Analytical Expressions Pertaining to the Concentration of Substrates and Product in Phenol-Polyphenol Oxidase System Immobilized in Laponite Hydrogels: A Reciprocal Competitive Inhibition Process

Abstract

Theoretical analysis corresponding to the diffusion and kinetics of substrate and product in an amperometric biosensor is developed and reported in this paper. The nonlinear coupled system of diffusion equations was analytically solved by Homotopy perturbation method. Herein, we report the approximate analytical expressions pertaining to substrate concentration, product concentration, and current response for all possible values of diffusion and kinetic parameters. The numerical solution of this problem is also reported using Scilab/Matlab program. Also, we found excellent agreement between the analytical results and numerical results upon comparison.

1. Introduction

Theoretical modeling of biosensors usually provides some important insight into understanding the functioning of a biosensor. Usually, with the aid of an analytical device, it is not possible to measure the concentration of substrates inside the enzyme membranes. As a result, theoretical model in biosensors has been developed and employed as an important tool to study the analytical characteristics of biosensors. Initially, Goldman et al. [1] reported the theoretical modeling on biosensor. In that report [1], they have published an extensive theoretical treatment corresponding to substrate and product distribution in membranes containing enzymes. Later, Sundaram and Laidler [2] reported equations that describe the kinetics of reaction in an enzyme membrane immersed in a substrate solution. Kasche and coworkers [3] presented a model and equations that describe steady-state catalysis by an enzyme immobilized in spherical gel particles. Furthermore, they have demonstrated that the catalysis by a bounded enzyme at low substrate concentrations differs from the catalysis brought about by an unbound enzyme. Blaedel et al. [4] derived equations for steady-state fluxes of substrates and product through a membrane in simple systems.

Catalytic biosensors are sensors that use enzymes which catalyse a specific conversion of analyte [5]. The analysis of the action of the biosensors containing one and two enzymes was performed under internal diffusion limitation in [6]. Gough et al. [7] have simulated the performance of a cylindrical biosensor employed for glucose monitoring at steady state. Jobst et al. [8] reported a finite difference scheme for the discretization of the model equation. Bacha et al. [9, 10] have developed a model that takes into account a variety of configuration designs to describe the behaviour of an amperometric biosensor for glucose monitoring. Recently, Baronas et al. [11–14] have developed a theoretical model to examine the dynamic response of amperometric biosensors under stirred and nonstirred solutions. Coche-Guerente et al. [15] analyzed the amperometric response of PPO-based rotating disk bioelectrodes based on the kinetic model by considering the internal and external mass transport effects and a CEC’ electroenzymatic mechanism.

Amperometric biosensors are rapid, sensitive, and highly stable candidates for environmental, clinical, and industrial applications. The general features of amperometric biosensors have been studied and analyzed extensively in the literature [16, 17]. However, due to the limited calculation possibilities and lack of model, the calculations were restricted by two critical concentrations of substrate (when enzyme acted at the first- and zero-order reaction conditions). Later, there are few reports that reported steady-state and nonstationary kinetics of amperometric biosensor response [17–21]. Baronas et al. [22] developed the model based on nonstationary diffusion equations containing a nonlinear term related to the enzymatic reaction. Ames [23] performed the digital simulation of the biosensor response with the implicit difference scheme.

Recently, Coche-Guerente et al. [24, 25] presented a theoretical analysis of the transport and kinetics of substrate and product in an immobilized enzyme layer involving an electrochemical substrate-recycling scheme. Baronas et al. [26] developed the mathematical model of the biosensor action to investigate the dynamics of the response of biosensors utilizing a triggered enzymatic reaction followed by the electrochemical and enzymatic product cyclic conversion scheme (CEC scheme). To the best of our knowledge, there are no analytical expressions corresponding to the steady-state substrate, product concentrations, and current for all values of parameters 𝜇₁,𝜇₂,𝛼₁, and 𝛼₂ at the rotating disk electrode for reciprocal competitive inhibition process [15]. These parameters are explained below (8). The purpose of this communication is to derive an analytical expression of steady-state concentration and current at electrodes for all values of parameters 𝜇₁,𝜇₂,𝛼₁, and 𝛼₂.

