Mathematical modeling in amperometric oxidase enzyme–membrane electrodes

Journal of Membrane Science 373 (2011) 20–28

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Mathematical modeling in amperometric oxidase enzyme–membrane electrodes S. Loghambal, L. Rajendran ∗ Department of Mathematics, The Madura College, Madurai 625011, Tamil Nadu, India

a r t i c l e

i n f o

Article history: Received 19 January 2011 Received in revised form 22 February 2011 Accepted 22 February 2011 Available online 1 March 2011 Keywords: Enzyme electrodes Membrane Biosensor Diffusion equations Oxidase

a b s t r a c t The theoretical analysis of the steady-state amperometric oxidase enzyme–membrane electrode is developed. The model is based on diffusion equations containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. We employ the homotopy perturbation method (HPM) to solve the system of coupled non-linear diffusion equations for the steady-state condition. Simple and approximate polynomial expressions of concentration of oxygen (mediator), substrate and flux are derived for all possible values of parameters  (Theiele modulus), BO (normalized surface concentration of mediator), and BS (normalized surface concentration of substrate). Furthermore, in this work the numerical solution of the problem is also reported using SCILAB program. The analytical results are compared with the numerical results and found to be in good agreement. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Problems of coupled diffusion and non-linear chemical reactions are found in practical situation [1–3]. In such systems the diffusion of the chemical species into a phase is accompanied by a chemical reaction either with species already presents in the phase, or catalysed by species within the phase. Examples include diffusion and reaction in immobilized enzyme membranes [4–6], diffusion into living cells and micro-organisms, and chemical reactions in high polymer substances. Biosensors, particularly enzyme-based amperometric sensors, have been studied extensively because of their scientific significance and commercial potential in both academic and applied fields [7,8]. In the first generation, enzymes were immobilized via membrane silica–gel (SiO2 + gelatin). This membrane creates a flexible matrix, negligible swelling in aqueous solution and thermal stability on the electrode [1]. In the second generation, glucose oxidases (GODs) were immobilized through a polyvinyl alcohol (PVA) layer and a Prussian blue (PB) mediator. In the last generation, GOD immobilization influence was also studied for the selfassembled monolayers (SAMs) of cysteamine onto the platinum surface [3]. The general principle of enzyme electrodes was introduced about three decades ago by Clark and Lyons [9]. Since then, many biosensors based on electrochemical enzyme electrodes have been described, the majority of these devices operating in an ampero-

∗ Corresponding author. Tel.: +91 0452 2673354; fax: +91 0452 2675238. E-mail address: raj [email protected] (L. Rajendran). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.02.033

metric mode. The development of models for enzyme electrodes provides a better understanding of the individual processes influencing the response of the device, and this information may be used as a guide for directions for improvement of the sensor design. Mathematical models can explain such regularities. The general features of amperometric response were analyzed in the publications of Mell and Maloy [10,11]. There have been many reports on models for enzyme electrodes. Schulmeister [12,13] has described models for multilayer and multienzyme electrodes; these models assumed operation of the electrode under diffusion control, such as the enzyme kinetics are linear with substrate. This allows the reaction and diffusion system to be described by a parabolic differential equation with linear inhomogeneities. A model for a two substrate enzyme electrode has been devised by Leypoldt and Gough [14] where the non-linear enzyme reaction was taken into account. The information gained from modeling can be useful in sensor design, optimization and prediction of the electrode’s response. However till date few of the models proposed for enzyme electrodes [15–19] have been presented with a specific view to electrode design. Cambiaso et al. [20] numerically obtained the transient behaviour of the electrode current in amperometric enzyme sensor for 4 different substrate concentrations using C language. Mackey et al. [21] compared the behaviour of an electrochemical enzyme biosensor with a theoretical analysis based on mathematical model and numerical simulation by the implicit Crank–Nicolson technique [22]. Jobst et al. [23] developed an implicit difference scheme for the simulation of the steady state and transient behaviour of multi-membrane multi-enzyme sensors. Some of the numerous algebraic solutions and simulations known to the literature

