Analysis of a pH‐Based Potentiometric Biosensor Using the Homotopy Perturbation Method

Abstract

A mathematical model of steady-state and non-steady-state responses of a pH-based potentiometric biosensor immobilizing organophosphorus hydrolase was developed. The model is based on non-stationary diffusion equations containing a nonlinear term related to the Michaelis-Menten kinetics of an enzymatic reaction. An analytical expression for the substrate concentration was obtained for all values of parameter a (Thiele modulus) using the homotopy perturbation method. From this result, the concentrations of the deprotonation products of an organophosphodiester (P h H, ZH and AH) were obtained. Our analytical results were compared with available simulation results. A satisfactory agreement with the simulation data is noted. IntroductionBiosensors are possible due to the improvements of the biological components and the implementation of microsystem technologies. Enzymes are still the most appropriate recognition elements because they combine high chemical specificity and inherent biocatalytic signal amplification. Biosensors are classified according to the nature of the physical transducer. The potentiometric biosensor has assumed great importance in both theoretical and applied work. Modeling of biosensors is of crucial importance to understand their behavior. Normally, it is not possible to measure the concentration of substrates inside the enzyme membrane with analytical devices. Thus, mathematical models of biosensors have been developed and used as an important tool to study their analytical characteristics.Enzyme-based electrochemical biosensors are attractive for the determination of organophosphorus pesticides (OP) [1][2][3][4][5]. Recently organophosphorus hydrolase (OPH) has been applied instead of acetylcholineesterase (AChE) or butyrylcholinesterase (BChE) in biosensors for the determination of OP. With the discovery of OPH, an enzyme that can hydrolyze a wide range of organophosphate pesticides releasing detectable products [6,7], several enzyme and microbial biosensors based on OPH for the rapid, simple, and selective monitoring of these neurotoxic pesticides, with the potential in the field analysis, have been reported [8][9][10][11][12]. Many analytical methods, including gas and liquid chromatographic technologies [13], immunoassays [14], and biosensors based on choline esterase or alkaline phosphatase inhibition [15], have been reported for the determination of OP.Theoretical models of OPH biosensors are necessary to optimize the design and predict the behavior of the system. Models of pH-based potentiometric enzyme electrodes immobilizing enzymes other than OPH have been reported [16,17]. Recently, Huanlin and coworkers [1] have developed a numerical model for predicting the steady-state response of a pH-based potentiometric biosensor immobilizing OPH. To our knowledge, no rigorous analytical solution for the concentration of the substrate has been reported. In this paper, the theoretical model of a pH-based electrode immobilizing OPH was developed. The purpose of this paper was to derive an ana...

A Theoretical Model of pH-Based Potentiometric Biosensor Based on Immobilized Enzyme Membrane

“…The complete description of the problem is given in [4] [10]. For the sake of completeness the brief description is given in this section and Appendix-A.…”

“…In the enzyme membrane, the reaction-diffusion equations for the concentration of species for non-steady state condition can be represented as follows [4] [10]. ,…”

Mathematical Modelling and Kinetics of Microchannel Reactor

“…Initially methods like Pade approximation method [6] and variational iteration method [7][8] have been widely used to study the nonlinear problems, later on Liao employed the fundamental ideas of homotopy in topology to propose a general analytic method for solving differential and integral equations, linear and non-linear, namely homotopy analysis method (HAM) [9]. A special case of HAM is homotopy perturbation method (HPM) [10] which was introduced by He, and Elçin Yusufoglu extended it to improved homotopy perturbation method [11], latter Rajendran introduced a new approach to homotopy perturbation method [12]. Adomian decomposition method [13] and modified Adomian decomposition method [14] are frequently used for solving non-linear problems till now.…”

“…In this appendix, we derive the general solution of non-linear reaction Eq. (9), (10), (11), (12) using Homotopy analysis method. To find the solution of Eq.…”

Hydrogen Production by a Photosynthetic Bacterium: Some Analytical Solutions

“…Solving systems of nonlinear differential equations has gained importance and popularity in recent years, mainly due to the necessity of analytical solutions for the large scale of nonlinear differential systems in diverse fields of science and engineering. Many authors have focused on studying the solutions of nonlinear ordinary differential equations by using various analytical methods like the homotopy perturbation method 21, the homotopy analysis method 22, the variational iteration method 23, the Laplace Adomian decomposition method 24, a new approach to the homotopy perturbation method 25, and others. Among these, the new approach to the homotopy perturbation method is employed to solve the nonlinear ordinary differential equations (1)–(3).…”