Analytical Expressions for Steady-State Concentrations of Substrate and Oxidized and Reduced Mediator in an Amperometric Biosensor

Abstract

A mathematical model of modified enzyme-membrane electrode for steady-state condition is discussed. This model contains a nonlinear term related to enzyme kinetics reaction mechanism. The thickness dependence of an amperometric biosensor is presented both analytically and numerically where the biological layer is immobilized between a solid substrate and permeable electrode. The analytical expressions pertaining to the concentration of species and normalized current are obtained using the Adomian decomposition method (ADM). Simple and approximate polynomial expressions of concentrations of an oxidized mediator, substrate, and reduced mediator are derived for all possible values of parameters 𝜙²_O (Thiele modulus), 𝐵_O (normalized surface concentration of oxidized mediator), and 𝐵_S (normalized surface concentration of substrate). A comparison of the analytical approximation and numerical simulation is also presented. A good agreement between theoretical predictions and numerical results is observed.

1. Introduction

In recent years, polymer membranes are widely used as carriers for immobilization of enzymes [1]. They have been utilized in biomaterials, bioseparators, and biosensors [2]. The membranes provide an ideal support for the immobilization of the biocatalyst. Substrate partition at the membrane/fluid interphase can be used to improve the selectivity of the catalytic reaction towards the desired products [3]. Recently a new method for enzyme immobilization [4], based on a molecular recognition process, has been successfully used for the building of enzymatic biosensors and also of a chemically active membrane [5]. In the recent three decades, much effort has been devoted to the development of various biosensors involving biologically sensitive component and transformers—devices with many fields of applications [6]. Changes in membrane chemistry have been demonstrated by Robeson [7]. The better way of changing the membrane geometry aims to increase the membrane area per volume, thereby speeding the separation. This increased surface area has recently been identified as a high priority research need for membranes [8].

A two-substrate model for enzyme electrode has been devised experimentally [10, 11] where the nonlinear enzyme reaction was taken into account. This model was employed to describe the behaviour of a glucose oxidize (GOx) electrode [12, 13]. It has been found that the mediators could not totally replace the natural cosubstrate when both were present in the assay solution so that here a three-substrate model would be required. In these cases, a complex calibration curve of the enzyme electrode was observed [14, 15].

Inspite of extensive experimental investigations for the design of bio-sensor, only a few studies concerned the modeling or theoretical design of such system. Recently Loghambal and Rajendran [16] have described the theoretical model in an amperometric oxidase enzyme-membrane electrode. Numerical solutions [9, 17, 18] were reported to a novel enzyme electrode. In those papers the enzyme electrode was modeled numerically using shooting method [9, 18] and Runge-Kutta method [17]. But, in this paper, the same system is modeled analytically. However, to the best of our knowledge, till date no general analytical results for the concentration of oxidized mediator, substrate, and reduced mediator for all values of the parameters 𝜙²_O, 𝐵_O, and 𝐵_S have been reported [9]. The purpose of this communication is to derive the closed-form of analytical expressions [19, 20] of concentrations of substrate and oxidized and reduced mediator by solving the system of nonlinear reaction-diffusion equations using the Adomian decomposition method (ADM) [21]. The theoretical models of enzyme electrodes give information about the mechanism and kinetics operating in the biosensor. Thus, the information gained from this modeling can be useful in sensor design, optimization, and prediction of the electrode’s response.

