# Modelling of Amperometric Biosensor Used for Synergistic Substrates Determination

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## Abstract

**:**In this paper the operation of an amperometric biosensor producing a chemically amplified signal is modelled numerically. The chemical amplification is achieved by using synergistic substrates. The model is based on non-stationary reaction-diffusion equations. The model involves three layers (compartments): a layer of enzyme solution entrapped on the electrode surface, a dialysis membrane covering the enzyme layer and an outer diffusion layer which is modelled by the Nernst approach. The equation system is solved numerically by using the finite difference technique. The biosensor response and sensitivity are investigated by altering the model parameters influencing the enzyme kinetics as well as the mass transport by diffusion. The biosensor action was analyzed with a special emphasis to the effect of the chemical amplification. The simulation results qualitatively explain and confirm the experimentally observed effect of the synergistic substrates conversion on the biosensor response.

## 1. Introduction

A biosensor is an electronic measuring device designed for measurement of the concentration of some specific substance (analyte) in a solution. The device specificity for a particular substance is achieved by some biological material, usually an enzyme [1–3]. An amperometric biosensor assesses the concentration of the analyte through the measurement of the current on a working electrode [4,5]. Biosensors are widely used in various applications that require fast quantitative analysis [6–10].

Oxidases are the kind of enzymes commonly used in biosensors. In the case of oxidases, molecules of an enzyme are reduced to their intermediate state and then oxidised by electron acceptors [5,11]. The reduction process as a rule is a specific reaction, while the oxidation of the reduced enzyme by electron acceptors is less specific. Many electron acceptors can be employed in this process [12].

The kinetic analysis of the reaction of a mixture of low and high reactive electron acceptors shows that the reduction of a low reactive electron acceptor may substantially increase if the rate of the cross reaction between these substances is high enough. This phenomenon is employed in biosensors for a high sensitivity determination of synergistic substrates [13–15].

The understanding of the kinetic peculiarities of the biosensors is of crucial importance for their design and optimization. To improve the productivity as well as the efficiency of the biosensor design, a model of the biosensor should be built [16,17]. Starting from the seventies various mathematical models have been widely used as important tools to study and optimize analytical characteristics of actual biosensors [18–22]. A comprehensive reviews on the modelling of the amperometric biosensors have been presented in [23,24]. Biosensors based on synergistic substrates were modelled at steady-state conditions a few years ago [25,26]. Very recently, a laccase-based biosensor utilizing simultaneous substrates conversion was numerically modelled at steady-state as well as transient conditions [27].

The goal of this investigation was to develop a computational model for an effective simulation of amperometric biosensors utilizing the synergistic substrates conversion as well as to investigate the influence of the physical and kinetic parameters on the biosensor response. The developed model is based on non-stationary reaction-diffusion equations [28,29]. The mathematical model comprises three compartments, a layer of enzyme solution entrapped on the electrode surface, a dialysis membrane covering the enzyme layer and an outer diffusion layer. By changing input parameters the output results were numerically analyzed at transition and steady-state conditions. The computational simulation was carried out using the finite difference technique [24,29,30]. The simulation results qualitatively explained and confirmed the experimentally observed effect of the synergistic substrates conversion on the biosensor response [13].

## 2. Mathematical Model

We consider the following kinetic scheme used for the determination of synergistic substrates [13]:

_{ox}and E

_{red}stand for oxidized and reduced enzyme forms, respectively, R is the enzyme reducer, S

_{1}and S

_{2}are the substrates, P, P

_{1}and P

_{2}are the products of the reactions, k

_{1}, k

_{2}, k

_{3}, k

_{4}are the reaction rate constants, and n

_{1}, n

_{2}are the numbers of electrons involved in a charge transfer at the electrode surface in reactions (1e) and (1f), respectively.

The reactions (1a) and (1b) may be considered as the reduction of a low reactive electron acceptor S_{1} by R and catalyzed by the enzyme. In the reaction (1a), the enzyme is reduced by the reducer R, and an intermediate form of enzyme (E_{red}) is formed. In the reaction (1b), the electron acceptor S_{1} is reduced by the E_{red}, the product P_{1} is formed and enzyme molecules regain their primary oxidized form E_{ox}.