S. Loghambal, L. Rajendran / Journal of Membrane Science 373 (2011) 20–28

describe the coupling of reaction and mass transport very versatile [24–26]. Sorochinskii and Kurganov analyzed the non-linear kinetics of cyclic conversions of the substrate in amperometric bienzyme sensors [27]. The transient kinetics of the biosensors was studied by Kulys et al. [28]. Rigorous analytical and numerical solution was reported by Manimozhi et al. [29] for a steady-state substrate concentration at the biosensor at mixed enzyme kinetics and external diffusion limitation in the case of substrate inhibition. Flexer et al. [30] applied relaxation and simplex mathematical algorithms to the study of steady-state electrochemical response of immobilized enzyme biosensors. Mathematical models for the description of the concentration profile measured by a single-layer, single-enzyme electrode placed in a single-line flow-injection system were developed by Kolev [31]. Also Kolev obtained the analytical solution in both the Laplace and the time domains for the special case of pseudo-first-order kinetics. Baronas et al. [32] analyzed a plate-gap model of a porous enzyme doped electrode covered by a porous inert membrane using finite-difference technique. Earlier, the response of an amperometric oxidase enzyme electrode, monitored by the consumption of oxygen, was numerically modeled using a two substrate model by Gooding and Hall [15]. However, to the best of our knowledge, till date no general analytical results for the concentration of mediator and substrate for all values of the parameters , BO and BS have been reported [15]. The purpose of this communication is to derive analytical expressions for concentration of the mediator (oxygen) and substrate in amperometric oxidase enzyme electrode. 2. Mathematical formulation of analysis and problems

Fig. 1. Schematic representation of a typical enzyme–membrane|electrode showing the processes is considered in the model. The model describes the mechanism by which an oxidase enzyme moves from the fully oxidised state to the fully reduced form and back to an oxidised state in a catalytic cycle as shown in the scheme in the text.

balance of oxygen and substrate, within the thickness of matrix may be written as [15] follows:

2.1. Mathematical formulation Building upon earlier work of this electrode, Gooding and Hall [15] presented a concise discussion and the derivation of mass transport equation for an amperometric enzyme electrode, which is summarized briefly for completeness. The model presented in this article is derived from a model developed by Parker and Schwartz [33] for a potentiometric sensor. This model is extended by Martens and Hall [18] for mediated amperometric sensors and Gooding and Hall [15] for oxidase enzyme electrode. The following assumptions include: steady-state conditions apply, diffusion in the matrix obeys Fick’s laws, the matrix is homogeneous, the reaction is isothermal and the rate constants of the immobilized glucose oxidase are the same as the soluble enzyme. 2.1.1. Schematic representation The model describes the mechanism by which an oxidase enzyme moves from the fully oxidised state to the fully reduced form and back to an oxidised state in a catalytic cycle which may be written as follows (see Fig. 1): k1

DO

EOX + S  ES−→Ered + P k−1

k3

Ered + O2 −→EOX + H2 O2 where km is the rate constant for the forward direction of the mth reaction and k−1 is the rate constant for the backward direction. The total enzyme concentration [ET ] at all times is, (1)

where [EOX ], [ES ] and [Ered ] are the oxidised, substrate and reduced mediator enzyme concentrations respectively. The diffusion of a substrate into the enzyme layer at steady-state is equal to the reaction rate of the substrate within the matrix. Thus the material

d2 [O2 ] k2 k1 = k3 [Ered ][O2 ] = [EOX ][S] k−1 + k2 dy2 =

k2 [ET ] (ˇS /[S]) + (ˇO /[O2 ]) + 1

(2)

