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 d₁, d₂ and d₃ 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 a₁, a₂ and a₃ be the distances between the electrode surface and one of those boundaries shown in Figure 1.

The diffusion layer (a₂ < x < a₃) may be treated as the Nernst diffusion layer [29]. According to the Nernst approach a layer of thickness d₃ remains unchanged with time. It was assumed that away from it the solution is uniform in concentration.

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 < a₁, t > 0): (2a) ∂ e o x ∂ t = − k 1 e o x r 1 + k 2 e red s 11 + k 3 e red s 21 (2b) ∂ e red ∂ t = k 1 e o x r 1 − k 2 e red s 11 − k 3 e red s 21 (2c) ∂ r 1 ∂ t = D R 1 ∂ 2 r 1 ∂ x 2 − k 2 e o x r 1 (2d) ∂ s 11 ∂ t = D S 1 1 ∂ s 2 11 ∂ x 2 − k 2 e red s 11 − k 4 s 11 p 21 (2e) ∂ s 21 ∂ t = D S 2 1 ∂ s 2 21 ∂ x 2 − k 3 e red s 21 + k 4 s 11 p 21 (2f) ∂ p 11 ∂ t = D P 1 1 ∂ 2 p 11 ∂ x 2 + k 2 e red s 11 + k 4 s 11 p 21 (2g) ∂ p 21 ∂ t = D P 2 1 ∂ 2 p 21 ∂ x 2 + k 3 e red s 21 − k 4 s 11 p 21where x stands for space, t is time, e_ox(x, t) and e_red(x, t) correspond to concentrations of oxidized (E_ox) and reduced (E_red) enzyme, respectively; s₁₁(x, t) and s₂₁(x, t) (p₁₁(x, t) and p₂₁(x, t)) are the concentrations of substrates (products) in the enzyme layer; r₁(x, t) is the reducer concentration in the enzyme layer, and D_R1, D_S11, D_S21, D_P11, D_P21 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): (3a) ∂ r i ∂ t = D R i ∂ 2 r i ∂ x 2 (3b) ∂ s 1 i ∂ t = D S 1 i ∂ 2 s 1 i ∂ x 2 − k 4 s 1 i p 2 i (3c) ∂ s 2 i ∂ t = D S 2 i ∂ 2 s 2 i ∂ x 2 + k 4 s 1 i p 2 i (3d) ∂ p 1 i ∂ t = D P 1 i ∂ 2 p 1 i ∂ x 2 + k 4 s 1 i p 2 i (3e) ∂ p 2 i ∂ t = D P 2 i ∂ 2 p 2 i ∂ x 2 − k 4 s 1 i p 2 iwhere i = 2 corresponds to the dialysis membrane, and i = 3 corresponds to the diffusion layer.

Let x = 0 represents the electrode surface, while x = a₁, x = a₂ and x = a₃ 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), (4a) r 1 ( x , 0 ) = s i 1 ( x , 0 ) = p i 1 ( x , 0 ) = 0 , 0 ≤ x ≤ a 1 , i = 1 , 2 (4b) e red ( x , 0 ) = 0 , e o x ( x , 0 ) = e 0 , 0 < x < a 1 (4c) r 2 ( x , 0 ) = s i 2 ( x , 0 ) = p i 2 ( x , 0 ) = 0 , a 1 ≤ x ≤ a 2 , i = 1 , 2 (4d) p i 3 ( x , 0 ) = 0 , a 2 ≤ x ≤ a 3 , i = 1 , 2 (4e) r 3 ( x 3 , 0 ) = s i 3 ( x , 0 ) = 0 , a 2 ≤ x < a 3 , i = 1 , 2 (4f) r 3 ( a 3 , 0 ) = r 0 , s 13 ( a 3 , 0 ) = s 10 (4g) s 23 ( a 3 , 0 ) = { 0 , T s 2 > 0 s 20 T s 2 = 0where e₀ stands for the total concentration of the enzyme in the enzyme layer (e₀ = e_ox(x, t) + e_red(x, t), ∀x, t : x ε (0, a₁), t > 0), r₀ is the reducer concentration in the bulk solution, s₁₀ and s₂₀ stand for substrate concentrations in the bulk solution, T_S2 is the moment of the substrate S₂ insertion into the bulk solution.

