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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.

The dependencies of the internal and external diffusion limitations on the response and sensitivity of amperometric biosensors are investigated computationally using a two-compartment model, and a framework for evaluating and selecting the biosensor configuration is proposed. Enzyme-based biosensors are mathematically modelled by a system of reaction-diffusion equations containing a nonlinear term related to Michaelis-Menten kinetics. The biosensor response, sensitivity and stability are numerically analysed at the transition and steady state conditions in a wide range of model parameters, and attractive biosensor configurations are derived. The performed calculations show nonlinear effects of internal and external diffusion limitations on the biosensor current, response time and sensitivity as well as on the linear range of the calibration curve. The phenomenon of the non-monotonicity of the biosensor response is also investigated.

The paper presents a non-linear mathematical model of the batch stirred tank reactor with spherical catalyst particles as microreactors. The different initial conditions are investigated. The influence of the diffusion modulus, the Biot number and the inverse adsorption capacity on the process duration has been numerically investigated. The simulation results showed that process durations might be different by magnitude of order just by different initial microreactor injections. The digital simulation was carried out using the finite difference technique.

This paper presents a method combining mathematical modeling, multi-objective optimization and multi-dimensional visualization intended for the design and optimization of amperometric biosensors. An approach for optimizing the biosensor parameters is based on the availability of mathematical model of a catalytic biosensor (bioelectrode). A multi-objective visualization of trade-off solutions and Pareto optimal decisions is applied for the selection of the most favorable decision by a human expert when designing the biosensor. The proposed method is applied to the industrially relevant optimization of a glucose dehydrogenase-based amperometric biosensor utilizing the synergistic substrates conversion. The following three objectives were optimized: the apparent Michaelis constant, the output current and the enzyme amount.

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 system-to-system communication involving stateful sessions between a clustered service provider and a service consumer is investigated in this paper. An algorithm allowing to decrease a number of calls to failed provider nodes is proposed. It is designed for a clustered client and is based on an asynchronous communication. A formal specification of the algorithm is formulated in the TLA\(^+\) language and was used to investigate the correctness of the algorithm.

Bioluminescence imaging experiments were carried out to characterize spatio-temporal patterns of bacterial self-organization in active suspensions (cultures) of bioluminescent Escherichia coli and its mutants. An analysis of the effects of mutations shows that spatio-temporal patterns formed in standard microtitre plates are not related to the chemotaxis system of bacteria. In fact, these patterns are strongly dependent on the properties of mutants that characterize them as self-phoretic (non-flagellar) swimmers. In particular, the observed patterns are essentially dependent on the efficiency of proton translocation across membranes and the smoothness of the cell surface. These characteristics can be associated, respectively, with the surface activity and the phoretic mobility of a colloidal swimmer. An analysis of the experimental data together with mathematical modelling of pattern formation suggests the following: (1) pattern-forming processes can be described by Keller–Segel-type models of chemotaxis with logistic cell kinetics; (2) active cells can be seen as biochemical oscillators that exhibit phoretic drift and alignment; and (3) the spatio-temporal patterns in a suspension of growing E. coli form due to phoretic interactions between oscillating cells of high metabolic activity.

The essential part of amperometric biosensor is an enzyme. It should be selective, i.e., react only with certain substrate. The selectivity of enzyme reduces the set of possible to use enzymes. This paper demonstrates that non selective enzymes (reacting with two substrates) can be used to determine concentrations of two substrates. For this purpose the steady-state current of two double biosensors was measured. The currents were used as input for an artificial neural network to determine concentrations of the substrates. The proposed approach was approved as the relative error of determined concentrations was relatively small. Paper analyses the influence of biosensor parameters to error values. The recommendations to error values minimisation were obtained.DOI: 10.15181/csat.v3i2.1109

