The enzyme in the oxidized form participates in the enzymatic Reaction (1) in both enzyme-loaded regions, Ω₁ and Ω₂. Additionally, the enzyme that is properly conjoined with the active sides of the CNTs is also involved in the electrochemical Reaction (2) and is re-oxidized. Due to the procedure of the mediatorless biosensor preparation, it was assumed that only a part of the whole enzyme has direct contact with the electro-active sides of the CNTs, and only that part of the enzyme is involved in the electrochemical Reaction (2) [19,20]. The dynamics of the concentration of the oxidized enzyme can be mathematically described as follows (t > 0): (4) ∂ E ox , i ∂ t = − k 1 E ox , i S i , x ∈ Ω i , i = 1 , 2 , (5) ∂ E e , ox , 2 ∂ t = − k 1 E e , ox , 2 S 2 + k 2 E e , red , 2 , x ∈ Ω 2 ,where x stands for the space, t is time, S_i = S_i(x, t) is the substrate concentration in the i^th layer Ω_i, E_ox,i = E_ox,i(x, t) is the concentration of the oxidized enzyme that is not involved in the Reaction (2). E_e,ox,2 = E_e,ox,2(x, t) and E_e,red,2 = E_e,red,2(x, t) are the concentrations of the enzyme involved in the Reaction (2) in the oxidized and the reduced forms, respectively.

The dynamics of the concentration of the reduced enzyme is practically opposite to that of the oxidized enzyme. E_red is produced in the enzymatic Reaction (1) in both the enzyme-loaded layers. The electrochemically active part of the reduced enzyme (E_e,red,2) is additionally re-oxidized in the electrochemical Reaction (2) (t > 0), (6) ∂ E red , i ∂ t = k 1 E ox , i S i , x ∈ Ω i , i = 1 , 2 , (7) ∂ E e , red , 2 ∂ t = − k 1 E e , ox , 2 S 2 − k 2 E e , red , 2 , x ∈ Ω 2 ,where E_red,i = E_red,i(x, t) is the concentration of the reduced enzyme that is not involved in the Reaction (2).

The molecules of both forms of the enzyme, E_ox and E_red, are considered as immobilized, and therefore there are no diffusion terms in the corresponding equations.

The substrate S diffuses from the bulk through the holes of the perforated membrane to inner layers of the biosensor. The substrate also participates in the enzymatic Reaction (1) in the enzyme-loaded layers (Ω₁ and Ω₂). The dynamics of the substrate concentration is described by the following reaction-diffusion equations: (8) ∂ S i ∂ t = { D S i ∂ 2 S i ∂ x 2 − k 1 E ox , i S i , for i = 1 , D S i ∂ 2 S i ∂ x 2 − k 1 E ox , i S i − − k 1 E e , ox , i S i , for i = 2 , , x ∈ Ω i , t > 0 , D S i ∂ 2 S i ∂ x 2 , for i = 3 , 4 , where D_S1 and D_S4 are the diffusion coefficients of the substrate S in the enzyme and the diffusion layers, respectively. The coefficients D_S2 and D_S3 stand for the effective diffusivity of the substrate in the CNT layer and the perforated membrane, respectively.

Assuming well stirred buffer solution leads to the constant thickness of the diffusion layer as well as the constant concentration above that layer, (9) S 4 | Γ 4 = S 0 , t > 0 ,where S₀ is the concentration of the substrate in the bulk solution.

The terylene membrane is placed between the enzyme and the insulating layer of the biosensor and plays a role of an insulating film immobilizing the enzyme. Assuming low volume of the membrane and the insulating layer behind it, the non-leakage condition is used for the substrate on the surface of the terylene membrane (Γ₀), (10) ∂ S 1 ∂ x | Γ 0 = 0 , t > 0.

On the boundaries between adjacent regions (Ω_i and Ω_i+1, i = 1,2,3), the merge conditions are defined for the substrate: (11) D S i ∂ S i ∂ x | Γ i = D S i + 1 ∂ S i + 1 ∂ x | Γ i , S i | Γ i = S i + 1 | Γ i , i = 1 , 2 , 3 , t > 0.

