Curvature-Tilt Theories of Lipid Membranes

On mesoscopic scales lipid membranes are well described by continuum theories whose main ingredients are the curvature of a membrane’s reference surface and the tilt of its lipid constituents. In particular, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] showed how to systematicallyderive such a tilt-curvature Hamiltonian, based on the elementary assumption of a thin fluid elastic sheet experiencing internal lateral pre-stress. Performing a dimensional reduction, they not only derived the basic form of the effective surface Hamiltonian, but also made connections between the trans-membrane moments of lower-level material parameters and the emergent elasticcouplings of surface energy. In the present thesis we argue, though, that their derivation unfortunately missed a coupling term between curvature and tilt. The origin of this term is the change of transverse distances due to the variation in the curvature along the membranes. This change gives rise to a contribution to the energy which was believed to be small, but nevertheless ends up contributing at the same (quadratic) order as other terms in their Hamiltonian. We show the immediate consequences of this novel coupling term by by deriving the monolayer and bilayerEuler-Lagrange equations for the tilt, as well as the power spectra of shape, tilt, and director fluctuations.We also obtain a novel set of terms, quadratic in both curvature and tilt, of which only two were part of the quadratic theory. These biquadratics manifest as geometry-dependent corrections to the tilt modulus, converting it into a position-dependent tilt modulus tensor. For typicalmaterial parameters, the resulting effective tilt modulus softens compared to the bare one, except within a small off-center domain of curvatures near the flat state. For sufficiently large curvatures, set by the characteristic length of tilt decay, the effective modulus even becomes negative.We show that biquadratics matter for strongly curved geometries, such as open edges, triple line junctions, fusion stalks, and even bicontinuous phases, and as an illustration we calculate the line tension of edges and junctions