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This paper presents a mathematical model of carbon nanotubes-based mediatorless biosensor. The developed model is based on nonlinear non-stationary reaction-diffusion equations. The model involves four layers (compartments): a layer of enzyme solution entrapped on a terylene membrane, a layer of the single walled carbon nanotubes deposited on a perforated membrane, and an outer diffusion layer. The biosensor response and sensitivity are investigated by changing the model parameters with a special emphasis on the mediatorless transfer of the electrons in the layer of the enzyme-loaded carbon nanotubes. The numerical simulation at transient and steady state conditions was carried out using the finite difference technique. The mathematical model and the numerical solution were validated by experimental data. The obtained agreement between the simulation results and the experimental data was admissible at different concentrations of the substrate.

Biosensors are devices mainly used to measure concentrations of substances (analytes) [

Amperometric biosensors measure the changes in the current on the working electrode due to the oxidation or reduction of the products of biochemical reactions [

Since the discovery of carbon nanotubes [

The development and optimization of new biosensors require a high number of experiments. Mathematical modelling is rather often used in order to decrease the number of physical experiments by replacing them with mathematical simulations [

A CNT-based biosensor was mathematically modelled by Lyons [

This paper presents a mathematical model of the mediatorless amperometric biosensor based on the enzyme-loaded CNT electrode deposited on the outer perforated membrane. The proposed model describes the unmediated operation of the biosensor having the geometrical structure similar to that of the already modelled mediated biosensor [

The investigated biosensor has the layered structure and is composed of different materials and sizes according to [

All electrochemical experiments were performed using a conventional three-electrode system containing a planar CNT electrode as a working electrode, a platinum wire as a counter electrode and an Ag/AgCl in saturated KCl as a reference electrode. The default buffer was 0.05 M acetate buffer (pH 6.0) containing 1 mM Ca^{2+}. Steady state currents of the biosensors were recorded at 0.4 V using a polarographic analyzer “PARSTAT 2273” (Princeton Applied Research, USA). Principal structure of the considered biosensor is shown in

Enzymatic reaction is employed in the biosensor to selectively detect the substrate (S) in the target analyte. The enzymatic reaction takes place in the regions of the biosensor filled with the enzyme,
_{1} is a constant of the enzymatic reaction rate. In the reaction, the substrate S reacts with the oxidized form of the enzyme (E_{ox}) and reduces it (E_{red}) producing the product P. The latter is considered as not impacting the processes in the biosensor and therefore is omitted in the following model.

The output current of the biosensor is generated due to the direct enzyme oxidation taking place in the layer of the carbon nanotubes,
_{2} is a constant of the electrochemical reaction rate and _{e} is the number of electrons released in one reaction event. The enzyme E_{red} is re-oxidized in the

The mathematical model for the biosensor is formulated as a system of non-linear reaction-diffusion equations with the corresponding initial and boundary conditions. The model is formulated in the one-dimensional space by applying the homogenization process to the perforated membrane and the mesh of carbon nanotubes [_{i}_{1} is the thickness of the enzyme layer existing between the CNTs and the terylene membrane, _{2} is the thickness of the CNT layer, _{3} is the thickness of the perforated membrane and _{4} stands for diffusion layer forming on the outer surface of the perforated membrane.

The enzyme in the oxidized form participates in the enzymatic _{1} and Ω_{2}. Additionally, the enzyme that is properly conjoined with the active sides of the CNTs is also involved in the electrochemical _{i}_{i}^{th} layer Ω_{i}_{ox,i} = _{ox,i}(_{e,ox,2} = _{e,ox,2}(_{e,red,2} = _{e,red,2}(

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 _{e,red,2}) is additionally re-oxidized in the electrochemical _{red,i} = _{red,i}(

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 _{1} and Ω_{2}). The dynamics of the substrate concentration is described by the following reaction-diffusion equations:
_{S1} and _{S4} are the diffusion coefficients of the substrate S in the enzyme and the diffusion layers, respectively. The coefficients _{S2} and _{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,
_{0} 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 (Γ_{0}),

On the boundaries between adjacent regions (Ω_{i}_{i}_{+1},

The diffusion layer Ω_{4} (_{3} < _{3}+_{4}) may be treated as the Nernst diffusion layer [_{4} remains unchanged with time.

The modelled experiment starts (_{4}, where the substrate concentration is considered equal to that in the bulk solution,
_{i}_{i}

At the beginning of the experiment all the enzyme is assumed to be in the oxidized form. The initial concentrations of the oxidized (_{ox,1}) and the reduced (_{red,1}) enzyme in the enzyme layer Ω_{1} are defined as follows:
_{0} is the enzyme concentration in the layer filled with the enzyme only (Ω_{1}).

