phase transition. Hence $L>0$ and no higher order terms in $\phi$ are needed. We include in the expansion only the two lowest coupling terms between the protein and lipid concentrations. The first is the bilinear coupling $\eta\phi$ and has an enthalpic origin. It reflects the overall preference of the protein to more condensed ($\lambda<0$) or more dilute ($\lambda>0$) regions of the lipid monolayer. The second coupling is the symmetric $\eta^2\phi$ term, which is invariant under $\eta \rightarrow -\eta$ transformation and provides the driving force for the localization of proteins at the LE-LC interface. In our mean-field model, taking into account only pair interactions, this coupling has a purely entropic origin. More generally, it can also include interaction terms of higher-order in a virial expansion. Finally, the higher-order coupling terms $\eta^2 \phi^2$ and $\eta^4 \phi$ are not considered here since we try to investigate the most simple and yet non-trivial type of coupling. A similar free energy coupling has been introduced in the context of polymer adsorption at liquid-liquid interfaces, where in analogy the polymer adsorbs preferentially at the interface from the bulk solution[@Pincus].

For the case where the proteins in the monolayer are in equilibrium with a solution of proteins 