On the boundary between two regions having different diffusivities, matching conditions have to be defined (t > 0, i = 1, 2, m = 1, 2), (5a) D R m ∂ r m ∂ x | x = a m = D R , m + 1 ∂ r m + 1 ∂ x | x = a m , r m ( a m , t ) = r m + 1 ( a m , t ) (5b) D S i m ∂ s i m ∂ x | x = a m = D S i , m + 1 ∂ s i , m + 1 ∂ x | x = a m , s i m ( a m , t ) = s i , m + 1 ( a m , t ) (5c) D P i m ∂ p i m ∂ x | x = a m = D P i , m + 1 ∂ p i , m + 1 ∂ x | x = a m , p i m ( a m , t ) = p i , m + 1 ( a m , t )where m = 1 corresponds to the boundary between the enzyme layer and the dialysis membrane, whereas m = 2 corresponds to the boundary between the dialysis membrane and the diffusion layer.

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.

In the bulk solution, concentrations of the reducer, substrates and products remain constant (t > 0), (6a) r 3 ( a 3 , t ) = r 0 (6b) s 13 ( a 3 , t ) = s 10 (6c) s 23 ( a 3 , t ) = { 0 , t < T S 2 s 20 , t ≥ T S 2 (6d) p i 3 ( a 3 , t ) = 0 , i = 1 , 2

The reaction products P₁ and P₂ 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 P₁ and P₂ at the electrode surface are permanently reduced to zero (t > 0) [13], (7a) p 11 ( 0 , t ) = 0 (7b) p 21 ( 0 , t ) = 0

Since the reaction (1e) produces as much S₁ as it consumes P₁, the concentration flux of P₁ on the electrode surface is equal to the concentration flux of S₁ but in opposite direction. The same is also assumed for the reaction (1f) and substances S₂ and P₂. These relations are expressed by the following boundary conditions (t > 0): (8a) D P 1 1 ∂ p 11 ∂ x | x = 0 = − D S 1 1 ∂ s 11 ∂ x | x = 0 (8b) D P 2 1 ∂ p 21 ∂ x | x = 0 = − D S 2 1 ∂ s 21 ∂ x | x = 0

The reducer R is electrode-inactive substance, thus its concentration flux on the electrode surface is equal to zero (t > 0), (9) D R 1 ∂ r 1 ∂ x | x = 0 = 0

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, (10a) j 1 ( t ) = n 1 F D P 1 1 ∂ p 11 ∂ x | x = 0 (10b) j 2 ( t ) = n 2 F D P 2 1 ∂ p 21 ∂ x | x = 0 (10c) j ( t ) = j 1 ( t ) + j 2 ( t )where j₁(t) and j₂(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 → ∞, (11a) j s t = lim t → ∞ j ( t ) (11b) j 1 s t = lim t → ∞ j 1 ( t ) (11c) j 2 s t = lim t → ∞ j 2 ( t )where j_st, j_1st and j_2st 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_1st to the density j_st of the overall steady-state current, (12) J 1 = j 1 s t j s t

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 S₂ [13]. The dimensionless biosensor sensitivity B to the concentration of the substrate S₂ was defined as follows: (13) B ( s 20 ) = d j s t ( s 20 ) d s 20 × s 20 j s t ( s 20 )where j_st(s₂₀) is the density of the steady-state biosensor current calculated at the concentration s₂₀ of the substrate S₂ in the bulk. B(s₂₀) 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 S₂-containing analyte to the response to the corresponding S₂-free analyte, (14) Amp ( s 20 ) = j s t ( s 20 ) j s t ( 0 )where Amp(s₂₀) is the dimensionless ratio of the signal amplification obtained by the insertion of the substrate S₂ of the concentration s₂₀ into the buffer solution [13,22].

