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A plate–gap model of a porous enzyme doped electrode covered by a porous inert membrane has been proposed and analyzed. The two–dimensional–in–space mathematical model of the plate–gap biosensors is based on the reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics. Using numerical simulation of the biosensor action, the influence of the geometry of the outer membrane on the biosensor response was investigated at wide range of analyte concentrations as well as of the reaction rates. The numerical simulation was carried out using finite–difference technique. The behavior of the plate–gap biosensors was compared with that of a flat electrode deposited with a layer of enzyme and covered with the same outer membrane.

A biosensor is a sensing device made up of a combination of a specific biological element, usually the enzyme that recognizes a specific analyte and the transducer that translates the changes in the bio–molecule into an electrical signal [

The amperometric biosensors are known to be reliable, cheap and highly sensitive for environment, clinical and industrial purposes [

The enzymatic layer of such biosensors was deposited as a thin layer of the enzyme gel on the surface of a flat metal electrode. An action of the biosensor of this type was described as a flat model in many papers [

Very recently a number of carbon paste based biosensors were created, and a plate–gap model of a porous electrode was proposed [

We investigate two types of amperometric biosensors. The first one is a porous carbon paste based biosensor with enzyme deposited in the pores of the electrode and covered with an inert membrane. We assume that the enzyme activity is gradually dispersed in the volume of porous electrode, and the distances between the enzymatic reaction sites and conducting walls of porous electrode are as short as an average radius of pores. According to this physical model, the enzyme activity is uniformly dispersed in the gap between two parallel conducting plates. The modeled physical system, in general, mimics the main features of the porous electrode. Firstly, the uniform dispersion of the enzyme activity is affirmed according to the definition of the modeled physical system. Secondly, the gap width dependent characteristic distances between the enzymatic reaction sites and conducting plates of the modeled system can be admitted to be similar to the average radius of pores in the porous electrode. In addition, the substrate or product molecules in the modeled plate–gap electrode may diffuse distantly in the directions, which are parallel to the surface of electrode, i.e. as it is in the three–dimensional network of porous electrode.

The biosensor of the second type is a flat electrode deposited with a layer of enzyme and covered with an inert membrane [

Consider a scheme where the substrate (S) binds to the enzyme (E) and is converted to the product (P) [

The mathematical model of the plate–gap biosensor with the outer membrane (

In the profile (

Let Ω_{1} and Ω_{2} be open regions corresponding to the enzyme region and the outer membrane, respectively, Γ_{1} – the outer membrane/bulk solution boundary, and Γ_{2} – the electrode border,

Let Ω̅_{1} and Ω̅_{2} denote the corresponding closed regions. Assuming a homogeneous distribution of the immobilized enzyme and coupling the enzyme–catalyzed reaction in the enzyme region with the two–dimensional–in–space mass transport by diffusion, described by Fick's law leads the system of the reaction–diffusion equations (_{e}(_{m}(_{e}(_{m}(_{Se}, _{Sm}, _{Pe}, _{Pm} are the diffusion coefficients, _{max} is the maximal enzymatic rate and _{M} is the Michaelis constant.

The biosensor operation starts when the substrate appears over the surface of the outer membrane. This is used in the initial conditions (_{0} is the concentration of substrate in the bulk solution.

The following boundary conditions express the symmetry of the biosensor (

In the scheme

If the bulk solution is well–stirred and in powerful motion then the diffusion layer remains at a constant thickness. The concentration of substrate as well as product over the outer membrane surface (bulk solution/membrane interface) remains constant while the biosensor keeps in touch with the substrate (

On the boundary between two adjacent regions Ω_{1} and Ω_{2} we define the matching conditions (

The measured current is accepted as a response of a biosensor in a physical experiment. The current depends upon the flux of the electro–active substance (product) at the electrode surface, i.e. on the border Γ_{2}. In the case of amperometry the biosensor current is directly proportional to the area of the electrode surface. Due to this we normalize the current with the area of the base of the biosensor. Consequently, the density _{g}(_{e} is a number of electrons involved in a charge transfer, and

We assume, that the system _{g} is the steady state current of the plate–gap biosensor.

The mathematical model of the flat biosensor (

Assuming

In the case the flat biosensor (_{f} of the biosensor current and the steady state current _{f} are described as follows:

The sensitivity is also one of the most important characteristics of biosensors. The sensitivity of a biosensor can be expressed as a gradient of the steady state current with respect to the substrate concentration. Since the biosensor current as well as the substrate concentration varies even in orders of magnitude, when comparing different sensors, another useful parameter to consider is the dimensionless sensitivity. The dimensionless sensitivity that varies between 0 and 1 is given by
_{Sg} and _{Sf} stand for the dimensionless sensitivities of the plate–gap and the flat biosensors, respectively.

