Analytical solution of the concentration and current in the electoenzymatic processes involved in a PPO-rotating-disk-bioelectrode
ABSTRACT
A mathematical model for electroenzymatic process of a rotating-disk-bioelectrode in which polyphenol oxidase occurs for all values of concentration of catechol substrate is presented. The model is based on system of reaction-diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical method (He’s Homotopy perturbation method) is used to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis-Menten kinetics law. Closed analytical expressions for substrate concentration, product concentration and corresponding current response have been derived for all values of parameter using 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.
A mathematical model for electroenzymatic process of a rotating-disk-bioelectrode in which polyphenol oxidase occurs for all values of concentration of catechol substrate is presented. The model is based on system of reaction-diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical method (He’s Homotopy perturbation method) is used to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis-Menten kinetics law. Closed analytical expressions for substrate concentration, product concentration and corresponding current response have been derived for all values of parameter using 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.
KEYWORDS
Mathematical Model; Polyphenol Oxidase; Steady-State; Homotopy Perturbation Method; Simulation
Mathematical Model; Polyphenol Oxidase; Steady-State; Homotopy Perturbation Method; Simulation
Cite this paper
Varadharajan, G. and Rajendran, L. (2011) Analytical solution of the concentration and current in the electoenzymatic processes involved in a PPO-rotating-disk-bioelectrode. Natural Science, 3, 1-8. doi: 10.4236/ns.2011.31001.
Varadharajan, G. and Rajendran, L. (2011) Analytical solution of the concentration and current in the electoenzymatic processes involved in a PPO-rotating-disk-bioelectrode. Natural Science, 3, 1-8. doi: 10.4236/ns.2011.31001.
References
[1] Desprez, V. and Labbe, P. (1996) A Kinetic model for the electroenzymatic processd involved in polyphenol-oxidase- based amperometric catechol sensors. Journal of Electroanalytical Chemistry, 415, 191-1995. doi:10.1016/S0022-0728(96)01011-X [2] Coche-Guerente, L., Labbe, P. and Mengeaud, V. (2001) Amplification of amperometric biosensor responses by electrochemical substrate recycling.3. theoretical and experimental study of the phenol-polyphenol oxidase system immoboilized in laponite hydrogels and layer- by-layer self-assembled structures. Anaytical chemistry, 73, 3206-3218. doi:10.1021/ac001534l [3] Kronkvist, K., Wallentin, K., Johansson, G. (1994) Selective enzyme amplification of NAD+/NADH using coimmobilized glycerol dehydrogenase and diaphorase with amperometric detection. Analytical chimica Acta, 290-335. doi:10.1016/0003-2670(94)80120-7 [4] Kotte, H., Grundig, B., Vorlop, K. D., Strehlitz, B. and Stottmeister, U. (1995) Methylphenazonium-modified enzyme sensor based on polymer thick films for subnanomolar detection of phenols. Analytical chemistry, 67, 65-70. doi:10.1021/ac00097a011 [5] Wang, J., Lu, J., Ly, S.Y., Vuki, M., Tian, B., Adeniyi, W.K. and Armendariz, R.A. (2000) Lab-on-a-Cable for electrochemical monitoring of phenolic contaminants. Analytical chemistry, 72, 2659-2663. doi:10.1021/ac991054y [6] Zhenjiu, L., Deng, J. and Li. D. (2000) A new tyrosinase biosensor based on tailoring the porosity of Al2O3 sol-gel to co- immobilize tyrosinase and the mediator. Analytical chimica Acta, 407, 87-96. doi:10.1016/S0003-2670(99)00807-7 [7] Russell, I. M. and Burton, S. G. (1999) Development and demonstration of an immobilised-polyphenol