$\phi_P$, $\phi_L$, and $\phi_V$, respectively, satisfying $\phi_P+\phi_L+\phi_V=1$. The vacancies are introduced in order to allow for independent variations of the protein and lipid concentrations, hence making coexistence of dilute and condensed regions of the monolayer possible. Inscribing the system on a lattice, with a lattice constant corresponding to the size of a lipid molecule, the free energy of mixing per lattice site within a mean field theory can be written for the three-component mixture as a sum of the enthalpy and entropy of mixing, ${\cal F} = {\cal U} -T{\cal S}$. The enthalpy of mixing includes all pair-wise interactions between the three species: $${\cal U}/T=
E_{LL} \phi_L^2 +E_{VV} \phi_V^2 +E_{PP} \phi_P^2
+E_{LV} \phi_L \phi_V +E_{PL} \phi_P \phi_L+E_{PV} \phi_P \phi_V$$ and the $E_{ij}$ are the dimensionless interaction parameters for all possible pairs. The entropy of mixing is related to the total number $\Omega$ of distinct microscopic configurations $${\cal S}=\frac{\log \Omega}{N}$$ where $N$ is the total number of lattice sites and the Boltzmann constant is set to unity ($k_B=1$). In the random-mixing approximation, $$\Omega=
\frac{(N/\alpha)!}{(N \phi_P /\alpha)! (N(1-\phi_P)/\alpha)!}
\frac{(N[\phi_L+\phi_V])!}{(N \phi_L)!(N \phi_V)!}$$ where the constant $\alpha>1$ denotes the ratio between the compact area occupied by a protein molecule and a lipid molecule at 