and DS

d2 [S] = k1 [EO ][S] − k−1 [ES] = dy2 =



k1 −

k−1 k1 k−1 + k2

k2 [ET ] (ˇS /[S]) + (ˇO /[O2 ]) + 1

 [EOX ][S] (3)

where DO and DS are the diffusion coefficients of oxygen and the substrate within the enzyme layer. [O2 ] and [S] are the concentration of the mediator (oxygen) and substrate at any position in the enzyme layer. Following the nomenclature of Athinson and Lester [34], we can write ˇS = (k−1 + k2 )/k1 and ˇO = k2 /k3 . It is important to observe that, from Eqs. (2) and (3), we get DO

k2

[ET ] = [EOX ] + [ES] + [Ered ]

21

d2 [O2 ] d2 [S] = DS 2 dy dy2

(4)

The implication of Eq. (4) is that there is only one independent variable which must be solved to obtain the concentration of the other species. We examine a planar matrix of thickness y = d where diffusion is considered in the y-direction only (edge effects are neglected). 2.1.2. Boundary conditions At the polymer/solution interface, y = d: [O2 ] = [O2 ]b = KO [O2 ]∞ [S] = [S]b = KS [S]∞

(5)

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Table 1 Comparison of the analytical result (Eq. (14)) with the numerical result of normalized oxygen concentration FO when BO = 0.0001, BS = 0.5 and for various values of . X

 = 0.1 Eq. (14)

=1 Numerical

0 0 0 0.2 0.2000 0.2000 0.4 0.4000 0.4000 0.6 0.6000 0.6000 0.8 0.8000 0.8000 1 1 1 Average deviation

=5

% deviation of Eq. (14)

Eq. (14)

Numerical

0 0 0 0 0 0 0

0 0 0.2000 0.2000 0.3999 0.3999 0.5999 0.5999 0.8000 0.8000 1 1 Average deviation

 = 10

% deviation of Eq. (14)

Eq. (14)

Numerical

0 0 0 0 0 0 0

0 0 0.1992 0.1992 0.3985 0.3986 0.5984 0.5984 0.7989 0.7989 1 1 Average deviation

% deviation of Eq. (14)

Eq. (14)

Numerical

0 0 0.0251 0 0 0 0.02319

0 0 0.1970 0.1968 0.3930 0.3945 0.5940 0.5937 0.7950 0.7952 1 1 Average deviation

% deviation of Eq. (14) 0.1015 0.3816 0.0505 0.0251 0 0.0931

Table 2 Comparison of the analytical result (Eq. (15)) with the numerical result of normalized substrate concentration FS when BO = 0.01, BS = 0.01 and for various values of . X

 = 0.1 Eq. (15)

0 0.9999 0.2 0.9999 0.4 0.9999 0.6 0.9999 0.8 0.9999 1 1 1 1 Average deviation

=1 Numerical

% deviation Eq. (15) of Eq. (15)

1 1 1 1 1 1 1

0.0100 0.0100 0.0100 0.0100 0.0100 0 0 0.0083

=5 Numerical

0.9988 0.9988 0.9989 0.9989 0.9989 0.9989 0.9991 0.9991 0.9994 0.9995 1 1 1 1 Average deviation

% deviation Eq. (15) of Eq. (15) 0 0 0 0 0.0100 0 0 0.00167

 = 10 Numerical

0.9717 0.9716 0.9719 0.9721 0.9739 0.9741 0.9786 0.9788 0.9871 0.9872 1 1 1 1 Average deviation

% deviation Eq. (15) of Eq. (15) 0.0062 0.0144 0.0216 0.0163 0.0086 0 0 0.00913

Numerical

0.8866 0.8928 0.8878 0.8945 0.8956 0.9019 0.9146 0.9197 0.9484 0.9513 1 1 1 1 Average deviation

% deviation of Eq. (15) 0.6993 0.7547 0.7079 0.5609 0.2996 0 0 0.5037

Table 3 Comparison of the analytical result (Eq. (14)) with the numerical result of normalized oxygen concentration FO when BO = 0.0005,  = 10 and for various values of normalized surface concentration of substrate BS . X

