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This paper presents a mathematical model of a batch stirred tank reactor based on an array of identical spherical porous microbioreactors loaded with non specific glucose dehydrogenase and oxygen reducing enzyme, i.e. laccase. The model was validated by experimental data. The microreactors (MR) are mathematically modeled by a two-compartment model, based on reaction–diffusion equations containing nonlinear terms related to the Michaelis–Menten kinetics of two enzymatic reactions with addition of the mass transport. The dynamics of oxygen concentration change is analysed numerically using the finite difference technique. The transient effectiveness factor and the process duration are investigated at different initial concentration of carbohydate (lactose) as well as at internal and external diffusion resistances. The simulation results show a non-monotonic effect of the initial concentration of lactose and nonlinear effects of the internal and external diffusion limitations on the transient effectiveness.

Coupling different enzymes either in sequence or in competition pathway is a way to enhance the range of analyte species accessible to measurement, the selectivity and the sensitivity of biosensors. In this chapter, mathematical models of several types of amperometric multi-enzyme biosensors utilizing consecutive or parallel substrates conversion are modeled and analysed at stationary and transient conditions. A biosensor based on bienzyme electrode with co-immobilized D-glucose oxidase and peroxidase is considered under stationary conditions at excess concentrations of oxygen and ferrocyanide. A trienzyme biosensor utilizing consecutive substrates conversion with three enzymes is modeled at internal diffusion limitation. A biosensor with dual catalase-peroxidase bioelectrode is mathematically modeled by nonlinear reaction–diffusion equations. Finally, multi-enzyme biosensors utilizing parallel and competitive multi-substrate conversion are analysed.

The additional application of catalytically inactive membranes can solve some drawbacks of biosensors, such as a relatively short linear range of the calibration graph, an instability and a low specificity. The selective membranes are usually used to increase the biosensors specificity. In this chapter, amperometric biosensors with inert and selective membranes are mathematically modeled by nonlinear reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction. At first, a biosensor, containing enzymatic and outer inert membranes, is mathematically and numerically modeled by a three-compartment model in one-dimensional space at transient conditions. Then, the model is extended to cover a transducer with an additional selective membrane permeable for the product of the enzymatic reaction, and the output results are numerically analysed with a special emphasis on the influence of the selective membrane to the biosensor response. And finally, the biosensor with selective and outer perforated membranes is modeled in two-dimensions. The biosensor response is analysed with a special focus on the geometry of the membrane perforation

The action of biosensors utilizing non-Michaelis–Menten kinetics is modeled at mixed enzyme kinetics and diffusion limitation in the cases of substrate and reaction product inhibition as well as of allostery at steady state and transient conditions. Computational modeling of the substrate inhibition at steady state shows multi-steady state concentrations of the substrate at the surface of the enzyme layer (membrane) when the diffusion module is much larger than one and the substrate bulk concentration is much higher than Michaelis–Menten constant. The multi-steady state concentration generates multi-response of the biosensor. At transient conditions, analytical systems are modeled by a two-compartment model comprising a mono-enzyme layer and an external Nernst diffusion layer. The complex enzyme kinetics produces different calibration curves for the response at the transition and the steady state. The cooperative phenomena of allosteric enzymes are modeled by applying the substrate uptake and the Hill equations. The positive cooperativity leads to a steady state current less than that when the biosensor action obeys the Michaelis–Menten kinetics, while negative cooperativity leads to increasing the biosensor response. The substrate concentration, at which the saturation curves of allosteric biosensors intersect, increases with increasing the external diffusion limitation.

This chapter introduces mathematical modeling of catalytic biosensors. After a brief tutorial consideration of kinetics of biocatalytic reactions, transducer function of biosensors and a general scheme of biosensor action, a detail mathematical model is then presented for an amperometric biosensor based on a mono-layer of an enzyme immobilized onto the surface of the electrode. The biosensor is modeled by two-component (substrate and product) reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzyme reaction. A few modifications of the mathematical model describing the action of potentiometric, optical and fluorescence biosensors are discussed, too. A special emphasis is placed on the modeling biosensors at steady state and internal or external diffusion limitation with a contribution to the modeling biosensors at non-stationary state at some critical concentrations of the substrate when analytical solution of the governing equations is performed. Using numerical simulation, the influence of the model parameters on the biosensor response is investigated. The simulation of the biosensor operation particularly showed a non-monotonous change of the steady state biosensor current versus the membrane thickness at the various maximal enzymatic rates.

