The Many-Body Problem for Anisotropic Membrane Inclusions and the Self-Assembly of “Saddle” Defects into an “Egg Carton”

Abstract

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.

Shape deformation of lipid membranes by banana-shaped protein rods: Comparison with isotropic inclusions and membrane rupture

“…Excluded-volume and other interactions between the proteins or between the protein and membrane can generate effective side spontaneous curvatures. When the rod and side curvatures are in opposite directions, saddle-shaped membranes, such as egg-carton [37,38], ring, and network structures [50], can be stabilized. Thus, anisotropic inclusions can induce much more variety in membrane structures than isotropic inclusions.…”

“…The classical Canham-Helfrich curvature free energy [32,33] has been extended to anisotropic curvatures [34][35][36]. To simplify the interactions, the protein and membrane underneath it have been often modeled together as an undeformable object with a fixed curved shape such as a point-like object with an anisotropic curvature [37,38] and a bent elliptical surface [39]. Furthermore, it has also been clarified that two undeformable parallel rods have an attractive interaction but the interaction is repulsive for a perpendicular orientation.…”

Indirect interactions of membrane-adsorbed cylinders

“…Static interactions are due to a perturbation of the equilibrium bilayer structure or equilibrium membrane shape by the embedded or adsorbed macromolecules. Examples for such macromolecules are trans-membrane proteins [20,21,22,23,24] and adsorbed molecules [25] causing a perturbation of the equilibrium membrane thickness, as well as conical or anisotropic inclusions [11,26,27,28,29,15,30] and membrane-anchored polymers [31,32] which cause a perturbation of the equilibrium membrane curvature.…”

Membrane Mediated Sorting

“…However, due to their short range, they do not provide a mechanism for some important biological functions like the recruitment of proteins to certain regions in the plasma membrane. Recently, attention was drawn to another type of interaction: membrane curvature mediated interactions [2][3][4][5][6][7][8][9][10][11][12]. These interactions are known to be long ranged [2] and non-pairwise additive [6].…”

“…Although the interactions are not pairwise additive, the qualitative dependence of V on the contact angles and inclusion distance does not change if more inclusions are added to the system [4,6]. It is therefore possible to use a mean-field description for a finite, closed system with many inclusions, from which the prefactor in Eq.…”