BS = 0.01 Eq. (14)

0 0 0.2 0.1872 0.4 0.3780 0.6 0.5754 0.8 0.7820 1 1 Average deviation

BS = 0.05 Numerical

% deviation Eq. (14) of Eq. (14)

0 0.1878 0.3790 0.5764 0.7826 1

0 0.3205 0.2645 0.1738 0.0767 0 0.1393

BS = 0.1 Numerical

0 0 0.1848 0.1856 0.3735 0.3748 0.5699 0.5712 0.7776 0.7784 1 1 Average deviation

% deviation Eq. (14) of Eq. (14) 0 0.4329 0.3373 0.2281 0.1029 0 0.1835

at the electrode, y = 0:

BS = 1 Numerical

0 0 0.1844 0.1852 0.3728 0.3741 0.5690 0.5704 0.7768 0.7777 1 1 Average deviation

% deviation Eq. (14) of Eq. (14) 0 0.412 0.3406 0.2390 0.1159 0 0.1846

Numerical

0 0 0.1841 0.1849 0.3722 0.3735 0.5682 0.5696 0.7762 0.7770 1 1 Average deviation

% deviation of Eq. (14) 0 0.4345 0.3493 0.2464 0.1031 0 0.1888

(6)

condition states that the oxygen and substrate in the matrix are in equilibrium with the surrounding solution. The second boundary condition states that at the electrode there is no flux of substrate and that all oxygen that reaches the electrode is consumed. KO and KS are the equilibrium partition coefficients for oxygen and the substrate respectively.

[O2 ]b and [S]b are the concentration of oxygen and substrate at the enzyme layer|electrode boundary, and [O2 ]∞ and [S]∞ are the bulk solution concentrations. The equivalent subscripts have the same meanings for the subtract concentration. The first boundary

2.1.3. Normalized form Non-dimensionalising Eqs. (2) and (3) and accompanying boundary conditions reduce the number of effective parameters

[O2 ] = 0 d[S] =0 dy

Table 4 Comparison of the analytical result (Eq. (15)) with the numerical result of normalized substrate concentration FS when BO = 0.05,  = 10 and for various values of normalized surface concentration of substrate BS . X

BS = 0.01 Eq. (15)

0 0.9329 0.2 0.9335 0.4 0.9378 0.6 0.9486 0.8 0.9686 1 1 Average deviation

BS = 0.05 Numerical

% deviation Eq. (15) of Eq. (15)

0.9346 0.9355 0.9397 0.9502 0.9696 1

0.1822 0.2142 0.2026 0.1687 0.1032 0 0.1452

BS = 0.1 Numerical

0.92078 0.9233 0.9214 0.9242 0.9260 0.9288 0.9382 0.9405 0.9617 0.9631 1 1 Average deviation

% deviation Eq. (15) of Eq. (15) 0.2737 0.2995 0.3013 0.2409 0.1414 0 0.2095

BS = 1 Numerical

0.9189 0.9215 0.9195 0.9225 0.9242 0.9271 0.9366 0.9390 0.9606 0.9620 1 1 Average deviation

% deviation Eq. (15) of Eq. (15) 0.2829 0.3208 0.3181 0.2562 0.1457 0 0.2207

Numerical

0.9170 0.9199 0.9177 0.9208 0.9224 0.9254 0.9350 0.9375 0.9596 0.9610 1 1 Average deviation

% deviation of Eq. (15) 0.3053 0.3378 0.3252 0.2642 0.1459 0 0.2297

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Fig. 2. Normalized concentration profiles of oxidised mediator FO for various values of (a) , (b) BO and (c) BS are plotted using Eq. (14).

Fig. 3. Normalized concentration profiles of substrate FS for various values of (a) , (b) BO and (c) BS are plotted using Eq. (15).

