Analytical expressions of steady-state concentrations of species in potentiometric and amperometric biosensor
Affiliation(s)
Department of Mathematics, Bharath Niketan Engineering College, Theni, India. Department of Mathematics, The Madura College, Madurai, India;.
Department of Mathematics, Bharath Niketan Engineering College, Theni, India. Department of Mathematics, The Madura College, Madurai, India;.
ABSTRACT
A mathematical model of potentiometric and amperometric enzyme electrodes is discussed. The model is based on the system of non-linear steady-state coupled reaction diffusion equations for Michaelis-Menten formalism that describe the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate, product and corresponding flux response have been derived for all values of parameters using Homotopy analysis method. The obtained solution allow a full characterization of the response curves for only two kinetic parameters (The Michaelis constant and the ratio of overall reaction and the diffusion rates). A simple relation between the concentration of substrate and products for all values of parameter is also reported. All the analytical results are compared with simulation results (Scilab/Matlab program). The simulated results are agreed with the appropriate theories. The obtained theoretical results are valid for the whole solution domain.
A mathematical model of potentiometric and amperometric enzyme electrodes is discussed. The model is based on the system of non-linear steady-state coupled reaction diffusion equations for Michaelis-Menten formalism that describe the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate, product and corresponding flux response have been derived for all values of parameters using Homotopy analysis method. The obtained solution allow a full characterization of the response curves for only two kinetic parameters (The Michaelis constant and the ratio of overall reaction and the diffusion rates). A simple relation between the concentration of substrate and products for all values of parameter is also reported. All the analytical results are compared with simulation results (Scilab/Matlab program). The simulated results are agreed with the appropriate theories. The obtained theoretical results are valid for the whole solution domain.
KEYWORDS
Enzyme Electrodes; Enzyme Reactors; Potentiometry; Amperometry; Homotopy Analysis Method; Biosensor
Enzyme Electrodes; Enzyme Reactors; Potentiometry; Amperometry; Homotopy Analysis Method; Biosensor
Cite this paper
Sivasankari, M. and Rajendran, L. (2012) Analytical expressions of steady-state concentrations of species in potentiometric and amperometric biosensor. Natural Science, 4, 1029-1041. doi: 10.4236/ns.2012.412132.
Sivasankari, M. and Rajendran, L. (2012) Analytical expressions of steady-state concentrations of species in potentiometric and amperometric biosensor. Natural Science, 4, 1029-1041. doi: 10.4236/ns.2012.412132.
References
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Analytical Chemistry, 47, 1359-1364. doi:10.1021/ac60358a075 [7] Morf, W.E. (1980) Theoretical evaluation of the performance of enzyme electrodes and of enzyme reactors. Microchimica Acta, 74, 317-332. doi:10.1007/BF01196457 [8] Morf, W.E. (1981) The principles of ion-selective electrodes and of membrane transport. Elsevier, New York. [9] Glab, S., Koncki, R. and Hulanicki, A. (1991) Kinetic model of pH-based potentiometric enzymic sensors. Part 1. Theoretical considerations. Analyst (London), 116, 453- 480. doi:10.1039/an9911600453 [10] Glab, S., Koncki, R. and Holona, I. (1992) Kinetic model of pH-based potentiometric enzymic sensors. Part 2. Method of fitting. Analyst (London), 117, 