A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method
ABSTRACT
A two parameter mathematical model was developed to find the concentration for immobilized enzyme systems in porous spherical particles. This model contains a non-linear term related to reversible Michaelies-Menten kinetics. Analytical expression pertaining to the substrate concentration was reported for all possible values of Thiele module φ and α . In this work, we report the theoretically evaluated steady-state effectiveness factor for immobilized enzyme systems in porous spherical particles. These analytical results were found to be in good agreement with numerical results. Moreover, herein we employ new “Homotopy analysis method” (HAM) to solve non-linear reaction/diffusion equation.
A two parameter mathematical model was developed to find the concentration for immobilized enzyme systems in porous spherical particles. This model contains a non-linear term related to reversible Michaelies-Menten kinetics. Analytical expression pertaining to the substrate concentration was reported for all possible values of Thiele module φ and α . In this work, we report the theoretically evaluated steady-state effectiveness factor for immobilized enzyme systems in porous spherical particles. These analytical results were found to be in good agreement with numerical results. Moreover, herein we employ new “Homotopy analysis method” (HAM) to solve non-linear reaction/diffusion equation.
Cite this paper
Joy, R. , Meena, A. , Loghambal, S. and Rajendran, L. (2011) A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method. Natural Science, 3, 556-565. doi: 10.4236/ns.2011.37078.
Joy, R. , Meena, A. , Loghambal, S. and Rajendran, L. (2011) A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method. Natural Science, 3, 556-565. doi: 10.4236/ns.2011.37078.
References
[1] Gomez J.L, Bodalo A., Gomez E., Bastida J. and Maximo M.F. (2003) Two-parameter model for evaluating effectiveness factor for immobilized enzymes with reversible Michaelis–Menten kinetics. Chemical Engineering Science, 58, 4287-4290. [2] Manjon A., Iborra J.L., Gomez J.L., Gomez E., Bastida J. and Bodalo A. (1987), Evaluation of the effectiveness factor along immobilized enzyme fixed-bed reactors: Design of a reactor with naringinase covalently immo- bilized into glycophase-coated porous glass. Biotech- nology and Bioengineering, 30, 491-497. doi:10.1002/bit.260300405 [3] Bodalo S.A., Gomez C.J.L., Gomez E., Bastida R.J. and Martinez M. E. (1993) Transient stirred-tank reactors operating with immobilized enzyme systems: Analysis and simulation models and their experimental checking. Biotechnology Progress, 9, 166-173. doi:10.1021/bp00020a008 [4] Bodalo A., Gomez J.L., Gomez E., Bastida J. and Maximo M.F. (1995) Fluidized bed reactors operating with immobilized enzyme systems: Design model and its experimental verification. Enzyme and Microbial Technology, 17, 915-922. doi:10.1016/0141-0229(94)00125-B [5] Carnahan B., Luther H.A. and Wilkes J.O. (1969) Applied numerical methods,Wiley, New York. [6] Villadsen J., Michelsen M.L. (1978) Solution of differential equation models by polynomial approximation. New York: Prentice-Hall,Englewood Cliffs. [7] Lee J. and Kim D.H. (2005) An improved shooting method for computation of effectiveness factors in porous catalysts. Chemical Engineering Science, 60, 5569-5573. doi:10.1016/j.ces.2005.05.027 [8] Lyons M.E.G., Greer J.C., Fitzgerald C.A., Bannon T. and Bartlett P.N. (1996) Reaction/Diffusion with