23

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and we arrive as follows:



2 O

=

 S2 =

d2 k2 [ET ] DO [O2 ]b d2 k2 [ET ] DS [S]b



(7)

 (8)

2 and 2 are the Thiele modulus for the oxygen and substrate O S which governs reaction/diffusion, BO = [O2 ]b /ˇO is the normalized surface concentration of oxygen, BS = [Sb ]/ˇS is the normalized surface concentration of the substrate, FO = [O2 ]/[O2 ]b is the normalized oxygen concentration in the matrix, FS = [S]/[S]b is the normalized substrate concentration in the matrix and X = y/d is the normalized distance. Considering O = S =  we get,



d2 FO = 2 dX 2 d2 FS = 2 dX 2



FO FS (FO /BS ) + (FS /BO ) + FO FS FO FS (FO /BS ) + (FS /BO ) + FO FS

 (9)

 (10)

The boundary conditions become, FO = 1,

FS = 1,

FO = 0,

dFS = 0, dX

when X = 1 when X = 0

(11) (12)

The non-linear two-point boundary problem defined by Eqs. (9) and (10) is solved analytically using HPM. The flux J at the electrode surface is given by, J=

 dF  O

dX

X=0

Fig. 4. Concentration profiles of oxidised mediator ( ) and substrate ) for BO = 0.1, BS = 0.05 and for various values of parameters are plotted. (

Appendix A), of the mediator (oxygen) and substrate as follows: FO = w2 [2BS2 [ln(BS ) − ln(w1 X + BS )] + {w2 −1 + 2w1 BS [ln(w1 + BS ) − ln(w1 X + BS )]

(13)

+ 2BS2 [ln(w1 + BS ) − ln(BS )] − w12 }X + w12 X 2 ] 2.2. Analytical solutions of concentrations of mediator and substrate under steady-state condition using the HPM Recently, various different analytical methods were applied to non-linear equations arising in engineering applications, such as the adomian decomposition method [35], homotopy perturbation method [36–39], and exp-function method [40,41]. He [42] solved the boundary value problems (BVPs) governing the motion and energy by an iterative scheme which produced successive approximations that converged rapidly to the desired solution. In this paper, the homotopy perturbation method [42–46] is applied and the obtained results show that the HPM is very effective and simple. The analytical expression of concentrations (refer

(14)

and FS = 1 − w2 [w12 + 2w1 BS [1 − ln(w1 + BS ) + ln(BS )] + 2BS2 [ln(w1 X + BS ) − ln(w1 + BS )] − 2w1 BS [1 − ln(w1 X + BS ) + ln(BS )]X − w12 X 2 ]

(15)

where w1 = BO (1 + BS ),

w2 =

2 BS 2 (1 + B )3 2BO S

(16)

From Eq. (14) we get the dimensionless flux J = 1 − w2 [w1 (w1 + 2BS ) + 2BS (w1 + BS )(ln BS − ln(w1 + BS )]

(17)

Fig. 5. The normalized three-dimensional (a) mediator (oxygen) concentration profile FO (Eq. (14)) and (b) substrate concentration profile FS (Eq. (15)) for  in the range of 1–5 are given.

S. Loghambal, L. Rajendran / Journal of Membrane Science 373 (2011) 20–28

25

Fig. 6. Variation of normalized steady-state flux J with (a) , (b) BS and (c) BO are given. The curves are computed using Eq. (17).

