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

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.


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][12][13][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.

Advances in Physical Chemistry
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 steadystate and nonstationary kinetics of amperometric biosensor response [17][18][19][20][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 steadystate substrate, product concentrations, and current for all values of parameters μ 1 , μ 2 , α 1 , and α 2 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 μ 1 , μ 2 , α 1 , and α 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 S 1 , S 2 and the product P 2 follows reciprocal competitive inhibition process.Therefore, enzymatic rate of formation of o-quinone from phenol and catechol substrates, V 1 and V 2 can be expressed as follows: , where [E T ] represent the total concentration of active enzyme.K 1 , K 2 , k 1 , and k 2 are the kinetic parameters.The nonlinear differential equations corresponding to the reciprocal competitive inhibition process with concentrations S 1 , S 2 and P 2 within the enzymatic layer [15,26] at steady state condition can be written as follows: where X represents the distance from the electrode surface, Λ 1 and Λ 2 are the reaction lengths related to phenol substrate S 1 , and catechol substrate S 2 .Further, , 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]: In addition, the mass conservation implies that 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 S 1 , S 2 , and P 2 on each side of the interface are equal [15].Next, we introduce the following set of dimensionless variables: 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: where where μ 1 is the ratio of the square of the thickness L and the reaction length Λ 1 ; μ 2 is the ratio of the square of the thickness L and the reaction length Λ 2 ; α 1 is the ratio of [S 1 ] ∞ and the kinetic parameter K 1 ; α 2 is the ratio of [S 1 ] ∞ and the kinetic parameter K 2 .Now the initial and boundary conditions are given by the following equation: Advances in Physical Chemistry 3 The dimensionless current is given by [15] where F is the Faraday constant.

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.

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.   (15).The steady-state current of membrane biosensors significantly depends on the thickness of the enzymatic layer.Figures 5(a From these figures it is evident that the current decreases when α 1 and α 2 increase.The current I is monotonously increasing function of both arguments μ 1 and μ 2 .In the case of CEC mode an application of an active enzyme [E T ] (μ 2 > 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 Step E

Current Response. The analytical expression of steadystate current
Step C E 1

D f δ layer
Step C  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.

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 μ 1 , μ 2 , α 1 and α 2 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.

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: The approximate solutions of (A.1) are The boundary conditions are Solving (A.3) using the boundary conditions (A.4), we can obtain

δ:
Thickness of the diffusion-convection layer (cm) D f : Diffusion coefficient in the enzymatic layer (cm 2 s −1 ) D e : Diffusion coefficient in the diffusion convection layer (cm 2 s −1 ) Λ 1 , Λ 2 : Reaction lengths (cm) K 1 , K 2 : Kinetic parameters (mole cm −3 ) k 1 , k 2 : Kinetic parameters (s −1 ) μ 1 : Ratio of the square of thickness of the enzymatic layer and the reaction length (None) μ 2 : Ratio of the square of thickness of the enzymatic layer and the reaction length (None) α 1 : Dimensionless parameter (None) α 2 : Dimensionless parameter (None) δ/L: Ratio of the thickness of the enzymatic layer and the thickness of the diffusionconvection 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 oquinone product (None) I: Dimensionless current (None).
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 μ 1 ≤ 0.1 for all values of parameters.Figures 4(a)-4(c) represent the normalized oquinone 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.

) and 5
(b)  show the dimensionless current I for all values of parameters μ 1 , μ 2 , α 1 and α 2 .From these figures it is inferred that the current increases gradually when μ 1 and μ 2 increase.When μ 2 → 0 at any μ 1 > 0 current I → 0 (refer to Figure5(b)).Figures5(c) and 5(d) represent the dimensionless current I for all values of kinetic parameters.

Figure 1 :
Figure1: Schematic representation of rotating disk electrode modified by a PPO enzymatic layer and principle of the bioelectrode functioning in the presence of phenol substrates S 1 , S 2 and product P 2[15].
(13)Concentration Profiles.Equations (11)-(13)are the new and simple analytical expressions of concentrations of phenol (substrate u(μ 1 , μ 2 , α 1 , α 2 , δ/L)), catechol (substrate v(μ 1 , μ 2 , α 1 , α 2 , δ/L)), and o-quinone (product w(μ 1 , μ 2 , α 1 , α 2 , 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 Figures2-4.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 sensi- tivity.The normalized concentration of phenol substrate u is represented in Figures2(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 (μ 1 , μ 2 increase) for all values of α 1 and δ/L.The concentration of phenol substrate u is equal to 1 when μ 1 ≤ 0.1 and u becomes zero when μ 1 ≥ 30 for all values α 1 and μ 2 .