Periodic band pattern as a dissipative structure in ion transport systems with cylindrical shape

A theory is presented for appearance of periodic band patterns of ion concentration and electric potential associated with electric current surrounding a unicellular or multicellular system of a cylindrical shape. A flux continuity at the membrane (or the surface) is reduced to a nonlinear equation expressing passive and active fluxes across the membrane and intracellular diffusion flux. It is shown that, when an external parameter is varied from the sub-critical region, i.e. the homogeneous flux state, a symmetry breaking along a longitudinal axis usually appears prior to the one along a circumferential direction. The spectrum analysis shows that the correlation length is longer in the longitudinal direction. Growth of the band pattern from a patch-shaped pattern is demonstrated by the use of numerical calculations of proton concentration on the two-dimensional space of cylindrical surface. An experimental example of formative process of H⁺ banding is given for the internodal cell ofChara. It is shown that small patches on the surface decline or are sometimes gathered to the band surrounding the circle. The resulting pattern is suggested as a kind of dissipative structure appearing far from equilibrium.