2. Mathematical Formulation and Analysis of the Problems

2.1. Mathematical Formulation

Building upon earlier work for these mechanisms, Gooding and Hall [9] presented a concise discussion and derivation of the dimensionless nonlinear mass transport equation for this model, which is summarized briefly for completeness. In the enzyme-membrane geometry, the biological layer is located between a solid substrate and an outer permeable electrode in contact with the sample. In this model the substrate and cosubstrate penetrate through a permeable electrode to the enzyme layer and then reduce to the form of cosubstrate which diffuses back to the electrode. The general reaction scheme for an immobilized oxidase in the presence of two oxidants can be written as follows [9]: E_OX+S⇄^𝑘1_𝑘−1ES−−→^𝑘2E_redE+P_red+O−−→₂^𝑘3E_OX+H₂O₂E_red+Med−−→_OX^𝑘4E_OX+Med_red (1)(2)(3) where 𝑘_𝑚 is the rate constant for the forward direction of the 𝑚th reaction and 𝑘₋₁ is the rate constant for the backward direction. If [E_𝑇] is the total enzyme concentration in the matrix, then at all times E_𝑇=E_OX+[ES]+E_red, (4) where [E_OX], [ES], and [E_red] are the oxidized mediator, enzyme-substrate complex, and reduced mediator enzyme concentrations, respectively. At steady state, the diffusion of a substrate into the enzyme layer is equal to the reaction rate of the substrate within the matrix. We examine a planar matrix of thickness 𝑦=𝑑, where diffusion is considered in the 𝑦-direction only (edge effects are neglected) (Figure 1). The corresponding governing equations in Cartesian coordinates, for the planar diffusion and reaction in the enzyme electrode, are [9]𝐷_M𝑑²Med_OX𝑑𝑦²=𝑘₄E_redMed_OX=E_OXS]𝑘[²𝑘₁𝑘₋₁+𝑘₂=𝑘₂E_𝑇𝛽_S[S]+𝛽_OMed_OX+1⁻¹,𝐷_S𝑑²[S]𝑑𝑦²=𝑘₁E_OX[S]−𝑘₋₁[]=EES_OX[S]𝑘₁−𝑘₋₁𝑘₁𝑘₋₁+𝑘₂=𝑘₂E_𝑇𝛽_S[S]+𝛽_OMed_OX+1⁻¹,𝐷_M𝑑²Med_red𝑑𝑦²=−𝑘₄E_redMed_OX=−𝑘₂E_𝑇𝛽_S[S]+𝛽_OMed_OX+1⁻¹, (5a)(5b)(5c)where 𝐷_M is the diffusion coefficient of the oxidized and reduced forms of the mediator (assumed to be equal) and 𝐷_S is the diffusion coefficient of substrate within the enzyme layer. [Med_OX], [Med_red], and [S] are the concentration of oxidized mediator, reduced mediator, and substrate at any position in the enzyme layer. 𝛽_S and 𝛽_O are the dimensionless rate constants (𝛽_S=(𝑘₋₁+𝑘₂)/𝑘₁ and 𝛽_O=𝑘₂/𝑘₄). Equations (5a)–(5c) are solved for the following boundary conditions.

Figure 1 

Schematic representation of a typical enzyme-membrane electrode geometry [9].

At the far wall, 𝑦=0𝑑Med_OX=𝑑[S]𝑑𝑦=𝑑𝑑𝑦Med_red𝑑𝑦=0 (6) and, at the electrode, 𝑦=𝑑Med_OX=Med_OXb=𝐾_OMed_OX∞,[S]=[S]_b=𝐾_S[S]_∞,Med_red=0. (7) The first boundary condition states that there is no flux of reactants and products at the far wall of the sensor. The second condition states that the substrates in the matrix are in equilibrium with the surrounding fluid at the surface of the electrode and all the hydrogen peroxide which reaches the electrode surface is oxidized. [Med_OX]_b and [S]_b are the concentration of oxidized mediator and substrate at the enzyme layer|electrode boundary, and [Med_OX]_∞ and [S]_∞ are the bulk solution concentrations. 𝐾_O and 𝐾_S are the equilibrium partition coefficients for oxidized mediator and the substrate, respectively.

2.2. Normalized Form

We make the nonlinear differential Equations (5a)–(5c) to dimensionless form by defining the following dimensionless variables: 𝐹_O=Med_OXMed_OXb,𝐹_S=[S][S]_b,𝐹_R=Med_redMed_redb𝑦,𝜒=𝑑,𝐵_O=Med_OXb𝛽_O,𝐵_S=[S]_b𝛽_S,𝜙²_O=𝑑²𝑘₂E_𝑇𝐷_MMed_OXb,𝜇_S=𝐷_MMed_OXb𝐷_S[S]_b, (8) where 𝐹_O, 𝐹_S, and 𝐹_R are the normalized concentration of oxidized mediator, substrate, and reduced mediator in the matrix, respectively, and 𝜒 is the normalized distance. 𝐵_O and 𝐵_S are the normalized surface concentration of oxidized mediator and substrate. 𝜙²_O is the Thiele modulus for the oxidized mediator which governs reaction/diffusion.