When the high reactive electron acceptor S_{2} is introduced, two more reactions, (1c) and (1d), start. The reaction (1c) represents the enzyme oxidation by the high reactive electron acceptor S_{2}. The reaction (1d) is called a cross reaction. In the latter reaction, the substrate S_{1} oxidizes the product P_{2} and regenerates the substrate S_{2}. This reaction also produces the product P_{1}.

Reactions (1e) and (1f) are electrochemical reactions that take place on the biosensor electrode. Both reactions are assumed fast and irreversible due to the high electrode potential.

[Fe(CN)_{6}]^{3−} ion may be employed as the low reactive electron acceptor S_{1} [13,15]. In this case, [Fe(CN)_{6}]^{4−} ion would be the reaction product P_{1}. Cation radicals of 1-(N,N-dimethylamino)-4-(4-morpholino)benzene, 5,10-dihydro-5,10-dimethylphenazine, phenoxazine-10-propionic acid or phenothiazine-10-propionic acid may be used as high reactive electron acceptors (S_{2}). The reduced forms of those substances would be considered as the corresponding products (P_{2}) [13].

During the physical experiment [13], the substrate S_{2} was introduced into the bulk solution at the moment when the steady-state response to the substrate S_{1} was established. The thickness of the diffusion layer was controlled by the rotating magnetic bar and it was estimated using the rotating disk electrode.

#### 2.1. Biosensor Structure

The biosensor is considered as an electrode with a relatively thin layer of enzyme solution trapped on the surface of the electrode by applying a dialysis membrane [13]. The biosensor model involves four regions: the enzyme layer where the enzymatic and chemical reactions as well as the mass transport by diffusion take place, a dialysis membrane, a diffusion limiting region where only chemical reactions as well as the mass transport by diffusion take place, and a convective region where the analyte concentration is maintained constant, see Figure 1. Assuming a homogeneous distribution of the enzyme in the enzyme layer of the uniform thickness and symmetrical geometry of the dialysis membrane leads to the mathematical model of the biosensor action defined in a one-dimensional-in-space domain [23,24].

Let us define d_{1}, d_{2} and d_{3} as the thicknesses of the enzyme, dialysis membrane and diffusion layers, respectively. We will also need values representing the distances between the electrode surface and the boundaries of the regions. Let a_{1}, a_{2} and a_{3} be the distances between the electrode surface and one of those boundaries shown in Figure 1.

The diffusion layer (a_{2} < x < a_{3}) may be treated as the Nernst diffusion layer [29]. According to the Nernst approach a layer of thickness d_{3} remains unchanged with time. It was assumed that away from it the solution is uniform in concentration.

#### 2.2. Governing Equations

The governing equations for a chemical reaction network can be formulated by the law of mass action [11,21]. Coupling reactions in the enzyme layer with the one-dimensional-in-space diffusion, described by the Fick's second law, leads to the following equations of the reaction–diffusion type (0 < x < a_{1}, t > 0):

_{ox}(x, t) and e

_{red}(x, t) correspond to concentrations of oxidized (E

_{ox}) and reduced (E

_{red}) enzyme, respectively; s

_{11}(x, t) and s

_{21}(x, t) (p

_{11}(x, t) and p

_{21}(x, t)) are the concentrations of substrates (products) in the enzyme layer; r

_{1}(x, t) is the reducer concentration in the enzyme layer, and D

_{R}

_{1}, D

_{S}

_{11}, D

_{S}

_{21}, D

_{P}

_{11}, D

_{P}

_{21}are the diffusion coefficients of the corresponding substances defined by the subscript. In the definition of the concentrations and diffusion coefficients, here and later in this paper the last numeric subscript label denotes the region of the model, particularly, 1 stands for the enzyme layer. The molecules of both enzyme forms, E

_{ox}and E

_{red}, are considered as immobilized, and therefore there are no diffusion terms in the corresponding equations.

The product P does not act as a reactant in any reaction, so its concentration is not used in any further calculations. Therefore, equation system (2) contains no equation for the product P.

