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The response of an amperometric biosensor based on a chemically modified electrode was modelled numerically. A mathematical model of the biosensor is based on a system of non-linear reaction-diffusion equations. The modelling biosensor comprises two compartments: an enzyme layer and an outer diffusion layer. In order to define the main governing parameters the corresponding dimensionless mathematical model was derived. The digital simulation was carried out using the finite difference technique. The adequacy of the model was evaluated using analytical solutions known for very specific cases of the model parameters. By changing model parameters the output results were numerically analyzed at transition and steady state conditions. The influence of the substrate and mediator concentrations as well as of the thicknesses of the enzyme and diffusion layers on the biosensor response was investigated. Calculations showed complex kinetics of the biosensor response, especially when the biosensor acts under a mixed limitation of the diffusion and the enzyme interaction with the substrate.

Biosensors are analytical devices converting a biochemical recognition reaction into a measurable effect [

A large part of commercially available and disposable biosensors is prepared by screen-printing technology [

The understanding of the kinetic peculiarities of biosensors is of crucial importance for their design. The mathematical modelling is rather widely used to improve the efficiency of the biosensors design and to optimize their configuration [

The goal of this investigation was to make a model allowing an effective computer simulation of amperometric biosensors based on CME as well as to investigate the influence of the physical and kinetic parameters on the biosensor response. An ordered ping-pong scheme of the enzyme catalysed substrate conversion in presence of a mediator is considered. The CME is considered as an electrode containing a relatively thin layer of the low soluble mediator and covered with an enzyme membrane. The developed model is based on non-stationary reaction-diffusion equations [

We consider an ordered ping-pong scheme of enzyme (E) catalysed substrate (S) conversion in presence of mediator (M),
_{ox}, E_{red} and ES are oxidized enzyme, reduced enzyme and enzyme substrate, respectively, P and P_{1} are the reaction products.

The reaction takes part on a chemically modified electrode (CME). The CME is considered as an electrode containing a relatively thin layer of the low soluble mediator and covered with an enzyme membrane. The model involves three regions: the enzyme layer where the enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the mass transport by diffusion takes place and a convective region where the analyte concentration is maintained constant.

Assuming the quasi steady state approximation, the concentration of the intermediate complex (ES) do not change and is usually neglected when simulating the biochemical behaviour of biosensors [_{e}_{e}_{e}_{e}_{se}_{me}_{pe}_{e}_{e}_{t}_{cat}_{cat}_{2}, _{red}_{red}_{1}_{2}/(_{−1} + _{2}), _{ox}_{ox}_{3}. The total sum _{t}_{t}_{ox}_{red}_{s}_{ox}_{red}_{s}_{ox}, E_{red}, ES, respectively. From _{d}_{d}_{d}_{d}_{sd}_{md}_{pd}

The diffusion layer (_{e}_{e}_{d}_{d}

Let _{e}_{0} is the concentration of the mediator at the boundary between electrode and enzyme layer, _{0} is the concentration of the substrate in the bulk solution.

On the boundary between two regions having different diffusivities, we define the matching conditions (

These conditions mean that fluxes of the substrate, mediator and product through the stagnant external layer equal to the corresponding fluxes entering the surface of the enzyme membrane. The partition of the substrate, mediator and product in the membrane versus the bulk is assumed to be equal.

In the bulk solution the concentrations of the substrate, mediator and product remain constant (

The concentration _{e}

The constant concentration _{0} of the mediator on the electrode can be achieved by permanent dissolution of adsorbed mediator. The direct measurements show that _{0} can be as low as 10^{-6} M [

The measured current is accepted as a response of an amperometric biosensor in physical experiments. The anodic current is directly proportional to the flux of the reaction product at the electrode surface [_{e}_{S}

The sensitivity is also a very important characteristic of biosensors [_{S}_{0}) stands for the dimensionless sensitivity of the biosensor at the concentration _{0} of the substrate in the bulk solution, _{S}_{0}) is the steady state current calculated at the substrate concentration _{0}.

