Analytical Expression for the Concentration of Substrate and Product in Immobilized Enzyme System in Biofuel / Biosensor

In this paper, an approximate analytical method to solve the non-linear differential equations in an immobilized enzyme film is presented. Analytical expressions for concentrations of substrate and product have been derived for all values of dimensionless parameter. Dimensionless numbers that can be used to study the effects of enzyme loading, enzymatic gel thickness, and oxidation/ reduction kinetics at the electrode in biosensor/biofuel cell performance were identified. Using the dimensionless numbers identified in this paper, and the plots representing the effects of these dimensionless numbers on concentrations and current in biosensor/biofuel cell are discussed. Analytical results are compared with simulation results and satisfactory agreement is noted.


Introduction
Biosensors and biofuel cells are commonly used for industrial, environmental and medical applications.However there are no clear guidelines for the design of electrochemical biosensors or biofuel cells employing immobilized enzymes that will produce a targeted linear range, limit of detection and sensitivity.Such guidelines can be provided using analytical simulation tools that assess sensor feasibility prior to extensive development.
Biosensors and biofuel cell face increasing demand for selective and sensitive detection of different molecules for industrial, environmental and clinical applications [1]- [4].There are many affordable alternatives to labora-tory techniques that require trained personnel, expensive equipment and possibly delayed response time.Electrochemical biosensors and biofuel cell especially desirable for use in field applications because of their compact design, ease of manufacture, real time response, sensitivity and selectivity [3]- [6].They are used in many applications ranging from glucose detection to detection of neurotoxic agents [1] [6] [7].Here we focus on biosensors and biofuel cell that employ immobilized enzymes and the electrochemical detection of the enzymatic reaction.Some important parameters that affect these goals are listed and include transport of the substrate and the product through the immobilized enzyme layer, oxidation/reduction kinetics at the electrode, enzyme activity and loading and operating conditions such as pH and temperature.Of these parameters optimizing the enzyme loading and activity has been a major challenge and it depends primarily on the enzyme immobilization method.Different methods such as chemical modification of the electrode surface, entrapment in a membrane and physical absorption are commonly used to create enzyme layers on electrodes [8].
A mathematical model considering reaction and diffusion processes in biofuel cell or biosensor, contains a system of non-linear partial differential equations.Numerical and analytical solutions to the reaction-diffusion equations have been presented for different cases by many authors [9]- [14].Analytical solutions are available for limiting cases, whereas numerical solutions were used to determine and optimize a wide range of experimental parameters [15].Many of the earlier studies have focused on optimizing glucose biosensors where the enzyme was entrapped in a redox hydrogel [16] [17].Simple Michaelis-Menten kinetics was used to model the enzyme kinetics, and first order kinetics between the mediator and the electrode were assumed [9] [17].The effects of experimental parameters on the response at steady state and during a transient were studied [12].Especially the behavior of the glucose sensor in the diffusion limited regime was analyzed since this leads to an extended linear range [16] [18].Substrate and product inhibition in an enzyme with first order reaction kinetics [19], diffusion through a semi-permeable outer membrane [20] [21] and data analysis to determine kinetic constants and enzyme activity [22] were also studied by different groups.
Sachin [23] used a finite difference method for electrochemical biosensors with an immobilized enzyme layer.Sachin described the general criteria using Michaelis-Menten rate equation and effect of gel thickness on the response of this biosensor.To our knowledge no rigorous analytical solutions for non-steady-state concentration and current have been reported.In this paper, we have derived the analytical expressions of concentration and current using a new approach of Homotopy perturbation method [24]- [27].The result of the Equations ( 2)-(3) in immobilized enzyme system is relevant because its solution describes important applications such as biosensors, bioreactors, and biofuel cells, among others.

