  Up-down asymmetric inclusions impose a local, spontaneous curvature to an
elastic membrane. When several of them are inserted in a same membrane, they
feel effective forces mediated by the membrane, both of elastic and entropic
nature. Following an approach initiated by Dommersnes and Fournier in the
vanishing tension case [Eur. Phys. J. B 12, 9 (1999)], and also using a
pseudo-analytical micellization theory, we derive the statistical mechanics of
asymmetric inclusion assemblies when they are also subject to an additional
short-range, attractive interaction. Our main conclusion is that generically,
when the membrane is under tension, these inclusions live in small clusters at
equilibrium, leading to local membrane invaginations. We also propose a novel
curvature-induced demixing mechanism: when inclusions imposing local curvatures
of opposite sign coexist, they tend to demix in distinct clusters under
realistic conditions. This work has potential implications in the context of
the thermodynamics of proteins embedded in biological lipid bilayers.
