phase-diagrams are obtained. For the study of protein profiles, one can further simplify this expression. First, for small values of the order parameters, i.e., relatively close to the critical point of demixing of the lipid and for small protein concentrations, it is legitimate to expand the free energy of mixing up to order ${\cal O}(\eta^4)$ and ${\cal O}(\phi^2)$. In addition, since typical proteins occupy a much larger area then lipids, the area ratio is in the range of $\alpha  \sim 50 - 100$, and the protein entropy terms (of order $1/\alpha$) can be neglected in (7). The validity of the latter ($\alpha \rightarrow \infty$) approximation will be reexamined in Sect. III[@note1]. With these simplifications, the approximated free energy density can be written as $${\cal F}^0/{T}=
-J \eta^2 +\frac{1}{12} \eta^4 +L \phi^2
+ \lambda \eta \phi +\frac{1}{2} \eta^2 \phi$$ where the simplified thermodynamic potential using (8) is given by ${\cal G}^0/{T}={\cal F}^0/{T}- \mu \phi -\mu_{\eta} \eta$. The free energy density (13) needs some further discussion. Coexistence between dense ($\eta >0$) and dilute regions ($\eta<0$) requires that $J>0$ and a positive fourth-order term $\eta^4$ is needed to stabilize the free energy. The protein itself is assumed not to be close to any 