The resultant expressions for the oxidized mediator, substrate, and reduced mediator in nondimensionalized form become as follows:𝑑²𝐹_O𝑑𝜒²=𝜙²_O𝐵_O𝐵_S𝐹_O𝐹_S𝐵_O𝐹_O+𝐵_S𝐹_S+𝐵_O𝐵_S𝐹_O𝐹_S,𝑑²𝐹_S𝑑𝜒²=𝜇_S𝜙²_O𝐵_O𝐵_S𝐹_O𝐹_S𝐵_O𝐹_O+𝐵_S𝐹_S+𝐵_O𝐵_S𝐹_O𝐹_S,𝑑²𝐹_R𝑑𝜒²=−𝜙²_O𝐵_O𝐵_S𝐹_O𝐹_S𝐵_O𝐹_O+𝐵_S𝐹_S+𝐵_O𝐵_S𝐹_O𝐹_S. (9a)(9b)(9c) The consumption of oxidized mediator, the consumption of substrate, and the consumption of reduced mediator are all related processes. So there is only one independent variable for which to solve 𝑑²𝐹_O𝑑𝜒²=1𝜇_S𝑑²𝐹_S𝑑𝜒²𝑑=−²𝐹_R𝑑𝜒²=𝜙²_O𝐵_O𝐵_S𝐹_O𝐹_S𝐵_O𝐹_O+𝐵_S𝐹_S+𝐵_O𝐵_S𝐹_O𝐹_S. (9d)The normalized boundary conditions are given by 𝐹^′_O(0)=0,𝐹^′_S(0)=0,𝐹^′_R(0)𝐹=0,_O(1)=1,𝐹_S(1)=1,𝐹_R(1)=0. (10) From (9d), we get 𝑑²𝐹_O𝑑𝜒²=1𝜇_S𝑑²𝐹_S𝑑𝜒²,𝑑²𝐹_O𝑑𝜒²𝑑=−²𝐹_R𝑑𝜒². (11)(12) Integrating (11) and (12) twice and applying the appropriate boundary conditions (10), we get 𝐹_S𝜒=𝜇_S𝐹_O𝜒𝐹−1+1,_R𝜒=1−𝐹_O𝜒. (13)(14) Substituting (13) into (9a) and rearranging, we get 𝑑²𝐹_O𝑑𝑋²=𝜙²𝐹²_O+𝐹_O1−𝜇_S𝜇_S×𝐹²_O+𝐹_O1𝐵_S𝜇_S+1𝐵_O+1−𝜇_S𝜇_S+1−𝜇_S𝜇_S𝐵_O⁻¹. (15) The normalized current response is given by the following expression: 𝐼=−𝑑𝐹_R𝑑𝜒_𝜒=1. (16)

3. Concentrations of Oxidized Mediator, Substrate, and Reduced Mediator under Steady-State Condition

3.1. Analytical Solution Using ADM

In this paper, the Adomian decomposition method (see Appendix A) is used to solve nonlinear differential equations. The ADM [21–25] yields, without linearization, perturbation, or transformation, an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. The basic principle of this method is described in Appendix A and detailed derivation of dimensionless concentration of oxidized mediator 𝐹_O(𝜒), from the nonlinear equation (15), is described in Appendix B. The analytical expression of concentration (see Appendix B) of the oxidized mediator is as follows: 𝐹_O𝜒𝜙=1+^2O𝐵_O𝐵_S2𝐵_O+𝐵_S+𝐵_O𝐵_S×5𝑤₁−1+1−6𝑤₁𝜒²+𝑤₁𝜒⁴, (17) where 𝑤₁=𝜙²_O𝐵_O𝐵_S(𝐵_S+𝐵_O𝜇_S)/12(𝐵_O+𝐵_S+𝐵_O𝐵_S)².

Using (17) (or the dimensionless concentration 𝐹_O(𝜒)), we can obtain the concentrations of substrate and reduced mediator from (13) and (14). From (14), (16), and (17) we get the dimensionless current 𝜙𝐼=^2O𝐵_O𝐵_S1−4𝑤₁𝐵_O+𝐵_S+𝐵_O𝐵_S. (18)

3.2. Numerical Simulation

The nonlinear diffusion equations ((9a)–(9c)) for the boundary condition (10) are also solved numerically. We have used the function pdex4 in Scilab/Matlab numerical software to solve numerically the initial-boundary value problems for parabolic-elliptic partial differential equations. This numerical solution is compared with our analytical results in Figures 2 and 3 and Table 1. The average relative error between our analytical result (17) and the numerical result of oxidized mediator concentration 𝐹_O is less than 0.3% for various values of 𝜙²_O. The normalized current 𝐼 is compared with simulation result in Table 3 for various values of parameters 𝜙²_O. The average relative error is less than 3%. From the table it is also inferred that the value of the current increases when the thickness of the membrane increases.