Michaelis- Menten kinetics in electroactive polymer films Part 1. The steady-state amperometric response. Analyst, 12, 1715. [9] Lyons M.E.G., Bannon T., Hinds G. and Rebouillat S. (1998) Reaction/Diffusion with Michaelis-Menten kinetics in electroactive polymer films Part 1. The transient amperometric response. Analyst, 123, 1947. doi:10.1039/a803274b [10] Lyons M.E.G., Murphy J., Bannon T. and Rebouillat S. (1999) Reaction, diffusion and migration in conducting polymer electrodes: Analysis of the steady-state amperometric response. Journal of Solid State Electrochemistry, 3, 154-162. doi:10.1007/s100080050142 [11] Moo-Young M. and Kobayashi T. (1972) Effectiveness factors for immobilized enzyme reactions. Canadian Journal of Chemical Engineering, 50, 162-167. doi:10.1002/cjce.5450500204 [12] Kobayashi T. and Laidler K. J. (1973) Effectiveness factor calculations for immobilized enzyme catalysts. Biochimica et Biophysica Acta, 1, 302-311. [13] Engasser J.M. and Horvath J.C. (1973) Metabolism: interplay of membrane transport and consecutive enzymic reaction. Journal of Theoretical Biology, 42, 137-155. doi:10.1016/0022-5193(73)90153-7 [14] Marsh D.R., Lee Y.Y. and Tsao G.T. (1973) Immobilized glucoamylase on porous glass. Biotechnology and Bioengineering, 15, 483-492. doi:10.1002/bit.260150305 [15] Hamilton B.K., Cardner C.R. and Colton C.K. (1974) Effect of diffusional limitations on lineweaver-burk plots for immobilized enzymes. AIChE Journal, 20, 503-510. doi:10.1002/aic.690200310 [16] Rovito B.J. and Kittrell J.R. (1973) Film and pore diffusion studies with immobilized glucose Oxidase. Biotechnology and Bioengineering, 15, 143-161. doi:10.1002/bit.260150111 [17] Engasser J.M. (1978) The experimental results accorded quantitatively with the theory of diffusion limitation. Biochimica et Biophysica Acta, 526, 301-310. [18] Aydogan O., Bayraktar E. and Mehmetoglu U. (2011) Determination of effective diffusion coefficient of acetophenone in carrageenan and asymmetric bioreduction in packed bed reactor. Journal of Molecular Catalysis B: Enzymatic, 72, 46-52. doi:10.1016/j.molcatb.2011.04.023 [19] Puida M., Malinauskas A., Ivanauskas F. (2011) Mode- ling of electrocatalysis at conducting polymer modified electrodes: Nonlinear current-concentration profiles. Journal of Mathematical chemistry, 49, 1151-1162. doi:10.1007/s10910-011-9802-y [20] Marc A. and Engasser J.M. (1982) Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions. Journal of Theoretical Biology, 94, 179-189. doi:10.1016/0022-5193(82)90339-3 [21] Goldman R., Keden O., Silman I.H., Caplan S.R. and Katchalski E. (1968) Papain-collodion membranes. I. Preparation and properties. Biochemistry, 7, 486-500. doi:10.1021/bi00842a002 [22] Engasser J.M. and Horvath C. (1974) Inhibition of bound enzymes. II. characterization of product inhibition and accumulation. Biochemistry, 133, 849-3854. [23] Ramachandran P.A. (1975) Solution of immobilized enzyme problems by collocation methods. Biotechnology and Bioengineering, 17, 211-226. doi:10.1002/bit.260170207 [24] Liao S.J. (1992) The proposed homotopy analysis technique for the solution of nonlinear problems, Ph. D. Thesis, Shanghai Jiao Tong University, Shanghai. [25] Awawdeh F., Jaradat H.M. and Alsayyed O. (2009) Solving System of DAEs by Homotopy Analysis. Chaos Solitons and Fractals, 42, 1422-1427. doi:10.1016/j.chaos.2009.03.057 [26] Jafari H., Chun C., Seifi S. and Saeidy M., (2009) Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method. Applied Mathmatics, 4, 