\frac{\partial {\cal F}}{\partial \phi} \right|_{\eta_1,\phi_1}=
\left. \frac{\partial {\cal F}}{\partial \phi} \right|_{\eta_2,\phi_2}$$ which correspond to a common-tangent construction. These equations can be easily solved numerically. In order to estimate the role of the protein-lipid area ratio, $\alpha$, and to compare the results with the calculations presented in the next section based on the simplified expression (13), where $\alpha \rightarrow \infty$, we restrict the numerical analysis to the values $J=1/10$ and $L=10$. The small value of $J$ means that one is close to the critical point of the lipid phase separation, and the expansion in powers of $\eta$, leading to (13), is appropriate. The large value of $L$ means that the protein concentration is rather small everywhere and can be treated as a small perturbation. We will need this assumption for the analytic solution of the Euler-Lagrange equations in Sect. IV. The parameter $\alpha$ will be scanned in a rather wide range. With this choice of $L$ and $J$, it is clear that the simplified free energy expression (13) is asymptotically obtained for $\alpha \rightarrow \infty$.

The protein concentrations in the coexisting dense and dilute lipid regions scan a whole range of different values, depending on the values of the remaining parameters 