# Theoretical Analysis of an Amperometric Biosensor Based on Parallel Substrates Conversion

Authors: T. Praveen; L. Rajendran
Journal: ISRN Electrochemistry
 (2014)

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## Abstract

The behaviour of an amperometric biosensor based on parallel substrates conversion for steady-state condition has been discussed. This analysis contains a nonlinear term related to enzyme kinetics. Simple and closed form of analytical expressions of concentrations and of biosensor current is derived. This model was originally reported by Vytautas Aseris and his team (2012). Concentrations of substrate and product are expressed in terms of single dimensionless parameter. A new approach to Homotopy perturbation method (HPM) is employed to solve the system of nonlinear reaction diffusion equations. Furthermore, in this work, the numerical solution of the problem is also reported using Matlab program. The analytical results are compared with the numerical results. The analytical result provided is reliable and efficient to understand the behavior of the system.


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## Body

## 1. Introduction

A biosensor is an analytical device, used for the detection of an analyte, which combines a biological component with a physiochemical detector [1]. Analyzed substrate is biochemically converted to a product when the biosensor operation is being undertaken. In most of the cases, biosensor response is directly proportional to the concentration of the reaction product [2]. The amperometric biosensors have proved to be reliable in various systems with applications in the field of medicine, food technology, and the environmental industry [3, 4]. The understanding of the kinetic peculiarities of biosensors is of crucial importance for their design. The mathematical modeling is rather widely used to improve the efficiency of the biosensors design and to optimize their configuration [5–8]. Since 1970s, various mathematical models of biosensors have been developed and used successfully to study and optimize analytical characteristics of biosensors [9–14]. Mathematical modeling of two-enzyme biosensors has been started in 1980s with the modeling of an amperometric monolayer enzyme electrode with two coimmobilized enzymes [15, 16]. Later, nonlinear mathematical models have been developed for amperometric two-enzyme biosensors with different enzymes [17, 18].The numerical method of solving partial differential equations is to make calculation at all intervals of substrates concentration and at different diffusion and enzymatic reaction rates. The nonstationary diffusion equations [19], containing a nonlinear term related to the enzymatic reaction, are carried out using the implicit difference scheme [20]. In recent years, analytical solutions are reported for various types of biosensors [21–27].To the best of our knowledge, there is no simple analytical expression for steady-state concentrations and current of amperometric-based biosensor utilizing parallel substrates conversion for all values of parameters. In this paper, we present the simple and closed form of the analytical expressions for the concentrations of substrate and of product using new Homotopy perturbation method (HPM) [28]. The current density was determined corresponding to all possible values of the parameters.


