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A phenomenological Landau elasticity for the shape, dilation, and lipid-tiltof bilayer membranes is developed. The shape mode couples with the sum of themonolayers' tilt, while the dilation mode couples with the difference of themonolayers' tilts. Interactions among membrane inclusions within regular arraysare discussed. Inclusions modifying the membrane thickness and/or inducing atilt-difference due to their convex or concave shape yield a dilation-inducedattraction and a tilt-difference-induced repulsion. The resulting interactioncan stabilize 2D crystal phases, with the possible coexistence of differentlattice spacings when the dilation-tilt-difference coupling is large.Inclusions favoring crystals are those with either a long-convex or ashort-concave hydrophobic core. Inclusions inducing a local membrane curvaturedue to their conical shape repel one another. At short inclusions separations,a tilt comparable with the inclusion's cone angle develops: it relaxes themembrane curvature and reduces the repulsion. At large separations the tiltvanishes, whatever the value of the shape-tilt coupling.Comment: 13 pages, 19 figure

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