Table 1 

Comparison of normalized analytical steady-state concentrations of oxidized mediator 𝐹_O (17) with the numerical results when 𝐵_O=0.1, 𝐵_S=0.01, and 𝜇_S=0.05 and for various values of 𝜙²_O.

Figure 2 

Dimensionless concentration of (a) oxidized mediator 𝐹O (17), (b) substrate 𝐹S (13), and (c) reduced mediator 𝐹R (14) versus normalized distance 𝜒 for 𝐵O=0.5, 𝐵S=0.0052, 𝜇S=0.05, and 𝜙2O=400. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution.

(a)

(b)

(c)

Figure 3 

Dimensionless concentration of (a) oxidized mediator 𝐹O (17), (b) substrate 𝐹S (13), and (c) reduced mediator 𝐹R (14) computed for some fixed values of parameters 𝐵O=0.1, 𝐵S=0.01, and 𝜇S=0.05 and various values of Thiele modulus 𝜙2O. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution. (i) 𝜙2O=0.01 (𝑑=0.7878 μM). (ii) 𝜙2O=1 (𝑑=7.878 μM). (iii) 𝜙2O=25 (𝑑=39.39 μM). (iv) 𝜙2O=100 (𝑑=78.80 μM). (v) 𝜙2O=225 (𝑑=118.17 μM). (vi) 𝜙2O=225 (𝑑=118.17 μM).

4. Results and Discussion

Equations (17), (13), and (14) represent the new closed and simple approximate analytical expressions of the normalized concentrations of oxidized mediator 𝐹_O, substrate 𝐹_S, and reduced mediator 𝐹_R, respectively, for all values of parameters 𝜙²_O, 𝐵_O, and 𝐵_S. The current response is given in (18). All possible numerical values of the dimensionless parameters used by Gooding and Hall [9] and in this work are given in Table 2.

Table 2 

Possible numerical values for dimensionless parameters used by Gooding and Hall [9] and in this work.

Table 3 

Comparison of normalized current 𝐼 (18) with the numerical results when 𝐵_O=0.1, 𝐵_S=0.01, and 𝜇_S=0.05 and for various values of 𝜙²_O.

4.1. Concentration Profiles

The concentration profiles for all the three species are shown in Figure 2. The condition that the oxidized mediator 𝐹_O (17) is consumed by the enzyme reaction, as the mediator moves inwards from the electrode interface, is established in Figure 2. We understand that when the substrate concentration 𝐹_S (13) is changed slightly across the matrix, the oxidized mediator is limiting the reaction under these conditions rather than the substrate itself. The reduced mediator concentration 𝐹_R (14) has the reciprocal variation as expected from (9d). The maximum reduced mediator concentration is noticed at the same position in the enzyme layer, as the oxidized species becomes completely consumed.

4.2. Variations in the Thiele Modulus

The Thiele modulus 𝜙²_O essentially compares the reaction and diffusion in the enzyme layer. We examine the rise and downfall of concentration profiles in two cases. (a) When 𝜙²_O is less than 1, the kinetics dominate and the uptake of oxygen and substrate is kinetically controlled. The overall kinetics are governed by the total amount of active enzyme. (b) The response is under diffusion control, when the Thiele modulus is large (𝜙²_O>1), which is observed at high catalytic activity and great membrane thickness or low diffusion coefficient values.

In Figure 3, under the consideration of lower 𝜙²_O and the sensor under kinetic control, the concentration profile varies slightly from nonenzyme linked oxygen. As 𝜙²_O increases, the oxidized mediator concentration is consumed in the enzyme reaction. Therefore, the profile deviates more from the linearity. Furthermore, all the oxygen within the polymer matrix is consumed well before reaching the electrode. Thus, the concentration gradient nears the electrode, and hence the flux of oxygen at the electrode surface decreases. We observe that, for a given layer and hence value of 𝜙²_O, the concentration profiles are also altered according to the bulk concentrations of the substrate and cosubstrate. Thus, with increasing [S]_b the concentration profile deviates from the response where no oxygen is consumed. Thus, an increased substrate concentration 𝐹_S causes a decrease in the flux to the electrode. The concentration of the reduced mediator 𝐹_R increases in direct proportion to the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix.

