the interface. The above expression is the product of the number of all protein configurations and the number of all lipid/vacancy configurations in the remaining area not taken up by the proteins. Using Stirling’s Formula in the thermodynamic limit, defined by $N \rightarrow \infty$, the expression for ${\cal S}$ can be simplified $${\cal S}=-\phi_L \log(\phi_L) -\phi_V \log(\phi_V) -\phi_P \log(\phi_P)/\alpha
-(1/\alpha -1)(1-\phi_P) \log (1-\phi_P)$$ It is convenient to define the thermodynamic potential $${\cal G}/T=
{\cal F}/T- \mu_P \phi_P -\mu_L(\phi_L-\phi_V)$$ where the chemical potentials $\mu_P$ and $\mu_L$ are coupled to the protein concentration $\phi_P$ and the difference between the lipid and vacancy concentrations, $\phi_L-\phi_V$, respectively.

In (1)-(5), long-range interactions between the proteins, such as electrostatic forces, are not taken into account. In addition, the free energy of mixing assumes a confinement of the protein and lipid to the two-dimensional plane of the air-water interface. In fact, the variation of the protein concentration perpendicular to the monolayer in the subphase can be taken into account approximately and leads to a renormalization of the parameters of the two-dimensional model, as shown in Sect. VIII.

The lipid order parameter $\eta$, corresponding to the density of lipid molecules, can be written as $$\eta \equiv \phi_L - 