It has been recently shown that bioluminescence imaging can be usefully applied to provide new insights into bacterial self-organization. In this work we employ bioluminescence imaging to record images of nutrient rich liquid cultures of the lux-gene reporter Escherichia coli in microtiter plate wells. The images show that patterns of inhomogenous bioluminescence form along the three-phase contact lines. The paper analyzes the dependencies of the average number of luminous aggregates (clouds) on various environmental factors. In particular, our results show that optimal (neutral) pH and high aeration rates determine the highest mean number of clouds, and that spatiotemporal patterns do not form in the pH buffered suspensions. In addition, a sigmoidal (switch-like) dependence of the number of aggregates on the rate of aeration was observed. The obtained bioluminescence imaging data was interpreted by employing the Keller-Segel-Fisher (KSF) model of chemotaxis and logistic growth, adapted to systems of metabolically flexible (two-state) bacteria. The modified KSF model successfully simulated the observed switch-like responses. The results of the microtiter plate tests and their simulations indicate that the segregation of bacteria with different activities proceeds in the three-phase contact line region.

Bacteria are a widespread group of organisms. On the microscale, bacteria colonies are of discrete nature. Looking from the mechanical point of view the suspension containing the bacteria may be considered as a system of living active ultrafine particles (size: 0.1–10 μm). In order to understand the mechanical behaviour of the bacteria system it is important to understand the behaviour of a single bacterium. The present paper proposes an adhesive interaction model for the simulation of bacteria cells, which can be also applied for the interaction for other biological cells. The main attention is given to describe and numericaly simulate the interaction of a bacterium with a flat surface within a liquid medium. In the simulations an adhesive force is taken into account. Adhesion plays a significant role, because it can keep a bacterium on the surface. It is described by the attractive van der Waals force. In order to achieve the stick of adhesive particles, additionally developed adhesive–dissipative models are presented. The bacterium interaction is described by an elastic–plastic model; these results are thereafter compared with results from an elastic model. Different known models such as Derjaguin, Muller and Toporov (DMT) and Derjaguin, Landau, Verwey, Overbeek (DLVO) models are considered. Obtained results show the approach and deformation process of an adhesive–dissipative bacterium by presenting force displacement diagrams. The bacterium–surface interaction model is developed in the framework of the discrete element method (DEM). The numerical experiments confirm that force–displacement plots exhibit a hysteresis similar to those observed in Atomic Force Microscopy (AFM) experiments. The proposed model can be applied for the numerical simulation of the interaction process of bacteria with a surface, as well as simulations of the sticking process.

A set of oxidized graphite samples have been newly synthesized using different protocols. Atomic force microscopy, Raman spectroscopy, thermal gravimetric analysis and Brunauer–Emmett–Teller analysis revealed the changes in structure and functionalities of obtained graphite oxidation products (GOPs) compared to pristine graphite. The substances have been tested as electrode materials applicable for bioelectrocatalytic systems using pyrroloquinoline quinone-dependent glucose dehydrogenase (PQQ-GDH). The application of GOPs allowed achieving the direct electron transfer (DET) from active site of PQQ-GDH to the electrode surface. Needless of additional electron transfer (ET) mediating compounds highly improved features of the biosensors. The efficiency of the biosensors has been evaluated for all types of biosensors varied from 32 μA/cm 2 to 64 μA/cm 2 using as electrode materials GOP1 and thermally reduced graphite oxide (TRGO), respectively. TRGO containing function groups (according TGA, ∼6% of the weight loss) and smallest particles (average diameter was ∼11 nm and the average height was ∼0.5 nm) exhibited the higher efficiency for ET acceleration in the biosensor acting on principle of DET.