oxidase bioprobe for the detection of phenolic pollutants in water. Analytical chimica Acta, 389, 161-170. doi:10.1016/S0003-2670(99)00143-9 [8] Cosnier. S., Fombon. J.J., Labbe, P. and Limosin, D. (1999) Development of a PPO-poly (amphiphilic pyrrole) electrode for on site monitoring of phenol in aqueous effluents. Sensors and Actuators B, 59, 134-139. [9] Nistor, C., Emneus, J., Gorton, L. and Ciucu, A. (1999) Improved stability and altered selectivity of tyrosinase based graphite electrodes for detection of phenolic compounds. Analytical chimica Acta, 387, 309-326. doi:10.1016/S0003-2670(99)00071-9 [10] Forzani, E.S., Rivas, G.A. and Solis, V.M. (1999) Kinetic behaviour of dopamine-polyphenol oxidase on electrodes of tetrathiafulvalenium tetracyanoquinodimethanide and tetracyanoquinodimethane species V. M. Journal of Electroanalytical Chemistry, 461, 174-183. doi:10.1016/S0022-0728(98)00119-3 [11] Moore, T.J., Nam, G.G., Pipes, L.C. and Coury Jr, L.A. (1994) Chemically amplified voltammetric enzyme electrodes for oxidizable pharmaceuticals. Analytical Chemistry, 66, 3158-3163. doi:10.1021/ac00091a026 [12] Lisdat, F., Wollenberg, U., Paeschke, M. and Scheller, F.W. (1998) Sensitive catecholamine measurement using a monoenzymatic recycling system. Analytical chimica Acta, 368, 233-241. doi:10.1016/S0003-2670(98)00221-9 [13] Forzani, E.S., Solis, V. and Calvo, E.J. (2000) Electrochemical behavior of polyphenol oxidase immobilized in self-assembled structures layer by layer with cationic polyallylamine. Analytical Chemisty, 72, 5300-5307. [14] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using PID neural networks,” International Journal of NonlinearScience and Numerical Simulation, 7, 177-182. [15] 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. [16] He, J.H, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, Vol. 178 (1999) 257-262. doi:10.1016/S0045-7825(99)00018-3 [17] He, J.H. (2003) Homotopy perturbation method: a new nonlinear analytical Technique. Applied Mathematics and Computation, 135, 73-79. doi:10.1016/S0096-3003(01)00312-5 [18] 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 [19] He, J.H. (2006) Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20, 1141-1199. doi:10.1142/S0217979206033796 [20] He, J.H., Wu, G.C. and Austin, F. (2010) The variational iteration method which should be followed. Nonlinear Science Letters A, 1, 1-30. [21] He, J.H. (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Internationl Journal of Nonlinear Mechanics, 35 37-43. doi:10.1016/S0020-7462(98)00085-7 [22] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Analytical investigation of hyperbolic Equations via He’s methods. American. 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(1988) The determination of p-cresol in chloroform with an enzyme electrode used in the organic phase. Analytical chimica Acta, 213, 113-119. doi:10.1016/S0003-2670(00)81345-8 [29] Cosiner, S. and Innocent, C. (1992) A novel biosensor elaboration by electropolymerization of an adsorbed amphiphilic pyrrole-tyrosinase enzyme layer. Journal of Electroanalytical Chemistry, 328, 361-366. http://dx.doi.org/10.1016/0022-0728(92)80195-A [30] Wang, J., Nasser, N., Kwon and Cho, M.Y. (1992) Tissue bioelectrode for organic-phase enzymatic assays, Analytical chimica Acta, 2649, 7-12. doi:10.1016/0003-2670(92)85290-M [31] Ortega, F., Dominguez, E., Pettersson J. and Gorton, L. (1993) Amperometric biosensor for the determination of phenolic compounds using a tyrosinase graphite electrode in a flow injection system. Journal of Biotechnology, 31, 289-300. doi:10.1016/0168-1656(93)90075-X [32] Besombes, J. L., Cosnier, S., Labbe, P and Reverdy, G. (1995) Determination of Phenol and Chlorinated Phenolic Compounds Based on a PPO-Bioelectrode and Its Inhibition, Analytical Letters, 28, 405-424. [33] Lutz, M., Burestedt, E., Emneus, J., Liden, H., Gobhadi, S., Gorton, L. and Marko-Varga, G. (1995) Effects of different additives on a tyrosinase based carbon paste electrode, Analytical chimica Acta, 305, 8-17. doi:10.1016/0003-2670(94)00573-5 [34] Onnerfjord, P., Emneus, J., Marko-Varga, G., Gorton, L., Ortega, F. and Dominguez, E. (1995) Tyrosinase graphite-epoxy based composite electrodes for detection of phenols, Biosensors Bioelectronics, 10, 607-619. doi:10.1016/0956-5663(95)96937-T