149-154. [27] Sohouli A.R., Famouri M., Kimiaeifar A. and Domairry G., (2010) Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Communications in Nonlinear Science and Numerical Simulation, 15, 1691-1699. doi:10.1016/j.cnsns.2009.07.015 [28] Domairry G. and Fazeli M. (2009) Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Communications in Nonlinear Science and Numerical Simulation, 14, 489-499. doi:10.1016/j.cnsns.2007.09.007 [29] Liao S.J. (2004) On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499-513. doi:10.1016/S0096-3003(02)00790-7 [30] Domairry G. and Bararnia H. (2008) An Approximation of the Analytic Solution of Some Nonlinear Heat Transfer Equations: A Survey by using Homotopy Analysis Method. Advanced studies in theoretical physics, 2, 507-518. [31] Liao S.J. (2003) Beyond Perturbation: Introduction to the Homotopy analysis method. Chapman and Hall, CRC Press, Boca Raton. doi:10.1201/9780203491164
[1] Gomez J.L, Bodalo A., Gomez E., Bastida J. and Maximo M.F. (2003) Two-parameter model for evaluating effectiveness factor for immobilized enzymes with reversible Michaelis–Menten kinetics. Chemical Engineering Science, 58, 4287-4290. [2] Manjon A., Iborra J.L., Gomez J.L., Gomez E., Bastida J. and Bodalo A. (1987), Evaluation of the effectiveness factor along immobilized enzyme fixed-bed reactors: Design of a reactor with naringinase covalently immo- bilized into glycophase-coated porous glass. Biotech- nology and Bioengineering, 30, 491-497. doi:10.1002/bit.260300405 [3] Bodalo S.A., Gomez C.J.L., Gomez E., Bastida R.J. and Martinez M. E. (1993) Transient stirred-tank reactors operating with immobilized enzyme systems: Analysis and simulation models and their experimental checking. Biotechnology Progress, 9, 166-173. doi:10.1021/bp00020a008 [4] Bodalo A., Gomez J.L., Gomez E., Bastida J. and Maximo M.F. (1995) Fluidized bed reactors operating with immobilized enzyme systems: Design model and its experimental verification. Enzyme and Microbial Technology, 17, 915-922. doi:10.1016/0141-0229(94)00125-B [5] Carnahan B., Luther H.A. and Wilkes J.O. (1969) Applied numerical methods,Wiley, New York. [6] Villadsen J., Michelsen M.L. (1978) Solution of differential equation models by polynomial approximation. New York: Prentice-Hall,Englewood Cliffs. [7] Lee J. and Kim D.H. (2005) An improved shooting method for computation of effectiveness factors in porous catalysts. Chemical Engineering Science, 60, 5569-5573. doi:10.1016/j.ces.2005.05.027 [8] Lyons M.E.G., Greer J.C., Fitzgerald C.A., Bannon T. and Bartlett P.N. (1996) Reaction/Diffusion with Michaelis- Menten kinetics in electroactive polymer films Part 1. The steady-state amperometric response. Analyst, 12, 1715. [9] Lyons M.E.G., Bannon T., Hinds G. and Rebouillat S. (1998) Reaction/Diffusion with Michaelis-Menten kinetics in electroactive polymer films Part 1. The transient amperometric response. Analyst, 123, 1947. doi:10.1039/a803274b [10] Lyons M.E.G., Murphy J., Bannon T. and Rebouillat S. (1999) Reaction, diffusion and migration in conducting polymer electrodes: Analysis of the steady-state amperometric response. Journal of Solid State Electrochemistry, 3, 154-162. doi:10.1007/s100080050142 [11] Moo-Young M. and Kobayashi T. (1972) Effectiveness factors for immobilized enzyme reactions. Canadian Journal of Chemical Engineering, 50, 162-167. doi:10.1002/cjce.5450500204 [12] Kobayashi T. and Laidler K. J. (1973) Effectiveness factor calculations for immobilized enzyme catalysts. Biochimica et Biophysica