1671-1674. doi:10.1039/an9921701671 [11] Glab, S., Koncki, R. and Hulanicki, A. (1992) Kinetic model of pH-based potentiometric enzymic sensors. Part 3. Experimental verifications. Analyst (London), 117, 1675- 1678. doi:10.1039/an9921701675 [12] Mell, L.D. and Maloy, J.T. (1975) A Model for the amperometric enzyme electrode obtained through digital simulation and applied to the immobilized glucose oxidase system. Analytical Chemistry, 47, 299-307. doi:10.1021/ac60352a006 [13] Mell, L.D. and Maloy, J.T. (1976) Amperometric response enhancement of the immobilized glucose oxidase enzyme electrode. Analytical Chemistry, 48, 1597-1601. doi:10.1021/ac50005a045 [14] Olsson, B., Lundback, H. and Johansson, G. (1986) Theory and application of diffusion-limited amperometric enzyme electrode detection in flow injection analysis of glucose. Analytical Chemistry, 58, 1046-1052. doi:10.1021/ac00297a014 [15] Albery, W.J. and Bartlett, P.N. (1985) Amperometric enzyme electrodes. Journal of Electroanalytical Chemistry, 194, 211-222. doi:10.1016/0022-0728(85)85005-1 [16] Albery, W.J., Bartlett, P.N., Driscoll, B.J. and Lennox, R.B. (1992) Amperometric enzyme electrodes. 5. The homogeneous mediated mechanism. Journal of Electroanalytical Chemistry, 323, 77-102. doi:10.1016/0022-0728(92)80004-N [17] Mackey, D., Killard, A.J., Ambrosi, A. and Smyth, M.R. (2007) Optimizing the ratio of horseradish peroxidase and glucose oxidase on a bienzyme electrode: comparison of a theoretical and experimental approach. Sensors & Actuators B, 122, 395-402. doi:10.1016/j.snb.2006.06.006 [18] Matsue, T., Yamada, H., Chang, H.C., Uchida, I., Nagata, K. and Tomita, K. (1990) Electron transferase activity of diaphorase (NADH: acceptor oxidoreductase) from Bacillus stearothermophilus. Biochimica et Biophysica Acta, 1038, 29-38. doi:10.1016/0167-4838(90)90006-2 [19] Yokoyama, K. and Kayanuma, Y. (1998) Cyclic voltammetric simulation for electrochemically mediated enzyme reaction and determination of enzyme kinetic constants. Analytical Chemistry, 70, 3368-3376. [20] Baronas, R., Kulys, J. and Ivanauskas, F. (2004) Modelling amperometric enzyme electrode with substrate cyclic conversion. 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(2010) Theoretical treatment and numerical simulation of potential and concentration profiles in extremely thin non-electroneutral membranes used for ion-selective electrodes. Journal of Electroanalytical Chemistry, 641, 45-56. doi:10.1016/j.jelechem.2010.01.001 [25] Sandifer, J.R. and Buck, R.P. (1975) Algorithm for simulation of transient and alternating current electrical properties of conducting membranes, junctions, and one-dimensional, finite galvanic cells. Journal of Physical Chemistry, 79, 384-391. doi:10.1021/j100571a019 [26] Brumleve, T.R. and Buck, R.P. (1978) Numerical-solution of nernst-planck and poisson equation system with applications to membrane electrochemistry and solid-state physics. Journal of Electroanalytical Chemistry, 90, 1-31. doi:10.1016/S0022-0728(78)80137-5 [27] Sokalski, T. and Lewenstam, A. (2001) Application of nernst-plank equation and poisson equations for interpretation of liquid junction and membran potential in real-time and space domains. Electrochemistry Communications, 3, 107-112. doi:10.1016/S1388-2481(01)00110-2 [28] Sokalski, T., Lingenfelter, P. and Lewenstam, A. (2003) Numerical solution of the coupled Nernst-Planck and Poisson equations for liquid-junction and ion-selective membrane potentials. Journal of Physical Chemistry B, 117, 2443-2452. doi:10.1021/jp026406a [29] Lingenfelter, P., Bedlechowicz-Silwachowska, I., Sokalski, T., Maj-Zurawska, M. and Lewenstam, A. (2006) Time-dependent phenomena in the potential response of ion-selective electrodes treated by the nernst-planckpoisson model. 