[1] Desprez, V. and Labbe, P. (1996) A Kinetic model for the electroenzymatic processd involved in polyphenol-oxidase- based amperometric catechol sensors. Journal of Electroanalytical Chemistry, 415, 191-1995. doi:10.1016/S0022-0728(96)01011-X [2] Coche-Guerente, L., Labbe, P. and Mengeaud, V. (2001) Amplification of amperometric biosensor responses by electrochemical substrate recycling.3. theoretical and experimental study of the phenol-polyphenol oxidase system immoboilized in laponite hydrogels and layer- by-layer self-assembled structures. Anaytical chemistry, 73, 3206-3218. doi:10.1021/ac001534l [3] Kronkvist, K., Wallentin, K., Johansson, G. (1994) Selective enzyme amplification of NAD+/NADH using coimmobilized glycerol dehydrogenase and diaphorase with amperometric detection. Analytical chimica Acta, 290-335. doi:10.1016/0003-2670(94)80120-7 [4] Kotte, H., Grundig, B., Vorlop, K. D., Strehlitz, B. and Stottmeister, U. (1995) Methylphenazonium-modified enzyme sensor based on polymer thick films for subnanomolar detection of phenols. Analytical chemistry, 67, 65-70. doi:10.1021/ac00097a011 [5] Wang, J., Lu, J., Ly, S.Y., Vuki, M., Tian, B., Adeniyi, W.K. and Armendariz, R.A. (2000) Lab-on-a-Cable for electrochemical monitoring of phenolic contaminants. Analytical chemistry, 72, 2659-2663. doi:10.1021/ac991054y [6] Zhenjiu, L., Deng, J. and Li. D. (2000) A new tyrosinase biosensor based on tailoring the porosity of Al2O3 sol-gel to co- immobilize tyrosinase and the mediator. Analytical chimica Acta, 407, 87-96. doi:10.1016/S0003-2670(99)00807-7 [7] Russell, I. M. and Burton, S. G. (1999) Development and demonstration of an immobilised-polyphenol oxidase bioprobe for the detection of phenolic pollutants in water. Analytical chimica Acta, 389, 161-170. doi:10.1016/S0003-2670(99)00143-9 [8] Cosnier. S., Fombon. J.J., Labbe, P. and Limosin, D. (1999) Development of a PPO-poly (amphiphilic pyrrole) electrode for on site monitoring of phenol in aqueous effluents. Sensors and Actuators B, 59, 134-139. [9] Nistor, C., Emneus, J., Gorton, L. and Ciucu, A. (1999) Improved stability and altered selectivity of tyrosinase based graphite electrodes for detection of phenolic compounds. Analytical chimica Acta, 387, 309-326. doi:10.1016/S0003-2670(99)00071-9 [10] Forzani, E.S., Rivas, G.A. and Solis, V.M. (1999) Kinetic behaviour of dopamine-polyphenol oxidase on electrodes of tetrathiafulvalenium tetracyanoquinodimethanide and tetracyanoquinodimethane species V. M. Journal of Electroanalytical Chemistry, 461, 174-183. doi:10.1016/S0022-0728(98)00119-3 [11] Moore, T.J., Nam, G.G., Pipes, L.C. and Coury Jr, L.A. (1994) Chemically amplified voltammetric enzyme electrodes for oxidizable pharmaceuticals. Analytical Chemistry, 66, 3158-3163. doi:10.1021/ac00091a026 [12] Lisdat, F., Wollenberg, U., Paeschke, M. and Scheller, F.W. (1998) Sensitive catecholamine measurement using a monoenzymatic recycling system. Analytical chimica Acta, 368, 233-241. doi:10.1016/S0003-2670(98)00221-9 [13] Forzani, E.S., Solis, V. and Calvo, E.J. (2000) Electrochemical behavior of polyphenol oxidase immobilized in self-assembled structures layer by layer with cationic polyallylamine. Analytical