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

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), BO (normalized surface concentration of oxidizedmediator), andBS (normalized surface concentration of substrate). A comparison of the analytical approximation andnumerical simulation is also presented. A good agreement between theoretical predictions andnumerical results is observed.


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 bio-sensors 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 field 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 [9,10] where the non-linear enzyme reaction was taken into account.This model was employed to describe the behaviour of a glucose oxidize (GOx) electrode [11,12].It has been found that the mediators could not totally replace the natural co-substrate 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 [13,14].
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 [15] have described the theoretical model in an amperometric oxidase enzyme-membrane electrode.Numerical solutions [16][17][18] were reported to a novel enzyme electrode.In those papers the enzyme electrode was modeled numerically using shooting method [17,18] and Runge-Kutta method [16].
But in this chapter, 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 ^^ > ^o ^^^ B^ have been reported [17].The purpose of this chapter is to derive the closed-form of analytical expressions [19,20] of concentrations of mediator, substrate and reduced mediator by solving the system of non-linear 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.

MATHEMATICAL FORMULATION
Building upon earlier work for these mechanisms, Gooding and Hall [17] presented a concise discussion and derivation of the dimensionless non-linear 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 co-substrate penetrate through a permeable electrode to the enzyme layer and then reduces to the form of co-substrate 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 [17]: ^" (5.1) where k^ is the rate constant for the forward direction of the m reaction and k_^ is the rate constant for the backward direction.If [E^ ] is the total enzyme concentration in the matrix then at all times, J (5.4) where [EQXJJ [ES] and [E^^j] 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) (Fig. 5.1).
The corresponding governing equations in Cartesian coordinates, for the planar diffusion and reaction in the enzyme electrode are [17]: [S] [Medox] j (5.5a) where D^ is the diffusion coefficient of the oxidized and reduced forms of the mediator (assumed to be equal) and D^ is the diffusion coefficient of substrate within the enzyme layer.We examine a planar matrix of thickness y~d, where diffusion is considered in the ydirection only (edge effects are neglected).The consumption of oxidised mediator (oxygen) and glucose, the production of hydrogen peroxide, are all related processes so there is only one independent variable for which to solve [25] d^ where Dy^ is the diffusion coefficient of the oxidised and reduced form of the mediator assumed to be equal) and D^ is the diffusion coefficient of substrate within the enzyme layer in the bio-recognition matrix P^ = (A:_, +k2)/k^ and /?Q ^k^lk^.
[Medox], [Med^^j] and [S] are the concentration of oxidised mediator, substrate and reduced mediator at any position in the enzyme layer.(5.9c) The consumption of oxidized mediator, substrate and reduced mediator, are all related processes.So there is only one independent variable for which to solve The normalized boundary conditions are given by: From Eq. (5.9d) we get, (5.13) Integrating Eqs.(5.12) and (5.13) twice and applying the appropriate boundary conditions Eqs. (5.10) and (5.11) we get, Substituting the Eq.(5.14) into Eq.(5.9a) and rearranging we get The normalized current response is given by the following expression: / ^dF,^ (5.17

NUMERICAL SIMULATION
The non-linear diffusion equations (Eqs.(5.9a) -(5.9c)) for the boundary conditions (Eqs.(5.10) and (5.11)) are also solved numerically.We have used the function pdex4 in Scilab 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 Figs.(5.2) and ( 5.3) and Table 5.1.
The average relative error between our analytical result (Eq.5.18) and the numerical result of oxidized mediator concentration F^ is less than 0.38% for various values of ^Q.AH possible numerical values of the dimensionless parameters used in Hall et.al [17] and in this work are given in Table 5.2.The normalized current / is compared with simulation result in Table 5.3 for various values of parameters ^Q .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.

CONCENTRATION PROFILES
The concentration profiles for all the three species are shown in Fig. 5.2.The condition that the oxidized mediator FQ (Eq.5.18) is consumed by the enzyme reaction, as the mediator moves inwards from the electrode interface is established in the Fig. 5.2.We understand that when the substrate concentration F^ (Eq.5.14) is changed slightly across the matrix, the oxidized mediator is limiting under these conditions rather than the substrate itself.The reduced mediator concentration Fp, (Eq.5.15) has the reciprocal variation as expected from Eq. (5.9d).The maximum reduced mediator concentration is noticed at the same position in the enzyme layer, as the oxidized species becomes completely consumed.