In Figure 4, the concentrations of the oxidized mediator, substrate, and reduced mediator are shown for different substrate concentrations in the bulk electrolyte. In the absence of substrate (𝐵_S≤0.001), a straight line is observed for the mediator profile. With increasing substrate concentration in the bulk electrolyte, the profile of reduced mediator bends upward as the generation of reduced mediator by the enzyme reaction increases. The profiles for oxidized mediator and substrate bend downwards.

(a)

(b)

(c)

Figure 4 

Dimensionless concentration of (a) oxidized mediator 𝐹O (17), (b) substrate 𝐹S (13), and (c) reduced mediator 𝐹R (14) for some fixed values of parameters 𝐵O=0.17 and 𝜇S=0.5, 𝜙2O=100 and various values of normalized surface concentration of substrate 𝐵S. (i) 𝐵S=0.001. (ii) 𝐵S=0.005. (iii) 𝐵S=0.01. (iv) 𝐵S=0.05. (v) 𝐵S=0.1. (vi) 𝐵S=0.5. (vii) 𝐵S=1.

4.3. Current Response

The parameter of the greatest interest in an amperometric biosensor is the current. For a particular membrane, the variation in 𝜙²_O can be achieved practically, by varying the thickness of the membrane 𝑑 or the loading of the enzyme [E_𝑇]. As can be seen in (8), the thickness appears as a squared term and thus small changes will have a much more pronounced effect on the response of the enzyme electrode than the enzyme loading. The variation in current for various values of 𝑑 and hence 𝜙²_O is shown in Figure 5. It is evident from the figure that the current 𝐼 (18) increases when 𝜙²_O increases. Furthermore, when 𝐵_O and 𝐵_S are greater than 0.5, all the curves reach the steady-state value. Electroanalytical systems for high spatial resolution scanning and fabrication of micronano sensors need the three-dimensional study of current response.

(a)

(b)

Figure 5 

Variation of normalized current 𝐼 with (a) normalized surface concentration of oxidized mediator 𝐵O and (b) normalized surface concentration of the substrate 𝐵S for various values of Thiele modulus 𝜙2O using (18). (i) 𝜙2O=50 (𝑑=55.71 μM) (ii) 𝜙2O=100 (𝑑=78.80 μM) (iii) 𝜙2O=200 (𝑑=111.42 μM) (iv) 𝜙2O=300 (𝑑=136.46 μM) (v) 𝜙2O=400 (𝑑=158.00 μM).

5. Conclusions

This paper presents a theoretical treatment of an oxidized enzyme-membrane electrode in an amperometric biosensor. In this paper, we have solved the nonlinear differential equations both analytically and numerically. The approximate analytical expressions for the steady-state concentrations of oxidised mediator, substrate, and reduced mediator for all values of parameters 𝜙²_O, 𝐵_O, and 𝐵_S at the enzyme-membrane electrode geometry are obtained using the Adomian decomposition method. A satisfactory agreement with the numerical result is noted. The analytical results obtained in this paper can be used to analyze the effect of different parameters such as membrane thickness, type of buffer in the external solution and enzyme loading in the membrane. This theoretical result is also useful for the optimization of the biosensor.

