The theoretical model for an amperometric glucose biosensor is discussed. In this model glucose oxidase enzyme is immobilized in conducting polypyrrole. This model contains a nonlinear term related to enzyme reaction kinetics. He’s homotopy perturbation method is used to find the approximate analytical solutions of coupled non-linear reaction diffusion equations. A closed-form expression of substrate and mediator concentration under non-steady-state conditions is obtained. A comparison of the analytical approximation and numerical simulation is also presented. An agreement between analytical expressions and numerical results is observed. 1. Introduction Since the second half of the last century, numerous efforts have been devoted to the development of insoluble immobilized enzymes for a variety of applications [1]. These applications can clearly benefit from use of the immobilized enzymes rather than the soluble counterparts, for instance as reusable heterogeneous biocatalysts, with the aim of reducing production costs by efficient recycling and control of the process [2], as stable and reusable devices for analytical and medical applications [3–9], as selective adsorbents for purification of proteins and enzymes [10], as fundamental tools for solid-phase protein chemistry [11, 12], and as effective microdevices for controlled release of protein drugs [13]. Immobilized enzymes are becoming increasingly popular as reusable, selective analytical chemical reagents in solid-phase flow-through reactors, as membranes in sensors, and as films in dry reagent kits. The attractions of immobilized enzymes from an analytical standpoint are primarily their reusability, and hence cost saving, and the greater efficiency and control of their catalytic activity [14] (e.g., potentially longer half-lives, predictable decay rates and more efficient multistep reactions). The immobilization of enzymes in conducting polymer [15] during electro-polymerization step has proved to be well suited to the preparation of biosensors [16–19]. This method is simple and easy to control. Another important advantage of this immobilization technique is the possibility of entrapping the mediator in the polymer as a dopant anion [20–23] or by covalent fixation on the pyrrole monomer [24]. Bartlett and Whitaker [25] have already presented a theoretical model for an amperometric polypyrrole + glucose oxidase (PPY + GOD) electrode. In their model PPY was considered as an insulating polymer, and the reduced mediator H2O2 was oxidized at the metallic surface after diffusion in the polymer.
References
[1]
H. Silman and E. Katchalski, “Waterinsoluble derivatives of enzymes, antigens and antibodies,” Annual Review of Biochemistry, vol. 35, pp. 837–877, 1966.
[2]
E. J. Vandamme, “Peptide antibiotic production through immobilized biocatalyst technology,” Enzyme and Microbial Technology, vol. 5, no. 6, pp. 403–416, 1983.
[3]
B. Schulze and M. G. Wubbolts, “Biocatalysis for industrial production of fine chemicals,” Current Opinion in Biotechnology, vol. 10, pp. 609–615, 1999.
[4]
L. C. Clark and C. Lyons, “Electrode systems for continuous monitoring in cardiovascular surgery,” Annals of the New York Academy of Sciences, vol. 102, pp. 29–45, 1962.
[5]
D. H. Campbell, F. L. Luescher, and L. S. Lerman, “Immologic adsorbents. I. Isolation of antibody by means of a cellulose-protein antigen,” Proceedings of the National Academy of Sciences, vol. 37, pp. 575–578, 1951.
[6]
S. Watanabe, Y. Shimizu, T. Teramatsu, T. Murachi, and T. Hino, “Application of immobilized enzymes for biomaterials used in surgery,” Methods in Enzymology, vol. 137, pp. 545–551, 1988.
[7]
T. M. S. Chang, Medical Application of Immobilised Enzymes and Proteins, vol. 1-2, Plenum Press, New York, NY, USA, 1977.
[8]
M. D. Klein and R. Langer, “Immobilised enzymes in clinical medicines: an emerging approaches to new drug therapies,” Trends in Biotechnology, vol. 4, pp. 179–186, 1986.
[9]
L. J. Kircka and G. H. G. Thorpe, “Immobilised enzymes in analysis,” Trends in Biotechnology, vol. 4, pp. 253–258, 1986.
[10]
B. R. Dunlap, Immobilised Chemicals and Affinity Chromatography, Plenum Press, New York, NY, USA, 1974.
[11]
G. F. Bickerstaff, “Application of immobilised enzymes to fundamental studies on enzyme structure and function,” in Topics in Enzyme and Fermentation Biotechnology, A. Wiseman, Ed., pp. 162–201, Ellis Horwood, Chichester, UK, 1984.
[12]
K. Martinek and V. V. Mozhaev, “Immobilization of enzymes: an approach to fundamental studies in biochemistry,” Advances in Enzymology and Related Areas of Molecular Biology, vol. 57, pp. 179–249, 1985.
[13]
C. Cristallini, L. Lazzeri, M. G. Cascone, G. Polacco, D. Lupinacci, and N. Barbani, “Enzyme-based bioartificial polymeric materials: the α-amylase-poly(vinyl alcohol) system,” Polymer International, vol. 44, no. 4, pp. 510–516, 1997.
[14]
P. J. Worsfold, “Classification and chemical characteristics of immobilized enzymes,” Pure and Applied Chemistry, vol. 67, no. 4, pp. 597–600, 1995.
