Colonies of mutant E. coli strains, when inoculated in the centre of a plate, form highly symmetric, stable spot patterns. These spot patterns are only observed in chemotactic E. coli strains, i.e. strains that bias their motion so that cell populations move up gradients of a chemical in the cells' local environment. It is an important question whether these patterns are due to genetic control or self-organization. Here we present a macroscopic continuum model of E. coli pattern formation that incorporates cell diffusion, chemotaxis, population growth and conversion to an inactive state. This model satisfactorily reproduces the observed spot patterns, supporting the view that these patterns are indeed a result of self-organization, and allows us to infer plausible minimal mechanisms that generate the observed patterns.

We examine the mean flux across a homogeneous membrane of a charged tracer subject to an alternating, symmetric voltage waveform. The analysis is based on the Nernst-Planck flux equation, with electric field subject to time dependence only. For low frequency electric fields the quasi steady-state flux can be approximated using the Goldman model, which has exact analytical solutions for tracer concentration and flux. No such closed form solutions can be found for arbitrary frequencies, however we find approximations for high frequency. An approximation formula for the average flux at all frequencies is also obtained from the two limiting approximations. Numerical integration of the governing equation is accomplished by use of the numerical method of lines and is performed for four different voltage waveforms. For the different voltage profiles, comparisons are made with the approximate analytical solutions which demonstrates their applicability.

A one-dimensional model of solute transport through the stratum corneum is presented. Solute is assumed to diffuse through lipid bi-layers surrounding impermeable corneocytes. Transverse diffusion (perpendicular to the skin surface) through lipids separating adjacent corneocytes, is modeled in the usual way. Longitudinal diffusion (parallel to the skin surface) through lipids between corneocyte layers, is modeled as temporary trapping of solute, with subsequent release in the transverse direction. This leads to a linear equation for one-dimensional transport in the transverse direction. The model involves an arbitrary function whose precise form is uncertain. For a specific choice of this function, closed form expressions for the Laplace transform of solute out-flux at the inner boundary, and for the time lag are obtained in the case that a constant solute concentration is maintained at the outer skin surface, with the inner boundary of the stratum corneum kept at zero concentration, and with the stratum corneum initially free of solute.