, P. V. Sundaram, ‘Estimation of intrinsic kinetic parameters in tubular enzyme reactors by a direct approach’, Applied Biochemistry and Biotechnology, 1985, Volume 11, Issue 1, pp 29-44.
10. V. Bales and P. Rajniak, ‘Mathematical simulation of fixed bed reactor using immobilized enzymes’, Chemical Papers, 40 (3), 329–338 (1986).
11. Messing, R. A., Immobilized Enzyme for Industrial Reactors, Academic Press, New York (1975).
12. Wilhelm Tischer, Frank Wedekind, ‘Immobilized Enzymes: Methods and Applications’, Topics in Current Chemistry, Vol. 200© Springer Verlag Berlin Heidelberg 1999.
13. Engasser, J. M. and Horvath, C.: Diffusion and kinetics with immobilized enzymes, p. 127-220.
14. Farhadkhorasheh, Azadehkheirolomoom, and Seyedalirezamireshghi, ‘application of an optimization algorithm for estimating intrinsic kinetic parameters of immobilized enzymes’, journal of bioscience and bioengineering, vol. 94, no. 1, l-7. 2002.
15. Houng, J. Y., Yu, H., Chen, K. C., and Tiu, C.: Analysis ofsubstrate protection of an immobilized glucose isomerasereactor. Biotechnol. Bioeng., 41, 451458 (1993).
16. G. Adomian, Convergent series solution of nonlinear equations, J. Comp. App. Math. 11(1984) 225-230.
17. N. A. Hassan Ismail et al, Comparison study between restrictive Taylor, restrictive Pade´approximations and Adomian decomposition method for the solitary wave solution of the General KdV equation, Appl. Math. Comp. 167 (2005) 849–869.
18. A. M. Wazwaz, A reliable modification of ADM, Appl. Math. Comp. 102 (1) (1999) 77-86.
19. Yahya Q. H., Liu M. Z., “Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method”, Commun. Nonlinear Sci. Numer. Simulat, doi: 10.1016/j.cnsns.2008.09.02 14 (2009) 2592–2596.
20. Yahya Q. H., “Modified Adomian decomposition method for second order singular initial value problems”, Advances in computational mathematics and its applications, vol. 1, No. 2, 2012.
21. B. Muatjetjeja, C. M. Khalique, Exact solutions of the generalized Lane–Emden equations of the first and second kind. Pramana 77, 545–554 (2011).
22. J.-S. Duan. R. Rach, A. M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method. J. Math. Chem., 53, 1054–1067 (2015).
23. R. Rach, J. S. Duan, A. M. Wazwaz, On the solution of non-isothermal reaction–diffusion model equations in a spherical catalyst by the modified Adomian method, Chem. Eng. Comm., 202(8), 1081–1088 (2015).
24. A. Saadatmandi, N. Nafar, S. P. Toufighi, Numerical study on the reaction– diffusion process in a spherical biocatalyst. Iran. J. Mathl. Chem., 5, 47–61 (2014).
25. V. Ananthaswamy, L. Rajendran, Approximate Analytical Solution of Non-Linear Kinetic Equation in a Porous Pellet, Global Journal of Pure and Applied Mathematics Volume 8, Number 2 (2012), pp. 101-111.
26. S. Sevukaperumal, L. Rajendran, Analytical solution of the Concentration of species using modified adomian decomposition method, International Journal of mathematical Archive-4(6), 2013, 107-117.
27. T. Praveen, Pedro Valencia, L. Rajendran, Theoretical analysis of intrinsic Reaction kinetics and the behavior of immobilized Enzymes system for steady-state conditions, Biochemical Engineering Journal, 91, 2014, pp. 129-139.
28. V. Meena, T. Praveen, and L. Rajendran, mathematical Modeling and analysis of the Molar Concentrations of Ethanol, Acetaldehyde and Ethyl Acetate Inside the Catalyst Particle. ISSN 00231584, Kinetics and Catalysis, 2016, Vol. 57, No. 1, pp. 125–134.
29. S. Liao, J. Sub, A. T. Chwang, Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int. J. Heat Mass Tran., 49, 2437–2445 (2006).
30. V. Ananthaswamy, R. Shanthakumari, M. Subha, Simple analytical expressions of the non–linear reaction diffusion process in an immobilized biocatalyst particle using the new homotopy perturbation method, Review of Bioinformatics and Biometrics. 3, 23– 28 (2014).
31. J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.
32. Q. K. Ghori, M. Ahmed, and A. M. Siddiqui, “ApplicationofHomotopy perturbation method to squeezing flow of anewtonian fluid,” International Journal of Nonlinear Sciencesand Numerical Simulation, vol. 8, no. 2, pp. 179–184, 2007.
33. S.-J. Li and Y.-X. Liu, “An improved approach to nonlineardynamical system identification using PID neural networks,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 2, pp. 177–182, 2006.
34. L. Rajendran and S. Anitha, “Reply to ‘Comments on analyticalsolution of amperometric enzymatic reactions based onHPM’, ElectrochimicaActa, vol. 102, pp. 474–476, 2013.
35. MATLAB 6. 1, The Math Works Inc., Natick, MA (2000),