[15]
N. C. Foulds and C. R. Lowe, “Enzyme entrapment in electrically conducting polymers. Immobilisation of glucose oxidase in polypyrrole and its application in amperometric glucose sensors,” Journal of the Chemical Society, Faraday Transactions 1, vol. 82, no. 4, pp. 1259–1264, 1986.
[16]
P. N. Bartlett and R. G. Whitaker, “Electrochemical immobilisation of enzymes. Part II. Glucose oxidase immobilised in poly-N-methylpyrrole,” Journal of Electroanalytical Chemistry, vol. 224, no. 1-2, pp. 37–48, 1987.
[17]
D. Bélanger, J. Nadreau, and G. Fortier, “Electrochemistry of the polypyrrole glucose oxidase electrode,” Journal of Electroanalytical Chemistry, vol. 274, no. 1-2, pp. 143–155, 1989.
[18]
P. Janda and J. Weber, “Quinone-mediated glucose oxidase electrode with the enzyme immobilized in polypyrrole,” Journal of Electroanalytical Chemistry, vol. 300, no. 1-2, pp. 119–127, 1991.
[19]
M. Marchesiello and E. M. Geniés, “Glucose sensor: polypyrrole-glucose oxidase elecrtode in the presence of p-benzoquinone,” Electrochimica Acta, vol. 37, no. 11, pp. 1987–1992, 1992.
[20]
Y. Kajiya, H. Sugai, C. Iwakura, and H. Yoneyama, “Glucose sensitivity of polypyrrole films containing immobilized glucose oxidase and hydroquinonesulfonate lons,” Analytical Chemistry, vol. 63, no. 1, pp. 49–54, 1991.
[21]
P. N. Bartlett, Z. Ah, and V. E. Field, “Electrochemical immobilisation of enzymes,” Journal of the Chemical Society, Faraday Transactions, vol. 88, no. 18, p. 2677, 1992.
[22]
Y. Kajiya, R. Tsuda, and H. Yoneyama, “Conferment of cholesterol sensitivity on polypyrrole films by immobilization of cholesterol oxidase and ferrocenecarboxylate ions,” Journal of Electroanalytical Chemistry, vol. 301, no. 1-2, pp. 155–164, 1991.
[23]
Y. Kajiya, H. Matsumoto, and H. Yoneyama, “Glucose sensitivity of poly(pyrrole) films containing immobilized glucose dehydrogenase, nicotinamide adenine dinucleotide, and β-naphthoquinonesulphonate ions,” Journal of Electroanalytical Chemistry, vol. 319, no. 1-2, pp. 185–194, 1991.
[24]
N. C. Foulds and C. R. Lowe, “Immobilization of glucose oxidase in ferrocene-modified pyrrole polymers,” Analytical Chemistry, vol. 60, no. 22, pp. 2473–2478, 1988.
[25]
P. N. Bartlett and R. G. Whitaker, “Electrochemical immobilisation of enzymes: part II. Glucose oxidase immobilised in poly-N-methylpyrrole,” Journal of Electroanalytical Chemistry, vol. 224, no. 1-2, pp. 37–48, 1987.
[26]
M. Marchesiello and E. Geniès, “A theoretical model for an amperometric glucose sensor using polypyrrole as the immobilization matrix,” Journal of Electroanalytical Chemistry, vol. 358, no. 1-2, pp. 35–48, 1993.
[27]
C. Bourdillon, C. Hervagault, and D. Thomas, “Increase in operational stability of immobilized glucose oxidase by the use of an artificial cosubstrate,” Biotechnology and Bioengineering, vol. 27, no. 11, pp. 1619–1622, 1985.
[28]
H. Naarmann, W.R. Salaneck, D.T. Clark, and E. J. Samuelsen, “Science and applications of conducting polymers,” in Proceedings of the 6th Europhysics Industrial Workshop, Adam Hiler, Lofthus, Norway, May 1990.
[29]
E. M. Geniès, M. Marchesiello, and G. Bidan, “Preparation and properties of polypyrrole made in the presence of biological buffers,” Electrochimica Acta, vol. 37, no. 6, pp. 1015–1020, 1992.
[30]
Q. K. Ghori, M. Ahmed, and A. M. Siddiqui, “Application of homotopy perturbation method to squeezing flow of a newtonian fluid,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 179–184, 2007.
[31]
T. ？zi？ and A. Yildirim, “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, vol. 8, no. 2, pp. 243–248, 2007.
[32]
S. J. Li and Y. X. Liu, “An improved approach to nonlinear dynamical system identification using PID neural networks,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 2, pp. 177–182, 2006.
[33]
M. M. Mousa and S. F. Ragab, “Application of the homotopy perturbation method to linear and nonlinear schr？dinger equations,” Z. Naturforsch, vol. 63a, pp. 140–144, 2008.
[34]
J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
[35]
J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
[36]
J. H. He, “A simple perturbation approach to Blasius equation,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 217–222, 2003.
[37]
J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
[38]
M. Ghasemi, M. Tavassoli kajani, and E. Babolian, “Numerical solutions of the nonlinear integro-differential equations: wavelet-Galerkin method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 450–455, 2007.
[39]
Z. Odibat and S. Momani, “A reliable treatment of homotopy perturbation method for Klein-Gordon equations,” Physics Letters A, vol. 365, no. 5-6, pp. 351–357, 2007.
