Analytical Expressions of Concentrations of Substrate and Hydroquinone in an Amperometric Glucose Biosensor

Abstract

The theoretical model for an amperometric glucose biosensor is discussed. In this model glucose oxidase enzyme is immobilized in conducting polypyrrole. This model contains a nonlinear term related to enzyme reaction kinetics. He’s homotopy perturbation method is used to find the approximate analytical solutions of coupled non-linear reaction diffusion equations. A closed-form expression of substrate and mediator concentration under non-steady-state conditions is obtained. A comparison of the analytical approximation and numerical simulation is also presented. An agreement between analytical expressions and numerical results is observed.

1. Introduction

Since the second half of the last century, numerous efforts have been devoted to the development of insoluble immobilized enzymes for a variety of applications [1]. These applications can clearly benefit from use of the immobilized enzymes rather than the soluble counterparts, for instance as reusable heterogeneous biocatalysts, with the aim of reducing production costs by efficient recycling and control of the process [2], as stable and reusable devices for analytical and medical applications [3–9], as selective adsorbents for purification of proteins and enzymes [10], as fundamental tools for solid-phase protein chemistry [11, 12], and as effective microdevices for controlled release of protein drugs [13].

Immobilized enzymes are becoming increasingly popular as reusable, selective analytical chemical reagents in solid-phase flow-through reactors, as membranes in sensors, and as films in dry reagent kits. The attractions of immobilized enzymes from an analytical standpoint are primarily their reusability, and hence cost saving, and the greater efficiency and control of their catalytic activity [14] (e.g., potentially longer half-lives, predictable decay rates and more efficient multistep reactions).

The immobilization of enzymes in conducting polymer [15] during electro-polymerization step has proved to be well suited to the preparation of biosensors [16–19]. This method is simple and easy to control. Another important advantage of this immobilization technique is the possibility of entrapping the mediator in the polymer as a dopant anion [20–23] or by covalent fixation on the pyrrole monomer [24]. Bartlett and Whitaker [25] have already presented a theoretical model for an amperometric polypyrrole + glucose oxidase (PPY + GOD) electrode. In their model PPY was considered as an insulating polymer, and the reduced mediator H₂O₂ was oxidized at the metallic surface after diffusion in the polymer. Marchesiello and Geniès [26] obtained the analytical expressions of concentrations of substrate and benzoquinone for low substrate and high benzoquinone concentration compared with respective Michaelis constants.

To my knowledge no rigorous analytical expressions of concentrations of substrate and mediator of ampereometric glucose under non-steady-state conditions for all values parameters 𝛼, 𝛼^′ and 𝑀 have been reported. These parameters are defined in (6). The purpose of this communication is to derive approximate analytical expressions for the non-steady-state concentrations and current using Homotopy perturbation method.

2. Mathematical Formulation and Analysis of the Problems

2.1. Mathematical Formulation

Building upon earlier work, Marchesiello and Geniès [26] presented a concise discussion and derivation of mass transport nonlinear equations in glucose biosensor, which is summarized briefly below. Figure 1 is a schematic representation of the PPY + GOD electrode working as a glucose sensor. The different steps which lead to the electrocatalytic current are as follows. (i) Glucose and benzoquinone diffusion from bulk of the solution to the electrode, (ii) glucose and benzoquinone diffusion into the conducting polymer, and (iii) the enzymatic reaction between glucose and benzoquinone: Glucose+GOD(FAD)→GODFADH₂+gluconolactone,Q+GODFADH₂→GOD(FAD)+H₂Q (1) (iv) The oxidation of hydroquinone: H₂Q→Q+2H⁺+2e⁻.

Figure 1 

Schematic diagram of the electrocatalytic oxidation of glucose at a GOD + PPY electrode in the presence of benzoquinone (Q).

