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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.

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.

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 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.

A bacterial self-organization in a rounded container as detected by bioluminescence imaging is mathematically modeled by applying the the Keller–Segel approach with logistic growth. The pattern formation in a colony of luminous Escherichia coli is numerically simulated by the nonlinear reaction-advection-diffusion equations. In this work, the pattern formation is studied in 3D and the results are compared with previous and new 2D and 1D simulations. The numerical simulation at transition conditions was carried out using the finite difference technique. The simulation results showed that the developed 3D model captures fairly well the sophisticated patterns observed in the experiments. Since the numerical simulation based on the 3D model is very time-consuming, the reduction of spatial dimension of the model for simulating 1D spatiotemporal patterns is discussed. Due to the accumulation of luminous cells near the top three-phase contact line the experimental patterns of the bioluminescence can be qualitatively described by 1D and 2D models by adjusting values of the diffusion coefficient and/or chemotactic sensitivity.

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.

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.

The optimization-based quantitative determination of multianalyte concentrations from biased biosensor responses is investigated under internal and external diffusion-limited conditions. A computational model of a biocatalytic amperometric biosensor utilizing a mono-enzyme-catalyzed (nonspecific) competitive conversion of two substrates was used to generate pseudo-experimental responses to mixtures of compounds. The influence of possible perturbations of the biosensor signal, due to a white noise- and temperature-induced trend, on the precision of the concentration determination has been investigated for different configurations of the biosensor operation. The optimization method was found to be suitable and accurate enough for the quantitative determination of the concentrations of the compounds from a given biosensor transient response. The computational experiments showed a complex dependence of the precision of the concentration estimation on the relative thickness of the outer diffusion layer, as well as on whether the biosensor operates under diffusion- or kinetics-limited conditions. When the biosensor response is affected by the induced exponential trend, the duration of the biosensor action can be optimized for increasing the accuracy of the quantitative analysis.

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.

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 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.

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 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. 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.

This paper presents a one-dimensional-in-space mathematical model of an amperometric biosensor based on a carbon nanotube electrode deposited on a perforated membrane. The developed model is based on nonlinear reaction-diffusion equations. The conditions at which the one-dimensional mathematical model can be applied to an accurate simulation of the biosensor response are investigated. The accuracy of the response simulated by using one-dimensional model is evaluated by the response simulated by the corresponding two-dimensional model. The mathematical model and the numerical solution are also validated by an experimental data. The obtained agreement between the simulation results and experimental data is admissible for different configurations of the biosensor operation. The numerical simulation was carried out using the finite difference technique.

This paper presents a one-dimensional-in-space mathematical model of a bacterial selforganization in a circular container along the contact line as detected by quasi-one-dimensional bioluminescence imaging. The pattern formation in a luminous Escherichia coli colony was modeled by the nonlinear reaction-diffusion-chemotaxis equations in which the reaction term for the cells is a logistic (autocatalytic) growth function. By varying the input parameters the output results were analyzed with a special emphasis on the influence of the model parameters on the pattern formation. The numerical simulation at transition conditions was carried out using the finite difference technique. The mathematical model and the numerical solution were validated by experimental data.

This paper presents one-dimensional (1-D) and two-dimensional (2-D) in-space mathematical models for amperometric biosensors with an outer perforated membrane. The biosensor action was modelled by reaction-diffusion equations with a nonlinear term representing the Michaelis-Menten kinetics of an enzymatic reaction. The conditions at which the 1-D model can be applied to simulate the biosensor response accurately were investigated numerically. The accuracy of the biosensor response simulated by using 1-D model was evaluated by the response simulated with the corresponding 2-D model. A procedure for a numerical evaluation of the effective diffusion coefficient to be used in 1-D model was proposed. The numerically calculated effective diffusion coefficient was compared with the corresponding coefficients derived analytically. The numerical simulation was carried out using the finite difference technique.

The paper presents a one-dimensional-in-space mathematical model of a peroxidase-based fluorescent biosensor. The mathematical model is based on a system of nonlinear reaction diffusion equations. The problem was solved numerically using finite difference technique. The model was used to study the effect of biosensor parameters, namely the thickness of the enzyme layer and the outer substrate concentration, on the response of the biosensor and it’s sensitivity. The performed calculations showed a complex influence of the substrate and hydrogen peroxide concentrations. INTRODUCTION Biosensors are analytical devices converting a biochemical recognition reaction into a measurable effect (Scheller and Schubert, 1992; Turner et al., 1987). They are applied widely to monitor chemical substances in the medicine, food technology and the environmental industry (Wollenberger et al., 1997). Optical biosensors based on the fluorescence detection generate a photoluminescent signal indicative of target analyte binding (Ligler and Taitt, 2002). Fluorescence emission is then detected and converted into an analytical signal (Knopf and Bassi, 2007). They have been used for the analysis of many chemical and biological substances, especially in cases limited by low analyte concentrations (Atwood and Steed, 2004). The mathematical model of optical biosensors has been very recently developed (Baronas et al., 2007a). However, the model was based upon absorbance measurements. In this report, a mathematical model for fluorescent biosensors is described. The model is used to predict the response of biosensor with varying physical and chemical characteristics.

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