Elastic energy of tilt and bending of fluid membranes

Tilt of hydrocarbon chains of lipid molecules with respect to membrane plan e is commonly considered to characterize the internal structure of a membra ne in the crystalline state. However, membranes in the liquid state can als o exhibit tilt resulting from packing constraints imposed on the lipid mole cules in diverse biologically relevant structures such as intermediates of membrane fusion, pores in lipid bilayers and others. We analyze the energet ics of tilt in liquid membranes and its coupling with membrane bending. We consider three contributions to the elastic energy: constant tilt, variatio n of tilt along the membrane surface and membrane bending. The major assump tion of the model is that the core of a liquid membrane has the common prop erties of an elastic continuum. We show that the variation of tilt and memb rane bending are additive and that their energy contributions are determine d by the same elastic coefficient: the Helfrich bending modulus, the modulu s of Gaussian curvature and the spontaneous curvature known from previous s tudies of pure bending. The energy of a combined deformation of bending and varying tilt is determined by an effective tensor accounting for the two f actors. In contrast, the deformation of constant tilt does not couple with bending and its contribution to the elastic energy is determined by an inde pendent elastic constant. While accurate determination of this constant req uires additional measurements, we estimate its value using a simplified app roach. We discuss the relationships between the obtained elastic Hamiltonia n of a membrane and the previous models of membrane elasticity.