This chapter presents two-dimensional and one-dimensional-in-space mathematical models of mediated and unmediated (mediatorless) amperometric biosensors based on an enzyme-loaded carbon nanotube (CNT) layer deposited on the perforated membrane. The models are based on nonlinear reaction–diffusion equations and involve four regions: the enzyme and the CNT regions where enzymatic reactions as well as the mass transport by diffusion take place, a diffusion limiting layer where only the mass transport by diffusion takes place and a convective region where the analyte concentration is maintained constant. By changing input parameters the output results are numerically analysed with an emphasis to the influence of the geometry and the catalytic activity of the biosensors to their response and sensitivity. The mediatorless transfer of the electrons in the region of enzyme-loaded carbon nanotubes is especially investigated. The numerical simulation at transient conditions was carried out using the finite difference technique. The mathematical models and the numerical solutions were validated by experimental data. The obtained agreement between the simulation results and the experimental data was admissible at different concentrations of the substrate.

The sensitivity of biosensors can be notably increased by a cyclic conversion of substrates or reaction products. In this chapter, mathematical models of different types of amperometric biosensors utilizing the cyclic conversion are modeled and analysed at transient conditions assuming an absence of outer diffusion limitations. A specific type of highly sensitive biosensors utilizing the substrate cyclic conversion in single enzyme membrane has been analytically modeled assuming the first-order reaction kinetics. The amplification of biosensor response by conjugated electrochemical and enzymatic substrate conversions is modeled by reaction–diffusion equations containing a nonlinear term related to Michaelis–Menten kinetic of the enzymatic reaction. The trigger of the response of biosensors utilizing substrate (analyte) conversion following the cyclic product conversion has been modeled and analysed computationally, too. The simulated response of the biosensors acting in two trigger schemes is compared with the response of a single enzyme biosensor utilizing Michaelis–Menten kinetics. The numerical experiments demonstrated significant gain in the biosensor sensitivity when the biosensor response was under diffusion control.

This chapter numerically investigates the sensitivity of an amperometric biosensor acting in the flow injection mode when the biosensor contacts an analyte for a short time. The analytical system is modeled by non-stationary reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction. At first, the biosensor action is modeled by a mono-layer mono-enzyme model assuming no external diffusion limitation. Then, the model is extended to a two-compartment model by adding an outer diffusion layer. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme. The apparent Michaelis constant is used as a main characteristic of the sensitivity and the calibration curve of the biosensor. The numerical simulation was carried out using the finite difference technique.

Biosensors containing glucose oxidase, carbohydrate oxidase and laccase and utilizing a few synergistic schemes of substrates conversion are modeled at steady state and transient conditions. A glucose dehydrogenase-based bioelectrocatalytical system, where ferricyanide is converted to ferrocyanide in the presence of highly reactive organic electron transfer compounds, and a laccase-based bioelectrode utilizing synergistic N-substituted phenothiazine and phenoxazine oxidation in the presence of hexacyanoferrate (II) are modeled mathematically by nonlinear reaction–diffusion equations. The modeling biosensors comprise three compartments, an enzyme layer, a dialysis membrane and an outer diffusion layer. The digital simulation was carried out using the finite difference technique. By changing the input parameters, the action of biosensors was analysed with a special emphasis to the influence of the kinetic constants and reagents concentrations on the synergy of the simultaneous substrates conversion. The digital simulation of the system confirmed that the high sensitivity of the bioelectrode achieved in the presence of organic mediators is due to the synergistic substrates conversion demonstrated experimentally.

This chapter deals with applying a multi-layer approach to the modeling of biosensors. The multi-layer mathematical models of amperometric biosensors are considered at stationary and transient conditions. First, a multi-layer approach is demonstrated for a biosensor having several mono-enzyme layers sandwich-likely applied onto the electrode surface. Then, a two-compartment model involving an enzyme layer, where the enzyme reaction as well as the mass transport by diffusion take place, and a diffusion limiting region, where only the mass transport by diffusion takes place, is analysed. The dependencies of the internal and external diffusion limitations on the response and sensitivity of amperometric biosensors are investigated, and the conditions when the mass transport outside the enzyme membrane may be neglected to simulate the biosensor response accurately in a well-stirred solution are considered. The mathematical models are based on the reaction–diffusion equations containing a nonlinear term related to Michaelis–Menten kinetics. The computer simulation was carried out using the finite difference technique.

