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

A theoretical model for the non steady-state response of a pH-based potentiometric biosensor immobilizing organophosphorus hydrolase (OPH) is discussed. The model is based on a system of five coupled nonlinear reaction-diffusion equations under non steady-state conditions for enzyme reactions occurring in potentiometric biosensor that describes the concentration of substrate and hydrolysis products within the membrane. New approximate analytical expressions for the concentration of the substrate (organophosphorus pesticides (OPs)) and products are derived for all values of Thiele modulus and buffer concentration using new approach of homotopy perturbation method. The analytical results are also compared with numerical ones and a good agreement is obtained. The obtained results are valid for the whole solution domain.


Introduction
A potentiometric biosensor is a type of chemical sensor that may be used to find the concentration of some components of the analyte.These sensors measure the electrical potential of an electrode when no voltage is present.The potentiometric biosensors have been widely used in environmental, medical and industrial applica-tions [1].Also potentiometric biosensor can be used for detection of all OPs but they don't have low enough limits of detection [2].
The theoretical modeling of biosensors involves solving the system of linear/non-linear reaction-diffusion equations for substrate and product with a term containing a rate of biocatalytical transformation of substrate.The complications of modeling arise due to solving the partially differential equations with non-linear reaction term and with complex initial and boundary conditions.The modeling of biosensor is analyzed by numerical [1] and analytical method [3] of partial differential equation with various boundary conditions.Recently Meena and Rajendran (2010) discussed a theoretical model of a pH-based potentiometric biosensor immobilizing organophosphorus hydrolase (OPH) for steady state conditions [4].
Rahamathunissa and Rajendran (2008) implemented He's variational iteration method in nonlinear boundaryvalue problems in enzyme substrate reaction diffusion processes in amperometric biosensor [5].Manimozhi et al. [6] presented the solution of steady-state substrate concentration in the action of biosensor response with mixed enzyme kinetics under a Michalis-Menten scheme.Analytical solutions for the steady-state current at a microdisk chemical sensor have been reported by Dong and Che [7] and by Lyons et al. [7] [8].Recently, Eswari and Rajendran [9] derived the concentration profile of the product of the enzyme reaction and the electrode current for all values of Michalis-Menten constant using the Homotopy perturbation method.
To our knowledge, no general analytical expressions of the concentrations of the substrate, hydrolysis products, added external buffer and hydrogen ions have been reported for all values of parameters.The purpose of this communication is to derive an analytical expression of non-steady state concentrations of OPs and the deprotonation products for all values of reaction parameter using new homotopy perturbation method.

Mathematical Formulation of the Problem
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.A schematic diagram of the pH-based potentiometric biosensor immobilizing organophosphorus hydrolase (OPH) is represented in Figure 1.
In this figure, S denotes the substrate of organophosphorus pesticides (OPs).h P H and ZH are represent the hydrolysis products of organophosphodiester and alcohol respectively.AH is the added external buffer and h P − , Z − , A − , H + are the deprotonation products.The general scheme that represents an enzyme-cata-lyzed reaction within enzyme membrane can be written as follows: .
The non-linear reaction-diffusion equations for non-steady state condition can be described as follows ( ) where i C is the concentration of species, i D is the diffusion coefficient and is the reaction rate.The reaction rate is a non-linear function of concentration of substrate.The reaction rate is non-linear with respect to substrate because of product inhibition, saturation of the enzyme with substrate, reverse reaction and enzyme loading.The nomenclature is also presented in Table 1.

Dimensionless Form
The dimensionless reaction-diffusion equations for non-steady state condition can be written as follows (Appendix A): where [ ] A denotes the sum of dissociated and undissociated concentrations of the species h P H , ZH , and AH respectively and T H +     is the concentration of hydrogen ions.Here the Thiele modulus a, which represents the ratio of the characteristic time of the enzymatic reaction to that of the substrate diffusion is The initial and boundary conditions for the above equations becomes A graphical representation of the boundary conditions of this system conditions can be seen in Figure 2.

