New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analysis Method

Seyyed Ali Madani Tonekaboni, Ali Shahbazi Mastan Abad, Shahab Karimi, Mitra Shabani

School of Mechanic, University of Waterloo, Ontario, Canada.
School of Mechanics, University of Tehran, Tehran, Iran.
School of Mechanics, University of Tehran, Tehran, Iran..

DOI: 10.4236/wjet.2013.13005                

Abstract

In this article, Discrete Homotopy Analysis Method (DHAM), as a new numerical method, is employed to investigate amperometric biosensor at mixed enzyme kinetics and diffusion limitation. Mathematical modeling of the problem is developed utilizing non-Michaelis-Menten kinetics of the enzymatic reaction. Different results are obtained for different values of the dimensionless parameters described in the paper. The presented solution is then compared with the available actual and simulated results.

Keywords

Discrete Homotopy Analysis Method; Amperometric Biosensor; Mathematical Modeling; Non-Michaelis-Menten Kinetics

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	F. Scheller and F. Schubert, “Biosensors,” Vol. 7, Elsevier, Amsterdam, 1988.
[2]	U. Wollenberger, F. Lisdat and F. W. Scheller, “Enzymatic Substrate Recycling Electrodes. Frontiers in Biosensorics. B and II, Practical Applications,” Birkhauser Verlag, Basel, 1997, pp. 45-70.
[3]	A. Baeumner, Analytical and Bioanalytical Chemistry, Vol. 377, 2003, pp. 434-445.
[4]	M. Pohanka, P. Skladal and M. Kroca, “Biosensors for Biological Warfare Agent Detection,” Defense Science Journal, Vol. 57, No. 3, 2007, pp. 185-93.
[5]	M. Pohanka, D. Jun and K. Kuca, “Mycotoxin Assay Using Biosensor Technology: A Review,” Drug and Chemical Toxicology, Vol. 30, No. 3, 2007, pp. 253-261. http://dx.doi.org/10.1080/01480540701375232
[6]	S. Haron and A. K. Ray, “Optical Biodetection of Cadmium and Lead Ions in Water,” Medical Engineering and Physics, Vol. 28, 2006, No. 10, pp. 978-981.
[7]	A. J. Baeumner, C. Jones, C.Y. Wong and A. Price, “A Generic Sandwich-Type Biosensor with Nanomolar Detection Limits,” Analytical and Bioanalytical Chemistry, Vol. 378, No. 6, 2004, pp. 1587-1593. http://dx.doi.org/10.1007/s00216-003-2466-0
[8]	T. Schulmeister, “Mathematical Modeling of the Dynamic Behavior of Ampero-Metric Enzyme Electrodes,” Selective Electrode Reviews, Vol. 12, 1990, pp. 203-260.
[9]	R. Aris, “The Mathematical Theory of Diffusion and Reaction in Permeable Cat-alysts. The Theory of the Steady State,” Clarendon Press, Oxford, 1975.
[10]	L. K. Bieniasz and D. Britz, “Recent Developments in Digital Simulation of Electroan-Alytical Experiments,” Polish Journal of Chemistry, Vol. 78, 2004, pp. 1195-1219.
[11]	J. D. Cole, “Perturbation Methods in Applied Mathematics,” Blaisdel, Waltham, 1968.
[12]	A. V. Karmishin, A. I. Zhukov and V. G. Kolosov, “Methods of Dynamics Calculation and Testing for Thin-Walled Structures,” Mashinostroyenie, Moscow, 1990.
[13]	G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994. http://dx.doi.org/10.1007/978-94-015-8289-6
[14]	G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, 1998, pp. 501-544. http://dx.doi.org/10.1016/0022-247X(88)90170-9
[15]	J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. http://dx.doi.org/10.1016/S0045-7825(99)00018-3
[16]	J. H. He, “An Approximate Solution Technique depEnding upon an Artificial Parameter,” Communications in Nonlinear Science and Numerical Simulation, Vol. 3, No. 2, 1998, pp. 92-97. http://dx.doi.org/10.1016/S1007-5704(98)90070-3
[17]	S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” PhD Thesis, Shanghai Jiao Tong University, 1992.
[18]	S. Abbasbandy, “The Application of Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letters A, Vol. 360, No. 1, 2006, pp. 109-113. http://dx.doi.org/10.1016/j.physleta.2006.07.065
[19]	S. J. Liao, “Beyond Perturbation: Introduction to Homotopy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2003. http://dx.doi.org/10.1201/9780203491164
[20]	J.-H. He, “Comparison of Homotopy Perturbation Method and Homotopy Analysis Method,” Applied Mathematics and Computation, Vol. 156, No. 2, 2004, pp. 527-539. http://dx.doi.org/10.1016/j.amc.2003.08.008
[21]	S. J. Liao, “A Short Review on the Homotopy Analysis Method in Fluid Mechanics,” Journal of Hydrodynamics, Ser. B, Vol. 22, No. 5, 2010, pp. 882-884.
[22]	S. J. Liao,” On the Homotopy Analysis Method for Nonlinear Problems”, Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 499-513. http://dx.doi.org/10.1016/S0096-3003(02)00790-7
[23]	A. Bratsos, M. Ehrhardt and I. T. Famelis, “A Discrete Adomian Decomposition Method for Discrete Nonlinear Schrodinger Equations,” Applied Mathematics and Computation, Vol. 197, No. 1, 2008, pp. 190-205. http://dx.doi.org/10.1016/j.amc.2007.07.055
[24]	H. Q. Zhu, H. Z. Shu and M. Y. Ding, “Numerical Solutions of Partial Differential Equations by Discrete Homotopy Analysis Method,” Applied Mathematics and Computation, Vol. 216, No. 12, 2010, pp. 3592-3605. http://dx.doi.org/10.1016/j.amc.2010.05.005
[25]	M. Daniel, “Dunlavy Ph.D. Candidacy Prospectus, A Homotopy Method for Predicting the Lowest Energy Conformations of Proteins,” April 18, 2003.
[26]	P. Manimozhia, A. Subbiahb and L. Rajendrana, “Solution of Steady-State Substrate Concentration in the Action of Biosensor Response at Mixed Enzyme Kinetics,” Sensors and Actuators B, Vol. 147, No. 1, 2010, pp. 290-297. http://dx.doi.org/10.1016/j.snb.2010.03.008
[27]	C. V. Pao, “Mathematical Analysis of Enzyme-Substrate Reaction Diffusion Insomebiochemical Systems,” Nonlinear Analysis: Theory, Vol. 4, No. 2, 1979, pp. 369-392. http://dx.doi.org/10.1016/0362-546X(80)90061-9