Acta, 1, 302-311. [13] Engasser J.M. and Horvath J.C. (1973) Metabolism: interplay of membrane transport and consecutive enzymic reaction. Journal of Theoretical Biology, 42, 137-155. doi:10.1016/0022-5193(73)90153-7 [14] Marsh D.R., Lee Y.Y. and Tsao G.T. (1973) Immobilized glucoamylase on porous glass. Biotechnology and Bioengineering, 15, 483-492. doi:10.1002/bit.260150305 [15] Hamilton B.K., Cardner C.R. and Colton C.K. (1974) Effect of diffusional limitations on lineweaver-burk plots for immobilized enzymes. AIChE Journal, 20, 503-510. doi:10.1002/aic.690200310 [16] Rovito B.J. and Kittrell J.R. (1973) Film and pore diffusion studies with immobilized glucose Oxidase. Biotechnology and Bioengineering, 15, 143-161. doi:10.1002/bit.260150111 [17] Engasser J.M. (1978) The experimental results accorded quantitatively with the theory of diffusion limitation. Biochimica et Biophysica Acta, 526, 301-310. [18] Aydogan O., Bayraktar E. and Mehmetoglu U. (2011) Determination of effective diffusion coefficient of acetophenone in carrageenan and asymmetric bioreduction in packed bed reactor. Journal of Molecular Catalysis B: Enzymatic, 72, 46-52. doi:10.1016/j.molcatb.2011.04.023 [19] Puida M., Malinauskas A., Ivanauskas F. (2011) Mode- ling of electrocatalysis at conducting polymer modified electrodes: Nonlinear current-concentration profiles. Journal of Mathematical chemistry, 49, 1151-1162. doi:10.1007/s10910-011-9802-y [20] Marc A. and Engasser J.M. (1982) Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions. Journal of Theoretical Biology, 94, 179-189. doi:10.1016/0022-5193(82)90339-3 [21] Goldman R., Keden O., Silman I.H., Caplan S.R. and Katchalski E. (1968) Papain-collodion membranes. I. Preparation and properties. Biochemistry, 7, 486-500. doi:10.1021/bi00842a002 [22] Engasser J.M. and Horvath C. (1974) Inhibition of bound enzymes. II. characterization of product inhibition and accumulation. Biochemistry, 133, 849-3854. [23] Ramachandran P.A. (1975) Solution of immobilized enzyme problems by collocation methods. Biotechnology and Bioengineering, 17, 211-226. doi:10.1002/bit.260170207 [24] Liao S.J. (1992) The proposed homotopy analysis technique for the solution of nonlinear problems, Ph. D. Thesis, Shanghai Jiao Tong University, Shanghai. [25] Awawdeh F., Jaradat H.M. and Alsayyed O. (2009) Solving System of DAEs by Homotopy Analysis. Chaos Solitons and Fractals, 42, 1422-1427. doi:10.1016/j.chaos.2009.03.057 [26] Jafari H., Chun C., Seifi S. and Saeidy M., (2009) Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method. Applied Mathmatics, 4, 149-154. [27] Sohouli A.R., Famouri M., Kimiaeifar A. and Domairry G., (2010) Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Communications in Nonlinear Science and Numerical Simulation, 15, 1691-1699. doi:10.1016/j.cnsns.2009.07.015 [28] Domairry G. and Fazeli M. (2009) Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Communications in Nonlinear Science and Numerical Simulation, 14, 489-499. doi:10.1016/j.cnsns.2007.09.007 [29] Liao S.J. (2004) On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499-513. doi:10.1016/S0096-3003(02)00790-7 [30] Domairry G. and Bararnia H. (2008) An Approximation of the Analytic Solution of Some Nonlinear Heat Transfer Equations: A Survey by using Homotopy Analysis Method. Advanced studies in theoretical physics, 2, 507-518. [31] Liao S.J. (2003) Beyond Perturbation: Introduction to the Homotopy analysis method. Chapman and Hall, CRC Press, Boca Raton. doi:10.1201/9780203491164