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, (15) t r min j ( t ) > 0 { t : t j ( t ) | d j ( t ) d t | < ɛ } , j ( t r ) ≈ j s twhere t_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]: k 1 = 1.25 × 10 4 M − 1 s − 1 , k 2 = 1.2 × 10 2 M − 1 s − 1 k 3 = 1.4 × 10 6 M − 1 s − 1 , k 4 = 1.4 × 10 6 M − 1 s − 1 e 0 = 0.04 mM , s 10 = 8 mM , r 0 = 40 mM D R 1 = D S i 1 = D P i 1 = 3.15 × 10 − 6 cm 2 / s , i = 1 , 2 D R 2 = D S i 2 = D P i 2 = 4.2 × 10 − 7 cm 2 / s , i = 1 , 2 D R 3 = D S i 3 = D P i 3 = 6.3 × 10 − 6 cm 2 / s , i = 1 , 2 d 1 = 23.3 μ m , d 2 = 18.6 μ m n 1 = n 2 = 1

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 S₂ was added into the bulk solution at the moment t = 130 s which corresponds to t = 100 s in the simulations. Thus T_S2 = 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, (16) j A = j A

In the physical experiment, the electrode with the surface area of A = 0.3 cm² 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_S2 (before insertion of S₂) is about 14%. This indicates that mathematical model quite accurately reflects the physical experiment. However, after the insertion of S₂, 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 k₃ and the diffusion layer thickness d₃. 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.

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 &upsilon;̄₁, &upsilon;̄₂, &upsilon;̄₃, &upsilon;̄₄ as the average rates of the reactions (1a), (1b), (1c), (1d) taking place in the enzyme layer, (17a) υ ¯ 1 = k 1 e ¯ o x r ¯ (17b) υ ¯ 2 = k 2 e ¯ red s ¯ 1 (17c) υ ¯ 3 = k 3 e ¯ red s ¯ 2 (17d) υ ¯ 4 = k 4 s ¯ 1 p ¯ 2where r̄, s̄₁, s̄₂, p̄₁, p̄₂, ē_ox and ē_red stand for the average concentrations of R, S₁, S₂, P₁, P₂, 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 S₂ is introduced into the bulk solution, a fast reaction (1c) starts, and a large amount of E_ox and P₂ 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 P₂ start to react with S₁ 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 P₂ and S₂).

The rate of the reaction (1c) is controlled by the concentration of S₂. 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 P₁ 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) = e₀ holds for ∀x ε [0, a₁] 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) = s₁₀ 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).

Figure 5 shows the dynamics of the biosensor current j_A simulated at two thicknesses (d₃) of the diffusion layer and three concentrations (s₂₀) of the substrate S₂. For simplicity, here and below, the substrate S₂ is introduced in the bulk solution in the beginning of the experiments (T_S2 = 0 s).

At low substrate concentration (s₂₀ = 4 μM) the steady-state current of the biosensor is not influenced by the thickness d₃ 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 (s₂₀ = 40 μM, s₂₀ = 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 s₂₀ of the substrate S₂ in the bulk solution. At higher substrate concentrations (s₂₀ = 40 μM, s₂₀ = 400 μM) it is also observable that steady-state current depends inversely on the thickness d₃ of the diffusion layer. Intensity of the solution stirring plays important role when working with solutions of high concentration of the substrate.

The dependence of the steady-state biosensor current on the concentration s₂₀ of the substrate S₂ was investigated at different diffusion layer thicknesses d₃ as well as at different values of the reaction rate constant k₃. The results of the numerical simulations are depicted in Figure 6.

The biosensor may reliably measure the concentration of substrate S₂ at the range of concentrations where the biosensor response is dependent on the concentration s₂₀ but not dependent on the diffusion layer thickness d₃. As one can observe from Figure 6 this range depends on the reaction rate constant k₃. The lower boundary of this range varies from s₂₀ ≈ 0.02 μM when k₃ is relatively high (curves 3, 6) to s₂₀ ≈ 0.1 μM when k₃ is relatively low (curves 1, 4). The upper boundary of this range varies from s₂₀ ≈ 16 μM when k₃ is relatively high (curves 3, 6) to s₂₀ ≈ 40 μM when k₃ is relatively low (curves 1, 4). At the concentrations s₂₀ ≳ 250 μM the biosensor response is not influenced by the concentration of the substrate S₂.

