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

Authors: Loghambal Shunmugham; L. Rajendran
Journal: International Journal of Electrochemistry
 (2013)

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## 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 parametersϕO2 (Thiele modulus), BO (normalized surface concentration of oxidized mediator), and BS (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.


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## Body

## 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 ϕO2, BO, and BS 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]:
(1)EOX+S⇄k1k-1ES→k2Ered+P(2)Ered+O2→k3EOX+H2O2(3)Ered+MedOX→k4EOX+Medred
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. If [ET] is the total enzyme concentration in the matrix, then at all times
(4)[ET]=[EOX]+[ES]+[Ered],
where [EOX], [ES], and [Ered] 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](5a)DMd2[MedOX]dy2=k4[Ered][MedOX]=[EOX][S]k2k1k-1+k2=k2[ET](βS[S]+βO[MedOX]+1)-1,(5b)DSd2[S]dy2=k1[EOX][S]-k-1[ES]=[EOX][S](k1-k-1k1k-1+k2)=k2[ET](βS[S]+βO[MedOX]+1)-1,(5c)DMd2[Medred]dy2=-k4[Ered][MedOX]=-k2[ET](βS[S]+βO[MedOX]+1)-1,where DM is the diffusion coefficient of the oxidized and reduced forms of the mediator (assumed to be equal) and DS is the diffusion coefficient of substrate within the enzyme layer. [MedOX], [Medred], and [S] are the concentration of oxidized mediator, reduced mediator, and substrate at any position in the enzyme layer. βS and βO are the dimensionless rate constants (βS=(k-1+k2)/k1 and βO=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=0(6)d[MedOX]dy=d[S]dy=d[Medred]dy=0
and, at the electrode, y=d(7)[MedOX]=[MedOX]b=KO[MedOX]∞,[S]=[S]b=KS[S]∞,[Medred]=0.
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. [MedOX]b and [S]b are the concentration of oxidized mediator and substrate at the enzyme layer|electrode boundary, and [MedOX]∞ and [S]∞ are the bulk solution concentrations. KO and KS 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:
(8)FO=[MedOX][MedOX]b,FS=[S][S]b,FR=[Medred][Medred]b,χ=yd,BO=[MedOX]bβO,BS=[S]bβS,ϕO2=d2k2[ET]DM[MedOX]b,μS=DM[MedOX]bDS[S]b,
where FO, FS, and FR are the normalized concentration of oxidized mediator, substrate, and reduced mediator in the matrix, respectively, and χ is the normalized distance. BO and BS are the normalized surface concentration of oxidized mediator and substrate. ϕO2 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:(9a)d2FOdχ2=ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS],(9b)d2FSdχ2=μSϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS],(9c)d2FRdχ2=-ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS].
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
(9d)d2FOdχ2=1μSd2FSdχ2=-d2FRdχ2=ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS].The normalized boundary conditions are given by
(10)FO′(0)=0,FS′(0)=0,FR′(0)=0,FO(1)=1,FS(1)=1,FR(1)=0.
From (9d), we get
(11)d2FOdχ2=1μSd2FSdχ2,(12)d2FOdχ2=-d2FRdχ2.
Integrating (11) and (12) twice and applying the appropriate boundary conditions (10), we get
(13)FS(χ)=μS(FO(χ)-1)+1,(14)FR(χ)=1-FO(χ).
Substituting (13) into (9a) and rearranging, we get
(15)d2FOdX2=ϕ2[(FO2+FO(1BSμS+1BO+1-μSμS)+1-μSμSBO)-1(FO2+FO[(1-μS)μS])ddddd×(FO2+FO(1BSμS+1BO+1-μSμS)+1-μSμSBO)-1].
The normalized current response is given by the following expression:
(16)I=-(dFRdχ)χ=1.