In order to understand the behaviour system of the bacteria it is important understand the behaviour of a single bacteria. The suspension forming bacteria may be considered as system of living active ultrafine particles (size 1 μm). Present investigation addresses to the simulation of the S. aureus bacterium-surface interactions in a framework of the Discrete Element Method (DEM). Bacterium is of the spherical shape, while the glass surface is flat and considered as elastic. In this work the theoretical model for bacteria is similar to that used for the ultrafine size stiff particles. We investigate the behaviour of the active particle by applying two known Derjaguin, Muller, Toporov (DMT) [1] and Derjaguin, Landau, Verwey, Overbeek (DLVO) [2,3] models, which are used for simulation of ultrafine size objects. These models are enhanced by applying suggested dissipation mechanism related to the adhesion. It was assumed that energy can be dissipated and the force-displacement hysteresis can occur through the adhesion effect, where an amount of dissipated energy is fixed and independent on initial kinetic energy. This force-displacement hysteresis was observed at the physical experiments with bacteria provided by the means of the atomic force microscopy (AFM), Ubbink and Schar-Zammaretti (2007) [4]. It was illustrated that the presented adhesive-dissipative model, which applies DEM, offers the opportunity to capture dissipation effect during the contact. The numerical experiments confirm that force-displacement plots exhibit hysteresis typical to those which are observed in AFM experiments. This model can be useful for numerical simulation of interaction of bacterium to the substrate.

This paper presents a mathematical model of a biosensor based on a chemically modified electrode. The model is solved numerically and results of the simulations are validated with analytical solutions for a specific set of parameter values. The mathematical model of the biosensor comprises three layers: an enzyme layer, a dialysis membrane and an outer diffusion layer. The dialysis membrane is merged with the diffusion layer in the mathematical model by introducing an effective diffusion coefficient. The main purpose of this work was to determine at which parameter values the three-layer model can be replaced with the two-layer model with desired accuracy. Additionally, a possibility to use the maximal gradient of the biosensor current density instead of the steady-state current density is investigated. Numerical experiments showed that using the maximal gradient of the output current density can be very attractive to improve biosensor sensitivity.

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).

In this paper, an amperometric glucose biosensor is modeled numerically. The model is based on non-stationary reaction-diffusion type equations. The model consists of four layers. An enzyme layer lies directly on a working electrode surface. The enzyme layer is attached to an electrode by a polyvinyl alcohol (PVA) coated terylene membrane. This membrane is modeled as a PVA layer and a terylene layer, which have different diffusivities. The fourth layer of the model is the diffusion layer, which is modeled using the Nernst approach. The system of partial differential equations is solved numerically using the finite difference technique. The operation of the biosensor was analyzed computationally with special emphasis on the biosensor response sensitivity to oxygen when the experiment was carried out in aerobic conditions. Particularly, numerical experiments show that the overall biosensor response sensitivity to oxygen is insignificant. The simulation results qualitatively explain and confirm the experimentally observed biosensor behavior.

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.

An original concept in technology of biosensitive electrodes based on single-wall carbon nanotubes (SWCNT) is proposed. According to a newly proposed technique the electrode surface consisting of SWCNT’s embedded in a porous polycarbonate film was preoxidized enzymatically using laccase from Basidiomycete Lac. An efficient electron transfer from the active centre of PQQ-dependent glucose dehydrogenase (s-PQQ-GDH) to the SWCNT electrode surface allowed the application of the s-PQQ-GDH as recognition element for the reagentless biosensor design. The residual activity of the biosensor was found to be 40 % after one month of operational working. A mathematical model of the biosensor has been developed and an admissible agreement between the simulation results and the experimental data was obtained. The model appears to be promising for optimization of amperometric analysis.

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.

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 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 modelled by non-stationary reaction-diffusion equations containing a non-linear term related to the Michaelis-Menten kinetics of an enzymatic reaction. The mathematical model involves three regions: the enzyme layer where enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region. 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.

A possible reason for the complexity of the signals produced by bioluminescent biosensors might be self-organization of the cells. In order to verify this possibility, bioluminescence images of cultures of lux gene reporter Escherichia coli were recorded for several hours after being placed into 8–10 mm diameter cylindrical containers. It was found that luminous cells distribute near the three-phase contact line, forming irregular azimuthal waves. As we show, space–time plots of quasi-one-dimensional bioluminescence measured along the contact line can be simulated by reaction–diffusion–chemotaxis equations, in which the reaction term for the cells is a logistic (autocatalytic) growth function. It was found that the growth rate of the luminous cells (~0.02 s−1) is >100 times higher than the growth rate of E. coli. We provide an explanation for this result by assuming that E. coli exhibits considerable respiratory flexibility (the ability of oxygen-induced switching from one metabolic pathway to another). According to the simple two-state model presented here, the number of oxic (luminous) cells grows at the expense of anoxic (dark) cells, whereas the total number of (oxic and anoxic) cells remains unchanged. It is conjectured that the corresponding reaction–diffusion–chemotaxis model for bioluminescence pattern formation can be considered as a model for the energy-taxis and metabolic self-organization in the population of the metabolically flexible bacteria under hypoxic conditions. Copyright © 2011 John Wiley & Sons, Ltd.

