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-tilt 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 regular 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 resulting interaction can stabilize 2D crystal phases, with the possible coexistence of different lattice spacings when the dilation-tilt-difference coupling is large. Inclusions favoring crystals are those with either a long-convex or a short-concave hydrophobic core. Inclusions inducing a local membrane curvature due to their conical shape repel one another. At short inclusions separations, a tilt comparable with the inclusion's cone angle develops: it relaxes the membrane curvature and reduces the repulsion. At large separations the tilt vanishes, whatever the value of the shape-tilt coupling.