Effect of surface charges on the curvature moduli of a membrane

The modification of the curvature moduli due to surface charges in lipid bilayers is analyzed using the nonlinear Poisson-Boltzmann equation for the relationship between the charge density and surface potential. An expansion in a small parameter $\epsilon ,$ which is the ratio of the Debye length and the radius of curvature, is used. At low charge densities, previous results obtained from the Debye-Huckel approximation are recovered. At high charge densities, the corrections to the mean and Gaussian curvature approach constant values. The total energy of curvature for a symmetrically charged membrane becomes negative when the charge density is increased beyond a critical value, indicating that the membrane spontaneously forms vesicles. An asymmetry in the charge densities on the two monolayers that form the bilayer results in a spontaneous curvature, and the radius of curvature could be large compared to the Debye length when the asymmetry is small. The case of adsorbed charges is also considered, where there is a reduction in the chemical energy when a charge is adsorbed on the surface. At low charge density, the mean and Gaussian curvature are equal in magnitude and opposite in sign to that for fixed charges, while at high charge density, the mean and Gaussian curvature approach values identical to that for a surface with fixed charges. Numerical calculations of the change in the curvature moduli with realistic parameter values indicate that these effects are likely to be of importance in the spontaneous formation of vesicles.