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.

In this paper the response of an amperometric biosensor at mixed enzyme kinetics and diffusion limitations is modelled in the case of the substrate and the product inhibition. The model is based on non-stationary reaction–diffusion equations containing a non-linear term related to non-Michaelis–Menten kinetics of an enzymatic reaction. A numerical simulation was carried out using a finite difference technique. The complex enzyme kinetics produced different calibration curves for the response at the transition and the steady-state. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor response change shows a maximal value. The dependence of the biosensor sensitivity on the biosensor configuration is also investigated. Results of the simulation are compared with known analytical results and with previously conducted researches on the biosensors.

This paper presents a mathematical model of the amperometric biosensor based on an electrode covered with an enzyme membrane. The model involves three regions: the enzyme layer where enzymatic reaction as well as mass transport by diffusion takes place, a diffusion-limiting region where only diffusion takes place, and a convective region, where the analyte concentration is maintained constant. Using computer simulation the influence of the biosensor geometry as well as the flow intensity on the biosensor response was investigated. This paper also deals with the conditions when the mass transport in the exterior region may be neglected to simulate the biosensor response assuming that the buffer solution is in intense flow and in powerful motion. The digital simulation was carried out using the finite difference technique.

Prokaryotic bacteria are a widespread group of organisms on Earth. Bacteria inhabit practically all environments where some liquid water is present. Bacteria form structures such as biofilms and play a significant role in infections in the human body. In order to understand the behaviour of bacteria systems it is important to understand the behaviour of a single bacterium first. A bacterium is considered as an active particle and the numerical task of describing the interaction of a bacterium with a substrate is solved by applying the discrete element method. One of the factors influencing the stability of a biofilm structure are adhesion forces which allow keeping the bacteria system sticked together. The present theoretical investigation gives a characterisation of the influence of adhesion on a bacterium. The adhesion force is described by the van der Waals force by applying a recently developed adhesive-dissipative model. The investigation presents the theoretical behaviour of the bacterium interaction while different models such as DMT and DLVO are used. Obtained results show the approach and deformation process of an adhesive-dissipative bacterium by presenting force displacement diagrams.

This paper presents a mathematical model of a potentiometric biosensor based on a potentiometric electrode covered with an enzyme membrane. The model is based on the reaction–diffusion equations containing a non-linear term related to theMichaelis–Menten kinetics of the enzymatic reaction. Using computer simulation the influence of the thickness of the enzyme membrane on the biosensor response was investigated. The digital simulation was performed using the finite difference technique. Results of the numerical simulation were compared with known analytical solutions.

Biosensor response, in case of multi-substrate mixture, has nonlinear dependence on substrate concentrations. This work investigates the possibility to approximate this dependency with artificial neural network. Also the influence of external diffusion layer to results of multi-substrate determination was investigated. The numerically modelled biosensor response was used as experimental data. The principal components analysis was used to reduce the dimension of biosensor response. Prefered method gives acceptable acuratnes on multi-substrate determination and it can be improved by relatively large external diffusion layer.

It has been recently shown that bioluminescence imaging can be usefully applied to provide new insights into bacterial self-organization. In this work we employ bioluminescence imaging to record images of nutrient rich liquid cultures of the lux-gene reporter Escherichia coli in microtiter plate wells. The images show that patterns of inhomogenous bioluminescence form along the three-phase contact lines. The paper analyzes the dependencies of the average number of luminous aggregates (clouds) on various environmental factors. In particular, our results show that optimal (neutral) pH and high aeration rates determine the highest mean number of clouds, and that spatiotemporal patterns do not form in the pH buffered suspensions. In addition, a sigmoidal (switch-like) dependence of the number of aggregates on the rate of aeration was observed. The obtained bioluminescence imaging data was interpreted by employing the Keller-Segel-Fisher (KSF) model of chemotaxis and logistic growth, adapted to systems of metabolically flexible (two-state) bacteria. The modified KSF model successfully simulated the observed switch-like responses. The results of the microtiter plate tests and their simulations indicate that the segregation of bacteria with different activities proceeds in the three-phase contact line region.

