Microscopic membrane elasticity and interactions among membrane inclusions: interplay between the shape, dilation, tilt and tilt-difference modes

A phenomenological Landau elasticity for the shape, dilation, and lipid-til t of bilayer membranes is developed. The shape mode couples with the sum of the monolayers' tilt, while the dilation mode couples with the difference of the monolayers' tilts. Interactions among membrane inclusions within reg ular arrays are discussed. Inclusions modifying the membrane thickness and/ or inducing a tilt-difference due to their convex or concave shape yield a dilation-induced attraction and a tilt-difference-induced repulsion. The re sulting interaction can stabilize 2D crystal phases, with the possible coex istence of different lattice spacings when the dilation-tilt-difference cou pling is large. Inclusions favoring crystals are those with either a long-c onvex or a short-concave hydrophobic core. Inclusions inducing a local memb rane curvature due to their conical shape repel one another. At short inclu sions separations, a tilt comparable with the inclusion's cone angle develo ps: it relaxes the membrane curvature and reduces the repulsion. At large s eparations the tilt vanishes, whatever the value of the shape-tilt coupling .