Hexagonal, square, and stripe patterns of the ion channel density in biomembranes.

Transmembrane ion flow through channel proteins undergoing density fluctuations may cause lateral gradients of the electrical potential across the membrane giving rise to electrophoresis of charged channels. A model for the dynamics of the channel density and the voltage drop across the membrane (cable equation) coupled to a binding-release reaction with the cell skeleton [P. Fromherz and W. Zimmerman, Phys. Rev. E 51, R1659 (1995)] is analyzed in one and two spatial dimensions. Due to the binding release reaction spatially periodic modulations of the channel density with a finite wave number are favored at the onset of pattern formation, whereby the wave number decreases with the kinetic rate of the binding-release reaction. In a two-dimensional extended membrane hexagonal modulations of the ion channel density are preferred in a large range of parameters. The stability diagrams of the periodic patterns near threshold are calculated and in addition the equations of motion in the limit of a slow binding-release kinetics are derived.