2. Mathematical Formulation of the Problem and Analysis

The complete mathematical formulation of this problem is described in Coche-Guerente et al. [15]. Figure 1 shows the schematic representation of a rotating disk electrode modified by a PPO enzymatic layer of thickness 𝐿 [15]. The enzymatic reaction of substrates 𝑆₁,𝑆₂ and the product 𝑃₂ follows reciprocal competitive inhibition process. Therefore, enzymatic rate of formation of o-quinone from phenol and catechol substrates, 𝑉₁ and 𝑉₂ can be expressed as follows: 𝑉₁=𝑘₁𝐸_𝑇𝑆₁𝑆₁+𝐾₁1+𝑆₂/𝐾₂,𝑉₂=𝑘₂𝐸_𝑇𝑆₂𝑆₂+𝐾₂1+𝑆₁/𝐾₁, (1) where [𝐸_𝑇] represent the total concentration of active enzyme. 𝐾₁, 𝐾₂, 𝑘₁, and 𝑘₂ are the kinetic parameters. The nonlinear differential equations corresponding to the reciprocal competitive inhibition process with concentrations 𝑆₁,𝑆₂ and 𝑃₂ within the enzymatic layer [15, 26] at steady state condition can be written as follows:𝜕²𝑆₁𝜕𝑋²−𝑆₁Λ²₁1+𝑆₁/𝐾₁+𝑆₂/𝐾₂𝜕=0,₂𝑆₂𝜕𝑋²−𝑆₂Λ²₂1+𝑆₁/𝐾₁+𝑆₂/𝐾₂𝜕=0,₂𝑃₂𝜕𝑋²+𝑆₁Λ²₁1+𝑆₁/𝐾₁+𝑆₂/𝐾₂+𝑆₂Λ²₂1+𝑆₁/𝐾₁+𝑆₂/𝐾₂=0, (2) where 𝑋 represents the distance from the electrode surface, Λ₁ and Λ₂ are the reaction lengths related to phenol substrate 𝑆₁, and catechol substrate 𝑆₂. Further, Λ₁=(𝐷_𝑓𝐾₁/𝑘₁[𝐸_𝑇])^1/2, Λ₂=(𝐷_𝑓𝐾₂/𝑘₂[𝐸_𝑇])^1/2, where 𝐷_𝑓 is the diffusion coefficient in the enzymatic layer. These equations are solved for the following initial and boundary conditions as described in the literature [15]:𝜕𝑆₁𝑆𝜕𝑋=0,₁+𝑆₂=𝑆_1∞,𝑃₂𝑆=0when𝑋=0,₁=𝑆_1∞,𝑆₂=0,𝑃₂=0when𝑋=𝐿+𝛿. (3) In addition, the mass conservation implies that [𝑆₁]+[𝑆₂]+[𝑃₂]=[𝑆₁]_∞ when 𝑋≥𝐿⁺, where 𝐿 is the thickness of the enzymatic layer and 𝛿 is the thickness of the diffusion convection. At 𝑋=𝐿, the fluxes 𝐽_𝑆1,𝐽_𝑆2, and 𝐽_𝑃1 of 𝑆₁,𝑆₂, and 𝑃₂ on each side of the interface are equal [15]. Next, we introduce the following set of dimensionless variables:𝑆𝑢=¹𝑆_1∞𝑆,𝑣=²𝑆_1∞𝑃,𝑤=²𝑆_1∞𝑋,𝑥=𝐿, (4) where 𝑢,𝑣, and 𝑤 represent dimensionless concentration and 𝑥 is the distance parameter. The dimensionless nonlinear equations for rotating disk electrode are as follows:𝜕²𝑢𝜕𝑥²−𝜇₁𝑢1+𝛼₁𝑢+𝛼₂𝑣𝜕=0,₂𝑣𝜕𝑥²−𝜇₂𝑣1+𝛼₁𝑢+𝛼₂𝑣𝜕=0,₂𝑤𝜕𝑥²+𝜇₁𝑢1+𝛼₁𝑢+𝛼₂𝑣+𝜇₂𝑣1+𝛼₁𝑢+𝛼₂𝑣=0, (5)(6)(7) where𝜇₁=𝐿²Λ²₁,𝜇₂=𝐿²Λ²₂,𝛼₁=𝑆_1∞𝐾₁,𝛼₂=𝑆_1∞𝐾₂, (8) where 𝜇₁ is the ratio of the square of the thickness 𝐿 and the reaction length Λ₁; 𝜇₂ is the ratio of the square of the thickness 𝐿 and the reaction length Λ₂; 𝛼₁ is the ratio of [𝑆₁]_∞ and the kinetic parameter 𝐾₁; 𝛼₂ is the ratio of [𝑆₁]_∞ and the kinetic parameter 𝐾₂. Now the initial and boundary conditions are given by the following equation:𝜕𝑢𝛿𝜕𝑥=0,𝑢+𝑣=1,𝑤=0when𝑥=0,𝑢=1,𝑣=0,𝑤=0when𝑥=1+𝐿=𝑚. (9) The dimensionless current is given by [15]𝑗𝐼=^𝑓2𝐹=−𝜕𝑣𝜕𝑥_𝑥=0, (10) where 𝐹 is the Faraday constant.