Analytical Expression of Concentration of Substrate and Products Using New Homotopy Perturbation Method (New HPM) and Laplace Transform Technique
With the rapid development of nonlinear science, there appears an ever-increasing interest of scientists and engineers in the approximate analytical asymptotic techniques for nonlinear problems [11].It is very difficult to solve nonlinear problems either numerically or theoretically.Perturbation methods provide the most versatile tools available in nonlinear analysis of engineering problems, and they are constantly being developed and applied to ever more complex problems.Homotopy perturbation method was first proposed by the He [12].Recently, a new approach to HPM is presented to solve the nonlinear problem and this gives a simple approximate solution in the zeroth iteration [13].By using this new homotopy perturbation method and Laplace transform technique (Appendix B), the concentrations of substrate and products can be obtained as follows:  , 1

Results and Discussion
Equations ( 15  3) Influence of added buffer concentration on the concentration of species.The influence of added buffer concentration on the concentration of the substrate for some values of other parameters is shown in Figures 5(a)-(e).From this figure, it is inferred that the concentration of the substrate increases when added buffer concentration increases.Solid lines represent the Equation (15) and the dotted lines represent the numerical simulation.Satisfactory agreement is noted.The MATLAB program also given in Appendix B.       [ ] error deviation increases, when the reaction diffusion parameter a increases.Similarly in Table 3, average percentage of error deviation decreases, when the time τ increases.In Table 2, the maximum average relative er- ror between the analytical results and numerical results is 0.53%.

Conclusion
A non-linear time dependent system of differential equation in pH-based potentiometric biosensor has been solved using the new HPM.New approximate analytical expressions for the concentrations of the substrate and hydrolysis products are derived.The time dependent substrate concentration profiles are also presented using SCILAB program.Concentration of substrate and product depends upon Thiele modulus and initial concentration of substrate which is discussed in this communication.

Appendix A. The Dimensionless Reaction-Diffusion Equations
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].

] [ ]
we obtain the dimensionless form of Equations ( 6)- (10) for the concentration of species which are given in the text.
The approximate solution of Equation The initial and boundary conditions for Equations ( 12)-( 14 where α is defined as in Equation (20).Now, by applying Laplace transform and complex inversion formula where the integration in Equation (C.1) is to be performed along a line s c = in the complex plane where s x iy = + .The real number c is chosen such that s c = lies to the right of all the singularities but is otherwise assumed to be arbitrary.In practice, the integral is evaluated by considering the contour integral presented on the right-hand side of Equation (C.1), which is then evaluated using the so-called Bromwich contour.The contour integral is then evaluated using the residue theorem which states, for any analytic function The second residue in Equation (B.17) is given by ( where n f is defined as in Equation (20).Here, we used

Figure 2 .
Figure 2. Boundary conditions employed in the pH-based potentiometric biosensor for the substrate [ ] S , products

Figure 3 .
Figure 3. (a)-(f) Plot of dimensionless non-steady state concentration profiles of the substrate [ ]S

Figure 4 .
Figure 4. (a)-(f) Plot of dimensionless non-steady state concentration profiles of the substrate [ ]S

Figure 5 .
Figure 5. (a)-(e) Plot of dimensionless non-steady state concentration profiles of the substrate [ ] S versus di- mensionless distance x for fixed values of a and time τ and various values of the parameters [ ] b S .Solid

Figures 6 (
Figures 6(a)-(h) show the dimensionless non-steady state concentration profiles of products

Figure 7 (
Figure 7(a) & Figure 7(b) show the dimensionless non-steady state concentration of substrate [ ] S and products T H +     , [ ] h T P

Figure 7 .
Figure 7. (a)-(b) Plot of dimensionless non-steady state concentration profiles of substrate [ ] S and products

Table 2 .
Comparison of analytical expression of concentration of the substrate [ ] S (Equation (15)) with the numerical re- sult for various values of parameter a.

( 10 )
Appendix C) to Equation (B.9) and to the conditions in Equations (B.6)-(B.8),we obtained the solution of Equation (Using residue theorem (Appendix C) we can obtain the Equation (15) in the text.

3 )
are computed at the poles of the function ( ) F z .Hence, from Equation (C.2), we note that From the theory of complex variables, we can show that the residue of a function order to invert Equation (B.10), we need to evaluate , there is a simple pole at 0 s = and there are infinitely many poles given by the solution of the equation

Table 3 .
Comparison of analytical expression of concentration of the substrate [ ] 1, 10, 1S (Equation (15)) with the numerical re- sult for various values of parameter τ .