the case of freely suspended membranes, our findings imply a simple mechanism for the localization of integral membrane proteins along the one-dimensional boundary between coexisting domains. The resulting enrichment of proteins might be a prerequisite for proper biological function in certain cases.

In the following sections we formulate the model (Sect. II), inspect the minima of the free energy (Sect. III), solve the corresponding Euler-Lagrange equations in the coexistence region (Sect. IV), and calculate profiles both for the lipid and the (coupled) protein densities (Sect. V). From the profiles we generate a general phase diagram featuring localized, semi-localized and delocalized protein phases. We also calculate the total amount of adsorbed protein, the protein excess $\Gamma$ (Sect. VI), and the line tension $\tau$ of the LE-LC line interface (Sect. VII). It turns out that the line tension is strongly reduced by the adsorption of proteins. A finite solubility of the proteins in the subphase is taken into account in Sect. VIII. Finally, the connection to experimentally measurable quantities, such as the surface pressure $\Pi$, is made in Sect. IX.

The mixed lipid and protein free energy
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Consider the air-water interface with proteins, lipid molecules, and artificial “vacancies”, with area fractions 