This paper presents a two-dimensional-in-space mathematical model of amperometric biosensors with perforated and selective membranes. The model is based on the diffusion equations containing a non-linear term of the Michaelis–Menten enzymatic reaction. Using numerical simulation of the biosensors action, the influence of the geometry of the perforated membrane on the biosensor response was investigated. The numerical simulation was carried out using finite-difference technique. The calculations demonstrated non-linear and non-monotonous change of the biosensor steady-state current at various degree of the surface of the perforated membrane covering. The non-monotonous behaviour of the biosensor response was also observed when changing the thickness of the perforated membrane.

This paper presents a two-dimensional-in-space mathematical model of a sensor system based an array of enzyme microreactors immobilised on a single electrode. The system acts under amperometric conditions. The model is based on the diffusion equations containing a non-linear term related to the Michaelis-Menten kinetics of the enzymatic reaction. The model involves three regions: an array of enzyme microreactors (cells) where enzyme reaction as well as mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region, where the analyte concentration is maintained constant. Using computer simulation the influence of the geometry of the enzyme cells and the diffusion region on the biosensor response was investigated. The digital simulation was carried out using the finite difference technique.

This paper presents a sensor system based on a combination of an amperometric biosensor acting in batch as well as flow injection analysis with the chemometric data analysis of biosensor outputs. The multivariate calibration of the biosensor signal was performed using artificial neural networks. Large amounts of biosensor calibration as well as test data were synthesized using computer simulation. Mathematical and corresponding numerical models of amperometric biosensors have been built to simulate the biosensor response to mixtures of compounds. The mathematical model is based on diffusion equations containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. The principal component analysis was applied for an optimization of calibration data. Artificial neural networks were used to discriminate compounds of mixtures and to estimate the concentration of each compound. The proposed approach showed prediction of each component with recoveries greater that 99% in flow injection as well as in batch analysis when the biosensor response is under diffusion control.

A mathematical model of amperometric enzyme electrodes in which chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetic of the enzymatic reaction. The digital simulation was carried out using the finite difference technique. The influence of the substrate concentration, the maximal enzymatic rate as well as the membrane thickness on the biosensor response was investigated. The numerical experiments demonstrate significant (up to dozens of times) gain in biosensor sensitivity at low concentrations of substrate when the biosensor response is under diffusion control.

A mathematical model of amperometric biosensors in which chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane has been developed. The model involves three regions: the enzyme layer where enzyme reaction as well as mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region where the analyte concentration is maintained constant. Using computer simulation the influence of the thicknesses of the enzyme layer and the diffusion region on the biosensor response was investigated. This paper deals with conditions when the mass transport in the exterior region may be neglected to simulate the biosensor response in a well-stirred solution. The digital simulation was carried out using the finite difference technique.

A model of the moisture movement in solid wood under isothermal conditions taking into consideration coating (insulation) of the surface of the material, is presented in a 2-D-in-space formulation. The validity of using a corresponding 1-D model on a 2-D problem is investigated. A measure of reliability of the 1-D model is introduced. This paper presents a new technique, which adjusts the width of a material as well as the degree of coating of the edges, to ensure the relative error of the problem solution is less than the required, resulting from reducing the model from 2-D to 1-D-in-space. This technique is based on the computer simulation of 2-D diffusion to estimate the reliability of the corresponding 1-D diffusion model.

A two-dimensional-in-space mathematical model of amperometric biosensors has been developed. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction. The model takes into consideration two types of roughness of the upper surface (bulk solution/membrane interface) of the enzyme membrane, immobilised onto an electrode. Using digital simulation, the influence of the geometry of the roughness on the biosensor response was investigated. Digital simulation was carried out using the finite-difference technique.