1. Intramembrane processes and selectivity. Analytical Chemistry, 78, 6783-6791. doi:10.1021/ac060264p [30] Bobacka, J., Ivaska, A. and Lewenstam, A. (2008) Potentiometric ion sensors. Chemical Reviews, 108, 329-351. doi:10.1021/cr068100w [31] Stryer, L. (1975) Biochemistry. Freeman, San Francisco. [32] Reithel, F.J. (1971) Ureases. In: Boyer, P.D. Ed., The Enzymes, 3rd Edition, Academic Press, New York. [33] Morf, W.E., van der Wal, P.D., Pretsch, E. and de Rooji, N.F. (2011) Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 2: Time-dependent concentration profiles, fluxes, and responses. Journal of Electroanalytical Chemistry, 657, 1-12. doi:10.1016/j.jelechem.2011.02.007 [34] Morf, W.E., van der Wal, P.D., Pretsch, E. and de Rooji, N.F. (2011) Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 1: Steady-state concentration profiles, fluxes, and responses. Journal of Electroanalytical Chemistry, 657, 13-22. doi:10.1016/j.jelechem.2011.02.006 [35] Liao, S.J. (2012) The homotopy analysis method in nonlinear differential equations. Springer and Higher Education press, New York. doi:10.1007/978-3-642-25132-0 [36] Liao, S.J. (2003) Beyond perturbation: An introduction to the homotopy analysis method. Chapman & Hall/CRC Press, Boca Raton. doi:10.1201/9780203491164 [37] Allan, F.M. (2007) Derivation of the adomian decomposition method using the homotopy analysis method. Applied Mathematics and Computation, 190, 6-14. doi:10.1016/j.amc.2006.12.074 [38] Liao, S.J. (2003) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai. [39] 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 [40] Loghambal, S. and Rajendran, L. (2011) Analytical expressions of concentration of nitrate pertaining to the electrocatalytic reduction of nitrate ion. Journal of Electro- analytical Chemistry, 661, 137-143. doi:10.1016/j.jelechem.2011.07.027 [41] Liao, S.J. (2009) The proposed homotopy analysis tecnique for the solution of nonlinear problems. Communications in Nonlinear Science and Numerical Simulation, 14, 983-997. doi:10.1016/j.cnsns.2008.04.013 [42] Liao, S.J. and Tan, Y. (2007) A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics, 119, 297-355. doi:10.1111/j.1467-9590.2007.00387.x [43] Liao, S.J. (2003) Beyond perturbation: Introduction to the homotopy analysis method. Chapman and Hall, CRC Press, Boca Raton, 336. [44] 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
[1] Heller, A. and Feldman, B. (2008) Electrochemical glucose sensors and their applications in diabetes management. Chemical Reviews, 108, 2482-2505. doi:10.1021/cr068069y [2] Urban, P.L., Goodall, D. M. and Bruce, N. C. (2006) Enzymatic microreactors in chemical analysis and kinetic studies. Biotechnology Advances, 24, 42-57. doi:10.1016/j.biotechadv.2005.06.001 [3] Blaedel, W.J., Kissel, T.R. and Boguslaski, R.C. (1972) Kinetic behavior of enzymes immobilized in artificial membranes. Analytical Chemistry, 44, 2030-2037. doi:10.1021/ac60320a021 [4] Guilbault, G.G.. and Nagy, G.G. (1973) Improved urea electrode. Analytical Chemistry, 45, 417-419. doi:10.1021/ac60324a053 [5] Hameka, H.F. and Rechnitz, G.A. (1983) Theory of the biocatalytic membrane electrode. Journal of Physical Chemistry, 87, 1235-1241. doi:10.1021/j100230a029 [6] Tranh-Minh, C. and Broun, G. (1975) Construction and study of electrodes using cross-linked enzymes. Analytical Chemistry, 47, 1359-1364. doi:10.1021/ac60358a075 [7] Morf, W.E. (1980) Theoretical evaluation of the performance of enzyme electrodes and of enzyme reactors. Microchimica Acta, 74, 317-332. doi:10.1007/BF01196457 [8] Morf, W.E. (1981) The principles of ion-selective electrodes and of membrane transport. Elsevier, New