As shown in Figure 1, this reaction occurs on the surface of the platinum electrode after diffusion in the polymer and in the conducting polymer itself. The rate of the enzymatic reaction between glucose and benzoquinone follows a ping-pong type mechanism [27] 𝑣=𝑘_cat[E]/⌊1+𝐾_Q/[Q]+𝐾_S/[S]⌋ where 𝑘_cat/s⁻¹ is the turnover number for GOD, 𝐾_S and 𝐾_Q are the Michaelis constants for glucose and benzoquinone, respectively, and [E], [Q], and [S] are the enzyme, benzoquinone, and glucose concentrations in the film. The third term is the H₂Q oxidation term. The rate of the electrochemical reaction in the conducting polymer can be written as 𝑣=𝐾Σ[H₂Q], where 𝐾/ms⁻¹ is the charge-transfer constant for the oxidation of H₂Q in the conducting polymer. It depends on the electrode potential. Σ is the PPY specific surface and it was estimated to be in the range 5–50 m²m⁻³ for a PPY electrode deposited film [28]. The differential equations describing the concentrations of S and H₂Q at steady state are as follows [26]: 𝐷_S𝑑²[S]𝑑𝑥²−𝑘_cat[E]1+𝐾_Q/[Q]+𝐾_S/[S]𝐷=0,_H2Q𝑑²H₂Q𝑑𝑥²+𝑘_cat[E]1+𝐾_Q/[Q]+𝐾_S/[S]=𝐾ΣH₂Q. (2) These nonlinear reaction/diffusion equations are solved for the following boundary conditions. The boundary conditions are as follows [27]: [S]=[S]₀,H₂Q,𝑑[S]=0,at𝑥=𝐿polymer/solutioninterfaceH𝑑𝑥=0,₂Q.=0,at𝑥=0platinum/polymerinterface (3) The total electrocatalytic current 𝐼_𝑇 is the sum of two currents. The current 𝐼_S=2FAD_H2Q𝑑H₂Q𝑑𝑥_𝑥=0. (4) For H₂Q oxidation on the platinum electrode, and 𝐼_𝑃=^𝐿₀𝑑𝐼=^𝐿₀2FAKΣH₂Q𝑑𝑥, (5) where 𝐿 is the film thickness of the reaction layer. We assume that there is no concentration polarization of S and H₂Q in the solution and that the partition coefficients are equal to unity. [S₀] is the concentration of glucose in the bulk of the solution.

2.2. Normalized Form

By introducing the following set of nondimensional variables, 𝑋=𝑥/𝐿,𝑈=[S]/[S]₀,𝑉=H₂Q/[S]₀,𝛼=𝐿/Λ,𝛼^′=𝐿/Λ^′,𝐷Λ=^S𝐾_S𝐾_cat[E]^1/2,Λ^′=𝐷_H2Q𝐾Σ^1/2,[S]𝑀=⁰𝐾_S+[S]₀𝐾_Q𝐾_S[Q]. (6)

The dimensionless parameters 𝛼 compares the enzymatic reaction rate with the substrate diffusion in the polymer. The dimensionless parameter 𝛼^′ compares the H₂Q oxidation rate in the conducting polymer with H₂Q diffusion in the polymer. For electrochemically inert polymer, 𝛼^′ has low value and for electrochemically active polymer, 𝛼^′ has high value. The parameter 𝑀 is a dimensionless constant.

The coupled reaction/diffusion equations (2) take the following normalized form: 𝑑²𝑈𝑑𝑋²−𝛼²𝑈𝑑𝑀𝑈+1=0,₂𝑉𝑑𝑋²+𝛼²𝑈𝑀𝑈+1=𝛼^′2𝑉, (7) where 𝑈 and 𝑉 are the dimensionless concentration of S and H₂Q.

The transformed boundary conditions are 𝑑𝑈𝑑𝑋=0,𝑉=0,when𝑋=0,𝑈=1,𝑉=0,when𝑋=1. (8) The dimensionless form of the current is given by 𝜓_S=𝐼_S2FAD_H2Q=𝑑[𝑉]𝑑𝑋_𝑋=0,𝜓_P=𝐼_P=2FAKΣ¹₀𝑉𝑑𝑋. (9)