The reduction of the device size, reagents and sample consumption is among the most important advantages of miniaturization of analytical systems. An integration of the systems with enzymatic microreactors proved to be a very suitable approach to the biosensor miniaturization. In this chapter, three types of amperometric biosensors are mathematically and numerically modeled in a two-dimensional space at transient conditions. The biosensing systems are modeled by reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction. A biosensor based on a carbon paste electrode encrusted with a single microreactor is modeled by a two-compartment model. The constructed biosensor explores an idea to separate the enzyme and the electron transfer components in a microreactor, the silica particle, and use the well-established carbon paste electrode. Then, a biosensing system based on an array of enzyme microreactors immobilized on a single electrode is modeled. Carbon paste porous electrodes are also modeled and investigated by applying a plate–gap model. Using the numerical simulation, the influence of the geometry of the microreactors as well as of the diffusion region on the output current and sensitivity is investigated.

This paper presents a model of a batch stirred tank reactor with spherical catalyst particles as microreactors. The model involves three regions: an array of porous enzyme-loaded microreactors where enzyme reaction as well as mass transfer by diffusion take place, a diffusion limiting region surrounding the particles and a convective region where the substrate is of uniform concentration. The microbioreactors are mathematically modeled by a two-compartment model based on reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten enzyme kinetics. The influence of the physical and kinetic parameters of the microbioreactors on the transient effectiveness of the bioreactor system is numerically investigated in a wide range of model parameters. The numerical simulation was carried out using the finite difference technique. The simulation results show non-monotonic effect of the initial substrate concentration and nonlinear effects of the internal and external diffusion limitations as well as adsorption capacity of the microreactors on the transient effectiveness.

N , N' -dimethyl-4,4'-azopyridinium methyl sulfate (MAZP) was characterized as an electron transfer mediator for oxidation reactions catalyzed by NAD + - and pyrroloquinoline quinone (PQQ)-dependent alcohol dehydrogenases. The bimolecular rate constant of NADH reactivity with MAZP was defined as (2.2 ± 0.1) × 10 5 M −1 s −1 , whereas the bimolecular rate constant of reactivity of the reduced form of PQQ-dependent alcohol dehydrogenase with MAZP was determined to be (4.7 ± 0.1) × 10 4 M −1 s −1 . The use of MAZP for the regeneration of the cofactors was investigated by applying the electrochemical oxidation of the mediator. The total turnover numbers of mediator MAZP and cofactor NADH for ethanol oxidation catalyzed by NAD + -dependent alcohol dehydrogenase depended on the concentration of the substrate and the duration of the electrolysis, and the yield of the reaction was limited by the enzyme inactivation and the electrochemical process. The PQQ-dependent alcohol dehydrogenase was more stable, and the turnover number of the enzyme reached a value of 2.3 × 10 3 . In addition, oxidation of 1,2-propanediol catalyzed by the PQQ-dependent alcohol dehydrogenase proceeded enantioselectively to yield l -lactic acid.

Specific wiring of biocatalysts par excellence, viz. redox enzymes, to an electrode can be exploited in the fabrication of high-performance bioelectronic devices. Here we report oxygen electroreduction catalysed by Didymocrea sp. J6 laccase wired to gold nanoparticles via the trinuclear copper cluster. Bypassing the intramolecular electron transfer, which under certain conditions is the rate-limiting step of oxygen bioelectroreduction, has resulted in the fabrication of a high current density biocathode based on high-redox-potential laccase, which is able to operate in electrolytes with a broad pH range in the presence of high fluoride concentrations.

Electron and proton transfer reactions in enzymes are enigmatic and have attracted a great deal of theoretical, experimental, and practical attention. The oxidoreductases provide model systems for testing theoretical predictions, applying experimental techniques to gain insight into catalytic mechanisms, and creating industrially important bio(electro)conversion processes. Most previous and ongoing research on enzymatic electron transfer has exploited a theoretically and practically sound but limited approach that uses a series of structurally similar (“homologous”) substrates, measures reaction rate constants and Gibbs free energies of reactions, and analyses trends predicted by electron transfer theory. This approach, proposed half a century ago, is based on a hitherto unproved hypothesis that pre-exponential factors of rate constants are similar for homologous substrates. Here, we propose a novel approach to investigating electron and proton transfer catalysed by oxidoreductases. We demonstrate the validity of this new approach for elucidating the kinetics of oxidation of “non-homologous” substrates catalysed by compound II of Coprinopsis cinerea and Armoracia rusticana peroxidases. This study – using the Marcus theory – demonstrates that reactions are not only limited by electron transfer, but a proton is transferred after the electron transfer event and thus both events control the reaction rate of peroxidase-catalysed oxidation of substrates.

