APPROXIMATE EXPRESSIONS FOR THE SURFACE-POTENTIALS OF CHARGED VESICLES

In the development of a theoretical description of the formation and s tability of spontaneous cationic-anionic vesicles, one requires a rapi d, yet reasonably accurate, computational method to calculate the elec trostatic free energy, g(elec), of a charged vesicle. In turn, the eva luation of g(elec) via the charging process requires knowledge of the surface potentials of the charged vesicle. With this in mind, we deriv e approximate expressions for the outer and inner sui face potentials of a charged vesicle using the nonlinear Poisson-Boltzmann (PB) equati on. The derivation is carried out in two stages; The first stage is ba sed on a generalization of the dressed-ionic micelle theory to the cas e of charged vesicles. Specifically, the PB equation is utilized to es timate the potential gradients at the outer and inner surfaces of the vesicle, which are then substituted in the two boundary conditions tha t describe the variation of the electric field across the boundaries. Combined with an expression relating the inner surface potential to th e center-point potential, a set of three algebraic equations is obtain ed. This set of equations can then be solved numerically to calculate the two surface potentials of the vesicle. In the second stage, by exp anding around the surface potentials which correspond to a vesicle hav ing an electrically neutral interior, the two surface potentials are e xpressed approximately in terms of the surface charge densities and ot her known vesicular characteristics such as the size of the vesicle. T he resulting surface potentials can then be estimated directly and ana lytically without resorting to any numerical procedure. In general, th e surface potentials obtained by using the equations derived in the tw o stages are found to agree well with those obtained by a direct numer ical integration of the PB equation. The:approximate expressions for t he vesicle surface potentials derived in this paper eliminate the need for a direct numerical integration of the PB equation, thus providing a much more efficient computational route. This in turn, greatly faci litates the evaluation of the electrostatic free energy of a charged v esicle via the charging process.