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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 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.

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.

A multi-cellular network of metabolically active E. coli as a weak gel of living Janus particles Remigijus Šimkus, * a Romas Baronas b and Žilvinas Ledas b Bioluminescence images of nutrient rich liquid cultures of lux-gene reporter Escherichia coli were recorded for several hours after being placed into small diameter cylindrical containers (glass tubes and microtiter plate wells). It was found that luminous cells distribute near the three-phase contact line forming an irregular array of clumps, channels and plumes at the solid-liquid interface. The experimentally observed quasi-2-dimensional spatiotemporal patterns ('venation patterns') were simulated fairly well by the mean field Keller-Segel equations of chemotactic aggregation. The equations use a logistic cell growth term whose carrying capacity depends on oxygen concentration. The experimental and numerical results are interpreted as follows: (1) the patterns of bioluminescence form due to arrested phase separation in the culture; (2) the luminous phase formed along the upper part of the solid-liquid interface is a weak gel-like network of metabolically active bacteria which exhibit ligand-receptor type cell-cell adhesion and high rates of oxidative phosphorylation; (3) active bacteria in this gel can be viewed as self-generated Janus particles, i.e., polarized particles whose surface is divided into two chemically varying regions, the adhering region and the non-adhering region; (4) the life-time of the metabolically active bacterium (living Janus particle) in the weak gel phase is estimated to be $25 s; (5) the observed network of luminous bacteria can be viewed as a biofilm, in which bacteria move due to self-phoresis by generating pH gradients via metabolic reactions; and (6) the reversible clustering of cells near interfaces is attributed to the bacterial energy-taxis phenomenon.