2.3. Analytical Expressions of Concentrations of Substrate and the Mediator under Steady-State Condition

In recent days, Homotopy perturbation method (HPM) is often employed to solve several analytical problems. In addition, several groups demonstrated the efficiency and suitability of the HPM for solving nonlinear equations and other electrochemical problems [29–32]. Mousa and Ragab [33] used HPM to solve the Lighthill equation, the Duffing equation [34], and the Blasius equation [35]. HPM has also been used to solve nonlinear boundary value problems [33], integral equation [36–38], Emden-Flower type equations [39], and several other problems. Detailed derivation of the concentrations using Homotopy perturbation method is described in Appendix 5. As a result, we can obtain the dimensionless concentrations of substrate and mediator as follows: 𝑈()=()𝑋,𝛼cosh𝛼𝑋(𝛼)+(()]cosh𝑀cosh𝛼𝑋)[cosh2𝛼−36cosh³(𝛼)−𝑀[cosh(2𝛼𝑋)−3]6cosh²(𝛼),𝑉()=𝛼𝑋,𝛼,𝑀²𝛼^′2()𝑒𝑀+11−^𝛼′𝑋+𝑒^{𝛼′(1−𝑋)}𝑒^𝛼′+1. (10)(11) The dimensionless current 𝜓_S and 𝜓_P are obtained using (9) and (11): 𝜓_S𝛼,𝛼^′=𝛼,𝑀²𝑒^𝛼′−1𝛼^′(𝑀+1)𝑒^𝛼′,𝜓+1_P𝛼,𝛼^′=𝛼,𝑀²𝛼^′𝑒^𝛼′−𝑒^𝛼′+𝛼^′+1𝛼^′3(𝑀+1)𝑒^𝛼′−𝛼+1₂𝑒^𝛼′−1𝛼^′3(𝑀+1)𝑒^𝛼′.+1 (12)(13) The electrocatalytic current 𝜓_𝑇 is the sum of two currents 𝜓_S and 𝜓_P𝜓_𝑇𝛼,𝛼^′=𝛼,𝑀²𝑒^𝛼′−1𝛼^′3(𝑀+1)𝑒^𝛼′×𝑒+1_𝛼′𝛼^′−1+𝛼^′𝛼^′+1. (14)

2.4. First-Order Catalytic Kinetics

In this case, 𝑀𝑈<1 or 1+𝐾_Q/[Q]≪𝐾_S/[S]. Now (7) reduces to the following forms: 𝑑²𝑈𝑑𝑋²−𝑈𝛼²𝑑=0,₂𝑉𝑑𝑋²+𝑈𝛼²=𝛼^′2𝑉. (15) The solutions of (15) are given as follows: 𝑈()=()𝑋,𝛼cosh𝛼𝑋(𝛼),𝑉cosh𝑋,𝛼,𝛼^′=𝛼²𝛼²−𝛼′²𝑒^𝛼′𝑋(𝛼)−()coshcosh𝛼𝑋(𝛼)−𝛼coshsinh^′𝑋sinh𝛼^′𝑒^𝛼′(𝛼),𝜓cosh−1_S𝛼,𝛼^′=𝛼^′𝛼²𝛼²−𝛼^′2×1(𝛼)−1coshsinh𝛼^′𝑒^𝛼′(𝛼),𝜓cosh−1_P𝛼,𝛼^′=𝛼²𝛼²−𝛼^′2×1(𝛼)−1cosh2sinh𝛼^′𝑒^𝛼′(𝛼)×1cosh−1𝛼^′𝑒^𝛼′+𝑒−1_−𝛼′−12𝛼^′sinh𝛼^′𝑒^𝛼′(𝛼)−(𝛼)cosh−1sinh𝛼²(𝛼),𝜓cosh_𝑇=𝜓_𝑆+𝜓_𝑃. (16)(17)(18)(19)(20)