Figure 1 

Schematic representation of rotating disk electrode modified by a PPO enzymatic layer and principle of the bioelectrode functioning in the presence of phenol substrates 𝑆1,𝑆2 and product 𝑃2 [15].

3. Solution of Boundary Value Problem Using the Homotopy Perturbation Method

The HPM [27] uses the embedded parameter 𝑝 as small, and only a few iterations are essential for an asymptotic solution. Using this method (see Appendix A), the solutions to (5)–(7) can be deduced as follows:𝑢(𝑥)=𝑎cosh√𝜇₁𝑥+𝛼₂𝐴₀2𝑎cosh√𝜇₂𝑚𝜇₂−2𝜇₁√𝜇₂×sinh√𝜇₁(𝑚−𝑥)+4𝑎𝜇₁√𝜇₂sinh√𝜇₁𝑚cosh√𝜇₁𝑥−𝜇₂+2√𝜇₁𝜇₂√𝜇₁sinh√𝜇₁𝑥+√𝜇₂()−𝜇𝑚−𝑥₂−2√𝜇₁𝜇₂√𝜇₁×sinh−√𝜇₁𝑥+√𝜇₂()+𝛼𝑚−𝑥₁𝑎²6𝑎cosh√𝜇₁𝑥cosh2√𝜇₁2√𝑚−3−cosh𝜇₁𝑥+3, (11)𝑣(𝑥)=(1−𝑎)𝑏sinh√𝜇₂(𝑚−𝑥)−𝛼₂𝐴₀sinh√𝜇₂()×√𝑚−𝑥2𝑎𝑏𝜇₂sinh√𝜇₁𝑚×cosh√𝜇₂𝑚𝜇₂−2𝜇₁+2𝜇₁−2√𝜇₁𝜇₂+𝐴₁2𝜇₁sinh√𝜇₂(𝑚−𝑥)+4𝑏√𝜇₁√𝜇₂sinh√𝜇₁𝑚×sinh√𝜇₂𝑥−𝜇₁+2√𝜇₁√𝜇₂×sinh√𝜇₁𝑥+√𝜇₂()−𝜇𝑚−𝑥₁−2√𝜇₁√𝜇₂sinh−√𝜇₁𝑥+√𝜇₂(𝑚−𝑥)−𝛼₁𝑎²𝑏sinh√𝜇₂()×𝑎2√𝑚−𝑥cosh𝜇₁𝑚−36+13+𝐴₂cosh2√𝜇₂𝑚+3sinh√𝜇₂()√𝑚−𝑥+4sinh𝜇₂𝑥−sinh√𝜇₂𝑚×cosh2√𝜇₂(𝑚−𝑥)+3, (12) where 𝑎=1/(cosh(√𝜇₁𝑚)); 𝑏=1/(sinh(√𝜇₂𝑚)); 𝐴₀=(𝑎(1−𝑎)𝑏√𝜇₁)/(2(𝜇²₂−4𝜇₁𝜇₂)); 𝐴₁=(𝛼₁𝜇₂𝑎𝑏(1−𝑎))/(2(𝜇²₁−4𝜇₁𝜇₂)); 𝐴₂=(𝛼₂(1−𝑎)²𝑏³)/6. Adding (5) and (6) using the boundary conditions (9), we can obtain 𝑤(𝑥)=1−𝑢(𝑥)−𝑣(𝑥). (13) Equations (11)–(13) are the analytical expressions corresponding to the dimensionless concentrations as a function of dimensionless distance 𝑥. These equations satisfy the boundary conditions (9). When 𝛼₁ and 𝛼₂→ 0 (𝐾₁ and 𝐾₂→∞), the concentrations 𝑢,𝑣, and 𝑤 become𝑢(𝑥)=𝑎cosh√𝜇₁𝑥,𝑣(𝑥)=(1−𝑎)𝑏sinh√𝜇₂(),𝑤(𝑥)(𝑥)(𝑥).𝑚−𝑥=1−𝑢−𝑣 (14) Using (10) the current density is𝐼=(1−𝑎)𝑏√𝜇₂cosh√𝜇₂𝑚−𝛼₂𝐴₀√𝜇₂cosh√𝜇₂𝑚×2𝑎𝑏√𝜇₂sinh√𝜇₁𝑚×cosh√𝜇₂𝑚𝜇₂−2𝜇₁+2𝜇₁−2√𝜇₁𝜇₂+𝐴₁2𝜇₁√𝜇₂cosh√𝜇₂𝑚−4𝑏√𝜇₁𝜇₂sinh√𝜇₁𝑚+𝜇₁+2√𝜇₁√𝜇₂cosh√𝜇₂𝑚√𝜇₁−√𝜇₂−𝜇₁−2√𝜇₁√𝜇₂cosh√𝜇₂𝑚√𝜇₁+√𝜇₂−𝛼₁𝑎²𝑏√𝜇₂cosh√𝜇₂𝑚×𝑎cosh2√𝜇₁𝑚−36+13+𝐴₂cosh2√𝜇₂𝑚+3√𝜇₂cosh√𝜇₂𝑚−4√𝜇₂−2√𝜇₂sinh√𝜇₂𝑚sinh2√𝜇₂𝑚. (15) Equation (15) represents the dimensionless current as a function of dimensionless parameters 𝜇₁,𝜇₂,𝛼₁,𝛼₂, and 𝑚.