The diffusion layer Ω₄ (x₃ < x < x₃+d₄) may be treated as the Nernst diffusion layer [33]. According to the Nernst approach a layer of thickness d₄ remains unchanged with time.

The modelled experiment starts (t = 0) when the substrate is poured into the buffer solution. At this time, the substrate is absent in the entire biosensor and the diffusion layer, except only the outer boundary Γ₄, where the substrate concentration is considered equal to that in the bulk solution, (12) S i = 0 , x ∈ Ω ¯ i \ Γ 4 , S 4 | Γ 4 = S 0 , i = 1 , 2 , 3 , 4 , t = 0 ,where Ω̄_i is the closed region corresponding to the open region Ω_i.

At the beginning of the experiment all the enzyme is assumed to be in the oxidized form. The initial concentrations of the oxidized (E_ox,1) and the reduced (E_red,1) enzyme in the enzyme layer Ω₁ are defined as follows: (13) E ox , 1 = E 0 , E red , 1 = 0 , x ∈ Ω 1 , t = 0 ,where E₀ is the enzyme concentration in the layer filled with the enzyme only (Ω₁).

Due to the procedure of the mediatorless biosensor preparation, the concentration of the enzyme in the CNT layer is assumed to be lower than in the bulk [19,20]. Moreover, only a part of the enzyme is properly conjoined with the CNTs to be electrochemically active. Assuming a uniform distribution of the enzyme in the CNT layer leads to following conditions: (14) E ox , 2 = ( 1 − α ) η E 0 , E e , ox , 2 = α η E 0 , E red , 2 = 0 , x ∈ Ω 2 , t = 0 ,where η (0 ≤ η ≤ 1) is the ratio of the enzyme concentration in the CNTs to that in the enzyme layer Ω₁, and α (0 ≤ α ≤ 1) is the volume fraction of the electrochemically active enzyme in the CNT layer.

The output current of the biosensor is generated due to the enzyme re-oxidation in the electrochemical Reaction (2) taking place only in the enzyme-loaded CNT electrode (region Ω₂). It was assumed that only the enzyme molecules properly attached to the CNTs (E_e,red,2) can be re-oxidized in the Reaction (2). The density j(t) of the output current is proportional to the rate of the electrochemical Reaction (2), (15) j ( t ) = n e F k 2 ∫ x 1 x 2 E e , red , 2 d x ,where F is the Faraday constant.

In practice, the current generated at a steady state is often used as an output of the biosensor. Assuming that the system approaches the steady state as t → ∞ leads to the following definition of the density J of the steady state output current of the considered biosensor: (16) J = lim t → ∞ j ( t ) .

The sensitivity is one of the most important characteristics of the biosensor operation [1,2]. The normalized sensitivity to the substrate concentration B_S was measured when investigating the impact of the model parameters on the behaviour of the biosensor, (17) B S ( S 0 ) = d J ( S 0 ) d S 0 × S 0 J ( S 0 ) .

The enzyme and the diffusion layers were assumed to be homogeneous. The diffusion coefficients of the substrate S in these layers are constant, (18) D S 1 = D e , D S 4 = D n ,where D_e is the diffusion coefficient of the substrate for the enzyme layer, and D_n is the corresponding coefficient for the bulk.

The CNT layer and the perforated membrane both are non-homogeneous mediums. Assuming these layers as periodic media, the volume averaging approach can be applied to estimate the effective (averaged) diffusion coefficients for these layers [34–36]. The CNT layer is considered as a composite of three compartments: the carbon nanotubes, the enzyme and the bulk. The perforated membrane is treated as a composite of the non-permeable material the membrane is built of and the bulk in the holes of the membrane together with the carbon nanotubes sunk in. Assuming relatively low volume fraction of the CNTs, the diffusion coefficients of the substrate can be expressed in terms of the volume fractions of the compounds, the diffusion coefficients in the specific compounds and the tortuosity: (19) D S 2 = θ 2 ( η D e + ( 1 − η ) D n ) , (20) D S 3 = θ 3 ρ D n ,where ρ stands for the perforation level of the membrane as the ratio of the volume occupied by the holes to the overall volume of the membrane, θ₂ and θ₃ are the tortuosities describing the structural properties of the corresponding medium, < 0 < θ₂, θ₃ < 1. Values of the tortuosities θ₂ and θ₃ mainly depend on the directional anisotropy of the CNTs, particularly on the nanotubes orientation [29]. A near-unity tortuosity corresponds to a fully vertical alignment of the CNTs (Figure 1). The effect of the directional anisotropy of the nanotubes has been already investigated for the corresponding mediated biosensor with the CNT electrode deposited on the perforated membrane [31].