2.5. Zero-Order Catalytic Kinetics

In this case, 𝑀𝑈>1 or 1+𝐾_Q/[Q]≫𝐾_S/[S]. Now (7) reduce to the following forms: 𝑑²𝑈𝑑𝑋²−𝛼²𝑀𝑑=0,₂𝑉𝑑𝑋²+𝛼²𝑀=𝛼′²𝑉. (21) The solution of (21) is given as follows: 𝑈()𝛼𝑋,𝛼,𝑀=1−²2𝑀1+𝑋²,𝑉𝑋,𝛼,𝛼^′=𝛼,𝑀²𝑀𝛼^′21−𝑀𝑒^𝛼′𝑋+𝑒^{𝛼′(1−𝑋)}𝑀𝑒^𝛼′+1. (22)(23) The current is given by 𝜓_S𝛼,𝛼^′=𝛼,𝑀²𝑀𝛼^′2𝑒^𝛼′𝑒+1𝑒^𝛼′+^𝛼′1−𝛼^′,𝜓_P𝛼,𝛼^′,𝑀=−2𝛼²1−𝑒^𝛼′𝑀𝛼^′31+𝑒^𝛼′,𝜓_T=𝜓_S+𝜓_P. (24)(25)(26)

3. Numerical Simulation

The nonlinear differential equations (7) are also solved numerically. The function pdepe in Matlab/Scilab software is used to solve boundary value problems (BVPs) for partial differential equations. The numerical solution is compared with our analytical results. The Matlab/Scilab program is also given in Table 1. Upon comparison it is evident that both results gives satisfactory agreement.

Table 1 

Summary of analytical expressions of concentrations and currents.

4. Discussion

Figures 2(a)–2(c) represents the normalized non-steady state concentration of substrate 𝑈 versus dimensionless distance 𝑋, for different values of dimensionless reaction diffusion parameter 𝛼=0.1,…,100 and 𝑀=0.001,…,1. From this figure, it is evident that the concentration of substrate increases when 𝛼 decreases and 𝑀 increases. From this Figures 1(a)–1(c), it is evident that the value of concentration is uniform when 𝛼≥0.5.

(a)

(b)

(c)

Figure 2 

Profile of the normalized concentrations of the glucose. The concentrations were computed for various values of 𝛼 and 𝑀. The curves are plotted using (10).

Figures 3(a)–3(c) shows the concentration of mediator 𝑉 versus dimensionless distance 𝑋, for various values of dimensionless diffusion parameter 𝛼(=0.1,…,10), 𝛼^′(=0.01,…,100) and 𝑀(=0.01,…,100). From this figure, it is inferred that the value of the concentration of mediator decreases abruptly when 𝛼 decreases and 𝑀 and 𝛼^′ increases. Also the concentration of mediator reaches the maximum value at the middle of membrane (𝑥=0.5) for all values of 𝛼, 𝛼^′, and 𝑀.

(a)

(b)

(c)

Figure 3 

Profile of the normalized concentration of the mediator. The concentrations were computed for various values of 𝛼, 𝛼′, and 𝑀. The curves are plotted using (11).

The normalized steady state current 𝜓_S(𝛼,𝛼^′,𝑀) and 𝜓_P(𝛼,𝛼^′,𝑀) as the function of the dimensionless parameter 𝛼^′ is given in Figures 4(a)–4(d). From this figures, it is obvious that the values of the current decreases slowly and reaches constant value when 𝛼^′≥40. Also values of the current decreases when 𝑀 increases.

(a)

(b)

(c)

(d)

Figure 4 

Profile of the normalized current. The current curves were computed for various values of 𝛼, 𝛼′, and 𝑀. The curves are plotted using (12) and (13).

In Figures 5(a)-5(b), profile of the normalized current 𝜓_T is shown. The current were computed for various values of 𝛼, 𝛼^′, and 𝑀. The curves are plotted using (12). The value of the current increases when 𝛼 increases. The normalized non-steady-state current for diverse values of dimensionless diffusion parameter 𝛼 is plotted in Figures 6(a)–6(c).

(a)

(b)

Figure 5 

Profile of the normalized current. The current curves were computed for various values of 𝛼, 𝛼′, and 𝑀. The curves are plotted using (14).

(a)

(b)

(c)

Figure 6 

Profile of the normalized current for first-order kinetics. The current curves were computed for various values of 𝛼, 𝛼′, and 𝑀. The curves are plotted using (18)–(20).