4. Numerical Simulation

The system of steady-state nonlinear differential equations (5)–(7) are also solved in numerical methods. The function pdex4 in Scilab/Matlab software which is a function of solving partial differential equations is used to solve these equations. Its numerical solution is compared with the analytical solution obtained by HPM. Satisfactory agreement is noted. The Scilab/Matlab program is also given in Appendix B.

5. Results and Discussion

5.1. Concentration Profiles

Equations (11)–(13) are the new and simple analytical expressions of concentrations of phenol (substrate 𝑢(𝜇₁,𝜇₂,𝛼₁,𝛼₂,𝛿/𝐿)), catechol (substrate 𝑣(𝜇₁,𝜇₂,𝛼₁,𝛼₂,𝛿/𝐿)), and o-quinone (product 𝑤(𝜇₁,𝜇₂,𝛼₁,𝛼₂,𝛿/𝐿)), respectively. All substrate molecules are enzymatically oxidized within the enzymatic layer before reaching the electrode surface. Thus, the high sensitivity observed with a PPO bioelectrode working at a cathodic potential can be explained by an efficient electrochemical recycling of substrate which can enter a new enzymatic step in the proximity of the enzymatic layer interface. This interpretation agrees well with the substrates and product concentration profiles in the enzymatic layer calculated on the basis of (11)–(13) and reproduced in Figures 2–4. In these figures, the profiles of substrates (phenol and catechol) as well as product (o-quinone) concentration in the enzyme layer are presented for biosensor acting in CEC mechanism. The thickness of the enzymatic layer constitutes an important parameter that governs the biosensor sensitivity. The normalized concentration of phenol substrate 𝑢 is represented in Figures 2(a)–2(c). From this figure, it is evident that the value of phenol substrate concentration 𝑢 decreases when the thickness of the enzymatic layer increases and enzyme activity increases (𝜇₁,𝜇₂ increase) for all values of 𝛼₁ and 𝛿/𝐿. The concentration of phenol substrate 𝑢 is equal to 1 when 𝜇₁≤0.1 and 𝑢 becomes zero when 𝜇₁≥30 for all values 𝛼₁ and 𝜇₂. Figures 3(a)–3(c) stand for the normalized concentration profile of catechol 𝑣. It is clear that the catechol concentration 𝑣 increases when the thickness of the enzymatic layer increases. The concentration of catechol substrate 𝑣 becomes zero when 𝜇₁≤0.1 for all values of parameters. Figures 4(a)–4(c) represent the normalized o-quinone concentration 𝑤. It is clear that the concentration of o-quinone 𝑤 increases when the thickness of the enzymatic layer increases. The concentration of o-quinone product 𝑤 reaches the maximum value when 𝑥=0.3 and then decreases to the minimum value zero when the diffusion-convection layer is equal to zero.