The concentrations _{e}_{d}

All the concentration functions (_{e}_{d}

Definite problems arise when solving analytically non-linear partial differential equations [

A semi-implicit linear finite difference scheme has been built as a result of the difference approxima-tion [

To have an accurate and stable result it was required to use very small step size in _{e}_{e}_{d}_{e}_{e}_{d}_{e}_{d}_{e}_{e}_{d}_{e}

Usually an implicit computational scheme does not restrict time increment [

The digital simulator has been programmed in JAVA language [

In digital simulation, the biosensor response time was assumed as the time when the absolute current slope value falls below a given small value normalized with the current value. In other words, the time needed to achieve a given dimensionless decay rate _{R}^{-4}. The response time _{R}_{R}_{0 5} be the time at which the reaction-diffusion process reaches the medium, called the half time of the steady state or, particularly, half of the time moment of occurrence of the steady state current, i.e. _{0.5}) = 0.5, _{0.5}) = 0.5_{R}

The adequacy of the mathematical model of the biosensor was evaluated using known analytical solution of a two compartment model of amperometric biosensors [_{0}) to the mediator (_{0}) concentrations combining them with the rates of the corresponding reactions

At relatively low concentrations of the substrate when Σ ≪ 1 (_{0}_{red}_{0}_{ox}

Assuming _{0} ≪ _{cat}_{red}_{t}k_{red}s_{t}k_{red}s_{S}

The dimensionless factor
_{t}k_{red}

The mathematical model as well as the numerical solution of the model was evaluated for different concentrations of the mediator (_{0}) and the substrate (_{0}). The following values of the model parameters were constant in the numerical simulation:

The numerical solution of the model _{e}_{d}_{0} = 1 M and _{0} = 1mM, the relative difference between the numerical and analytical solutions was about 0.5%.

At relatively low concentrations of the mediator when Σ ≫ 1 (_{0}_{ox}_{0}_{red}

Assuming _{0} ≪ _{cat}_{ox}_{d}_{S}

At _{e}_{d}_{0} = 1 _{0} = 0.1 mM, the relative difference between the numerical and analytical solutions was about 1%.

The number _{t}_{red}_{ox}_{red}_{ox}_{red}_{ox}

It is rather well known that an ordinary enzyme electrode acts under diffusion limitation when the diffusion modulus is much greater than unity [

In the case of CM electrode, the kinetics of the enzymatic reaction was expressed by two rates: _{red}_{ox}_{red}_{ox}_{red}_{ox}_{ox}_{red}

In order to define the main governing parameters of the mathematical model we introduce the following dimensionless parameters:
_{0}, _{0} are the dimensionless concentrations. The dimensionless thickness of enzyme membrane equals one.

The governing

The governing

The initial conditions

The matching

The dimensionless current (flux) _{S}

Assuming the same diffusion coefficients for the all three species, only the following dimensionless parameters remain in the dimensionless mathematical model _{0} - the substrate concentration in the bulk solution, _{0} - the mediator concentration at the elec-trode surface, _{ox}_{red}_{rel}_{rel}_{sd}_{se}_{md}_{me}_{pd}_{pe}_{rel}_{ox}_{red}

Using numerical simulation, peculiarities of the biosensor action has been investigated at different values of the model parameters.

_{e}_{e}_{e}_{d}_{e}_{d}_{0}) as well as two concentrations (10^{-5} and 10^{-3} M) of the mediator (_{0}). The corresponding dimensionless concentrations of the substrate (_{0}) as well as of the mediator (_{0}) are: 0.1 and 10. Values of all other parameters are as defined in _{R}_{0.5} when 50% of the steady state current has been reached. At values

As one can see in _{0} of the mediator and low concentration _{0} of the substrate. At those conditions (_{0} ≫ _{0}, Σ ≪ 1), the rate of the enzymatic reaction depends practically only on the substrate concentration as defined in _{0} ≫ _{0}, Σ ≫ 1). Additional numerical experiments approved that a shoulder in the profile of the mediator concentration appears only in the cases when Σ ≪ 1.

_{0} = 0.1 and _{0} = 10, at which

One can see in _{0} and _{0}. The current grows notably faster at higher concentration _{0} (curves 3 and 4) of the substrate rather than at lower one (curves 1 and 2). The effect of concentration _{0} of the mediator on the biosensor response becomes notable with some delay. The mediator diffuses from the CME into the enzyme layer in a sufficient for reaction amount very quickly while the substrate has to diffuse across the Nernst diffusion and enzyme layers. Therefore, at the very beginning of the biosensor operation, the biosensor acts under a limitation of the substrate diffusion.

