Many-body interactions between curvature-inducing membrane inclusions with arbitrary cross-sections

By means of a multipolar expansion, we study analytically and numerically the interaction, in tensionless membranes, between multiple identical curvature-inducing membrane inclusions having arbitrary cross sections but uniform small detachment angles. In particular, for N circular inclusions forming regular polygons, we obtain analytical expressions for the total asymptotic interaction, up to N = 6, and we numerically compute the different multi-body contributions at arbitrary separations. We find that the latter are comparable to the sum of the two-body contributions. For N = 5 inclusions, the analytical asymptotic interaction scales as the inverse sixth power of the nearest neighbors distance d, weaker than the d^-4 power for N ≠ 5. The analytical interactions are always repulsive and in good agreement with the numerical results. In the case of noncircular cross sections, we consider the case of two identical inclusions having a given number of equally shaped lobes. Depending on the number of lobes and their amplitude, we find that the interaction is asymptotically either repulsive as d^-4 or attractive as d^-2, and always repulsive at short distances. We also characterize how the interaction depends on the inclusion rotation angles in the membrane plane.