Theory of electric dissipative structure in Characean internode

A band-type alternating pattern of acidic and alkaline regions formed along the Characean cell wall is discussed theoretically. The model system is constructed from linear diffusion equations for the concentration of H⁺ outside the internode and in the protoplasm. The plasmalemma is taken as a boundary transporting H⁺ under energy supply by light. The sizes of the protoplasm and extracellular water phase are taken into account explicitly in the present model system to reproduce qualitatively the characteristics observed in various types of experiments. Theoretical analysis shows that the band pattern belongs to dissipative structures emerging far from equilibrium, and is stabilized through the electric current loops produced by locally activated electrogenic H⁺ pumps and spatially separated passive H⁺ influx (or OH^- efflux) across the membrane. Both the numerical calculation and the theoretical analysis using a generalized time-dependent Ginzburg-Landau equation reveal the following points: (i) the intemodal cell with a larger vacuole in a smaller size of the extracellular water phase tends to exhibit a clearer band pattern; (ii) the increase in viscosity of the external aqueous medium makes the bands appear more easily and, furthermore, distinctly; (iii) the change in size of the extracellular water phase significantly affects the kinetics of the pattern- formation process. These results are interpreted reasonably by taking account of the electric current circulating between the acidic and alkaline regions.