Eq. (17) is the new approximate expression of flux. 2.2.1. For smaller values of BS and BO When BS is small, the dimensionless concentration of mediator (oxygen), substrate and flux are as follows: FO ≈ X + 0.52 X(X − 1)BS FS ≈ 1 + 0.52 (X 2 − 1)BS J ≈ 1 + 0.52 (2X − 1)BS The concentration of mediator (oxygen), substrate and flux for small values BO are as follows: FO ≈ X + 0.672 X(X 2 − 1)BO FS ≈ 1 + 0.672 (X 3 − 1)BO

4. Results and discussion Eqs. (14) and (15) represent the closed and simple approximate analytical expressions of the normalized concentration of mediator (oxygen) and substrate for all values of parameters , BO and BS . Eq. (17) represents the closed and simple analytical expression of flux. The parameter Thiele modulus  can be varied by changing either the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix. This parameter describes the relative importance of diffusion and reaction in the enzyme layer. When 2 is small, the kinetics is the dominant resistance; the overall uptake of mediator and substrate in the enzyme matrix is kinetically controlled. Under these conditions, the substrate concentration profile across the membrane is essentially uniform. The overall kinetics is determined by the total amount of active enzyme. In contrast, when the Thiele module  is large, diffusion limitations are the principal determining factor. It is noted that 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.

J ≈ 1 + 2 (3X 2 − 1)BO /6 4.1. Comparison of analytical and numerical results 3. Numerical simulation The function pdex4 in SCILAB software which is a function of solving the initial-boundary value problems for parabolic-elliptic partial differential equations is used to solve Eqs. (9) and (10). Tables 1–4 illustrate the comparison of analytical result obtained in this work with the numerical result. Upon comparison, it is evident that both the results give satisfactory agreement. The SCILAB program is also given in Appendix B.

Table 1, represents the comparison of analytical result with the numerical result of normalized mediator concentration FO for various values of  = [0.1,1,5,10] when BO = 0.001, BS = 0.5 using Eq. (14). From the table, it is inferred that the concentration FO decreases when  increases or the thickness of the electrode increases. The maximum percentage of relative error is 0.0931 when  = 10. Normalized substrate concentration FS (Eq. (15)) for various values of  and some fixed values of BO and BS is compared with the simulation results in Table 2. The average relative error for FS is

26

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Fig. 7. The normalized three-dimensional steady-state flux J versus  and (a) BO (b) BS are plotted using Eq. (17).

0.0083%, 0.00167%, 0.00913% and 0.5037% when  = 0.1, 1, 5 and 10 respectively. From this table, it is known that of normalized substrate concentration FS ≈ 1 when the value of  ≤ 5. Similarly the normalized concentration of oxygen FO (Table 3) and concentration of substrate FS (Table 4) are compared numerically for various values of normalized surface concentration of substrate BS . In all the cases, the average relative error is less than 0.23%. From these tables, we conclude that there is no significant difference in normalized concentration of mediator FO when Thiele module  ≤ 10. 4.2. Determination of FO and FS Fig. 2 represents the normalized concentration profiles of oxidised mediator FO for various values of , BO and BS using Eq. (14). It is clear that for the increasing values of all the parameters , BO and BS the profile deviates more from the linear profile as oxygen is consumed in the enzyme reaction. From these figures it is inferred that the concentration of mediator increases when , BO and BS decreases. Normalized concentration profiles of substrate FS for various values of , BO and BS are plotted using Eq. (15) in Fig. 3. From these figures, it is evident that the value of concentration is very close to the electrode surface (FS ≈ 1) for the small values of all the parameters , BO and BS . The concentration profiles for the oxidised mediator FO and substrate FS for various values of  are shown in Fig. 4. Oxidised mediator FO is consumed by the enzyme reaction as the mediator moves inwards from the electrode interface. The minor change in substrate concentration across the matrix indicates that the oxidised mediator is limit under these conditions rather than the substrate itself. Also FS = 1 and FO is linear when  ≤ 1. Also from this figure, it is inferred that the small change in  produces significant change only in the concentration of substrate FS . The concentration of mediator FO is same where  ≤ 1. The normalized three-dimensional mediator (oxygen) concentration profile FO and substrate concentration profile FS are plotted in Fig. 5 where the data given by Figs. 2–4 is verified. 4.3. Determination of J The parameter of greatest interest in an amperometric biosensor is the current, which is related to the flux of electroactive material to the electrode surface. Eq. (17) is a simple closed form of analytical expression of the flux. Fig. 6 describes variation of normalized flux response J for various values of parameters , BO and BS . From these figures, it is confirmed that the value of flux decreases with