## 2. Mathematical Formulation of the Problem and Analysis

The reaction scheme employing two parallel substrates (S1 and S2), two enzymes (E1 and E2), and two reaction products (P1 and P2) in biosensor may be as follows:
(1)S1→E112P1,S1+S2→E2P2.First phase of the biosensor action contains hydrogen peroxide (S1) only. The biosensor response reaches the steady-state at the end of the first phase. The second phase of the biosensor action begins at the second substrate (S2) poured into the solution. The relative changes between the responses in both phases are measured as the final response of the biosensor. The schematic diagram of the system is provided in Figure 1. The above reaction in the enzymatic layer with the one-dimensional-in-space diffusion, described by Fick’s law, leads to the following nonlinear transient reaction-diffusion equations [29]:
(2)DS1,e∂2S1,e∂x2-k1e1S1,e-k21k22e2S1,eS2,ek21S1,e+k22S2,e=0,DS2,e∂2S2,e∂x2-k21k22e2S1,eS2,ek21S1,e+k22S2,e=0,DP1,e∂2P1,e∂x2+k1e1S1,e2=0,DP2,e∂2P2,e∂x2+k21k22e2S1,eS2,ek21S1,e+k22S2,e=0,
where x denotes space, S1,e, S2,e are the molar concentration of the substrates, and P1,e, P2,e are the molar concentration of products. DS1,e, DS2,e, DP1,e, and DP2,e are the corresponding diffusion coefficients and k1, k21, and k22 are the reaction rate constants. Graphical representation of the products P1,e, P2,e and substrates S1,e, S2,e, with steady-state boundary conditions employed in the simulated multilayered biosensor is provided in Figure 2. The mass transport by diffusion of the substrates and products takes place outside the enzyme layer. The diffusion layer is treated as the Nernst diffusion layer. The boundary conditions for the above system of nonlinear equation are as follows:
(3)DCe∂ce∂x|x=0=0,P1,e(x=0)=0,c=S1,S2,P2Si,e(x=de+dd)=Si0,Pi,e(x=de+dd)=0,i=1,2,
where de and dd denote the thickness of the enzyme and diffusion layer. For the steady-state condition, the following sets of dimensionless variables are introduced:
(4)S1N=S1,eS10,S2N=S2,eS20,P1N=P1,eS10,P2N=P2,eS10,y=xd,γS1=d2k1e1DS1,e,γS2=d2k22e2DS1,e,γS3=d2k22e2DS2,e,γP1=d2k1e1DP1,e,γP2=d2k1e1DP2,e,α=S10S20,β=k22k21,
where d=de+dd. The governing nonlinear reaction diffusion equations (2) are expressed in the following steady-state nondimensional form:
(5)d2S1Ndy2=γS1S1N+γS2S1NS2NαS1N+βS2N,(6)d2S2Ndy2=γS3S1NS2NαS1N+βS2N,(7)d2P1Ndy2=-γP1S1N2,(8)d2P2Ndy2=-γP2S1NS2NαS1N+βS2N,
where S1N and S2N represent the dimensionless substrate concentrations, P1N and P2N represent the dimensionless product concentrations, and y represents the dimensionless distance. The reaction/diffusion parameters of substrates and products are considered to be γS1, γS2, γS3, γP1, and γP2. The boundary conditions are represented as follows:
(9)dS1Ndy=dS2Ndy=dP2Ndy=0,P1N=0wheny=0,S1N=S2N=1,P1N=P2N=0wheny=1.
The dimensionless current density is given by
(10)ψ=isdneFDP1,eAS10=dP1Ndy|y=0.Figure 1
Schematic representation of biosensor utilising parallel substrate conversion with enzymatic layer (of thicknessde) and the diffusional layer (of thickness dd) the bioelectrode functioning in the presence of hydrogen peroxide (S1) or catalyses (S2). E1 and E2 represent the catalase and peroxidase activities. P1 and P2 are products.Figure 2
Graphical representation of steady-state boundary conditions of the productsP1,e, P2,e and substrates S1,e, S2,e, for multilayered biosensor. Here, x=0 is taken at the surface of the electrode and x=de+dd is taken as the surface of the diffusional layer.


## 3. A New Approach to Homotopy Perturbation Method and Its Advantages

The Homotopy perturbation method is first introduced to solve nonlinear wave equations. The advantage of the method is that it does not need a small parameter in the system, leading to wide application in nonlinear wave equations [30]. Recently, many authors have used HPM to various problems and reported the efficiency of the HPM for handling nonlinear engineering problems [31–34]. This method is a combination of Homotopy in topology and classic perturbation techniques. He used the HPM to solve the Light hill equation [35]; this wide variety of applications shows the power of the HPM to solve functional equations. The HPM has uniqueness in its applicability, accuracy, and efficiency. Recently, a new approach to HPM is introduced to solve the nonlinear problem, in which one will get better simple approximate solution in the zeroth iteration [28].


## 4. Analytical Expression for the Concentration of Substrate and Product Using New Homotopy Perturbation Method

In this paper, a new approach to Homotopy perturbation method is used (AppendixA) to solve the nonlinear differential equations (5)–(8). Using this method, the analytical expression of the two substrate concentrations (S1N and S2N) and two product concentrations (P1N and P2N) can be obtained as follows:(11)S1N(y)=cosh⁡⁡(Ay)cosh⁡⁡(A),(12)S2N(y)=cosh⁡⁡(By)cosh⁡⁡(B),(13)P1N(y)=γP1[1-cosh⁡⁡(Ay)+ycosh⁡⁡(A)-y]2A2cosh⁡⁡(A)(14)P2N(y)=γP2γS3[1-cosh⁡⁡(By)cosh⁡⁡(B)],
where
(15)A=γS1(α+β)+γS2α+β,B=γS3α+β.
Equations (11)–(14) satisfy the boundary conditions (9). These equations represent the simple and closed form of analytical expressions of concentrations of substrate and product for all possible values of the parameters. Using (6) and (8) with boundary conditions (9), the following relation is also obtained:
(16)P2N(y)=γP2γS3[1-S2N(y)].
The analytical expression of the current using (10) is as follows:
(17)ψ=idneFADP1,eS10=γP1[cosh⁡⁡(A)-1]2A2cosh⁡⁡(A).
When A→0, the above expression can be simplified as ψ=γP1/4.