(a)

(b)

(c)

(d)

Figure 2 

The profile of the normalized concentration of phenol substrate 𝑢 versus the normalized distance 𝑥. The concentrations were computed using (11) when the parameters 𝜇2=1,𝛼1=0.1, 𝛼2=0.1 and for various values of 𝜇1 and 𝛿/𝐿. The key to the graph (stacked line) represents (11) and (dotted line) represents the numerical solution. (a) 𝛿/𝐿=0, (b)𝛿/𝐿=0.5, (c) 𝛿/𝐿=1, (d) 𝜇1=0.1.

(a)

(b)

(c)

(d)

Figure 3 

The profile of the normalized concentration of catechol substrate 𝑣 versus the normalized distance 𝑥. The concentrations were computed using (12) when the parameters 𝜇2=10, 𝛼1=0.1, 𝛼2=0.1 and for various values of 𝜇1 and 𝛿/𝐿. The key to the graph (stacked line) represents (12) and (dotted line) represents the numerical solution. (a) 𝛿/𝐿=0, (b)𝛿/𝐿=0.5, (c) 𝛿/𝐿=1, (d)𝜇1=0.1.

(a)

(b)

(c)

(d)

Figure 4 

The profile of the normalized concentration of o-quinone product 𝑤 versus the normalized distance 𝑥. The concentrations were computed using (13) when the parameters 𝜇2=10, 𝛼1=0.1, 𝛼2=0.1 and for various values of 𝜇1 and 𝛿/𝐿. The key to the graph (stacked line) represents (13) and (dotted line) represents the numerical solution. (a) 𝛿/𝐿=0, (b)𝛿/𝐿=0.5, (c) 𝛿/𝐿=1, (d) 𝜇1=0.1.

5.2. Current Response

The analytical expression of steady-state current 𝐼(𝜇₁,𝜇₂,𝛼₁,𝛼₂,𝛿/𝐿) is given in (15). The steady-state current of membrane biosensors significantly depends on the thickness of the enzymatic layer. Figures 5(a) and 5(b) show the dimensionless current 𝐼 for all values of parameters 𝜇₁,𝜇₂,𝛼₁ and 𝛼₂. From these figures it is inferred that the current increases gradually when 𝜇₁ and 𝜇₂ increase. When 𝜇₂→0 at any 𝜇₁>0 current 𝐼→0 (refer to Figure 5(b)). Figures 5(c) and 5(d) represent the dimensionless current I for all values of kinetic parameters. From these figures it is evident that the current decreases when 𝛼₁ and 𝛼₂ increase. The current 𝐼 is monotonously increasing function of both arguments 𝜇₁ and 𝜇₂. In the case of CEC mode an application of an active enzyme [𝐸_𝑇] (𝜇₂>0) stimulates an increase of the biosensor current. The steady-state current of membrane biosensor significantly depends on the thickness of the enzyme layer (𝐿). In view of biosensor optimizations, it can be seen that the current 𝐼 (refer to Figure 5(e)) is higher for low electrode rotation rate and small thickness of enzymatic layer.

(a)

(b)

(c)

(d)

(e)

Figure 5 

The steady state-current 𝐼 versus dimensionless parameters. The current is computed using (15). (a)𝛼1=0.1, 𝛼2=0.1, 𝛿/𝐿=1, (b) 𝛼1=0.1, 𝛼2=0.1, 𝛿/𝐿=1, (c) 𝜇1=100, 𝜇2=0.01, 𝛿/𝐿=0, (d) 𝜇1=100, 𝜇2=0.01, 𝛿/𝐿=0, (e) 𝜇1=0.1, 𝛼1=0.1, 𝛼2=0.1.

6. Conclusions

This paper reports a mathematical treatment for analyzing amperometric enzymatic reactions. In this paper, we have evaluated a theoretical model for a rotating disk electrode based on an immobilized enzyme layer. The novelty of this paper is an application of approximate method to the second-order nonlinear partial differential equations. The approximate analytical expressions for the steady state substrate concentrations and product concentration profiles for all values of 𝜇₁,𝜇₂,𝛼₁ and 𝛼₂ at the rotating disk electrode were obtained using Homotopy perturbation method. Furthermore, an analytical expression corresponding to the steady state current response is also presented. A satisfactory agreement with the existing limiting cases results is noted. The results presented in the paper extend to many applications related to biosensors modeling.