(a)

(b)

(c)

Figure 7 

Variation of normalized non-steady-state current response for zero-order kinetics. The current curves were computed for various values of 𝛼, 𝛼′ and 𝑀. The curves are plotted using (24)–(26).

5. Conclusions

The steady state nonlinear reaction/diffusion equations in an amperometric glucose sensor have been solved analytically. An approximate analytical expression for the concentrations and current for an amperometric glucose sensor are obtained by using the Homotopy perturbation method. The primary result of this work is simple approximate calculation of concentration profiles and current for all values of fundamental parameters 𝛼 and 𝛼^′. The analytical results obtained can be used for the optimization of the design of glucose sensor.

Appendix

Approximate Analytical Expression of the Normalized Concentration of Substrate and H₂Q

Using Homotopy perturbation method, we construct a homotopy for (7) as follows: 𝑑1−𝑝²𝑈𝑑𝑋²𝑑+𝑝²𝑈𝑑𝑋²−𝛼²𝑈𝑑𝑀𝑈+1=0,1−𝑝²𝑉𝑑𝑋²−𝛼^′2𝑉𝑑+𝑝²𝑉𝑑𝑋²−𝛼^′2𝛼𝑉+²𝑈𝑀𝑈+1=0. (A.1)(A.2) The approximate solution of (A.1) is given by 𝑈=𝑈₀+𝑝𝑈₁+𝑝²𝑈₂+𝑝³𝑈₃+⋯. (A.3) Substituting (A.3) into (A.1) and comparing the coefficients of like powers of 𝑝: 𝑝⁰∶𝑑²𝑈₀𝑑𝑋²−𝛼²𝑈₀𝑝=0,₁∶𝑑²𝑈₁𝑑𝑋²−𝛼²𝑈₁+𝑀𝑈₀𝑑²𝑈₀𝑑𝑋²=0. (A.4) The boundary conditions for the above equations are 𝑑𝑈₀𝑑𝑋=0at𝑋=0,𝑈₀=1,at𝑋=1,𝑑𝑈₁𝑑𝑋=0at𝑋=0,𝑈₁=0,at𝑋=1. (A.5) Using this boundary conditions the solution of (A.4) becomes as follows: 𝑈₀=cosh(𝛼𝑋)(𝛼),𝑈cosh₁=𝑀cosh(𝛼𝑋)(cosh(2𝛼)−3)6cosh³(𝛼)−𝑀(cosh(2𝛼𝑋)−3)6cosh²(𝛼). (A.6) Adding (A.6) we get (10) in the text

The approximate solution of (A.2) is given by 𝑉=𝑉₀+𝑝𝑉₁+𝑝²𝑉₂+𝑝³𝑉₃+⋯. (A.7) Substituting (A.3) and (A.2) and arranging the coefficients of like powers of 𝑝, we have 𝑝⁰∶𝑑²𝑉₀𝑑𝑋²−𝛼^′2𝑉₀𝑝=0,₁∶𝑑²𝑉₁𝑑𝑋²−𝛼^′2𝑉₁+𝛼²𝑈₀𝑀𝑈₀+1=0. (A.8)(A.9) The boundary conditions for the above equations are as follows 𝑉₀=0,𝑉₁𝑉=0,when𝑋=0,₀=0,𝑉₁=0,when𝑋=1. (A.10) Upon solving (A.7) and (A.8), and using this boundary condition, we get 𝑉₀(𝑋)𝑉=0,₁(𝑋)=𝛼²𝛼^′2()𝑒𝑀+11−^𝛼′𝑋+𝑒^{𝛼′(1−𝑋)}𝑒^𝛼′.+1 (A.11) Adding (A.11) we get, (11) in the text.

Acknowledgemnts

This work was supported by the University Grants Commission (F. No. 39-58/2010(SR)), New Delhi, India and Council of Scientific and Industrial Research (CSIR No.: 01(2442)/10/EMR-II), New Delhi, India. The authors are thankful to Dr. R. Murali, The Principal, The Madura College, Madurai and The Secretary, Madura College Board, Madurai for their encouragement.

Supplementary Materials

Matlab/Scilab program to find the numerical solution of Eqn. (7)

1. Supplementary Material