an increase in , BS and BO . Also the value of the flux increases when thickness of the electrode decreases. Fig. 7 characterizes the normalized three-dimensional steadystate flux response J. The flux reaches the steady-state value when the thickness of the membrane is less than or equal to one. These figures confirm the results given by Fig. 6. 5. Conclusions A theoretical model for a two substrate has been described and applied for the simulation of a mediated enzyme electrode in the presence of oxygen. In this work, system of coupled, steady-state non-linear reaction/diffusion equation has been formulated and solved analytically. Approximate analytical expressions for the concentrations and flux in amperometric oxidase enzyme electrode for homogeneous catalytic reactions with equal diffusion coefficients at a planar enzyme–membrane electrode under steady-state conditions are obtained using the homotopy perturbation method. The primary result of this work is simple approximate calculation of concentration profiles and flux for small values of fundamental parameters. The small variation in Thiele modulus (or thickness of enzyme electrode) caused a significant change in both the magnitude of the current response and the general behaviour of the system. The homotopy perturbation method is extremely simple and promising to solve other non-linear equations. This work can be easily extended to find the solution for other enzymatic systems in which the described boundary conditions and the homogeneous media for the membrane could be applied. Acknowledgements It is our pleasure to thank the referees for their valuable comments. 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 and Dr. T.V. Krishnamoorthy, Principal, S. Thiagarajan, Head of the Department of Mathematics, The Madura College, Madurai, India for their constant encouragement. Appendix A. Approximate analytical solutions of the oxygen and substrate concentration Using homotopy perturbation method, we construct a homotopy for the Eqs. (9) and (10) as follows



(1 − p)

d2 FO dX 2





d2 FO +p − 2 dX 2



FO FS (FO /BS ) + (FS /BO ) + FO FS



= 0(A1)

S. Loghambal, L. Rajendran / Journal of Membrane Science 373 (2011) 20–28

and



(1 − p)

2

d FS dX 2



 +p

2

d FS − 2 dX 2



FO FS (FO /BS ) + (FS /BO ) + FO FS



Appendix B. = 0(A2) function pdex4

The approximate solution of (A1) is as follows: FO = FO,0 + pFO,1 + p2 FO,2 + . . .

(A3)

The approximate solution of (A2) is as follows: FS = FS,0 + pFS,1 + p2 FS,2 + . . .

(A4)

Substituting Eqs. (A3) and (A4) into Eq. (A1) and arranging the coefficients of p powers, we have p0 : p1 :

d2 FO,0 dX 2 d2 FO,1 dX 2

=0

(A5)

 − 2

FO,0 FS,0 (FO,0 /BS ) + (FS,0 /BO ) + FO,0 FS,0

 =0

(A6)

Substituting Eqs. (A3) and (A4) into Eq. (A2) and arranging the coefficients of p powers, we have p0 : p1 :

d2 FS,0 dX 2 d2 FS,1 dX 2

=0

(A7)

 − 2

FO,0 FS,0 (FO,0 /BS ) + (FS,0 /BO ) + FO,0 FS,0

 =0

(A8)

The initial approximations are as follows: FO,0 (X = 1) = 1,

FO,0 (X = 0) = 0

(A9)

and FO,i (X = 1) = 0,

FO,i (X = 0) = 0

FS,0 (X = 1) = 1,

dFS,0 (X = 0) =0 dX

for all i = 1, 2, 3, . . .