Appendices

A. Solution of (5)–(7) Using Homotopy Perturbation Method

In this appendix, we indicate how (11)–(13) in this paper are deduced. In order to obtain the solution of (5)–(7), we first construct a Homotopy as follows:𝜕1−𝑝²𝑢𝜕𝑥²−𝜇₁𝑢𝜕+𝑝²𝑢𝜕𝑥²−𝜇₁𝑢+𝛼₁𝑢𝜕²𝑢𝜕𝑥²+𝛼₂𝑣𝜕²𝑢𝜕𝑥²𝜕=0,1−𝑝²𝑣𝜕𝑥²−𝜇₂𝑣𝜕+𝑝²𝑣𝜕𝑥²−𝜇₂𝑣+𝛼₁𝑢𝜕²𝑣𝜕𝑥²+𝛼₂𝑣𝜕²𝑣𝜕𝑥²=0. (A.1) The approximate solutions of (A.1) are𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯,𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+𝑝³𝑣₃+⋯. (A.2) Substituting (A.2) into (A.1) and comparing the coefficients of like powers of 𝑝𝑝⁰∶𝜕²𝑢₀𝜕𝑥²−𝜇₁𝑢₀𝑝=0,₁∶𝜕²𝑢₁𝜕𝑥²−𝜇₁𝑢₁+𝛼₁𝑢₀𝜕²𝑢₀𝜕𝑥²+𝛼₂𝑣₀𝜕²𝑢₀𝜕𝑥²𝑝=0,₀∶𝜕²𝑣₀𝜕𝑥²−𝜇₂𝑣₀𝑝=0,₁∶𝜕²𝑣₁𝜕𝑥²−𝜇₂𝑣₁+𝛼₁𝑢₀𝜕²𝑣₀𝜕𝑥²+𝛼₂𝑣₀𝜕²𝑣₀𝜕𝑥²=0. (A.3) The boundary conditions are𝜕𝑢₀𝜕𝑥=0,𝑢₀+𝑣₀𝑢=1when𝑥=0,₀=1,𝑣₀𝛿=0when𝑥=1+𝐿=𝑚,𝜕𝑢₁𝜕𝑥=0,𝑢₁+𝑣₁𝑢=0when𝑥=0,₁=0,𝑣₁𝛿=0when𝑥=1+𝐿=𝑚. (A.4) Solving (A.3) using the boundary conditions (A.4), we can obtain𝑢₀=𝑎cosh√𝜇₁𝑥,𝑢₁=𝛼₂𝐴₀2𝑎cosh√𝜇₂𝑚𝜇₂−2𝜇₁×√𝜇₂sinh√𝜇₁(𝑚−𝑥)+4𝑎𝜇₁√𝜇₂sinh√𝜇₁𝑚cosh√𝜇₁𝑥−𝜇₂+2√𝜇₁𝜇₂√𝜇₁sinh√𝜇₁𝑥+√𝜇₂()−𝜇𝑚−𝑥₂−2√𝜇₁𝜇₂√𝜇₁×sinh−√𝜇₁𝑥+√𝜇₂()+𝛼𝑚−𝑥₁𝑎²6𝑎cosh√𝜇₁𝑥cosh2√𝜇₁2√𝑚−3−cosh𝜇₁𝑥+3, (A.5)𝑣₀=(1−𝑎)𝑏sinh√𝜇₂(),𝑣𝑚−𝑥₁=−𝛼₂𝐴₀sinh√𝜇₂()×√𝑚−𝑥2𝑎𝑏𝜇₂sinh√𝜇₁𝑚×cosh√𝜇₂𝑚𝜇₂−2𝜇₁+2𝜇₁−2√𝜇₁𝜇₂+𝐴₁2𝜇₁sinh√𝜇₂()√𝑚−𝑥+4𝑏𝜇₁√𝜇₂sinh√𝜇₁𝑚sinh√𝜇₂𝑥−𝜇₁+2√𝜇₁√𝜇₂sinh√𝜇₁𝑥+√𝜇₂()−𝜇𝑚−𝑥₁−2√𝜇₁√𝜇₂sinh−√𝜇₁𝑥+√𝜇₂(𝑚−𝑥)−𝛼₁𝑎²𝑏sinh√𝜇₂(×𝑎2√𝑚−𝑥)cosh𝜇₁𝑚−36+13+𝐴₂cosh2√𝜇₂𝑚+3sinh√𝜇₂()√𝑚−𝑥+4sinh𝜇₂𝑥−sinh√𝜇₂𝑚×cosh2√𝜇₂(𝑚−𝑥)+3. (A.6) According to the HPM, we can conclude that𝑢=lim_𝑝→1𝑢(𝑥)≈𝑢₀+𝑢₁,𝑣=lim_𝑝→1𝑣(𝑥)≈𝑣₀+𝑣₁. (A.7)(A.8) After putting (A.5) into (A.7) and (A.6) into (A.8), the final results can be described in (11) and (12) in the text.

