N-body Study of Anisotropic Membrane Inclusions: Membrane Mediated Interactions and Ordered Aggregation

We study the collective behavior of inclusions inducing local anisotropic curvatures in a flexible fluid membrane. The N-body interaction energy for general anisotropic inclusions is calculated explicitly, including multi-body interactions. Long-range attractive interactions between inclusions are found to be sufficiently strong to induce aggregation. Monte Carlo simulations show a transition from compact clusters to aggregation on lines or circles. These results might be relevant to proteins in biological membranes or colloidal particles bound to surfactant membranes. PACS. 87.15.Kg – 64.60.Cn – 24.10.Cn The interplay between structural features and N -body interactions is a general physical problem arising in many different contexts, e.g., crystals structure [1], magnetic atom clusters [2,3], colloids in charged fluids [4], polyelectrolyte condensation [5,6], and protein aggregation in biological membranes [7]. N -body interactions can sometimes yield spectacular effects: non-pairwise summability of charge fluctuation forces can dramatically affect the stability of polyelectrolyte bundles [5]; three-body elastic interactions may induce aggregation of membrane inclusions, although two-body elastic interactions are repulsive [7]. In a system able to kinetically achieve equilibrium, the clusters formed are usually compact, however certain interactions may favor tenuous clusters. For instance, a recent N -body study has shown that above a critical strength of three-body interactions, the state of minimum energy is one in which all the particles are on a line [8]. It has also been observed recently that membrane mediated interactions can induce one-dimensional ring-like aggregates of colloidal particles bound to fluid vesicle membranes [9]. Manifolds embedded in a correlated elastic medium can impose boundary conditions, or modify the elastic constants. This usually gives rise to mean-field forces, which are due to the elastic deformation of the medium, and to Casimir forces, which are due to the modification of its thermal fluctuations. Such interactions are generally non pairwise additive [10]. The elastic interactions between defects in solids [11] or in liquid crystals [12] are well known examples of mean-field forces. Casimir forces exist between manifolds embedded in correlated fluids, such as liquid crystals and superfluids [13,14,10], or critical mixtures [15]. Another interesting example is the interaction between inclusions in flexible membranes [16]: it has been shown that cone shaped membrane inclusions experience both long range attractive Casimir interactions and repulsive elastic interactions falling of as R with separation R [17]. In this Rapid Note, following Netz [18], we give exact results concerning the long range multi-body interactions among membrane inclusions that break the bilayer’s updown symmetry. However, rather than supposing that the inclusions simply induce a local spontaneous curvature, we assume that the inclusions set a preferred curvature tensor [17,19]. This model is more realistic: the “preference” of a conically shaped inclusion is c1 = c2 = c0, where c1 and c2 are the membrane principal curvatures, rather than the weaker condition c1 + c2 = 2c0 assumed in Ref. [18]. In addition, the imposed curvature tensor can be anisotropic, thus describing inclusions that break the in-plane symmetry. In a first part we calculate the exact Casimir and mean-field twoand three-body interactions between such anisotropic inclusions. Then, the collective behavior of identical inclusions is investigated by means of a Monte Carlo (MC) simulation, using the full N -body interaction energy plus a hard-core repulsion modeling the simplest repulsive short-range interactions [20]. Our results could be relevant to understanding the aggregation and organization of proteins in biological membranes, or colloidal particles bound to surfactant membranes [9]. Let us consider a system of N anisotropic inclusions embedded in a flexible fluid membrane, in which they are free to diffuse laterally. In many situations, the surface tension is negligible and the membrane shape is governed by the Helfrich curvature energy h0 = 1 2 κ (c1 + c2) 2 + κ̄ c1c2 [21], where κ is the bending rigidity, and κ̄ the Gaussian modulus. For biological membranes, κ ∼ 30T , while for surfactant membranes it can be as small as a few T (T will denote throughout the temperature in energy units). We model the membrane shape by a simple parametric 2 P. G. Dommersnes, J.-B. Fournier: Anisotropic inclusions surface (r, u(r)), where r is a vector in the (x, y) plane and u(r) the normal displacement field along z. To quadratic order in u, the Helfrich Hamiltonian takes the form