### 4.1. Limiting Case

The mathematical as well as the corresponding computational models of the biosensor were evaluated. Assume that zero concentration of the second substrate (S20=0) leads to reducing the governing equations (5)–(8) to the following equations:
(18)d2S1Ndy2=γS1S1N,d2P1Ndy2=-γP1S1N2.
The steady-state solution of the above equations for the boundary condition (9) is given by
(19)S1,N=cosh⁡⁡(γS1y)cosh⁡⁡(γS1),P1N(y)=γP1[1-cosh⁡⁡(γS1y)+ycosh⁡⁡(γS1)-y]2γS1cosh⁡⁡(γS1).
The dimensionless steady-state current using (10) is given by
(20)ψ=idneFADP1,eS10=γP1[cosh⁡⁡(γS1)-1]2A2cosh⁡⁡(γS1).
Equation (20) is identical to (17), when S20=0 (i.e., α=∞) in (17).


## 4.1. Limiting Case

The mathematical as well as the corresponding computational models of the biosensor were evaluated. Assume that zero concentration of the second substrate (S20=0) leads to reducing the governing equations (5)–(8) to the following equations:
(18)d2S1Ndy2=γS1S1N,d2P1Ndy2=-γP1S1N2.
The steady-state solution of the above equations for the boundary condition (9) is given by
(19)S1,N=cosh⁡⁡(γS1y)cosh⁡⁡(γS1),P1N(y)=γP1[1-cosh⁡⁡(γS1y)+ycosh⁡⁡(γS1)-y]2γS1cosh⁡⁡(γS1).
The dimensionless steady-state current using (10) is given by
(20)ψ=idneFADP1,eS10=γP1[cosh⁡⁡(γS1)-1]2A2cosh⁡⁡(γS1).
Equation (20) is identical to (17), when S20=0 (i.e., α=∞) in (17).


## 5. Comparison of Analytical Expressions with Experimental Work and Numerical Simulation

To show the efficiency, the steady-state result (11)–(14) of the reaction diffusion equations is compared with the numerical solution of (5)–(8). The numerical solutions are found using pdepe (finite element method) in Matlab [36]. This Matlab program [37] is also given in Appendix B. The comparison is shown in Figures 3–6 and it gives satisfactory agreement.Figure 3
Normalized substrates and products concentration versus dimensionless spacey were computed using (11)–(14) for various values of reaction/diffusion parameters and for some fixed values of the other existing parameters. The key to the plot: (—) (11)–(14) and (∙∙∙) and (+++) numerical simulation of (5)–(8).Normalized substrate concentrationS1N versus dimensionless space y was computed using (11) for various values of reaction/diffusion parameter γS1, dimensionless parameter α and for some fixed values of the parameters γS1, γS2, α, β. The key to the plot: (—) (11) and (∙∙∙) numerical simulation of (5).
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(d)Normalized substrate concentrationS2N versus dimensionless space y was computed using (12) for various values of reaction/diffusion parameter γS3, dimensionless parameter α and for some fixed values of the parameters α, β, γS3. The key to the plot: (—) (12) and (∙∙∙) numerical simulation of (6).
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(d)Normalized product concentrationP1N versus dimensionless space y was computed using (13) for various values of reaction/diffusion parameter γP1, γS1 and for some fixed values of the parameters γS1, γS2, γP1, α, β. The key to the plot: (—) (13) and (∙∙∙) numerical simulation of (7).
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## 6. Discussion