Mathematical Formulation of the Problem
The chemical reactions in the layer are where E refers to the enzyme, S is the substrate, ES is a transitory complex assumed to be at a steady concentration, and P is the product.The schematic of the system modeled in this study is shown in Figure 1.An aqueous drop containing substrate (S) is placed on the electrode with an immobilized enzyme layer.As the substrate diffuses through the enzyme layer it reacts with the enzyme to form the product (P).The product then diffuses through the layer, and if it is electroactive, is oxidized or reduced at the electrode.When modeling this system, we used Michaelis-Menten equation to describe the kinetics within the enzyme layer and coupled it with Fick's law to describe the diffusion of the substrate and product as shown in Equations ( 2)-( 3): [ ] where P c , S c , P D and S D represent the concentrations and diffusion coefficients of the product and the substrate, respectively.cat k is the catalytic rate constant in the Michaelis-Menten mechanism, [E] is enzyme loading, and S K is Michaelis constant for the substrate.In the above equations the initial and boundary condi- tions are given by where z is the distance from the electrode surface and L is the enzyme layer thickness.

Sbulk c
represents the concentration of substrate in bulk solution.Current i occurring at the electrode surface due to reduction or oxidation of P is given by Equations ( 2)-(3) were made dimensionless using the following dimensionless parameters: The Equations ( 2)-( 3) in dimensionless form becomes as follows: From the Equation (4), the initial and boundary conditions in dimensionless form are given by Dimensionless current density becomes

General Analytical Expression of Concentration of Substrate and Product under Non-Steady State Condition Using Homotopy Perturbation Method (HPM)
In recent days, HPM is often employed to solve several analytical problems.In addition, several groups demonstrated the efficiency and suitability of the HPM for solving nonlinear equations in electrochemical problems [28]- [31].He et al. [24], used HPM to solve the Lighthill equation, the Duffing equation [25] and the Blasius equation [26].HPM has also been used to solve non-linear boundary value problems [27], integral equation [32]- [34], Klein-Gordon and Sine-Gordon equations [35], Emden-Flower type equations [36] and several other problems.Laplace transform and Homotopy perturbation method are used to solve the non-linear differential Equations ( 7)-( 8) (Appendix A).The analytical expressions of non-steady state concentrations are as follows: where ( ) Using ( 10) and ( 12), the current is given by When * t → ∞ (steady state), the above equation becomes ( )

Discussion
Equations (11) ( 12) and ( 14) are the new and simple analytical expressions of concentrations of substrate, product and current respectively.To show the efficiency of our non-steady-state result, it is compared with numerical solution in Figure 2   decreases.

Estimation of Kinetic Parameters
The current is dependent upon the parameters Thiele module s  15) can be written as ( ) Substituting the value of µ and k in the above equation, we get

Conclusion
The theoretical behavior of biofuel cell/biosensor was analyzed.The coupled time dependent non-steady state non-linear diffusion equations in biosensor or biofuel cell have been solved analytically and numerically.These analytical results will be used in determining the kinetic characteristics of the biofuel cell or biosensor.The analytical expressions for substrate, product concentration and transient current response are obtained using the method of Laplace transformation and HPM.A good agreement with numerical simulation data is noticed.Concentration of substrate, product and current depends upon Thiele modulus s φ and initial concentration of substrate which is discussed in this communication.Evaluation of kinetic parametr from the response of the steady-state current is also completely discussed.The theoretical model presented here can be used for the optimization of the design of the biosensor.The poles are obtained from cosh 0 s s a + =.Hence there is a simple pole at 0 s = and there are infinitely many poles given by the solution of the equation cosh 0 s a + = and so ( )

& Figure 3 .
Satisfactory agreement is noted.The SCILAB/MATLAB program is also given in Appendix B. Figure 2 shows the time-dependent normalized concentration profiles for the substrate s c * in the enzyme membrane.Figures 2(a)-(c) show dimensionless concentration s c * versus the dimensionless

Figure 2 .Figure 3 .
Figure 2. Dimensionless substrate concentration s c * versus distance from electrode surface z * using Equation (11) for vari- ous values of parameters Φ s , t and c s0 .distance * z .The concentration of substrate s c * depends upon the dimensionless parameter "a".The

Figure 4 .as shown in Figure 5 .
Figure 4. Dimensionless current density i/nFD p versus time t using Equation (14) for various values of parameters Φ s , c s0 and r.

Figure 5 .
Figure 5.A plot of tan h −1 (i/nFD p ) −2 versus initial substrate concentration C Sbulk using Equation (17) to estimate the kinetic parameters.