## 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]:
(1)EOX+S⇄k1k-1ES→k2Ered+P(2)Ered+O2→k3EOX+H2O2(3)Ered+MedOX→k4EOX+Medred
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. If [ET] is the total enzyme concentration in the matrix, then at all times
(4)[ET]=[EOX]+[ES]+[Ered],
where [EOX], [ES], and [Ered] 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](5a)DMd2[MedOX]dy2=k4[Ered][MedOX]=[EOX][S]k2k1k-1+k2=k2[ET](βS[S]+βO[MedOX]+1)-1,(5b)DSd2[S]dy2=k1[EOX][S]-k-1[ES]=[EOX][S](k1-k-1k1k-1+k2)=k2[ET](βS[S]+βO[MedOX]+1)-1,(5c)DMd2[Medred]dy2=-k4[Ered][MedOX]=-k2[ET](βS[S]+βO[MedOX]+1)-1,where DM is the diffusion coefficient of the oxidized and reduced forms of the mediator (assumed to be equal) and DS is the diffusion coefficient of substrate within the enzyme layer. [MedOX], [Medred], and [S] are the concentration of oxidized mediator, reduced mediator, and substrate at any position in the enzyme layer. βS and βO are the dimensionless rate constants (βS=(k-1+k2)/k1 and βO=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=0(6)d[MedOX]dy=d[S]dy=d[Medred]dy=0
and, at the electrode, y=d(7)[MedOX]=[MedOX]b=KO[MedOX]∞,[S]=[S]b=KS[S]∞,[Medred]=0.
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. [MedOX]b and [S]b are the concentration of oxidized mediator and substrate at the enzyme layer|electrode boundary, and [MedOX]∞ and [S]∞ are the bulk solution concentrations. KO and KS 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:
(8)FO=[MedOX][MedOX]b,FS=[S][S]b,FR=[Medred][Medred]b,χ=yd,BO=[MedOX]bβO,BS=[S]bβS,ϕO2=d2k2[ET]DM[MedOX]b,μS=DM[MedOX]bDS[S]b,
where FO, FS, and FR are the normalized concentration of oxidized mediator, substrate, and reduced mediator in the matrix, respectively, and χ is the normalized distance. BO and BS are the normalized surface concentration of oxidized mediator and substrate. ϕO2 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:(9a)d2FOdχ2=ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS],(9b)d2FSdχ2=μSϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS],(9c)d2FRdχ2=-ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS].
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
(9d)d2FOdχ2=1μSd2FSdχ2=-d2FRdχ2=ϕO2[BOBSFOFSBOFO+BSFS+BOBSFOFS].The normalized boundary conditions are given by
(10)FO′(0)=0,FS′(0)=0,FR′(0)=0,FO(1)=1,FS(1)=1,FR(1)=0.
From (9d), we get
(11)d2FOdχ2=1μSd2FSdχ2,(12)d2FOdχ2=-d2FRdχ2.
Integrating (11) and (12) twice and applying the appropriate boundary conditions (10), we get
(13)FS(χ)=μS(FO(χ)-1)+1,(14)FR(χ)=1-FO(χ).
Substituting (13) into (9a) and rearranging, we get
(15)d2FOdX2=ϕ2[(FO2+FO(1BSμS+1BO+1-μSμS)+1-μSμSBO)-1(FO2+FO[(1-μS)μS])ddddd×(FO2+FO(1BSμS+1BO+1-μSμS)+1-μSμSBO)-1].
The normalized current response is given by the following expression:
(16)I=-(dFRdχ)χ=1.


## 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 AppendixA) 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 FO(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:
(17)FO(χ)=1+ϕO2BOBS2(BO+BS+BOBS)×[5w1-1+(1-6w1)χ2+w1χ4],
where w1=ϕO2BOBS(BS+BOμS)/12(BO+BS+BOBS)2.Using (17) (or the dimensionless concentration FO(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
(18)I=ϕO2BOBS(1-4w1)(BO+BS+BOBS).


### 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 FO is less than 0.3% for various values of ϕO2. The normalized current I is compared with simulation result in Table 3 for various values of parameters ϕO2. 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 mediatorFO (17) with the numerical results when BO=0.1, BS=0.01, and μS=0.05 and for various values of ϕO2.
χ
ϕ
O
2
=
0.01
ϕ
O
2
=
1
ϕ
O
2
=
25
ϕ
O
2
=
100
ϕ
O
2
=
225
(d=0.7878 μM)
(d=7.878 μM)
(d=39.39 μM)
(d=78.80 μM)
(d=118.17 μM)
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
0
0.9999
1
0.01
0.9955
0.9955
0.00
0.8889
0.8889
0.00
0.5801
0.5773
0.46
0.1895
0.1908
0.69
0.2
0.9999
1
0.01
0.9957
0.9957
0.00
0.8933
0.8933
0.00
0.5965
0.5939
0.43
0.2197
0.2200
0.14
0.4
0.9999
1
0.01
0.9962
0.9962
0.00
0.9066
0.9066
0.00
0.6462
0.6440
0.34
0.3115
0.3101
0.45
0.6
0.9999
1
0.01
0.9971
0.9971
0.00
0.9288
0.9288
0.00
0.7295
0.7279
0.22
0.4686
0.4664
0.47
0.8
0.9999
1
0.01
0.9984
0.9984
0.00
0.9599
0.9599
0.00
0.9484
0.8471
0.85
0.6962
0.6947
0.21
1
1
1
0.0
1
1
0.00
1
1
0.00
1
1
0.00
1
1
0.00
Average deviation
0.01
Average deviation
0.00
Average deviation
0.00
Average deviation
0.38
Average deviation
0.33Figure 2
Dimensionless concentration of (a) oxidized mediatorFO (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 ϕO2=400. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution.Dimensionless concentration of (a) oxidized mediatorFO (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 ϕO2. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution. (i) ϕO2=0.01 (d=0.7878 μM). (ii) ϕO2=1 (d=7.878 μM). (iii) ϕO2=25 (d=39.39 μM). (iv) ϕO2=100 (d=78.80 μM). (v) ϕO2=225 (d=118.17 μM). (vi) ϕO2=225 (d=118.17 μM).
(a)
(b)
(c)