B. Scilab/Matlab Program to Find the Numerical Solutions for (5), (6), and (7)

function pdex4m = 0;x = linspace(0,1);t = linspace(0,100000);sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);u1 = sol(:,:,1);u2 = sol(:,:,2);u3 = sol(:,:,3);figureplot(x,u1(end,:))title(“u1(x,t)”)xlabel(“Distance x”)ylabel(“u1(x,2)”)figureplot(x,u2(end,:))title(“u2(x,t)”)xlabel(“Distance x”)ylabel(“u2(x,2)”)figureplot(x,u3(end,:))title(“u3(x,t)”)xlabel(“Distance x”)ylabel(“u3(x,2)”)function [c,f,s] = pdex4pde(x,t,u,DuDx)c = [1; 1; 1];f = [1; 1; 1].* DuDx;M = 100;N = 0.1;P = 1;Q = 0.1;F = −M*u(1)/(1 + N*u(1) + Q*u(2));F1 = −P*u(2)/(1 + N*u(1) + Q*u(2));F2 = −F − F1;s = [F; F1; F2];function u0 = pdex4ic(x);u0 = [0; 1; 0];function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)pl = [0; ul(2) – 1 + ul(1); ul(3)];ql = [1; 0; 0];pr = [ur(1) − 1; ur(2); ur(3)];qr = [0; 0; 0].

Nomenclature and Units

[𝑆1]:	Concentration of the phenol substrate (mole cm−3)
[𝑆2]:	Concentration of the catechol substrate (mole cm−3)
[𝑃2]:	Concentration of the 𝑜-quinone product (mole cm−3)
[𝑆1]∞:	Bulk concentration of the phenol substrate (mole cm−3)
[𝑆2]∞:	Bulk concentration of the catechol substrate (mole cm−3)
[𝑃2]∞:	Bulk concentration of the 𝑜-quinone product (mole cm−3)
[𝐸𝑇]:	Enzyme concentration (mole cm−3)
𝐿:	Thickness of the enzymatic layer (cm)
𝛿:	Thickness of the diffusion-convection layer (cm)
𝐷𝑓:	Diffusion coefficient in the enzymatic layer (cm2 s−1)
𝐷𝑒:	Diffusion coefficient in the diffusion convection layer (cm2 s−1)
Λ1,Λ2:	Reaction lengths (cm)
𝐾1,𝐾2:	Kinetic parameters (mole cm−3)
𝑘1,𝑘2:	Kinetic parameters (s−1)
𝜇1:	Ratio of the square of thickness of the enzymatic layer and the reaction length (None)
𝜇2:	Ratio of the square of thickness of the enzymatic layer and the reaction length (None)
𝛼1:	Dimensionless parameter (None)
𝛼2:	Dimensionless parameter (None)
𝛿/𝐿:	Ratio of the thickness of the enzymatic layer and the thickness of the diffusion-convection layer (None)
𝐹:	Faraday constant (None)
𝑢:	Normalized concentration of the phenol substrate (None)
𝑣:	Normalized concentration of the catechol substrate (None)
𝑤:	Normalized concentration of the o-quinone product (None)
𝐼:	Dimensionless current (None).

Acknowledgments

This work was supported by the University Grants Commission (F. no. 39-58/2010 (SR)), New Delhi, India. The authors are thankful to Dr. R. Murali, The Principal, The Madura College, Madurai, and Mr. M. S. Meenakshisundaram, The Secretary, Madura College Board, Madurai, for their encouragement. They also thank the reviewers for their valuable comments to improve the quality of the paper. The author K. Indira is very thankful to the Manonmaniam Sundaranar University, Tirunelveli, for allowing to do the research work.