No enzyme molecules appear in the dialysis membrane as well as in the diffusion layer. Only the reaction (1d) as well as the mass transport by diffusion of the reducer, substrates and products take place in these two regions. The governing equations for both these layers are represented as follows (a_{i}_{−1} < x < a_{i}, t > 0, i = 2, 3):

#### 2.3. Initial Conditions

Let x = 0 represents the electrode surface, while x = a_{1}, x = a_{2} and x = a_{3} represent the boundaries between the adjacent regions as described in Section 2.1 and shown in Figure 1. The biosensor operation starts when the reducer and the substrates appear in the bulk solution. This is used in the initial conditions (t = 0),

_{0}stands for the total concentration of the enzyme in the enzyme layer (e

_{0}= e

_{ox}(x, t) + e

_{red}(x, t), ∀x, t : x ε (0, a

_{1}), t > 0), r

_{0}is the reducer concentration in the bulk solution, s

_{10}and s

_{20}stand for substrate concentrations in the bulk solution, T

_{S}

_{2}is the moment of the substrate S

_{2}insertion into the bulk solution.

#### 2.4. Matching Conditions

On the boundary between two regions having different diffusivities, matching conditions have to be defined (t > 0, i = 1, 2, m = 1, 2),

These conditions mean that fluxes of the reducer, substrates and products through one region are equal to the corresponding fluxes entering the surface of the neighboring region. Concentrations of the reducer, substrates and products in one region versus the neighboring region are assumed to be equal.

#### 2.5. Boundary Conditions

In the bulk solution, concentrations of the reducer, substrates and products remain constant (t > 0),

The reaction products P_{1} and P_{2} take part in the electrochemical reactions (1e) and (1f), respectively, at the electrode surface (x = 0). Rates of those reactions are so high that the concentrations of P_{1} and P_{2} at the electrode surface are permanently reduced to zero (t > 0) [13],

Since the reaction (1e) produces as much S_{1} as it consumes P_{1}, the concentration flux of P_{1} on the electrode surface is equal to the concentration flux of S_{1} but in opposite direction. The same is also assumed for the reaction (1f) and substances S_{2} and P_{2}. These relations are expressed by the following boundary conditions (t > 0):

The reducer R is electrode-inactive substance, thus its concentration flux on the electrode surface is equal to zero (t > 0),

#### 2.6. Biosensor Response

The measured current is usually assumed as the response of an amperometric biosensor in physical experiments. When modelling the biosensor action, due to the direct proportionality of the current to the area of the electrode surface, the current is often normalized with that area [23,31]. The biosensor current density j(t) at time t was expressed explicitly from Faraday's and Fick's laws,

_{1}(t) and j

_{2}(t) are the faradaic current densities generated by the electrochemical reactions (1e) and (1f), respectively, F is the Faraday constant, F = 96, 486 C/mol.

We assume that the system approaches a steady state as t → ∞,

_{st}, j

_{1}

_{st}and j

_{2}

_{st}are the steady-state biosensor current densities.

Since the current is generated due to two electrochemical reactions (1e) and (1f), it is important to investigate the influence of each of them to the overall biosensor response. The contribution of the electrochemical reaction (1e) into the overall biosensor response was expressed as the ratio of the density of the steady-state current j_{1}_{st} to the density j_{st} of the overall steady-state current,

The sensitivity is also one of the most important characteristics of biosensors [1,2]. The biosensor sensitivity is usually expressed as the gradient of the biosensor current with respect to the concentration of the substrate in the bulk. Since the biosensor current as well as the substrate concentration varies even in orders of magnitude, a dimensionless expression of the sensitivity is preferable [31,32].

The biosensor based on synergistic reactions is designed to measure concentration of the substrate S_{2} [13]. The dimensionless biosensor sensitivity B to the concentration of the substrate S_{2} was defined as follows:

_{st}(s

_{20}) is the density of the steady-state biosensor current calculated at the concentration s

_{20}of the substrate S

_{2}in the bulk. B(s

_{20}) denotes the biosensor sensitivity at that concentration.

The chemical signal amplification caused by a synergistic effect is another important characteristic of this type of biosensors [13]. A measure Amp of the synergistic effect was expressed as the ratio of the biosensor response to a substrate S_{2}-containing analyte to the response to the corresponding S_{2}-free analyte,

_{20}) is the dimensionless ratio of the signal amplification obtained by the insertion of the substrate S

_{2}of the concentration s

_{20}into the buffer solution [13,22].

## 3. Numerical Simulation of Biosensor Action

#### 3.1. Simulating the Biosensor Operation

Analytical solutions are not usually possible when solving non-linear partial differential equations [19,23,29]. Therefore, the biosensor action was simulated numerically. The simulation was carried out using the finite difference technique [29,30]. An explicit finite difference scheme was built on a uniform discrete grid with 200 points in space direction for each modelled layer corresponding to a certain time moment. The simulator has been programmed by authors in C programming language [33].

In the numerical simulation, the biosensor response time was assumed as the time when the change of the biosensor current remains very small during a relatively long term. A special dimensionless decay rate ε was used,

_{r}is the biosensor response time. The decay rate value ε = 10

^{−3}was used in calculations.

The following values of the model parameters were constant in all numerical simulations, unless stated otherwise [13]:

#### 3.2. Experimental Validation

The numerical solution of the model (2)–(10) was compared with the experimental data of the research conducted by one of the authors [13]. The results are depicted in Figure 2. The experimental data were shifted by −30 s in time axis and by −1.9 μA in current axis [13]. The correction in time axis was applied in order to coincide the addition of reducer R (glucose in this particular case) into the bulk solution at the moment t = 0. The correction in current axis was applied in order to eliminate the background current influence to the overall biosensor response.

During physical experiment, the substrate S_{2} was added into the bulk solution at the moment t = 130 s which corresponds to t = 100 s in the simulations. Thus T_{S}_{2} = 100 s value was used in the numerical simulations. Having simulated density j of the biosensor current, the biosensor current j_{A} was calculated by multiplying the density by the area A of the electrode surface,

In the physical experiment, the electrode with the surface area of A = 0.3 cm^{2} was used [13].

The curve 2 in Figure 2 corresponds to the simulation carried out at the parameters equal to the parameters of the physical experiment [13]. The relative difference between the numerical solution and the experiment data at the moment T_{S}_{2} (before insertion of S_{2}) is about 14%. This indicates that mathematical model quite accurately reflects the physical experiment. However, after the insertion of S_{2}, the accuracy deteriorates. The simulated steady-state current is about 2.5 times higher than the experimental one (Figure 2). This indicates that experiment is really more complex than as defined by the mathematical model. Presumably, the reaction of the reduced enzyme with oxygen, which was unforeseen in the mathematical modelling, is a main factor determining this difference [13]. The difference may be also caused by the limited stability of oxidized heterocyclic compounds and some external diffusion limitation of substrates [13].

The curves 3 and 4 of Figure 2 demonstrate that simulation results may be fitted with the experimental ones by varying the value of the reaction rate constant k_{3} and the diffusion layer thickness d_{3}. However the measured values of the parameters will be used in this paper in further numerical simulations.

Despite this inadequacy with the experiment, the model (2)–(10) seems to be suitable for investigating the kinetic peculiarities and optimizing the configuration of biosensors utilizing the synergistic effect of substrates.

#### 3.3. Simulating the Synergistic Effect

The understanding the nature of the synergistic effect is of crucial importance for preparation of highly sensitive biosensors [13,15,26]. For a kinetic analysis of the biosensor operation, let us introduce ῡ_{1}, ῡ_{2}, ῡ_{3}, ῡ_{4} as the average rates of the reactions (1a), (1b), (1c), (1d) taking place in the enzyme layer,

_{1}, s̄

_{2}, p̄

_{1}, p̄

_{2}, ē

_{ox}and ē

_{red}stand for the average concentrations of R, S

_{1}, S

_{2}, P

_{1}, P

_{2}, E

_{ox}and E

_{red}in the enzyme layer.

Figures 3 and 4 show how the synergistic effect occurs. The biosensor parameters were kept the same as in Figure 2.

When S_{2} is introduced into the bulk solution, a fast reaction (1c) starts, and a large amount of E_{ox} and P_{2} molecules are produced (see Figure 3). In the meantime the increased concentration of E_{ox} initiates an increase in the rate of the reaction (1a), molecules of P_{2} start to react with S_{1} in reaction (1d) (see Figure 4). A decrease of the reducer R concentration is the second indication of the increase in the reaction (1a) rate as the reaction consumes more reactant molecules (see Figure 3). The reactions (1a) and (1c) are very strongly interrelated because E_{ox} is the product of reaction (1c), and at the same time it is the reactant of the reaction (1a) and vice versa with E_{red} molecule. Similar situation appears with the reactions (1c) and (1d) (molecules of P_{2} and S_{2}).

The rate of the reaction (1c) is controlled by the concentration of S_{2}. Reactions (1a) and (1d) are accelerated by the reaction (1c) and their rates tend to reach the rate of the reaction (1c). Hence the rates of reactions (1a), (1c) and (1d) are approximately equal during almost all the time of biosensor operation as seen in Figure 4. The rate of the reaction (1b) remains almost unaffected.

The reactions (1b) and (1d) produce the product P_{1} which is responsible for the biosensor current. Since the rate of the reaction (1b) remains almost unchanged and the rate of the reaction (1d) is greater than the rate of reaction (1b) by two orders of magnitude, the biosensor response is also increased by two orders of magnitude. In other words, the signal is amplified chemically. This is the desirable outcome of synergistic effect in this type of biosensors [13].

If the synergistic effect would not occur during the biosensor operation, that would mean that biosensor signal is not chemically amplified. A biosensor with no chemical amplification could be too weak for reliable measurement. That could mean the narrower range of biosensor application because biosensor would only work at relatively high concentrations of substrate. So, it is important to choose the correct biosensor parameters that support the occurrence of the synergistic effect.

The law of mass conservation was applied for an additional validation of the numerical solution of the initial boundary value problem Equations (2)–(9). Since no mass transport of the enzyme takes place in the enzyme layer, the equality e_{ox}(x, t) + e_{red}(x, t) = e_{0} holds for ∀x ε [0, a_{1}] and ∀t ≥ 0. At the steady-state conditions when ∂s_{im}/∂t= ∂p_{im}/∂t = 0, the equality s_{im}(x, t) + p_{im}(x, t) = s_{10} holds for ∀x ε [a_{m−1}, a_{m}] and t → ∞, i = 1, 2, m = 1,2,3. A fulfillment of these conditions can be observed in Figure 3 (please note that concentrations are provided in a logarithmic scale).

#### 3.4. Dynamics of Biosensor Current

Figure 5 shows the dynamics of the biosensor current j_{A} simulated at two thicknesses (d_{3}) of the diffusion layer and three concentrations (s_{20}) of the substrate S_{2}. For simplicity, here and below, the substrate S_{2} is introduced in the bulk solution in the beginning of the experiments (T_{S2} = 0 s).

At low substrate concentration (s_{20} = 4 μM) the steady-state current of the biosensor is not influenced by the thickness d_{3} of the diffusion layer. However at this concentration, a change in the thickness of the diffusion layer slightly changes the curvature of “current vs. time” curve and the biosensor with thinner diffusion layer reaches the steady state earlier. This is also correct in cases of higher substrate concentrations (s_{20} = 40 μM, s_{20} = 400 μM).

From the curves depicted in Figure 5 one can also observe another very important relationship: the steady-state current directly depends on the concentration s_{20} of the substrate S_{2} in the bulk solution. At higher substrate concentrations (s_{20} = 40 μM, s_{20} = 400 μM) it is also observable that steady-state current depends inversely on the thickness d_{3} of the diffusion layer. Intensity of the solution stirring plays important role when working with solutions of high concentration of the substrate.

## 4. Results and Discussion

#### 4.1. Biosensor Response vs. Substrates Concentrations

The dependence of the steady-state biosensor current on the concentration s_{20} of the substrate S_{2} was investigated at different diffusion layer thicknesses d_{3} as well as at different values of the reaction rate constant k_{3}. The results of the numerical simulations are depicted in Figure 6.

The biosensor may reliably measure the concentration of substrate S_{2} at the range of concentrations where the biosensor response is dependent on the concentration s_{20} but not dependent on the diffusion layer thickness d_{3}. As one can observe from Figure 6 this range depends on the reaction rate constant k_{3}. The lower boundary of this range varies from s_{20} ≈ 0.02 μM when k_{3} is relatively high (curves 3, 6) to s_{20} ≈ 0.1 μM when k_{3} is relatively low (curves 1, 4). The upper boundary of this range varies from s_{20} ≈ 16 μM when k_{3} is relatively high (curves 3, 6) to s_{20} ≈ 40 μM when k_{3} is relatively low (curves 1, 4). At the concentrations s_{20} ≳ 250 μM the biosensor response is not influenced by the concentration of the substrate S_{2}.