## 3.1. Analytical Solution Using ADM

In this paper, the Adomian decomposition method (see AppendixA) 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 FO(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:
(17)FO(χ)=1+ϕO2BOBS2(BO+BS+BOBS)×[5w1-1+(1-6w1)χ2+w1χ4],
where w1=ϕO2BOBS(BS+BOμS)/12(BO+BS+BOBS)2.Using (17) (or the dimensionless concentration FO(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
(18)I=ϕO2BOBS(1-4w1)(BO+BS+BOBS).


## 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 FO is less than 0.3% for various values of ϕO2. The normalized current I is compared with simulation result in Table 3 for various values of parameters ϕO2. 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 mediatorFO (17) with the numerical results when BO=0.1, BS=0.01, and μS=0.05 and for various values of ϕO2.
χ
ϕ
O
2
=
0.01
ϕ
O
2
=
1
ϕ
O
2
=
25
ϕ
O
2
=
100
ϕ
O
2
=
225
(d=0.7878 μM)
(d=7.878 μM)
(d=39.39 μM)
(d=78.80 μM)
(d=118.17 μM)
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
(17)
Num
Error %
0
0.9999
1
0.01
0.9955
0.9955
0.00
0.8889
0.8889
0.00
0.5801
0.5773
0.46
0.1895
0.1908
0.69
0.2
0.9999
1
0.01
0.9957
0.9957
0.00
0.8933
0.8933
0.00
0.5965
0.5939
0.43
0.2197
0.2200
0.14
0.4
0.9999
1
0.01
0.9962
0.9962
0.00
0.9066
0.9066
0.00
0.6462
0.6440
0.34
0.3115
0.3101
0.45
0.6
0.9999
1
0.01
0.9971
0.9971
0.00
0.9288
0.9288
0.00
0.7295
0.7279
0.22
0.4686
0.4664
0.47
0.8
0.9999
1
0.01
0.9984
0.9984
0.00
0.9599
0.9599
0.00
0.9484
0.8471
0.85
0.6962
0.6947
0.21
1
1
1
0.0
1
1
0.00
1
1
0.00
1
1
0.00
1
1
0.00
Average deviation
0.01
Average deviation
0.00
Average deviation
0.00
Average deviation
0.38
Average deviation
0.33Figure 2
Dimensionless concentration of (a) oxidized mediatorFO (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 ϕO2=400. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution.Dimensionless concentration of (a) oxidized mediatorFO (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 ϕO2. Solid lines represent the analytical solution whereas the dotted lines represent the numerical solution. (i) ϕO2=0.01 (d=0.7878 μM). (ii) ϕO2=1 (d=7.878 μM). (iii) ϕO2=25 (d=39.39 μM). (iv) ϕO2=100 (d=78.80 μM). (v) ϕO2=225 (d=118.17 μM). (vi) ϕO2=225 (d=118.17 μM).
(a)
(b)
(c)