Appendices

A. Basic Concept of the Adomian Decomposition Method (ADM)

Adomian decomposition method [26–31] depends on decomposing the nonlinear differential equation 𝐹𝜒,𝑦𝜒=0 (A.1) into the two components 𝐿𝑦𝜒+𝑁𝑦𝜒=0, (A.2) where 𝐿 and 𝑁 are the linear and the nonlinear parts of 𝐹, respectively. The operator 𝐿 is assumed to be an invertible operator. Solving for 𝐿(𝑦(𝜒)) leads to 𝐿𝑦𝜒=−𝑁𝑦𝜒. (A.3) Applying the inverse operator 𝐿 on both sides of (A.3) yields 𝑦𝜒=𝜑𝜒−𝐿⁻¹𝑁𝑦𝜒, (A.4) where 𝜑(𝜒) is a function that satisfies the condition 𝐿(𝜑(𝜒))=0. Now assuming that the solution 𝑦 can be represented as infinite series of the form, ^∞_𝑛=0𝑦_𝑛𝜒=𝜑𝜒−𝐿^−1∞_𝑛=0𝐴_𝑛𝜒, (A.5) where ^∞_𝑛=0𝑦_𝑛𝜒𝜒,𝐴=𝑦_𝑛𝜒=1𝑑𝑛!𝑑𝜆^𝑛𝑛𝑁^∞_𝑖=0𝜆^𝑖𝑦_𝑖𝜒_𝜆=0,_∞𝑛=0𝐴_𝑛(𝑥)=𝑁𝑦𝜒𝑛≥0. (A.6) Then equating the terms in the linear system of (A.5) gives the recurrent relation 𝑦₀=𝜑𝜒,𝑦_𝑛+1=−𝐿⁻¹𝐴_𝑛𝑛≥0. (A.7) However, in practice all the terms of series in (A.5) cannot be determined, and the solution is approximated by the truncated series ∑^𝑁_𝑛=0𝑦_𝑛(𝜒).

B. Analytical Expression of Concentrations of Oxidized Mediator Using the Adomian Decomposition Method

To solve (15) using the Adomian decomposition method [21–23], we write (15) in the operator form 𝐿𝐹_O𝜒=𝜙²𝑁𝐹_O𝜒, (B.1) where 𝑑𝐿=²𝑑𝜒²,𝑁𝐹_O𝜒=𝐹²_𝑂+𝐹_O1−𝜇_S𝜇_S×𝐹²_O+𝐹_O1𝐵_S𝜇_S+1𝐵_O+1−𝜇_S𝜇_S+1−𝜇_S𝜇_S𝐵_O⁻¹. (B.2) Applying the inverse operator 𝐿⁻¹ on both sides of (B.1) yields 𝐹_O𝜒=𝐴𝜒+𝐵+𝜙²𝐿⁻¹𝑁𝐹_O𝜒. (B.3) According to the ADM, the solution 𝐹_O(𝜒) can be elegantly computed by using the recurrence relation 𝐹_O,0𝜒𝐹=𝐴𝜒+𝐵,_𝑂,𝑛+1𝜒=𝜙²𝐿⁻¹𝑁𝐹_O𝜒=𝜙²𝐿⁻¹𝐴_𝑛𝜒𝑛≥0, (B.4) where 𝐴_𝑛 are the Adomian polynomials of 𝐹_O,1,𝐹_O,2,…,𝐹_O,𝑛. We can find the first few Adomian polynomial coefficients 𝐴_𝑛 using (A.6) as follows: 𝐴₀𝜒=𝑁𝐹_O,0=𝐹²_O,0+𝐹_O,01−𝜇_S𝜇_S×𝐹²_O,0+𝐹_O,01𝐵_S𝜇_S+1𝐵_O+1−𝜇_S𝜇_S+1−𝜇_S𝜇_S𝐵_O⁻¹,𝐴₁𝜒=𝑑𝑑𝜆𝑁(𝐹_O,0+𝜆𝐹_O,1)_𝜆=0=𝐹_O,1𝐵_O×𝐹²_O,0𝐵_O1𝐵_S𝜇_S+1𝐵_O+21−𝜇_S𝜇_S𝐹_O,0+1−𝜇_S𝜇_S²×𝐹²_O,0+𝐹_O,0𝐵_O1𝐵_S𝜇_S+1𝐵_O+1−𝜇_S𝜇_S+1−𝜇_S𝜇_S𝐵_O²⁻¹. (B.5) The remaining polynomials 𝐴_𝑖(𝜒) can be generated easily, using (A.6). Apply the following boundary conditions: 𝐹^′_O,0(0)=0,𝐹_O,0(1)𝐹=1,_′O,𝑖(0)=0,𝐹_O,𝑖(1)=0for𝑖≥1. (B.6) From (B.4) and using the above conditions we obtain the following results: 𝐹_O,0𝜒𝐹=1,_O,1𝜒=𝜙²_O𝐿⁻¹𝐴_O𝜒=𝜙²_O𝜒²−1211+𝐵_S+1𝐵_O⁻¹,𝐹_O,2𝜒=𝜙²_O𝐿⁻¹𝐴₁𝜒=𝜙⁴_O𝐵_S+𝐵_O𝜇_S𝜒⁴−6𝜒²+524𝐵_O𝐵_S1+1/𝐵_S+1/𝐵_O³. (B.7)(B.8)(B.9) Adding (B.7)–(B.9), we get the concentration of oxidized mediator (17) in the text. From this solution of the oxidized mediator, the concentration profiles for the substrate and reduced mediator can be obtained using (13) and (14).

