Partial Localization, Lipid Bilayers, and the Elastica Functional

TL;DR

This paper analyzes an energy functional modeling lipid bilayer membranes, showing that low-energy states are partially localized into thin structures with properties akin to cell membranes, and establishes a Gamma-convergence result to the Elastica energy.

Contribution

It provides a rigorous mathematical analysis of partial localization in lipid bilayers and connects the energy functional to classical elastica curves via Gamma-convergence.

Findings

Density fields of moderate energy are partially localized.

Deviation from uniform thickness incurs an energy penalty.

The zero-thickness limit converges to the Elastica energy of curves.

Abstract

Partial localization is the phenomenon of self-aggregation of mass into high-density structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial localization and displays the `solid-like' behavior of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, i.e. resemble thin structures. Deviation from a specific uniform thickness, creation of `ends', and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the zero-thickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and non-fracture; if these conditions are met, then the limit value is given by the classical Elastica (bending) energy of the curve.