## 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 FO, substrate FS, and reduced mediator FR, respectively, for all values of parameters ϕO2, BO, and BS. 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.
Parameters
Numerical values
Gooding and Hall [9]
This work
ϕ
O
2
=
d
2
k
2
[
E
T
]
/
D
M
[
Med
OX
]
b
10–400
0.01–400
B
O
=
[
Med
OX
]
b
/
β
O
0.00195
0-1
B
S
=
[
S
]
b
/
β
S
0.018, 0.035, 0.07, 0.14
0-1
μ
S
=
D
M
[
Med
OX
]
b
/
D
S
[
S
]
b
0.94736
0.05, 0.5Table 3
Comparison of normalized currentI (18) with the numerical results when BO=0.1, BS=0.01, and μS=0.05 and for various values of ϕO2.
ϕ
O
2
Normalized currentI
Error %
Analytical
Numerical
1
0.009
0.009
0
25
0.223
0.223
0
50
0.441
0.441
0
75
0.655
0.652
0.5
100
0.864
0.838
3


### 4.1. Concentration Profiles

The concentration profiles for all the three species are shown in Figure2. The condition that the oxidized mediator FO (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 FS (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 FR (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 modulusϕO2 essentially compares the reaction and diffusion in the enzyme layer. We examine the rise and downfall of concentration profiles in two cases. (a) When ϕO2 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 (ϕO2>1), which is observed at high catalytic activity and great membrane thickness or low diffusion coefficient values.In Figure3, under the consideration of lower ϕO2 and the sensor under kinetic control, the concentration profile varies slightly from nonenzyme linked oxygen. As ϕO2 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 ϕO2, 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 FS causes a decrease in the flux to the electrode. The concentration of the reduced mediator FR increases in direct proportion to the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix.In Figure4, 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 (BS≤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.Dimensionless concentration of (a) oxidized mediatorFO (17), (b) substrate FS (13), and (c) reduced mediator FR (14) for some fixed values of parameters BO=0.17 and μS=0.5, ϕO2=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.
(a)
(b)
(c)


### 4.3. Current Response

The parameter of the greatest interest in an amperometric biosensor is the current. For a particular membrane, the variation inϕO2 can be achieved practically, by varying the thickness of the membrane d or the loading of the enzyme [ET]. 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 ϕO2 is shown in Figure 5. It is evident from the figure that the current I (18) increases when ϕO2 increases. Furthermore, when BO and BS 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.Variation of normalized currentI with (a) normalized surface concentration of oxidized mediator BO and (b) normalized surface concentration of the substrate BS for various values of Thiele modulus ϕO2 using (18). (i) ϕO2=50 (d=55.71 μM) (ii) ϕO2=100 (d=78.80 μM) (iii) ϕO2=200 (d=111.42 μM) (iv) ϕO2=300 (d=136.46 μM) (v) ϕO2=400 (d=158.00 μM).
(a)
(b)


## 4.1. Concentration Profiles

The concentration profiles for all the three species are shown in Figure2. The condition that the oxidized mediator FO (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 FS (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 FR (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 modulusϕO2 essentially compares the reaction and diffusion in the enzyme layer. We examine the rise and downfall of concentration profiles in two cases. (a) When ϕO2 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 (ϕO2>1), which is observed at high catalytic activity and great membrane thickness or low diffusion coefficient values.In Figure3, under the consideration of lower ϕO2 and the sensor under kinetic control, the concentration profile varies slightly from nonenzyme linked oxygen. As ϕO2 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 ϕO2, 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 FS causes a decrease in the flux to the electrode. The concentration of the reduced mediator FR increases in direct proportion to the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix.In Figure4, 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 (BS≤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.Dimensionless concentration of (a) oxidized mediatorFO (17), (b) substrate FS (13), and (c) reduced mediator FR (14) for some fixed values of parameters BO=0.17 and μS=0.5, ϕO2=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.
(a)
(b)
(c)


## 4.3. Current Response

The parameter of the greatest interest in an amperometric biosensor is the current. For a particular membrane, the variation inϕO2 can be achieved practically, by varying the thickness of the membrane d or the loading of the enzyme [ET]. 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 ϕO2 is shown in Figure 5. It is evident from the figure that the current I (18) increases when ϕO2 increases. Furthermore, when BO and BS 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.Variation of normalized currentI with (a) normalized surface concentration of oxidized mediator BO and (b) normalized surface concentration of the substrate BS for various values of Thiele modulus ϕO2 using (18). (i) ϕO2=50 (d=55.71 μM) (ii) ϕO2=100 (d=78.80 μM) (iii) ϕO2=200 (d=111.42 μM) (iv) ϕO2=300 (d=136.46 μM) (v) ϕO2=400 (d=158.00 μM).
(a)
(b)


## 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 parametersϕO2, BO, and BS 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.



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*Source: 812856-2013-12-24.xml*