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Using digital simulation, the influence of the thickness of enzyme membrane on the biosensor response was investigated. The digital simulation of the biosensor operation showed the non-monotonous change of the maximal biosensor current versus the membrane thickness at the various maximal enzymatic rates. Digital simulation was carried out using the finite difference technique. Results of the numerical simulation was compared with known analytical solutions. This paper presents a framework for selection of the membrane thickness, ensuring the sufficiently stable sensitivity of a biosensor in a required range of the maximal enzymatic rate.

A mathematical model of amperometric biosensors has been developed to simulate the biosensor response in stirred as well as non stirred solution. The model involves three regions: the enzyme layer where enzyme reaction as well as mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region, where the analyte concentration is maintained constant. Using computer simulation the influence of the thickness of the enzyme layer as well the diffusion one on the biosensor response was investigated. The computer simulation was carried out using the finite difference technique.

Summary A model of wood drying under isothermal conditions taking into consideration coating of the surface of a specimen is presented in this paper in a two-dimensional formulation. The influence of the surface coating degree as well as geometrical shape of a wood specimen on the dynamics of drying is investigated. Exponentials, describing the dependence of the halfdrying time on the degree of coating of the edges, as well as on the ratio of the width to the thickness of the transverse section of specimens from the northern red oak (Quercus rubra), are presented for drying from above the fiber saturation point. This paper describes the conditions of usage of the two-dimensional moisture transfer model in contrast to the one-dimensional model for accurate prediction of the drying process taking into consideration the coating of edges of specimens having a rectangular transverse section. A measure of reliability of the one-dimensional model to predict the wood drying process of sawn boards is introduced in this paper.

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis–Menten kinetic of the enzymatic reaction. Using digital simulation, the influence of the substrate concentration as well as maximal enzymatic rate on the biosensor response was investigated. The digital simulation was carried out using the finite difference technique. The model describes the biosensor action in batch and flow injection regimes.

The paper presents nonlinear model which stands for effective digital simulation of electrochemical behavior of partially blocked electrodes under linear potential sweep and cyclic voltammetry conditions. The model is based on a system of diffusion equations, also involving the Nernst diffusion layer. The mass transport is assumed to be regular in the entire diffusion space. The influence of the thickness of the resist layer on the behavior of the partially blocked electrodes is investigated. The agreement between the theoretical results and experimental ones is obtained to be admirable for several model electrodes with different blocking degree.

Modelling of the amperometric biosensors based on carbon paste electrodes encrusted with a single heterogeneous microreactor is analyzed. The microreactor was constructed from CPC‐silica carrier and was loaded with glucose oxidase. The model is based on non‐stationary diffusion–reaction equations containing a non‐linear term related to the enzymatic reaction. A homogenization process having an effective algorithm for the digital modelling of the operation of the microreactor is proposed. The influence of the size, geometrical form, and the position of a microreactor on the operation of biosensors are investigated.