York. [9] Glab, S., Koncki, R. and Hulanicki, A. (1991) Kinetic model of pH-based potentiometric enzymic sensors. Part 1. Theoretical considerations. Analyst (London), 116, 453- 480. doi:10.1039/an9911600453 [10] Glab, S., Koncki, R. and Holona, I. (1992) Kinetic model of pH-based potentiometric enzymic sensors. Part 2. Method of fitting. Analyst (London), 117, 1671-1674. doi:10.1039/an9921701671 [11] Glab, S., Koncki, R. and Hulanicki, A. (1992) Kinetic model of pH-based potentiometric enzymic sensors. Part 3. Experimental verifications. Analyst (London), 117, 1675- 1678. doi:10.1039/an9921701675 [12] Mell, L.D. and Maloy, J.T. (1975) A Model for the amperometric enzyme electrode obtained through digital simulation and applied to the immobilized glucose oxidase system. Analytical Chemistry, 47, 299-307. doi:10.1021/ac60352a006 [13] Mell, L.D. and Maloy, J.T. (1976) Amperometric response enhancement of the immobilized glucose oxidase enzyme electrode. Analytical Chemistry, 48, 1597-1601. doi:10.1021/ac50005a045 [14] Olsson, B., Lundback, H. and Johansson, G. (1986) Theory and application of diffusion-limited amperometric enzyme electrode detection in flow injection analysis of glucose. Analytical Chemistry, 58, 1046-1052. doi:10.1021/ac00297a014 [15] Albery, W.J. and Bartlett, P.N. (1985) Amperometric enzyme electrodes. Journal of Electroanalytical Chemistry, 194, 211-222. doi:10.1016/0022-0728(85)85005-1 [16] Albery, W.J., Bartlett, P.N., Driscoll, B.J. and Lennox, R.B. (1992) Amperometric enzyme electrodes. 5. The homogeneous mediated mechanism. Journal of Electroanalytical Chemistry, 323, 77-102. doi:10.1016/0022-0728(92)80004-N [17] Mackey, D., Killard, A.J., Ambrosi, A. and Smyth, M.R. (2007) Optimizing the ratio of horseradish peroxidase and glucose oxidase on a bienzyme electrode: comparison of a theoretical and experimental approach. Sensors & Actuators B, 122, 395-402. doi:10.1016/j.snb.2006.06.006 [18] Matsue, T., Yamada, H., Chang, H.C., Uchida, I., Nagata, K. and Tomita, K. (1990) Electron transferase activity of diaphorase (NADH: acceptor oxidoreductase) from Bacillus stearothermophilus. Biochimica et Biophysica Acta, 1038, 29-38. doi:10.1016/0167-4838(90)90006-2 [19] Yokoyama, K. and Kayanuma, Y. (1998) Cyclic voltammetric simulation for electrochemically mediated enzyme reaction and determination of enzyme kinetic constants. Analytical Chemistry, 70, 3368-3376. [20] Baronas, R., Kulys, J. and Ivanauskas, F. (2004) Modelling amperometric enzyme electrode with substrate cyclic conversion. Biosensors & Bioelectronics, 19, 915-922. doi:10.1016/j.bios.2003.08.022 [21] Morf, W.E., Pretsch, E. and de Rooij, N.F. (2007) Computer simulation of ion-selective membrane electrodes and related systems by finite-difference procedutres. Journal of Electroanalytical Chemistry, 602, 43-54. doi:10.1016/j.jelechem.2006.11.025 [22] Morf, W.E., Pretsch, E. and de Rooij, N.F. (2008) Theory and computer simulation of the time-dependent selectiveity behavior of polymeric membrane ion-selective electrodes. Journal of Electroanalytical Chemistry, 614, 15- 23. doi:10.1016/j.jelechem.2007.10.027 [23] Morf, W.E., Pretsch, E. and de Rooij, N.F. (2009) Memory effects of ion-selective electrodes: Theory and computer simulation of the time-dependent potential response to multiple sample changes. Journal of Electroanalytical Chemistry, 633, 137-145. doi:10.1016/j.jelechem.2009.05.004 [24] Morf, W.E., Pretsch, E. and de Rooij, N.F. (2010) Theoretical treatment and numerical simulation of potential and concentration profiles in extremely thin non-electroneutral membranes used for ion-selective electrodes. Journal of Electroanalytical Chemistry, 641, 