C. Scilab/Matlab Program

Scilab/Matlab program for the numerical solution of the systems of nonlinear equations (9a)–(9c) is given below. function pdex4 m = 0; x = 00.20.40.60.81; t = 0246810; sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u1 = sol(:,:,1); u2 = sol(:,:,2); u3 = sol(:,:,3); figure plot(x,u1(end,:)) xlabel(‘Distance x’) ylabel(‘u1(x,2)’) figure plot(x,u2(end,:)) xlabel(‘Distance x’) ylabel(‘u2(x,2)’) figure plot(x,u3(end,:)) xlabel(‘Distance x’) ylabel(‘u3(x,2)’) % ––––––––––––––––––––––––––––––– function [c,f,s] = pdex4pde(x,t,u,DuDx) Bs = 0.01; Bo = 0.1; A = 10; us = 0.05; up = 1; c = [1;1;1]; f = [1;1;1].* DuDx; F1 =-A∧2*(u(1)*u(2)/(u(1)/Bs+u(2)/Bo+(u(1)*u(2)))); F2 =-us*A∧2*(u(1)*u(2)/(u(1)/Bs+u(2)/Bo+(u(1)*u(2)))); F3 =up*A∧2*(u(1)*u(2)/(u(1)/Bs+u(2)/Bo+(u(1)*u(2)))); s = [F1; F2; F3]; % ––––––––––––––––––––––––––––––– function u0 = pdex4ic(x); u0 = [1; 0; 1]; % ––––––––––––––––––––––––––––––– function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [0;0;0]; ql = [1;1;1]; pr = [ur(1)-1; ur(2)-1; ur(3)]; qr = [0;0;0];

D. Nomenclature

Symbols

[E𝑇]:	Total enzyme concentration (mM)
[EOX]:	Enzyme concentration of the oxygen (mM)
[ES]:	Enzyme concentration of the substrate (mM)
[Ered]:	Reduced enzyme concentration (mM)
[MedOX]:	Concentration of oxidized mediator at any position in the enzyme layer (mM)
[Medred]:	Concentration of reduced mediator at any position in the enzyme layer (mole cm−3)
𝐷M:	Diffusion coefficient of oxidized mediator (cm2s−1)
𝐷S:	Diffusion coefficient of substrate (cm2s−1)
𝑑:	Thickness of the planar matrix (cm)
[MedOX]b:	Oxidized mediator concentration at the enzyme layer ∣electrode boundary (mM)
[MedOX]∞:	Oxidized mediator concentration in bulk solution (mM)
[S]:	Concentration of substrate at any position in the enzyme layer (mM)
[S]b:	Substrate concentration at the enzyme layer|electrode boundary (mM)
[S]∞:	Substrate concentration in bulk solution (mM)
𝑘1, 𝑘3, 𝑘4:	Rate constants (M−1s−1)
𝑘−1, 𝑘2:	Rate constants (s−1).

Greek Symbols

𝜙2O:	Thiele modulus for the oxidized mediator (none).

Subscripts

OX:	Oxidized
𝑇:	Total
red:	Reduced
O:	Oxygen
S:	Substrate
P:	Product
R:	Reduced
b:	Boundary
∞:	Bulk.

Acknowledgments

This work was supported by the Council of Scientific and Industrial Research (CSIR no.: 01(2442)/10/EMR-II), Government of India. The authors also thank Mr. M. S. Meenakshisundaram, Secretary, The Madura College Board, Dr. R. Murali, the Principal, and Professor S. Thigarajan, HOD, Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India, for their constant encouragement. The author S. Loghambal is very thankful to the Manonmaniam Sundaranar University and V V college of Engineering, Tirunelveli, for allowing the research work.