Biojutikliai yra analitiniai įrenginiai, skirti medžiagų koncentracijoms matuoti. Kuriant naujus biojutiklius reikia atlikti daug eksperimentų. Siekiant sumažinti atliekamų fizinių eksperimentų skaičių taikomas kompiuterinis biojutiklių veiksmo odeliavimas, kai įprastai kiekvienam struktūriškai naujam biojutikliui yra sudaromas matematinis modelis, tuomet jis keičiamas skirtuminiu, o jo lygčių sistemos sprendimas įgyvendinamas sudarant kompiuterinį modelį. Kiekvienas žingsnis reikalauja atidos ir turėtų būti automatizuotas. Straipsnyje yra pateikiamas biojutiklio metamodelis, leidžiantis formuluoti biojutiklių modelius dalykinės srities sąvokomis. Pasiūlytasis metamodelis aprašo biojutiklių modelius, formuluojamus dvimatėje erdvėje, apimančius biojutiklio struktūros, jo geometrinių savybių, biojutikliuose vykstančių reakcijų ir difuzijos procesų aprašus. Sudarius metamodelį, buvo sukurta programinė įranga, automatiškai sukonstruojanti kompiuterinį biojutiklio modelį pagal metamodeliosąvokomis išreikšto biojutiklio aprašą. Metamodelis ir programinė įranga buvo taikoma realiam biojutiklio modeliui sudaryti ir jo veiksmui modeliuoti kompiuteriniu būdu.", t. y. ištrinti žodžius "biojutiklių veiksmo.Computer-Aided Modeling of Biosensors with a Complex Geometrical StructureRomas Baronas, Karolis Petrauskas SummaryBiosensors are analytical devices used to measure the concentration of substances. When developing new biosensors, a lot of experiments are needed to be performed. Mathematical modeling of biosensors is used to decrease the number of physical experiments. Models of biosensors are usually created for each structurally unique biosensor by defining its mathematical model and the corresponding numerical approximation. Equations of the numerical model are then solved using computer programs, usually created for a particular model of the biosensor. Each of these steps requires a great attention and should be automated. The article presents a metamodel for a biosensor, enabling one to define models of biosensors in domain-specific terms. The proposed metamodel describes biosensor models, defined in the two-dimensional space and including definitions of the structure of a biosensor, its geometrical properties, reactions and diffusion processes taking place in it. Upon defining the metamodel, we compiled the computer software able to create computer models for biosensors from the models formulated according to the proposed metamodel. The metamodel was practically used to define a model for a real biosensor, and the biosensor modeling software was used to simulate its operation.

Šiame straipsnyje yra tiriamas amperometrinis biojutiklis, kuriame biojutiklio atsakas yra stiprinamas chemiškai – sinerginiais substratais. Tokiuose biojutikliuose, be substrato, kurio koncentracija matuojama, naudojamas ir pagalbinis substratas, reikalingas substratų sinergetikai. Biojutiklis yra modeliuojamas naudojant nestacionarias netiesines reakcijos-difuzijos lygtis. Modeliuojami keturi biojutiklio sluoksniai: fermento sluoksnis, kuriame vyksta visos biocheminės reakcijos ir difuzija, dializėsmembrana ir difuzijos sluoksnis, kuriuose vyksta tik difuzija ir reakcijos, kuriose nedalyvauja fermentas, o ketvirtasis sluoksnis yra tirpalo dalis, kurioje palaikoma vienoda medžiagų koncentracija. Lygčių sistema sprendžiama skaitiškai, naudojant baigtinių skirtumų metodą. Tiriama biojutiklio atsako bei jautrio priklausomybė nuo substratų koncentracijų ir nuo difuzijos sluoksnio storio.Modelling Amperometric Biosensors with Synergistic Substrate AmplificationDainius Šimelevičius, Romas Baronas SummaryComputational modelling of a biosensor in which chemical amplification by synergistic substrates takes place was investigated in this study. In the biosensors of this type, in addition to the substrate (analyte), another auxiliary substrate is used. It is necessary to achieve the substrates synergy. The operation of the biosensor is modelled using non-stationary reactiondiffusion equations. The model involves four regions: the enzyme layer where the enzymatic reactions as well as the mass transport by diffusion take place, the dialysis membrane and the diffusion limiting region where the mass transport by diffusion and non-enzymatic reactions take place, and the convective region in which the analyte concentration is maintained constant. The equation system is solved numerically using the finite difference technique. The biosensor response dependency on substrate concentrations and the diffusion layer thickness, as well as the biosensor sensitivity dependence on the same parameters have been studied."line-height: 18px;">

The amperometric biosensors have proved to be reliable and low-cost in various analytical systems with applications in biotechnology, medicine and environmental monitoring [106, 218, 229, 246, 275]. However, amperometric biosensors possess a number of serious drawbacks. One of the main reasons restricting wider use of the biosensors is a relatively short linear range of the calibration curve. Increasing the concentration range of detectable analyte, the sensitivity and specificity of the detection event improves the prospects for commercialising biosensors [176, 196, 217, 228, 246].