(A10) Nomenclature (A11)

and FS,i (X = 1) = 0,

dFS,i (X = 0) =0 dX

m = 0; x = [0 0.2 0.4 0.6 0.8 1]; t = [0246810]; sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u1 = sol(:,:,1); u2 = sol(:,:,2); figure plot(x,u1(end,:)) title(’Solution at t = 2’) xlabel(’Distance x’) ylabel(’u1(x,2)’) figure plot(x,u2(end,:)) title(’Solution at t = 2’) xlabel(’Distance x’) ylabel(’u2(x,2)’) % ————————————————————– function [c,f,s] = pdex4pde(x,t,u,DuDx) Bs = 0.5; Bo = 0.001; A = 10; c = [1;1]; f = [1; 1].* DuDx; ˆ F1 = -A2*(u(1)*u(2)/(u(1)/Bs + u(2)/Bo + (u(1)*u(2)))); ˆ F2 = -A2*(u(1)*u(2)/(u(1)/Bs + u(2)/Bo + (u(1)*u(2)))); s = [F1; F2]; % ————————————————————– function u0 = pdex4ic(x); u0 = [1; 0]; % ————————————————————– function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [ul(1); 0]; ql = [0; 1]; pr = [ur(1)-1; ur(2)-1]; qr = [0; 0];

for all i = 1, 2, 3, . . .

(A12)

From Eq. (A5) we get FO,0 = X

(A13)

From Eq. (A7) we get FS,0 = 1

(A14)

Symbols [ET ] [EO ] [ES] [Ered ] DO DS d [O2 ] [O2 ]b

Substituting Eqs. (A13) and (A14) in Eq. (A6) we obtain the solution of Eq. (A6),

[O2 ]∞ [S]

FO,1 = w2 [2BS2 [ln(BS ) − ln(w1 X + BS )]

[S]b

+ {2w1 BS [ln(w1 + BS ) − ln(w1 X + BS )] + 2BS2 [ln(w1 + BS ) − ln(BS )] − w12 }X + w12 X 2 ]

(A15)

Using Eqs. (A13) and (A14) in Eq. (A8) and then solving we get, FS,1 = −w2 [w12 + 2w1 BS [1 − ln(w1 + BS ) + ln(BS )]

KS BO BS

+ 2BS2 [ln(w1 X + BS ) − ln(w1 + BS )] − 2w1 BS [1 − ln(w1 X + BS ) + ln(BS )]X − w12 X 2 ]

[S]∞ KO

(A16)

FO FS

Adding (A13) and (A15) we get Eq. (14) in the text. Similarly adding (A14) and (A16) we get, Eq. (15) in the text.

total enzyme concentration in the matrix (␮M) enzyme concentration of the oxygen (␮M) enzyme concentration of the substrate (␮M) reduced enzyme concentration (␮M) diffusion coefficient of oxygen (cm2 s−1 ) diffusion coefficient of substrate (cm2 s−1 ) thickness of the planar matrix (cm) concentration of oxygen at any position in the enzyme layer (mol cm−3 ) oxygen concentration at the enzyme layer|electrode boundary (mM) oxygen concentration in bulk solution (mol cm−3 ) concentration of substrate at any position in the enzyme layer (mol cm−3 ) substrate concentration at the enzyme layer|electrode boundary (mM) substrate concentration in bulk solution (mol cm−3 ) equilibrium partition coefficients for the oxygen (dimensionless) equilibrium partition coefficients for the substrate normalized surface concentration of mediator (dimensionless) normalized surface concentration of the substrate (dimensionless) normalized concentration of the mediator (dimensionless) normalized concentration of the substrate (dimensionless)

27

28

S. Loghambal, L. Rajendran / Journal of Membrane Science 373 (2011) 20–28

X J k1 , k3 k−1 , k2

normalized distance (dimensionless) dimensionless flux (dimensionless) rate constants (M−1 s−1 ) rate constants (s−1 )

Greek symbols 2 O Thiele modulus for the mediator (dimensionless) S2 Thiele modulus for the substrate (dimensionless) Subscripts OX oxidised T total red reduced o oxygen s substrate b boundary ∞ bulk

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