To investigate the dependence of the biosensor sensitivity on the substrate concentration, the biosensor response was simulated at the same values of the diffusion layer thickness d_{3} and the reaction rate constant k_{3} as in Figure 6. The calculation results are depicted in Figure 7.

As one can observe from Figure 7, the biosensor shows the highest values of the sensitivity when the concentration of the substrate S_{2} is moderate (1 μM ≤ s_{20} ≤ 10 μM). At low substrate S_{2} concentrations (0.01 μM ≤ s_{20} < 1 μM) as well as at high concentrations of S_{2} (10 μM < s_{20} ≤ 1,000 μM) biosensor exhibits poor sensitivity.

At low substrate concentrations, a higher k_{3} yields a higher biosensor sensitivity, whereas the thickness d_{3} of the diffusion layer does not influence the biosensor response at this concentration range. At high substrate concentrations, k_{3} has the opposite effect on the biosensor sensitivity compared with a low substrate concentration range. A higher k_{3} yields a lower biosensor sensitivity. At this concentration range, the thickness d_{3} of the diffusion layer does influence the biosensor sensitivity. A thicker diffusion layer diminishes the value of biosensor sensitivity.

Although this type biosensors are designed to measure the concentration s_{20} of the substrate S_{2}, it is also important to investigate the biosensor response dependency on the concentration s_{10} of the substrate S_{1} in order to optimize the biosensors. The dependence of the steady-state biosensor current j_{st} on the concentration s_{10} was investigated at different values of diffusion layer thickness d_{3} as well as at different concentrations (s_{20}) of the substrate S_{2}. The simulation results are depicted in Figure 8.

As we can observe from the curves in Figure 8, the biosensor response directly depends on the concentration s_{10}. In cases of low and moderate substrate concentrations of S_{2}, this dependency holds through the whole range of investigated s_{10} concentrations. At high concentrations of S_{2} (s_{20} = 100 μM), this dependency is valid only for low concentrations of S_{1} (s_{10} < 1mM). At high concentrations of S_{1} (s_{10} > 1 mM) the biosensor response becomes practically insensitive to the change of s_{10}.

The thickness d_{3} of the diffusion layer has no noticeable influence on the biosensor response at the majority of the biosensor parameters investigated. Only at high concentrations of S_{1} and high concentrations of S_{2} the effect of the diffusion layer thickness becomes noticeable.

#### 4.2. The Composition of Biosensor Response

The biosensor response is determined by two electrochemical reactions (1e) and (1f). It is important to understand the role of each process at certain values of the biosensor parameters.

The dependence of the biosensor response composition on the reaction rate constant k_{4} was investigated at different diffusion layer thicknesses (d_{3}) as well as at different values of the substrate concentration s_{20}. Results of the numerical simulation are depicted in Figure 9.

As one can see in Figure 9, the ratio J_{1} of the density j_{1}_{st} of the steady-state current generated by the electrochemical reactions (1e) to the density j_{st} of the overall steady-state current directly depends on the reaction rate constant k_{4}. This can be explained by the reaction (1d). The faster the reaction (1d) the more P_{2} consumes, and molecules of P_{2} do not reach electrode surface.

At low s_{20} concentrations, a value of k_{4} only slightly influences the composition of the biosensor response (curves 1 and 4). The biosensor response is almost entirely generated by the faradaic process (1e) at such concentrations of S_{2}. As one can see from the reaction scheme (1), a low concentration of S_{2} directly cause a low concentration of P_{2}. However at a high concentration s_{20} of S_{2}, a value of the reaction rate constant k_{4} makes a major impact on the composition of the biosensor response. When k_{4} is small enough, the ratio J_{1} is less than 0.5, i.e., the main part of the biosensor response is generated by the faradaic process (1f).

Figure 9 also shows that the thickness d_{3} of the diffusion layer only slightly influences the composition of the biosensor response.

#### 4.3. The Role of Substrates Synergistic Conversion in the Biosensor Response

The synergistic effect was analyzed with respect to the chemical amplification Amp. The dependence of the chemical amplification on the reaction rate constant k_{3} was investigated at different diffusion layer thicknesses (d_{3}) as well as at different values of the substrate concentration s_{20}. Results of the numerical simulation are depicted in Figure 10.