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 [_{1}, and

The output current of the biosensor is generated due to the enzyme re-oxidation in the electrochemical _{2}). It was assumed that only the enzyme molecules properly attached to the CNTs (_{e,red,2}) can be re-oxidized in the

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

The sensitivity is one of the most important characteristics of the biosensor operation [_{S}

The enzyme and the diffusion layers were assumed to be homogeneous. The diffusion coefficients of the substrate S in these layers are constant,
_{e} is the diffusion coefficient of the substrate for the enzyme layer, and _{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 [_{2} and _{3} are the tortuosities describing the structural properties of the corresponding medium, < 0 < _{2}, _{3} < 1. Values of the tortuosities _{2} and _{3} mainly depend on the directional anisotropy of the CNTs, particularly on the nanotubes orientation [

Similar approach to estimating the effective diffusion coefficients was applied to the two-dimensional model for the mediated biosensor based on a CNT electrode [

The formulated mathematical

In this work the method of finite differences was used to solve numerically the system of the proposed model [_{i}

In the numerical simulations, the steady state was assumed at the time, when the increase of the output current becomes small enough,
_{R} is the time when the steady state results in the output current _{R}

The following parameter values were used as a basic configuration of the considered biosensor and were kept constant in all the experiments:

The mathematical

It is difficult to experimentally measure the concentration of the enzyme involved in the electrochemical _{2} of the electrochemical _{2}-parameters. These two parameters were fitted by minimizing the relative error between the simulated and experimental responses. A satisfactory match was obtained at _{2} = 550 s^{−1}, when only 0.5% of the enzyme is assumed as participating in the electrochemical

As it can be observed in _{0} of the substrate: _{0} = 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 _{2} suitably matches with reported values of apparent heterogeneous electron transfer rate constants [

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

When modelling the corresponding mediated biosensor, no similar parameter was used [

The behaviour of the response of the mediatorless biosensor was investigated by performing numerical simulations based on the developed mathematical _{2} and the concentration _{0} of the enzyme.

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 _{0} on the behaviour of the biosensor action, the biosensor responses were simulated in wide ranges of the enzyme (_{0}) as well as the substrate (_{0}) concentrations.

The biosensor current varies in orders of magnitude as the concentrations of these species change, therefore the dimensionless sensitivity _{S}

As one can see in _{0} 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 _{S}_{0} > 40mol m^{−3}) the length of the linear part of the calibration curve is practically proportional to the enzyme concentration _{0}, while at the low enzyme concentrations (_{0} < 4mol m^{−3}) the sensitivity of the biosensor loosely depends on the concentration _{0}. The especially drastic change in the sensitivity appears at the moderate values of _{0}. ^{−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 (^{2} essentially compares the rate of the enzyme Reaction (_{1}_{0}) with the mass transport through the enzyme-loaded layer
^{2} ≫ 1. In the very opposite case, when ^{2} ≪ 1, the enzyme kinetics predominates in the response. The diffusion module ^{2} approaches one at the concentration _{0} of the electrochemically active enzyme approximately equal to 0.75 mol m^{−3}. This value of _{0} favorably matches with the enzyme concentrations at which the behaviour of the biosensor sensitivity distinctly changes (see

Mathematical models for amperometric biosensors are usually formulated assuming electrochemical reactions to be extremely fast when all the available reagent is immediately consumed [^{−1} of the _{2}-parameter was fitted by minimizing the relative error between the simulated and experimental responses.

In order to show the impact of the rate constant _{2} of the electrochemical _{S}_{2}-parameter by changing the substrate concentrations _{0} in a wide range. The simulation results are depicted in

As one can see in _{2} proportionally shifts the curve representing the sensitivity _{S}_{2} proportionally prolongs the linear part of the biosensor calibration curve. The shape of the curve remains unchanged for all _{2} used in the investigation.

The developed mathematical

The numerical simulation showed that only a fraction of the enzyme participates in the direct electron transfer at the nanoscale (_{2} was introduced into the mathematical

The sensitivity of the mediatorless biosensor based on the enzyme-loaded CNT electrode can be noticeably increased and the linear part of the calibration curve can be prolonged by increasing the volume fraction

The biosensor sensitivity can be also significantly increased by increasing the enzyme concentration. The linear part of the calibration curve is longer when the biosensor acts in the diffusion-limiting mode rather than in the enzyme reaction-controlled mode (

To prove the conclusions made, physical experiments are running using biosensors acceptable for detection of a wide range of carbohydrates with a tunable selectivity.

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.

Principal structure of the active surface of the biosensor. The figure is not to scale.

The densities (_{2} = 550 s^{−1}, _{0} = 0.0455 mol m^{−3} and three values of the ratio

The dynamics of the density _{0}): 0.099 (1a, 1b), 0.99 (2a, 2b), 1.96 (3a, 3b) and 2.92 mol m^{−3} (4a, 4b). The simulation was performed at

The dimensionless biosensor sensitivity _{S}_{0} at _{2} = 550 s^{−1}, _{0}: 0.0455 (1), 0.455 (2), 4.55 (3), 45.5 (4), 455 (5) and 4, 550mol m^{−3} (6). Other parameters are as defined in

The dimensionless biosensor sensitivity _{S}_{0} at _{0} = 0.0455 mol m^{−3} and four values of the electrochemical reaction rate constant _{2}: 5.5 (1), 55 (2), 550 (3) and 5,500 s^{−1} (4). Other parameters are as defined in