The maximal gradient of the biosensor current calculated with respect to the time is another common characteristic of the biosensor action. Since the biosensor current as well as the time varies even in orders of magnitude, the dimensionless maximal gradient is used to compare different sensors. The dimensionless maximal gradient that varies between 0 and 1 is given by
_{Gg} and _{Gf} stand for the dimensionless maximal gradient of the biosensor current with respect to the time calculated for the plate–gap and the flat biosensors, respectively, _{Rf} and _{Rg} are the response times.

Definite mathematical solutions are not usually possible when analytically solving multi–dimensional non–linear partial differential equations with complex boundary conditions [

We introduced an uniform discrete grid in all directions:

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 _{Rf} and _{Rg} are assumed as the response times and _{Rf}, _{Rg} are assumed as the approximate steady state biosensor currents. In calculations, we used ^{-5}. However, the response time _{Rα} as an approximate steady–state time is very sensitive to the decay rate _{Rα} → ∞ when _{α}(_{0.5α} 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 maximal current, i.e.,

The mathematical model as well as the numerical solution of the model was evaluated for different values of the maximal enzymatic rate _{max}, the substrate concentration _{0} and the geometry of the enzyme–filled gaps as well as the thickness of the outer membrane. The following values of the parameters were employed in the numerical simulation of all the experiments:

The adequacy of the mathematical model of the flat biosensor was evaluated using known analytical solution of a two–layer model of amperometric biosensors [_{0} ≪ _{M}, the steady state biosensor current can be calculated as follows [^{2} is known as the diffusion modulus (Damköhler number) [^{2} essentially compares the rate of enzyme reaction (_{max}/_{M}) with the diffusion through the enzyme layer (^{2}/_{Se}). The response of the enzyme membrane biosensor is known to be under diffusion control when ^{2} ≫ 1. If ^{2} ≪ 1 then the enzyme kinetics predominates in the response.

The numerical solution of the model of the flat biosensor was compared with the analytical one _{max} (0.1, 1, 10 mM/s), the substrate concentration _{0} (1μM = 0.001_{M}) and _{m} = 0.1_{e}. In all these cases, the relative difference between the numerical and analytical solutions was less than 0.2%.

Assuming _{g} → _{f} when _{0}, _{max} the same as above and

Using numerical simulation, the influence of the thickness and of the permeability of the outer membrane as well as of the geometry of the enzyme region on the biosensor steady state current was investigated. In terms of the mathematical model _{Sm} and _{Pm}.

To investigate the effect of the thickness _{max} and substrate concentration _{0}. The steady state biosensor current is very sensitive to changes of _{max} and _{0} [_{f}(_{g}(_{f}(0) and _{g}(0) correspond to the steady state currents of the biosensors having no outer membrane, i.e. _{δ}_{f} and _{δ}_{g} as the steady state currents of the biosensors, having outer membrane divided by the steady state currents of the corresponding biosensors having no outer membrane,

_{m} = 0.1μm^{2}/s = 0.1 _{e} and the following values of the domain geometry:

One can see in _{δ}_{g} and _{δ}_{f} (as well as of the non–normalized ones _{g} and _{f}) is very sensitive to changes of the maximal enzymatic rate _{max} and substrate concentration _{0}. _{δ}_{g} and _{δ}_{f} are monotonous decreasing functions of the outer membrane thickness _{max} (10 mM/s) and relatively low values of _{0} (0.1 and 1 mM) (curves 3 and 6). _{δ}_{g} and _{δ}_{f} are monotonous increasing functions of _{max} (0.1 and 1.0 mM/s) and a high value of _{0} (10 mM) (curves 7 and 8).

Very similar behavior of the biosensor response was observed when modeling one–layer biosensors acting in a non–stirred analyte [^{2} > 1). In the cases when the enzyme kinetics controls the biosensor response (^{2} < 1), the steady state current increases with increase of the thickness of the diffusion layer. Thus the steady state current varied up to several times. When ^{2} ≈ 1, the variation of the steady state current is rather small. Let us notice that in the cases presented in ^{2} = 0.16 at _{max} = 0.1 mM/s and ^{2} = 1.6 at _{max} = 10 mM/s.

When comparing simulation results of biosensors of different types, we can see in _{0} > _{M}) the response of the plate–gap biosensor is more stable to changes of the thickness

As one can see in

On the other hand, at high values of _{max} and _{0} (curve 9), the sensitivity of the plate–gap biosensor (

The main physical reason of the superior behavior of the plate–gap biosensors vs. the flat ones is that the product of the enzymatic reaction is better (more completely) converted into the biosensor current. The product, which is electro–active substance, is better captured, i.e. it has less time to diffuse away before it is electro–oxidized or – reduced, in the plate–gap model rather than in flat one.

To investigate the dependence of the biosensor response on the diffusivity _{m} = _{Sm} = _{Pm} of the outer membrane the biosensor responses were calculated at constant thickness _{m} from 1 to 0.025 μm^{2}/s, i.e. from _{e} to 1/50 _{e}. In this case the current was normalized with respect to the maximal value _{e} of the diffusivity the outer membrane to be analyzed,
_{f}(_{m}) and _{g}(_{m}) are the steady state currents calculated at the diffusivity _{m} the outer membrane for the flat and plate–gap biosensors, respectively. Results of the calculations are depicted in _{m} notably depends on the maximal enzymatic rate _{max} and substrate concentration _{0}. Although the shapes of curves in _{m} of the membrane is very similar to that of the membrane thickness _{0} > _{M}) conditions.

The similarity between the effects of the outer membrane thickness _{m} is also notable when comparing _{m} of the outer membrane (curves 5, 6, 9 in

When calculating the maximal gradients _{Gg} and _{Gf} of the biosensor responses, no notable difference was found changing the substrate concentration _{0} and maximal enzymatic rate _{max}. Changing _{0} and _{max} in several orders of magnitude, values of the gradients _{Gg} and _{Gf} varied less than 1%. However, the effect of the thickness _{m} of the outer membrane on the biosensor response was substantial. As one can see in _{m}. However, the shape of curves differs. The maximal gradient is practically linear function of _{m}. The maximal gradient method of evaluation of biosensor response usually is used in bioanalytical instruments, when the time of the measurement cycle is necessary to reduce, thereby, to increase the speed of the analysis. Another feature – after the biosensor response passes maximal gradient, probe can be removed or replaced by buffer, and thereby the biosensor operates at lower concentrations of the substrate inside of the membrane and products as well. In some cases it is important for the stability of the biosensor, because the product of the enzymatic reaction can be chemically active and destroy the membrane (for example, a number of biosensors, containing oxidases and producing hydrogen peroxide;. This positive feature compensates the worse stability of biosensor concerning the sensitivity to the fluctuations of the membrane thickness.

_{Gf} – _{Gg} < 3.6. The maximal gradient of the response of plate–gap biosensor is lower than that of the corresponding flat one. Additional numerical experiments at other values of the parameters approved this feature. This can be explained by difference in the geometry of the electrodes. When the enzyme reaction starts, the gradient of the current gains the maximum immediately after some product touches the electrode surface, i.e. at the very beginning of the biosensor operation. The delay time depends mainly on the rate of the diffusion through the enzyme. In the case of the flat biosensor the touch occurs in one time at entire surface of the electrode, while in the case of the plate–gap biosensor, the current arises very gradually: firstly on the sides of gaps (from outside to inside the biosensor) and only then on the bottom of gaps. The current gradient is greater when the current arises like avalanche, i.e. in the case of the flat biosensor.

For the plate–gap biosensors the model parameter _{m} = 0.1μm^{2}/s of the outer membrane changing _{0} of _{f}(_{g}(_{0} = 2 μm.

As it is possible to notice in _{cg} and _{cf} are practically constant functions of _{max} (10 mM/s) and relatively low values of _{0} (0.1 and 1 mM) (curves 3 and 6).

Let us notice, that these properties are valid at values of

_{max} and _{0} (curve 9).

To investigate the dependence of the biosensor response on the width of the gaps we calculated the biosensor response at a constant distance 2(_{g} → _{f} when _{g}(_{f} is the steady state current of the corresponding flat biosensor. _{max} and _{0}. As one can see in _{ag}(_{g}(_{f} does not exceed 20% (_{ag}_{≥}_{max} (10 mM/s) and low values of the concentration _{0} (0.1 and 1 mM) (curves 3 and 6) _{g}(_{f} notable faster than at other values of _{max} and _{0}.

An increase in the width as well as in the depth of the gaps increases the volume of the enzyme used in plate–gap biosensors. Summarizing the results presented in _{max} rather than at higher ones and at higher values of _{0} rather than at lower ones.