Chemisty, 72, 5300-5307. [14] Li, S.J. and Liu, Y.X. (2006) An improved approach to nonlinear dynamical system identification using PID neural networks,” International Journal of NonlinearScience and Numerical Simulation, 7, 177-182. [15] 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. [16] He, J.H, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, Vol. 178 (1999) 257-262. doi:10.1016/S0045-7825(99)00018-3 [17] He, J.H. (2003) Homotopy perturbation method: a new nonlinear analytical Technique. Applied Mathematics and Computation, 135, 73-79. doi:10.1016/S0096-3003(01)00312-5 [18] 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 [19] He, J.H. (2006) Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20, 1141-1199. doi:10.1142/S0217979206033796 [20] He, J.H., Wu, G.C. and Austin, F. (2010) The variational iteration method which should be followed. Nonlinear Science Letters A, 1, 1-30. [21] He, J.H. (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Internationl Journal of Nonlinear Mechanics, 35 37-43. doi:10.1016/S0020-7462(98)00085-7 [22] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Analytical investigation of hyperbolic Equations via He’s methods. American. Journal of Engineering and Applied Science, 1, 399-407. doi:10.3844/ajeassp.2008.399.407 [23] Bartlett, P.N. and Whitaker, R.G. (1987) Electrochemical immobilisation of enzymes: Part I. Theory, Journal of Electroanalytical Chemistry, 224, 27-35. doi:10.1016/0022-0728(87)85081-7 [24] Toyota, T., Kuan, S.S. and Guilbault, G.G. (1985) Determination of total protein in serum using a tyrosinase enzyme electrode, Anaytical chemistry, 57, 1925-1928. doi:10.1021/ac00286a030 [25] Kulys, J. and Schmid, R.D. (1990) A Sensitive Enzyme Electrode for Phenol Monitoring. Analytical Letters, 23, 589-597. [26] Wang, J. and Varughese, K. (1990) Polishable and robust biological electrode surfaces. Analytical chemistry,, 62 318-320. doi:10.1021/ac00202a019 [27] Skladal, P. (1991) Mushroom tyrosinase-modified carbon paste electrode as an amperometric biosensor for phenols. Collection of Czechoslovak Chemical Communications, 569, 1427-1433. [28] Hall, G.F., Best, D.J. and Turner, A.P.F. (1988) The determination of p-cresol in chloroform with an enzyme electrode used in the organic phase. Analytical chimica Acta, 213, 113-119. doi:10.1016/S0003-2670(00)81345-8 [29] Cosiner, S. and Innocent, C. (1992) A novel biosensor elaboration by electropolymerization of an adsorbed amphiphilic pyrrole-tyrosinase enzyme layer. Journal of Electroanalytical Chemistry, 328, 361-366. http://dx.doi.org/10.1016/0022-0728(92)80195-A [30] Wang, J., Nasser, N., Kwon and Cho, M.Y. (1992) Tissue bioelectrode for organic-phase enzymatic assays, Analytical chimica Acta, 2649, 7-12. doi:10.1016/0003-2670(92)85290-M [31] Ortega, F., Dominguez, E., Pettersson J. and Gorton, L. (1993) Amperometric biosensor for the determination of phenolic compounds using a tyrosinase graphite electrode in a flow injection system. Journal of Biotechnology, 31, 289-300. doi:10.1016/0168-1656(93)90075-X [32] Besombes, J. L., Cosnier, S., Labbe, P and Reverdy, G. (1995) Determination of Phenol and Chlorinated Phenolic Compounds Based on a PPO-Bioelectrode and Its Inhibition, Analytical Letters, 28, 405-424. [33] Lutz, M., Burestedt, E., Emneus, J., Liden, H., Gobhadi, S., Gorton, L. and Marko-Varga, G. (1995) Effects of different additives on a tyrosinase based carbon paste electrode, Analytical chimica Acta, 305, 8-17. doi:10.1016/0003-2670(94)00573-5 [34] Onnerfjord, P., Emneus, J., Marko-Varga, G., Gorton, L., Ortega, F. and Dominguez, E. (1995) Tyrosinase graphite-epoxy based composite electrodes for detection of phenols, Biosensors Bioelectronics, 10, 607-619. doi:10.1016/0956-5663(95)96937-T