Similar approach to estimating the effective diffusion coefficients was applied to the two-dimensional model for the mediated biosensor based on a CNT electrode [31] and the one-dimensional model for the biosensor with the outer perforated membrane [37].

In this work the method of finite differences was used to solve numerically the system of the proposed model [32]. The model domain was discretized by applying the regular mesh to each space region Ω_i(i = 1,2,3,4), and a constant step was used to discretize the time. The model equations were approximated by semi-implicit finite difference schemes. The diffusion terms were approximated by the backward differences, while the reaction terms were approximated by the forward differences [38]. The resulting system of linear tri-diagonal algebraic equations was solved effectively. The computer simulation was carried out using a software developed by the authors in C++ programming language [39].

In the numerical simulations, the steady state was assumed at the time, when the increase of the output current becomes small enough, (21) T R = min j ( t ) > 0 { t : d j ( t ) d t × t j ( t ) < ε } ,where T_R is the time when the steady state results in the output current j(T_R), and ε is the decay rate of the output current to be considered in the simulations. In all the simulations presented in this paper ε = 0.01 was used.

The following parameter values were used as a basic configuration of the considered biosensor and were kept constant in all the experiments: (22) d 1 = 10 − 7 m , d 2 = 4 × 10 − 1 m , d 3 = 10 − 5 m , d 4 = 3 × 10 − 4 m , D e = 3 × 10 − 10 m 2 s − 1 , D n = 2 D e , η = 0.5 , θ 2 = 1 / 3 , θ 3 = 0.5 , ρ = 0.0625 n e = 2 , k 1 = 6.9 × 10 2 m 3 mol − 1 s − 1 .

The mathematical Model (4)–(16) and the corresponding numerical solution were validated by experimental data. The experimental data along with the corresponding simulated responses of the biosensor are shown in Figures 2 and 3.

It is difficult to experimentally measure the concentration of the enzyme involved in the electrochemical Reaction (2). The volume fraction α of the electrochemically active enzyme highly depends on the properties of carbon nanotubes and the technology of the electrode preparation [20]. The rate constant k₂ of the electrochemical Reaction (2) is also very specific for the developed CNT electrode. Multiple numerical simulations were performed in order to estimate the values of the α- and k₂-parameters. These two parameters were fitted by minimizing the relative error between the simulated and experimental responses. A satisfactory match was obtained at α = 0.005 and k₂ = 550 s⁻¹, when only 0.5% of the enzyme is assumed as participating in the electrochemical Reaction (2) in the CNT layer. The steady state responses of the simulated as well as physical experiments are shown in Figure 2.

As it can be observed in Figure 2, the responses obtained in numerical experiments using α = 0.005 are close to the experimental data. At that value of α, the relative error of the simulated steady state responses is less than 10%, except two concentrations S₀ of the substrate: S₀ = 0.199 and 0.299 mol m⁻³. At low substrate concentrations the relative error reaches almost 20%. Taking into account relatively low biosensor currents and possible measuring errors in the physical experiments, the error of the simulated responses can be considered admissible. The fitted value 550 s⁻¹ of the electrochemical reaction rate constant k₂ suitably matches with reported values of apparent heterogeneous electron transfer rate constants [9].

Figure 2 also shows how sensitive is the biosensor response to the α-parameter. The steady state current directly depends on the α. Increasing the volume fraction α of the electrochemically active enzyme increases the biosensor current and prolongs the linear part of the biosensor calibration curve.

The proposed model was also validated at the transient conditions. The simulation was carried out using the same biosensor configuration as in the simulations shown in Figure 2. The dynamics of the simulated density j(t) of the biosensor current along with the corresponding experimental data are shown in Figure 3. The oscillations in the experimental data shown in Figure 3 are caused by the measurement errors. Due to relatively low output currents, the errors are relatively high. Taking into account the measurement errors, the simulated responses can be assumed adequate to that observed in the physical experiments.

When modelling the corresponding mediated biosensor, no similar parameter was used [31]. The response of the mediated biosensor based on an enzyme-loaded CNTs was modelled assuming whole enzyme involved in the corresponding mediated electrochemical reaction. Multiple numerical simulations of the response of the mediatorless biosensor showed that involving the whole enzyme into the electrochemical Reaction (2) leads to the steady-state currents being in about two order greater than the experimental ones. Because of this, the volume fraction α of the electrochemically active enzyme is of crucial importance when modelling the mediatorless biosensor based on enzyme loaded CNT electrode.

The proper conjunction of the enzyme with the active sites of the CNTs is essential for the mediatorless biosensor operation. The proposed model of the biosensor assumes the proportional dependence between the total concentration of the enzyme and the concentration of the enzyme capable to participate in the electrochemical Reaction (2). To investigate the impact of the enzyme concentration E₀ on the behaviour of the biosensor action, the biosensor responses were simulated in wide ranges of the enzyme (E₀) as well as the substrate (S₀) concentrations.

The biosensor current varies in orders of magnitude as the concentrations of these species change, therefore the dimensionless sensitivity B_S was considered instead of the density J of the steady state current. The simulation results are provided in Figure 4.

As one can see in Figure 4, an increase in the enzyme concentration E₀ noticeably prolongs the linear part of the biosensor calibration curve, ensuring qualitative determination of higher substrate concentrations. The linear part of the biosensor calibration curve corresponds to B_S ≈ 1. At the high enzyme concentrations (E₀ > 40mol m⁻³) the length of the linear part of the calibration curve is practically proportional to the enzyme concentration E₀, while at the low enzyme concentrations (E₀ < 4mol m⁻³) the sensitivity of the biosensor loosely depends on the concentration E₀. The especially drastic change in the sensitivity appears at the moderate values of E₀. Figure 4 shows that a tenfold increase of the enzyme concentration from 45.5 (curve 4) to 455mol m⁻³ (curve 5) leads to an approximately 50 times longer linear part of the biosensor calibration curve.

The noticeable change in the behaviour of the biosensor sensitivity (Figure 4) at the moderate enzyme concentrations could be explained by the transition from the kinetic-limited to the diffusion-controlled mode of the biosensor action [26–28]. The diffusion module σ² essentially compares the rate of the enzyme Reaction (k₁αE₀) with the mass transport through the enzyme-loaded layer ( D S 2 / d 2 2 ), σ 2 = k 1 α E 0 d 2 2 / D S 2. The biosensor response is known to be under diffusion control when the diffusion module σ² ≫ 1. In the very opposite case, when σ² ≪ 1, the enzyme kinetics predominates in the response. The diffusion module σ² approaches one at the concentration αE₀ of the electrochemically active enzyme approximately equal to 0.75 mol m⁻³. This value of αE₀ favorably matches with the enzyme concentrations at which the behaviour of the biosensor sensitivity distinctly changes (see Figure 4).

Mathematical models for amperometric biosensors are usually formulated assuming electrochemical reactions to be extremely fast when all the available reagent is immediately consumed [27,28]. In modelling the mediatorless biosensor, it was assumed that only a small fraction of the enzyme participates in the electrochemical reaction taking place in the CNT electrode. Because of the relatively spare sites, where the direct electron transfer is possible, the electrochemical Reaction (2) cannot be assumed very fast. The appropriate value 550 s⁻¹ of the k₂-parameter was fitted by minimizing the relative error between the simulated and experimental responses.

In order to show the impact of the rate constant k₂ of the electrochemical Reaction (2) on the biosensor response, the response was numerically simulated and the corresponding dimensionless biosensor sensitivity B_S was calculated at different values of the k₂-parameter by changing the substrate concentrations S₀ in a wide range. The simulation results are depicted in Figure 5.

As one can see in Figure 5 an increase in the electrochemical reaction rate constant k₂ proportionally shifts the curve representing the sensitivity B_S to the right. Thus an increase in the coefficient k₂ proportionally prolongs the linear part of the biosensor calibration curve. The shape of the curve remains unchanged for all k₂ used in the investigation.