The many-body problem for anisotropic membrane inclusions and the self-assembly of saddle defects into an egg carton

We calculate the many-body, nonpairwise interaction between N rigid, anisotropic membrane inclusions by modeling them as point-like constraints on the membrane's curvature tensor and by minimizing the membrane's curvature energy. Because multipolar distortions of higher-order decay on very short distances, our calculation gives the correct elastic interaction energy for inclusions separated by distances of the order of several times their size. As an application, we show by thermally equilibrating the many-body elastic energy using a Monte Carlo algorithm, that inclusions shaped as saddles attract each other and build an egg-carton structure. The latter is reminiscent of some patterns observed in membranes obtained from biological extracts, the origin of which is still mysterious.