The direct influence of the reaction rate constant k_{3} on the Amp is the main dependency that can be observed in Figure 10. At a higher rate constant (k_{3}) of the biochemical reaction (1c) more molecules of P_{2} are produced. Because of this, the rate of the reaction (1d) increases, and eventually the amount of P_{1} also increases. Product P_{1} is the main generator of the current as was shown in the Section 4.2.

As one can also see in Figure 10, a higher concentration s_{20} of S_{2} corresponds to a higher chemical amplification Amp. At low values of k_{3} (k_{3} < 10^{3} M^{−1} s^{−1}) this effect is rather weak, whereas at moderate values of k_{3} (from 10^{4} to 10^{6} M^{−1} s^{−1}) this effect is strongly expressed. However, at high values of the reaction rate constant k_{3} (k_{3} > 10^{7} M^{−1} s^{−1}) this effect noticeably diminishes again. A positive influence of s_{20} on the amplification can be explained by the increase of the rate of reaction (1c) that is caused by the increase of the concentration of one of the reactants. At low values of k_{3} this is not so obvious because the increase in the reactant concentration does not counterweigh low values of k_{3}. At high values of k_{3} the system arrives at the point where the chemical amplification Amp is not dependent on the reaction rate constant k_{3}. The influence of the diffusion layer thickness d_{3} appears noticeable only at high values of k_{3}.

Similar limitations of the rate of the synergistic process have been also observed experimentally using glucose oxidase biosensors utilizing the synergistic substrates conversion [13].

The amplification of the biosensor current was also investigated with respect to the substrate S_{1} concentration. Numerical experiments were conducted at different diffusion layer thicknesses (d_{3}) as well as at different concentrations (s_{20}). The results of the numerical simulation are depicted in Figure 11.

As one can see in Figure 11 the biosensor exhibits the highest values of the chemical signal amplification Amp at low concentrations of substrate S_{1} and the lowest values of Amp at high values of s_{10}. So, the influence of s_{10} on the biosensor response that was observed in Figure 8 is diminished by the chemical amplification, because the magnitude of the amplification buffers the impact of the substrate S_{1} concentration change. When the concentration s_{10} is low, the response of the biosensor is highly amplified by the synergistic effect and vice versa in the case of high s_{10}.

The thickness of the diffusion layer only slightly influences the chemical amplification throughout the range of the concentration s_{10} that was used in this investigation. A slight influence of the diffusion layer thickness d_{3} is observed only in the case of high concentrations of both substrates, S_{1} and S_{2}.

## 5. Conclusions

The developed mathematical model (2)–(10) is suitable for the modelling of an amperometric biosensor with synergistic substrates conversion. The model can be used as a tool for the biosensor optimization prior to the development stage of an actual biosensor.

The investigation of the biosensor response showed that the range of concentrations, where the biosensor reliably operates, depends on the reaction rate constant k_{3}. In cases when k_{3} is lower the biosensor may be used to measure higher concentrations s_{20}. In cases when k_{3} is higher the biosensor may be used to measure lower concentrations s_{20} (Figure 6).

The biosensor has the highest sensitivity at moderate concentrations of the substrate S_{2} (Figure 7). At low and high concentrations of S_{2}, the biosensor sensitivity is quite low.

The biosensor response is directly dependent on the concentration s_{10} of the substrate S_{1}. Low and moderate concentrations of S_{1} are preferable, because at these concentrations the biosensor response is practically independent from the diffusion layer thickness (Figure 8).

The biosensor response almost entirely depends on the electrochemical reaction (1e) (Figure 9). The electrochemical reaction (1f) may generate a significant current only in cases when the reaction rate constant of cross reaction (1d) is low and the concentration s_{20} of S_{2} is moderate or high (Figure 9).

The biosensor exhibits the highest values of the chemical amplification in cases when the reaction rate constant k_{3} as well as the concentration s_{20} are the high (Figure 10). The diffusion has a minor effect on the rate Amp of the chemical amplification. A thicker diffusion layer weakens the chemical amplification only at high values of the reaction rate constant k_{3} and high concentrations of S_{2}. The amplification ratio Amp inversely depends on the concentration s_{10} of the substrate S_{1} (Figure 11). This effect is somewhat opposite to the s_{10} effect on the biosensor response.

