Transmembrane ion flow through channel proteins undergoing densityfluctuations may cause lateral gradients of the electrical potential across themembrane giving rise to electrophoresis of charged channels. A model for thedynamics of the channel density and the voltage drop across the membrane (cableequation) coupled to a binding-release reaction with the cell skeleton (P. Fromherz and W. Zimmerman, Phys. Rev. E 51, R1659 (1995)) is analyzed in oneand two spatial dimensions. Due to the binding release reaction spatiallyperiodic modulations of the channel density with a finite wave number arefavored at the onset of pattern formation, whereby the wave number decreaseswith the kinetic rate of the binding-release reaction. In a two-dimensionalextended membrane hexagonal modulations of the ion channel density arepreferred in a large range of parameters. The stability diagrams of theperiodic patterns near threshold are calculated and in addition the equationsof motion in the limit of a slow binding-release kinetics are derived.
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