2 the bending free energy of a curved surface of finite thickness is examined and the model is reviewed for the case of a curved lipid leaflet in the inverse hexagonal phase of lipids. To further motivate the need for an unambiguous theory for calculating the local pressure, in Sec. 4, 5, 6, 7 Given a practical theory of the tension of curved systems applicable to detailed biophysical models, simulations could provide a description of bending free energies of the various biological substituents under controlled conditions. For example, the spontaneous curvature of a lipid leaflet in a bilayer may influence transmembrane protein conformation, spacing, and aggregation, as well as the energy of tubes or intermediates in endo- and exocytosis. The question of how to model curved surfaces is important in a biophysical context largely due to the interest in how the properties of the cell membrane affect and are affected by biological machinery. The method presented herein relies on the connection between pressure differences and tensions described by the Young-Laplace equation (YL), so that quantifying the tension of a curved surface is reduced to calculating the pressure at the interior of that curved surface. This paper develops a practical solution to the problem of the tension of curved interfaces by examining the local pressure at the center of an enclosed surface (a quantity that can be measured experimentally) using virtual mechanical deformations. Quantification of the energy of deformation of a curved surface then depends on how that surface is defined. The problem is compounded by the fact that the area of a curved molecular surface is rarely well defined, unlike the projected area of a planar interfacial system. For a spherical drop, no simple ensemble exists to keep the area constant i.e., an ensemble that would be constrained by a measurable external force. 1 For planar systems with periodic boundary conditions, the tension can be computed by examining the external force on the simulation cell necessary to keep the system at constant area, see, e.g., Refs.

Evaluating the surface tension of curved interfaces from simulation presents theoretical and practical problems not encountered for planar systems.

Forces on curved survaces