In this paper we report the development of a highly sensitive amperometric glycerol biosensor based on alcohol dehydrogenase from Pseudomonas putida immobilized on graphite electrode modified with carbon nanotubes and a redox mediator tetrathiafulvalene. The designed biosensor demonstrates very high sensitivity towards glycerol (29.2 ± 0.9 µA mM–1 cm–2), low limit of detection (18 µM), linear range from 0.05 to 1.0 mM, high selectivity and satisfactory stability. Biosensor has been successfully used for the determination of glycerol concentration in buffer solutions as well as in the human urine samples. Received results shows a satisfactory agreement with the control measurements carried out using colorimetric commercially available glycerol determination assay kit, thus developed biosensor can be successfully applied for measurements of glycerol concentration in human urine and may be a fast, attractive and non-invasive tool for the determination of glycerol.

The action of bacterial glucose dehydrogenase (GDH) from Ewingella americana on the gold nanoparticles (AuNPs) modified with 4-aminothiophenol (4-ATP) and 4-mercaptobenzoic acid (4-MBA) was studied. GDH adsorbed on these nanostructured electrodes catalysed glucose oxidation in a mediator-free manner. The voltammetric investigations suggested the direct electron transfer between the enzyme active centre and the electrode. The maximum biocatalytic current of 10 mA cm−2 and low onset potential (–0.22 V vs. SCE) was observed in assays containing AuNPs modified with 4-ATP. The turnover number of the adsorbed enzyme was determined as 28 ± 5 s−1 and is comparable to catalytic constant (36 ± 5 s−1) of the native enzyme in the solution. The kinetic model of the action of AuNPs-GDH system comprising Michaelis-Menten and Butler-Volmer equations was analysed and the analytical solution of the dependence of the biocatalytic current on kinetic parameters of the enzyme was derived.

A model for the numerical simulation of the action of microbioreactor acting in the continuous flow mode was developed. The microbioreactor system was mathematically modelled by a two-compartment model based on transient reaction-diffusion equations containing a non-linear term related to the Michaelis–Menten kinetics of the enzymatic reaction. The effectiveness of microbioreactor and the process duration were numerically and partially analytically analysed at transition and steady-state conditions in a wide range of model parameters. The computational simulation was carried out using the finite difference technique. The performed calculations showed nonlinear effects of the internal and external diffusion limitations on the effectiveness and process duration.

A mediatorless electron transfer in the chain of glucose dehydrogenase (GDH) and laccase (LAC) catalysing the oxidation of glucose by molecular oxygen was studied. To demonstrate mediatorless processes, the GDH from Ewingella americana was adsorbed on single-walled carbon nanotubes (SWCNT). The effective mediatorless oxidation of glucose proceeded at 0.2–0.4 V vs. SCE. The electrode was most active at pH 6.1, and generated 0.8 mA cm−2 biocatalytic current in the presence of 50 mM glucose. The electrode showed a bell-shaped pH dependence with pKa values of 4.1 and 7.5. LAC from Trichaptum abietinum adsorbed on SWCNT exhibited mediatorless oxygen reduction at electrode potential less than 0.65 V. The electrode was most active at pH 3.0–4.0 and generated 1.1 mA cm−2 biocatalytic current in the presence of 0.254 mM oxygen, with an apparent pKa of 1.0 and 5.4. The electrodes prepared by simultaneous adsorption of GDH and LAC on SWCNT exhibited glucose oxidation at a potential higher than 0.25 V. The oxygen consumption in the chain was demonstrated using a Clark-type oxygen electrode. The dependence of oxygen consumption on glucose and lactose concentrations as well as activity of the system on pH were measured. A model of the pH dependence as well as mediatorless consecutive glucose oxidation with oxygen catalysed by LAC/GDH system is presented. This work provides a novel approach towards the synthesis of artificial multi enzyme systems by wiring oxidoreductases with SWCNT, and offers a better understanding of natural electron transfer networks and chains.

