Inclusions embedded in lipid membranes

We study the equilibrium of Lipid membranes embedding a protein, conceived as a rigid inclusion. By use of a two-dimensional model, we are able to determine the exact equilibrium shape of the membrane, which is idealized as a closed curve. We also discuss the role played by the anchoring of the protein to the membrane. When the anchoring is strong, we are always able to iind different classes of equilibrium shapes and, within them, to assess the uniqueness of regular equilibrium configurations. In contrast, the presence of a weak anchoring yields a loss of uniqueness within each class, and even a loss of any regular equilibrium configuration, if the spontaneous curvature exceeds a critical value, which increases with the anchoring strength. In this case, suitable singular solutions are found, where the protein is segregated from the rest of the membrane. These solutions mimic the behaviour of coat proteins in lipid membranes, suggesting that weak anchoring could be invoked as a possible mechanism to induce protein segregation. A resort to numerical techniques provides us with a comprehensive view of the problem.