_{Sg} of the plate–gap biosensor on the width of the gaps. As it is possible to notice in _{Sg} is practically constant function of the width _{0} upon the biosensor sensitivity _{Sg}. At all the values of _{0} corresponds to the higher sensitivity _{Sg}. This feature of the biosensor is very well known [_{max} on the biosensor sensitivity is more or less notable only in the cases when the substrate concentration _{0} varies about the Michaelis constant _{M}, i.e. when the enzyme kinetics changes from zero order to the first order across the enzyme region, _{0} ≈ _{M}.

To investigate the dependence of the biosensor response on the substrate concentration the response was simulated at wide range of the concentrations _{0}.

As one can see in _{g} of the steady state current of the plate–gap biosensor is slightly less than that of the flat one. However, when comparing the simulation results of the biosensors of different types one can see very similar shape of all curves. Very similar shapes of all curves we can see also in

As it is possible to notice in _{0.5g} and _{0.5f} are monotonous decreasing functions of _{0} at the maximal enzymatic rate _{max} of 10 as well as of 100 mM/s. At _{0} being between 0.1 and 10 mM (between _{M} and 100 _{M}) shoulders on the curves appears for _{max} = 1 mM/s. It seems possible that the shoulders on the curves arise because of high maximal enzymatic rate _{max} at the substrate concentrations at which the kinetics changes from zero order to first order across the enzyme region. At substrate concentration _{0} ≫ _{M} the reaction kinetics for _{0} ≪ _{M} the kinetics is first order throughout. At intermediate values of _{0} the kinetics changes from zero order to first order across the membrane. Similar effect of the substrate concentration on the response time was noticed in the cases of an amperometric biosensor based on an array of enzyme microreactors [_{0} = 100 mM (_{0} = 100 _{M}), the catalytic reaction makes no notable effect on the biosensors response time.

The mathematical model

At low maximal enzymatic rates (_{max}) and high substrate concentrations (_{0} > _{M}) the response of the plate–gap biosensor is more resistant to changes of the thickness of the outer membrane than the response of the corresponding flat one (

The response of the biosensors of two considered types: plate–gap and flat, both with the outer membrane, is more resistant to changes in volume of the enzyme at lower values of _{max} rather than at higher ones and at higher values of _{0} rather than at lower ones (

The maximal gradient of the current of plate–gap biosensor is lower than that of the corresponding flat one. In both cases, the maximal gradient is practically linear increasing function of thickness of the outer membrane and it is non linear monotonously decreasing function of the diffusivity of the membrane (

Work is now in progress to compare the simulations obtained for plate–gap biosensors with similar experimental studies [

This work was partially supported by Lithuanian State Science and Studies Foundation, project No. C–03048.

A principal structure of a plate–gap biosensor (a) and the corresponding flat one (b), both with the outer membrane. The figure is not to scale.

A profile of the unit cell of the plate–gap biosensor.

The normalized steady state current vs. the thickness _{max}: 0.1 (1, 4, 7), 1 (2, 5, 8), 10 (3, 6, 9) mM/s and three substrate concentrations _{0}: 0.1 (1–3), 1 (4–6), 10 (7–9) mM, _{m} = 0.1μm^{2}/s,

The normalized biosensor sensitivity vs. the thickness

The normalized steady state current vs. the diffusivity _{m} of the outer membrane in the cases of plate–gap (a) and flat (b) biosensors at the thickness

The normalized biosensor sensitivity vs. the diffusivity _{m} of the outer membrane in the cases of plate–gap (a) and flat (b) biosensors,

The normalized maximal gradient vs. the thickness _{m} (b) of the outer membrane, _{max} = 1 mM/s, _{0} = 1 mM/s, _{m} = 0.1μm^{2}/s (a),

The normalized steady state current vs. the gap depth

The normalized biosensor sensitivity vs. the gap depth

The normalized steady state current (a) and the biosensor sensitivity (b) vs. the gap width

The steady state current vs. the substrate concentration _{0} in the cases of plate–gap (a) and flat (b) biosensors at three maximal enzymatic rates _{max}: 0.1 (1, 4, 7), 1 (2, 5, 8), 10 (3, 6. 9) mM/s and three values of c: 2 (1–3), 4 (4–6), 6 (7–9) μm, _{m} = 0.1μm2/s,

The normalized biosensor sensitivity vs. the substrate concentration _{0} in the cases of plate–gap (a) and flat (b) biosensors. The parameters and notations are the same as in

The half time of the steady state biosensor response vs. the substrate concentration _{0} in the cases of plate–gap (a) and flat (b) biosensors. The parameters and notations are the same as in