Equations (11)–(14) are the new simple and closed approximate analytical expressions for the dimensionless substrate concentrations (S1N and S2N) and product concentrations (P1N and P2N) in terms of A or B. The analytical results (11)–(14) are compared with numerical solution, which is provided in Figures 3–7. Figure 3 shows the plots of all the concentration versus dimensionless distance y.Normalized product concentrationP2N versus dimensionless space y was computed using (14) for various values of reaction/diffusion parameter γP2, dimensionless parameter α and for some fixed values of the parameters γP1, γS3, α, β. The key to the plot: (—) (14) and (∙∙∙) numerical simulation of (8).
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(d)The kinetic response of a biosensor depends on the concentration species. The concentration depends on the following factors:γS1, γS2, γS3, γP1, and γP2. The diffusion parameters γS1 and γS3 represent the ratio of the characteristic time of the enzymatic reaction to that of concentration diffusion. These parameters can be varied by changing the thickness of the enzyme or diffusional layer. These parameters describe the relative importance of diffusion and reaction in both layers. When γS1 or γS3 is small, the kinetics is dominant resistance; the uptake of concentration species is kinetically controlled. Under these conditions, the concentration species across both the layers is essentially uniform. When the diffusion parameter γS1 or γS3 is large, diffusion limitations are the principal determining factor.Figures4 and 5 show the dimensionless concentration of substrates (S1N and S2N) versus dimensionless distance y. Figures 3 and 4(a) and 4(b) are plotted for various values of diffusion parameters γS1 and γS3 for some fixed values of α,β and γS2. From these figures, it is clear that the concentration of substrates S1N and S2N decreases when the value of γS1 and γS3 increases, respectively. The concentration is approximately equal to one when γS1≤0.01 and γS3≤0.01. Figures 4(c) and 5(c) are plotted for various values of dimensionless parameter α for some fixed values of other existing parameters. From these figures, it is clear that concentration increases for large values of α. Figures 4(d) and 5(d) are plotted for various values of A and B, respectively. It is evident that the concentration is uniform, when γS1(α+β)+γS2<(0.01)(α+β) (or A<0.1) and γS3<(0.01)(α+β) (or B<0.1).Figures6(a)–6(f) show the dimensionless product concentration profile P1N versus dimensionless distance y using (13). Figures 6(a) and 6(b) are plotted for all various values of γP1 for some fixed values of α, β, γS1 and γS2. Thus, it is concluded that there is a simultaneous increase in the values of the concentration P1N as well as in γP1. Figure 6(c) is plotted for various values of α and the fixed value of γS1, γS2, γP1, and β. It shows that the concentration is always minimum for all values of α. Concentration versus distance is plotted in Figures 6(d) and 6(e) for various values of γS1 and for some fixed values of other existing parameters. Thus, it is concluded that the concentration is always minimum and also there is a simultaneous decrease in the values of the concentration P1N as the value of γS1 increases. Figure 6(f) is plotted for various values of A, and from these graphs, it is inferred that concentration is minimum and uniform when A≥0.Figures7(a)–7(d) are plotted for the dimensionless product concentration P2N as ordinate versus dimensionless space y as abscissa using (14). From Figures 7(a) and 7(b), it is evident that the concentration of P2N increases as the diffusivity parameter γP2 increases. The concentration is approximately equal to zero, that is, (P2N≈0) when γP2≤0.1 for all possible values of the parameters α, β and γS3. From Figure 7(c), it is inferred that concentration P2N decreases, when the value of α increases and P2N≈0 when α≥10. Figure 7(d) is plotted for various values of B; it is evident that the concentration increases as the value of B increases. The concentration increases, when γS3>(0.01)(α+β) or B>0.1.Figures8(a)–8(e) are plotted using (17), which shows the dynamics of normalized steady-state current ψ versus dimensionless parameters γP1, (α+β), γS1, γS2, and A, respectively. From Figure 8(a), it is noted that, for small values of γS1(γS1≤0.1) and for any values of α and β, the value of the current is significantly high. The current grows notably faster at higher values of γS1 but in the case of γS1≥10 and α=β=0.1, the current does not increase. Figure 8(b) is plotted for various values of γP1 and for some fixed values of γS1 and γS2. It is observed that the current increases rapidly and reaches the steady-state when (α+β)≥0.5. From Figures 8(c) and 8(d), it is evident that the current is uniformly decreasing for large values of γP1, that is, (γP1≥5), and constant elsewhere. Figure 8(e) is plotted for various values of the reaction/diffusion parameter γP1 as abscissa and normalized steady-state current ψ as ordinate. It is observed that the current increases as the value of γP1 increases. Also current decreases for all values of the reaction/diffusion parameter γP1 and reaches zero when A≥10.Dimensionless currentψ versus dimensionless parameters γP1, (α+β), γS1, γS2 and A for some fixed values of the other parameters using (17).
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## 7. Conclusion

Mathematical model, describing the steady-state response of an amperometric biosensor based on parallel substrate conversion (catalase-peroxidase biochemical reactions), is presented. A nonlinear time independent coupled system of partial differential equations has been solved analytically using a new Homotopy perturbation method. Analytical expressions for substrate and product concentrations as function of single dimensionless parameterA or B are derived. The optimum value of A or B is also reported. The analytical expressions provided are much simpler and more efficient to understand and to predict the behavior of the system. Our analytical solution for substrates and products are compared with numerical solution providing consistent results. Concentration of substrates, products, and current depends upon reaction/diffusion parameters γS1, γS2, γS3, γP1, and γP2. The response of the steady-state current with respect to all parameters is discussed. For all different concentrations, the analytical results matched well with the simulated result. The close matching of simulated result and the analytical data shows that our proposed solution is able to simulate the dynamic performance of sensing system using the parameters. The analytical results provided here are useful in sensor design, optimization, and prediction of electrode response.



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