To prove conclusions made the physical experiments are running using glucose dehydrogenase as well as laccase-based biosensors with different configurations. A more precise computational model implying the reaction of the reduced enzyme with oxygen in addition to the reactions (1e) and (1f) is now under development for a more accurate simulation of the biosensor action.

## Acknowledgments

This research was funded by the European Social Fund under the Global Grant Measure, Project No. VP1-3.1-ŠMM-07-K-01-073/MTDS-110000-583.

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**Figure 2.**Model validation with experimental data; (1) experimental data, (2)–(4) simulation results. Simulation was carried out at two reaction rate constants (k

_{3}) of the reaction (1c) (1.4 × 10

^{6}M

^{−1}s

^{−1}(2) and 0.35 × 10

^{6}M

^{−1}s

^{−1}(3, 4)) and at two thicknesses (d

_{3}) of the diffusion layer (223 μm (2, 3) and 116.5 μm (4)); s

_{20}= 39 μM.

**Figure 3.**Average concentrations r̄ (1), s̄

_{1}(2), s̄

_{2}(3), p̄

_{1}(4), p̄

_{2}(5), ē

_{ox}(6) and ē

_{red}(7).

**Figure 4.**Average rates ῡ

_{1}(1), ῡ

_{2}(2), ῡ

_{3}(3), ῡ

_{4}(4) of the reactions (1a)–(1d) taking place in the enzyme layer (curves 3 and 4 coincide).

**Figure 5.**The evolution of the biosensor current j

_{A}at different thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and concentrations (s

_{20}) of the substrate S

_{2}(4 μM (1, 4), 40 μM (2, 5) and 400 μM (3, 6).

**Figure 6.**The dependence of the steady-state current density j

_{st}on the substrate concentration s

_{20}at two thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and three reaction rate constants (k

_{3}) of the reaction (1c) (7 × 10

^{5}M

^{−1}s

^{−1}(1, 4), 1.4 × 10

^{6}M

^{−1}s

^{−1}(2, 5) and 2.8 × 10

^{6}M

^{−1}s

^{−1}(3, 6)).

**Figure 7.**The dependence of the biosensor sensitivity B on the substrate concentration s

_{20}. Values of all parameters are the same as in Figure 6.

**Figure 8.**The dependence of the steady-state current density j

_{st}on the substrate concentration s

_{10}at two thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and three concentrations (s

_{20}) of the substrate S

_{2}(1 μM (1, 4), 10 μM (2, 5) and 100 μM (3, 6)).

**Figure 9.**The dependence of the ratio J

_{1}on the reaction rate k

_{4}at two thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and three concentrations (s

_{20}) of the substrate S

_{2}(0.1 μM (1, 4), 10 μM (2, 5) and 1, 000 μM (3, 6)).

**Figure 10.**The dependence of the chemical amplification Amp on the reaction rate constant k

_{3}at two thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and three concentrations (s

_{20}) of S

_{2}(1 μM (1, 4), 10 μM (2, 5) and 100 μM (3, 6)).

**Figure 11.**The dependence of the chemical amplification Amp on the substrate concentration s

_{10}at two thicknesses (d

_{3}) of the diffusion layer (116.5 μm (1, 2, 3) and 233 μm (4, 5, 6)) and three concentrations (s

_{20}) of the substrate S

_{2}(1 μM (1, 4), 10 μM (2, 5) and 100 μM (3, 6)).

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Simelevicius, D.; Baronas, R.; Kulys, J.
Modelling of Amperometric Biosensor Used for Synergistic Substrates Determination. *Sensors* **2012**, *12*, 4897-4917.
https://doi.org/10.3390/s120404897

**AMA Style**

Simelevicius D, Baronas R, Kulys J.
Modelling of Amperometric Biosensor Used for Synergistic Substrates Determination. *Sensors*. 2012; 12(4):4897-4917.
https://doi.org/10.3390/s120404897

**Chicago/Turabian Style**

Simelevicius, Dainius, Romas Baronas, and Juozas Kulys.
2012. "Modelling of Amperometric Biosensor Used for Synergistic Substrates Determination" *Sensors* 12, no. 4: 4897-4917.
https://doi.org/10.3390/s120404897