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 d₃ and the reaction rate constant k₃ 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 S₂ is moderate (1 μM ≤ s₂₀ ≤ 10 μM). At low substrate S₂ concentrations (0.01 μM ≤ s₂₀ < 1 μM) as well as at high concentrations of S₂ (10 μM < s₂₀ ≤ 1,000 μM) biosensor exhibits poor sensitivity.

At low substrate concentrations, a higher k₃ yields a higher biosensor sensitivity, whereas the thickness d₃ of the diffusion layer does not influence the biosensor response at this concentration range. At high substrate concentrations, k₃ has the opposite effect on the biosensor sensitivity compared with a low substrate concentration range. A higher k₃ yields a lower biosensor sensitivity. At this concentration range, the thickness d₃ 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 s₂₀ of the substrate S₂, it is also important to investigate the biosensor response dependency on the concentration s₁₀ of the substrate S₁ in order to optimize the biosensors. The dependence of the steady-state biosensor current j_st on the concentration s₁₀ was investigated at different values of diffusion layer thickness d₃ as well as at different concentrations (s₂₀) of the substrate S₂. 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 s₁₀. In cases of low and moderate substrate concentrations of S₂, this dependency holds through the whole range of investigated s₁₀ concentrations. At high concentrations of S₂ (s₂₀ = 100 μM), this dependency is valid only for low concentrations of S₁ (s₁₀ < 1mM). At high concentrations of S₁ (s₁₀ > 1 mM) the biosensor response becomes practically insensitive to the change of s₁₀.

The thickness d₃ 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 S₁ and high concentrations of S₂ the effect of the diffusion layer thickness becomes noticeable.

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 k₄ was investigated at different diffusion layer thicknesses (d₃) as well as at different values of the substrate concentration s₂₀. Results of the numerical simulation are depicted in Figure 9.

As one can see in Figure 9, the ratio J₁ of the density j_1st 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 k₄. This can be explained by the reaction (1d). The faster the reaction (1d) the more P₂ consumes, and molecules of P₂ do not reach electrode surface.

At low s₂₀ concentrations, a value of k₄ 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 S₂. As one can see from the reaction scheme (1), a low concentration of S₂ directly cause a low concentration of P₂. However at a high concentration s₂₀ of S₂, a value of the reaction rate constant k₄ makes a major impact on the composition of the biosensor response. When k₄ is small enough, the ratio J₁ 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 d₃ of the diffusion layer only slightly influences the composition of 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 k₃ was investigated at different diffusion layer thicknesses (d₃) as well as at different values of the substrate concentration s₂₀. Results of the numerical simulation are depicted in Figure 10.

The direct influence of the reaction rate constant k₃ on the Amp is the main dependency that can be observed in Figure 10. At a higher rate constant (k₃) of the biochemical reaction (1c) more molecules of P₂ are produced. Because of this, the rate of the reaction (1d) increases, and eventually the amount of P₁ also increases. Product P₁ 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 s₂₀ of S₂ corresponds to a higher chemical amplification Amp. At low values of k₃ (k₃ < 10³ M⁻¹ s⁻¹) this effect is rather weak, whereas at moderate values of k₃ (from 10⁴ to 10⁶ M⁻¹ s⁻¹) this effect is strongly expressed. However, at high values of the reaction rate constant k₃ (k₃ > 10⁷ M⁻¹ s⁻¹) this effect noticeably diminishes again. A positive influence of s₂₀ 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 k₃ this is not so obvious because the increase in the reactant concentration does not counterweigh low values of k₃. At high values of k₃ the system arrives at the point where the chemical amplification Amp is not dependent on the reaction rate constant k₃. The influence of the diffusion layer thickness d₃ appears noticeable only at high values of k₃.

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 S₁ concentration. Numerical experiments were conducted at different diffusion layer thicknesses (d₃) as well as at different concentrations (s₂₀). 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 S₁ and the lowest values of Amp at high values of s₁₀. So, the influence of s₁₀ 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 S₁ concentration change. When the concentration s₁₀ is low, the response of the biosensor is highly amplified by the synergistic effect and vice versa in the case of high s₁₀.

The thickness of the diffusion layer only slightly influences the chemical amplification throughout the range of the concentration s₁₀ that was used in this investigation. A slight influence of the diffusion layer thickness d₃ is observed only in the case of high concentrations of both substrates, S₁ and S₂.