The dimensionless model _{red}_{ox}_{red}_{ox}_{red}_{ox}_{red}_{ox}_{S}_{0.5} on _{0}) and three concentrations of the mediator (_{0}) changing exponentially the thickness de of the enzyme layer from 0.3 _{ox}^{–7} Ms, and _{red}^{–4} Ms.

As one can see in _{ox}_{S}_{red}_{S}_{S}_{S}_{ox}_{ox}_{red}_{red}_{S}_{red}_{S}

The complex effect of the diffusion modulus on the biosensor response can be seen also in _{ox}^{–7} Ms), the biosensor sensitivity _{S}_{e}_{t}_{S}^{−7} to 10^{−4} Ms, i.e. when _{ox}_{red}_{red}^{–4} Ms), the biosensor sensitivity slightly increases with increase in

_{ox}^{–7} Ms). This linearity can also be noticed in _{0.5} is distinctly a linear function of the diffusion modulus when _{ox}_{0.5} changes non-linearly.

The dependence of the biosensor response on the thickness of the external diffusion layer is shown in the _{d}_{d}_{e}

As one can see in _{S}_{S}_{S}_{S}

_{S}_{S}_{S}_{S}

A relatively short linear range of the calibration curve is one of serious drawbacks restricting wider use of biosensors [

The dependence of the biosensor response on the dimensionless ratio Σ of the substrate and mediator concentrations is depicted in _{0} in the bulk solution and keeping the mediator concentration _{0} constant.

One can see in _{0} ≈ 0.1, _{0} ≈ 10 mM). The dependence of the steady state current on the ratio Σ is noticeably affected by the diffusion modulus. The current is directly proportional to
_{S}_{S}_{S}

^{–2}) and they are of very low sensitivity a high concentrations of the substrate (Σ > 1). This effect can also be noticed in

One can see no notable difference between the shapes of curves 1 and 2 in _{S}

_{0.5}. No notable effect is observed at low values of Σ. The value of Σ at which _{0.5} starts to decrease depends on the diffusion modulus. A constant range of _{0.5} increases with an increase in the diffusion modulus.

The mathematical model

The biosensor current grows notable faster at higher substrate concentrations in the bulk solution than at lower ones (

A value of the diffusion modulus substantially determines the behaviour of the response and sensi-tivity of the biosensor. The steady state biosensor current is a nonsmonotonous function of the diffusion modulus (

The ratio of the enzyme-substrate reaction rate to the diffusion rate (the diffusion modulus
_{S}_{S}_{S}

This work was partially supported by Lithuanian State Science and Studies Foundation, project registration No N-08007. The authors express sincere gratitude also to prof. Feliksas Ivanauskas for his valuable contribution into modelling of biosensors.

Profiles of the normalized concentrations of the substrate (1, 4), mediator (2, 5) and product (3, 6) in the enzyme layer _{R}_{0.5} = 1.86 (4-6), _{0}= 0.1, _{0} = 10.

Profiles of the normalized concentrations of the substrate (1, 4), mediator (2, 5) and product (3, 6) in the enzyme and diffusion layers at approximate steady state dimensionless time _{R}_{0.5} = 0.435 (4-6), _{0} = 10, _{0} = 0.1.

The dynamics of the dimensionless biosensor current _{0}: 0.1 (1, 2), 10 (3, 4) and two concentrations of the mediator _{0}: 0.1 (1, 3), 10 (2, 4). Other parameters are the same as in

The steady state dimensionless current _{S}_{0}: 0.1 (1), 1 (2, 4, 5), 10 (3), _{0}: 0.1 (4), 1 (1-3), 10 (5). Other parameters are the same as in

The biosensor sensitivity _{S}

The steady state biosensor current _{S}

The dimensionless half-time _{0.5} versus the reduced diffusion modulus

The steady state dimensionless current _{S}^{–4} (1), 10^{–3} (2), 10^{–2} (3), 0.1 (4), 1 (5), 10 (6).

The biosensor sensitivity _{S}

The steady state dimensionless current _{S}^{−4}(1), 10^{−3}(2), 10^{−2}(3), 0.1 (4), 1 (5), 10 (5), keeping constant concentration _{0} = 1 of the mediator.

The biosensor sensitivity BS versus the ratio Σ of the substrate and mediator concentrations at different values of the diffusion modulus. The parameters and notation are the same as in 10.

The dimensionless half-time _{0.5} versus the ratio Σ of the substrate and mediator concentra-tions at different values of the diffusion modulus. The parameters and notation are the same as in