A scheme of a glucose dehydrogenase-based bioelectrocatalytical system where ferricyanide is converted to ferrocyanide in the presence of highly reactive organic electron transfer compounds is investigated digitally. Glucose dehydrogenase (GDH) is highly reactive towards organic electron acceptors compared to the low reactivity of GDH with inorganic electron acceptors. In this bioelectrocatalytical scheme, the enzyme efficiently catalyzes biocatalytical conversion of an organic mediator, which is followed by the fast chemical cross reaction of the product in the presence of ferricyanide excess. By changing input parameters a special emphasis was directed towards the influence of the kinetic constants and reagents concentration on the sensitivity of the bioelectrode. The digital simulation of the system confirmed that the high sensitivity of the bioelectrode achieved in the presence of organic mediators is due to the synergistic substrates conversion demonstrated experimentally. in (Electroanalysis, 2012 (24) 273-277).

Bioelectrocatalytic reduction of oxygen to water at electrodes modified with gold nanoparticles and a new laccase from Trichaptum abietinum (TaLc) was studied. The bioelectrocatalytic current was found to be much higher at TaLc modified electrodes than at similarly prepared electrodes modified with a broadly used laccase from Trametes hirsuta (ThLc). To explain this difference the bioelectrocatalysis was described in terms of kinetic rate constants based on simultaneous cyclic voltammetry and quartz crystal microbalance measurements. From analysis of the rate constants both laccases appeared to possess similar rates (k0) of direct electron transfer. However, the enzyme turnover (kcat) was about three-fold higher for gold nanoparticle bound TaLc than for ThLc, calculated using surface concentration of enzyme established by QCM-D. Near reversible potential-induced reorientation of adsorbed proteins was observed by surface enhanced Raman spectroscopy.

The kinetics of laccase-catalyzed NAD(P)H oxidation in the presence of syringaldazine (4-hydroxy-3,5-dimethoxybenzaldehyde azine) was investigated spectrophotometrically and by measuring the consumption of oxygen. Laccases from Didymocrea sp. (DsL) and Trichaptum abietinum (TaL) were used. These enzymes showed different pH profiles towards syringaldazine. DsL showed a maximum activity at pH 8.0 that is unusual for fungal laccases. The maximal activity of TaL was observed at pH 5.5. A kinetic mechanism for the laccase-catalyzed syringaldazine oxidation and disproportionation of reaction products as well as their reaction with NAD(P)H was suggested. Since TaL was slightly more reactive with syringaldazine than DsL, the calculated constants of enzymatic reaction were different for the investigated laccases. The constant of NAD(P)H oxidation by oxidized syringaldazine (tetramethoxy azobismethylene quinine, TMAMQ) is 1.0 × 10 4 M −1 s −1 , which was sufficient for effective NAD(P)H oxidation. The production of an enzymatically active NAD + was proved by coupling the laccase/mediator system with alcohol oxidation catalyzed by yeast alcohol dehydrogenase.

We develop a general algorithm for fitting the biochemical kinetics data. The developed algorithm searches and analyzes numerous minima. This approach allows us to analyze biochemical data without a priori quasi-steady-state assumptions. The algorithm allows us to treat the biochemical kinetics data that has varying degree of steadiness. We test the approach by analyzing experiment data from 4-nitrophenyl phosphate hydrolysis with alkaline phosphatase.

A bioelectrode containing immobilized catalase and peroxidase was built using a Clark-type oxygen electrode. The bioelectrode responded to hydrogen peroxide (H2O2) as well as to acetaminophen (Ac). The sensitivity of the bioelectrode for H2O2 was 0.35 mM O2/mM H2O2 and for Ac it was 0.23–1.05 µM O2/µM Ac at pH 6.6 and 25 °C. The limit of detection of Ac varied from 12 to 44 µM. The half-time of the bioelectrode response to hydrogen peroxide was 36 s. The modeling of the bioelectrode action was performed digitally at transition and steady-state conditions using finite difference technique. The calculated half-time of the bioelectrode response to hydrogen peroxide was 53 % larger and the steady-state response 11 % less than experimentally determined. The response to Ac was 2–3 times smaller in comparison to the experimental values. The calculated response change correlated with the experimentally determined when the catalase and peroxidase concentrations in the biocatalytical membrane changed 3–4 orders of magnitude. The simulations of the bioelectrode response revealed that the bioelectrode acts in diffusion limiting conditions at almost all enzymes concentrations. The model appears to be promising for optimization of the bioelectrode response.