45-56. doi:10.1016/j.jelechem.2010.01.001 [25] Sandifer, J.R. and Buck, R.P. (1975) Algorithm for simulation of transient and alternating current electrical properties of conducting membranes, junctions, and one-dimensional, finite galvanic cells. Journal of Physical Chemistry, 79, 384-391. doi:10.1021/j100571a019 [26] Brumleve, T.R. and Buck, R.P. (1978) Numerical-solution of nernst-planck and poisson equation system with applications to membrane electrochemistry and solid-state physics. Journal of Electroanalytical Chemistry, 90, 1-31. doi:10.1016/S0022-0728(78)80137-5 [27] Sokalski, T. and Lewenstam, A. (2001) Application of nernst-plank equation and poisson equations for interpretation of liquid junction and membran potential in real-time and space domains. Electrochemistry Communications, 3, 107-112. doi:10.1016/S1388-2481(01)00110-2 [28] Sokalski, T., Lingenfelter, P. and Lewenstam, A. (2003) Numerical solution of the coupled Nernst-Planck and Poisson equations for liquid-junction and ion-selective membrane potentials. Journal of Physical Chemistry B, 117, 2443-2452. doi:10.1021/jp026406a [29] Lingenfelter, P., Bedlechowicz-Silwachowska, I., Sokalski, T., Maj-Zurawska, M. and Lewenstam, A. (2006) Time-dependent phenomena in the potential response of ion-selective electrodes treated by the nernst-planckpoisson model. 1. Intramembrane processes and selectivity. Analytical Chemistry, 78, 6783-6791. doi:10.1021/ac060264p [30] Bobacka, J., Ivaska, A. and Lewenstam, A. (2008) Potentiometric ion sensors. Chemical Reviews, 108, 329-351. doi:10.1021/cr068100w [31] Stryer, L. (1975) Biochemistry. Freeman, San Francisco. [32] Reithel, F.J. (1971) Ureases. In: Boyer, P.D. Ed., The Enzymes, 3rd Edition, Academic Press, New York. [33] Morf, W.E., van der Wal, P.D., Pretsch, E. and de Rooji, N.F. (2011) Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 2: Time-dependent concentration profiles, fluxes, and responses. Journal of Electroanalytical Chemistry, 657, 1-12. doi:10.1016/j.jelechem.2011.02.007 [34] Morf, W.E., van der Wal, P.D., Pretsch, E. and de Rooji, N.F. (2011) Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 1: Steady-state concentration profiles, fluxes, and responses. Journal of Electroanalytical Chemistry, 657, 13-22. doi:10.1016/j.jelechem.2011.02.006 [35] Liao, S.J. (2012) The homotopy analysis method in nonlinear differential equations. Springer and Higher Education press, New York. doi:10.1007/978-3-642-25132-0 [36] Liao, S.J. (2003) Beyond perturbation: An introduction to the homotopy analysis method. Chapman & Hall/CRC Press, Boca Raton. doi:10.1201/9780203491164 [37] Allan, F.M. (2007) Derivation of the adomian decomposition method using the homotopy analysis method. Applied Mathematics and Computation, 190, 6-14. doi:10.1016/j.amc.2006.12.074 [38] Liao, S.J. (2003) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai. [39] 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 [40] Loghambal, S. and Rajendran, L. (2011) Analytical expressions of concentration of nitrate pertaining to the electrocatalytic reduction of nitrate ion. Journal of Electro- analytical Chemistry, 661, 137-143. doi:10.1016/j.jelechem.2011.07.027 [41] Liao, S.J. (2009) The proposed homotopy analysis tecnique for the solution of nonlinear problems. Communications in Nonlinear Science and Numerical Simulation, 14, 983-997. doi:10.1016/j.cnsns.2008.04.013 [42] Liao, S.J. and Tan, Y. (2007) A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics, 119, 297-355. doi:10.1111/j.1467-9590.2007.00387.x [43] Liao, S.J. (2003) Beyond perturbation: Introduction to the homotopy analysis method. Chapman and Hall, CRC Press, Boca Raton, 336. [44] 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