Calculating Minimal Energy Shapes of Fusion Pores

Abstract

We have developed a new computational model to calculate the shape of fusion pores that are in a state of minimal elastic energy as a function of pore length and lumen radius. A Helfrich Hamiltonian accounting for splay and lipid tilt was used for calculations. Minimal-energy shapes were obtained by numerically solving steepest descent partial differential equations derived from the Hamiltonian. The energy landscape was calculated by describing the bilayer as a single surface--the midplane between monolayers--or by describing the bilayer as two abutted monolayers, each with a neutral surface. The constraint imposed by mathematically placing two monolayers in apposition causes minimal energy to be larger than that predicated by (incorrectly) assuming that the elastic properties of a bilayer can be quantitatively captured through a single surface. Independent of pore size, the deformation of tilt did not appreciably affect elastic energies; in other words, membrane splay dominates elastic energies. For small radii, shapes of minimal energy were close to the shape of a catenoid. For large pores, however, deviations of minimal energy shapes from catenoids were large, resulting from the necessity that the membranes be parallel and the separation between them fixed at distances far from the rim of the pore. Energies for minimal shapes were 15-60kT less than the energy of the toroidal shape for pore radii in the range of 2-16 nm and for initially parallel membranes that were separated by 2-4 nm. For the smallest pore possible (i.e., an initial pore), a toroidal geometry overestimated the minimal energy by 30 kT. For pores with radius larger than length, membrane separation near the rim of the pore exceeds the distance between the parallel membranes. These shapes of minimal elastic energy can now be used to calculate fusion pore dynamics.