Two mathematical models of an amperometric bienzyme biosensor are analysed digitally. The models hold different boundary conditions describing the singularity of the electrode (transducer). The influence of the initial and boundary conditions on the biosensors action at different sets of parameters is analysed.The digital simulation at the transient and steady-state conditions was carried out by using finite difference technique. The comparison of the simulation results revealed that some of the calculated parameters, i.e. response and sensitivity is the same, while the others, i.e. half-time of the steady-state is significantly different for distinct models.

The peculiarities of an amperometric biosensor based on parallel substrates conversion are investigated digitally. The mathematical model of the biosensor is based on a system of non-linear reaction–diffusion equations. The model involving the enzyme and diffusion layers is defined in a one-dimensional domain. By changing input parameters the biosensor action is analysed numerically with a special emphasis to the influence of the catalytic activity and the geometry of the biosensor on its response and sensitivity. The digital simulation at the transition and steady-state conditions was carried out by using the finite difference technique. Calculations showed that the half maximal effective concentration of the substrate varies by several orders of magnitude by changing the concentrations of enzymes, the hydrogen peroxide concentration as well as the enzyme layer thickness. The behaviour of the biosensor sensitivity and the half maximal effective concentration is especially complex at large concentrations of the catalase as well as low concentrations of the peroxidase.

In this paper the operation of an amperometric biosensor producing a chemically amplified signal is modelled numerically. The chemical amplification is achieved by using synergistic substrates. The model is based on non-stationary reaction-diffusion equations. The model involves three layers (compartments): a layer of enzyme solution entrapped on the electrode surface, a dialysis membrane covering the enzyme layer and an outer diffusion layer which is modelled by the Nernst approach. The equation system is solved numerically by using the finite difference technique. The biosensor response and sensitivity are investigated by altering the model parameters influencing the enzyme kinetics as well as the mass transport by diffusion. The biosensor action was analyzed with a special emphasis to the effect of the chemical amplification. The simulation results qualitatively explain and confirm the experimentally observed effect of the synergistic substrates conversion on the biosensor response.

A Bioelectrode utilizing a synergistic scheme of substrate conversion was built using glucose dehydrogenase from Acinetobacter calcoaceticus immobilized on the surface of a graphite electrode. At saturated glucose concentration the bioelectrode responded to the low reactive substrate hexacyanoferrate(III) with a sensitivity of 0.0035 µA/µM cm2. The response of the bioelectrode increased up to the 3.4×104 fold in the presence of high reactive organic electron acceptors (mediators). The increase of the response depended on the concentration of the mediators and their chemical nature. The sensitivity of the bioelectrode to mediators reached 7.3–77 µA/µM cm2. The comparison of the bioelectrode sensitivity with kinetic parameters of enzyme action in homogeneous solution revealed good correlation between the sensitivity of the bioelectrode and the predicted value from the kinetic scheme of the reactivity of mediators. This confirms a synergistic scheme of bioelectrode action.

The response of a laccase-based amperometric biosensor that acts in a synergistic manner was modelled digitally. A mathematical model of the biosensor is based on a system of non-linear reaction diffusion equations. The modelling biosensor comprises three compartments, an enzyme layer, a dialysis membrane and an outer diffusion layer. By changing input parameters the biosensor action was analysed with a special emphasis to the influence of the species concentrations on the synergy of the simultaneous substrates conversion. The digital simulation was carried out using the finite difference technique.

This paper presents a two-dimensional-in-space mathematical model of an amperometric biosensor based on an enzyme-loaded carbon nanotubes layer deposited on a perforated membrane. The developed model is based on non-linear non-stationary reaction-diffusion equations. By changing input parameters the output results are numerically analysed with a special emphasis to the influence of the geometry and the catalytic activity of the biosensor to its response. The numerical simulation at transition 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 experimental data was admissible at different concentrations of the substrate and the mediator.

Bioelectrodes employing the conversion of N-substituted phenothiazine/hexacyanoferrate(II) and N-substituted phenoxazine/hexacyanoferrate(II) that act as synergistic substrates were built using graphite electrode and recombinant laccase. The response of the bioelectrodes to low reactive substrate (hexacyanoferrate(II)) increases 2.9–97 times in presence of high reactive phenothiazine or phenoxazine mediator. The sensitivity of the bioelectrodes reaches 0.5–14.3 A/M⋅cm2. To background a scheme of bioelectrodes action the kinetics of laccase-catalyzed synergistic substrates oxidation was investigated in homogenous solution. A satisfactory agreement of experimentally determined rate constants and calculated using parameters of bioelectrocatalytical system confirms the scheme of bioelectrodes action.