VARIATIONS IN THE THIELE MODULUS
The Thiele modulus (j)^, essentially compares the reaction and diffusion in the enzyme layer.We examine the rise and downfall of concentration profiles in two cases, (a) When ^^ is less than 1, the kinetics dominate and the uptake of oxygen and substrate is kinetically controlled.The overall kinetics is governed by the total amount of active enzyme, (b) The response is under diffusion control, when the Thiele modulus is large (^Q > 1) , which is observed at high catalytic activity and great membrane thickness or low diffusion coefficient values.
In Fig. 5.3, under the consideration of lower (^^ and the sensor under kinetic control, the concentration profile varies slightly from non-enzyme linked oxygen.As ^Q 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 <^Q , the concentration profiles are also altered according to the bulk concentrations of the substrate, and co-substrate.Thus with increasing [S];, the concentration profile deviates from the response where no oxygen is consumed.Thus an increased substrate concentration Fg causes a decrease in the flux to the electrode.The concentration of the reduced mediator F^ increases in direct proportion to the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix.(5.B1) where Fig. 5.1 Schematic representation of a typical enzyme-membrane electrode geometry [17].
5a)-(5.5c) are solved for the following boundary conditions: At the far wall, y = 0 d[Medox]/dy = d[S]/dy = d[Med,,J/dy = 0 (5.6)At the electrode, y -d [Medox] = [Medo^], = ZJMedoxl.,[S] = [S], = ^s[S]", [Med,,J = 0 (5.7)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.[Medoxlb and [S]j, are the concentration of oxidized mediator and substrate at the enzyme layer|electrode boundary, and [MedQ^l.^and [S]^ are the bulk solution concentrations.KQ and K^ are the equilibrium partition coefficients for oxidized mediator and the substrate respectively.5.2.3 NORMALISED FORM We make the non-linear differential Eqs.(5.5a)-(5.5c) to dimensionless form by defining the following dimensionless variables, Fo = [Medox]/[Medox]b, ^s = [S]/[S]b, F^ = [Med,,,]/[Med,,Jb, Z = y/d, Bo= [MedoxlbMo > -^s = [SJb/y^s, ^l=d^k^[E-,]/Du[Medo^]^ and //g = ^^[Medoxlb/^sLSlt (5.8)where F^, F^ and Fj^are the normalized concentration of oxidized mediator, substrate and reduced mediator in the matrix respectively and x is the normalized distance.BQ and B^ are the normalized surface concentration of oxidized mediator and substrate.^^ is the Thiele modulus for the oxidized mediator which governs reaction/diffusion.The resultant expressions for the oxidized mediator, substrate and reduced mediator in non-dimensionalized form become as follows: In this chapter, the Adomian decomposition method (see Appendix 5.A) is used to solve non-linear 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 5.A and detailed derivation of dimensionless concentration of oxidized mediator FQ(X), jfrom the non-linear Eq.(5.16) is described in Appendix 5.B.The analytical expression of concentration of the oxidized mediator is as follows:Fniz) = 1 + -r ^^^^ rf5w, -1 + (1 -6w,)z' + w^] , ^^^ ^^^ dimensionless n{B^+B,+B^Bj concentration FQ(Z)]we can obtain the concentrations of substrate and reduced mediator from the Eqs.(5.14) and (5.15).From Eqs. (5.15), (5.17) and (5 Fig. 5.2 Dimensionless concentration of (a) oxidized mediator F^ (Eq.5.18), Fig. 5.3 Concentration profiles of (a) oxidised mediator F^ (Eq.5.18) Fig. 5.4 Concentration proiiles of (a) oxidised mediator F^ (Eq.5.18) Fig, 5.5 Variation of normalised current / response with (a) normalised

Table 5
substrate F^ and reduced mediator F,^ respectively for all values of parameters ^Q, BQ and B^.The current response is given in Eq. (5.19).

Table 5 .3 Comparison of normalized current / (Eq. 5.19) with the numerical results when
BQ = 0.1, B^ = 0.01, ju^ = 0.05 and for various values of ^Q .