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We discuss the electrostatic contribution to the elastic moduli of a cell orartificial membrane placed in an electrolyte and driven by a DC electric field.The field drives ion currents across the membrane, through specific channels,pumps or natural pores. In steady state, charges accumulate in the Debye layersclose to the membrane, modifying the membrane elastic moduli. We first study amodel of a membrane of zero thickness, later generalizing this treatment toallow for a finite thickness and finite dielectric constant. Our resultsclarify and extend the results presented in [D. Lacoste, M. CosentinoLagomarsino, and J. F. Joanny, Europhys. Lett., {\bf 77}, 18006 (2007)], byproviding a physical explanation for a destabilizing term proportional to$\kps^3$ in the fluctuation spectrum, which we relate to a nonlinear ($E^2$)electro-kinetic effect called induced-charge electro-osmosis (ICEO). Recentstudies of ICEO have focused on electrodes and polarizable particles, where anapplied bulk field is perturbed by capacitive charging of the double layer anddrives flow along the field axis toward surface protrusions; in contrast, wepredict "reverse" ICEO flows around driven membranes, due to curvature-inducedtangential fields within a non-equilibrium double layer, which hydrodynamicallyenhance protrusions. We also consider the effect of incorporating the dynamicsof a spatially dependent concentration field for the ion channels.

Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane