\phi_V$$ Using that $\phi_P + \phi_L +\phi_V=1$, and defining the protein concentration as $\phi \equiv \phi_P$, the free energy ${\cal F}$ and the potential ${\cal G}$ can be rewritten as $$\begin{aligned}
{\cal F}/T &=&
-(J+1/2) \eta^2 +L \phi^2
+ \lambda \eta \phi+ \\ \nonumber
&& (1+\eta-\phi)\log[(1+\eta-\phi)/2]/2+
   (1-\eta-\phi)\log[(1-\eta-\phi)/2]/2+\\ \nonumber
&& \phi \log[\phi]/\alpha +(1/\alpha-1) (1-\phi) \log[1-\phi]\end{aligned}$$ and $${\cal G}/T= {\cal F}/T- \mu_{\eta} \eta-(\mu +\log 2 ) \phi$$ where constant terms have been omitted and linear terms in $\eta$ and $\phi$ have been dropped out from ${\cal F}$ for convenience. They merely contribute a constant shift to $\mu$ and $\mu_{\eta}$ in ${\cal G}$. The reduced interaction parameters: $J$, $L$, $\mu$ and $\lambda$ are related to the original $E_{ij}$ and $\mu_P$ in the following way $$-J \equiv 
\frac{1}{4}(E_{LL} + E_{VV} - E_{LV})+\frac{1}{2}$$ $$L\equiv \frac{1}{4}(E_{LL} + E_{VV} +  E_{LV})+ 
E_{PP}-\frac{1}{2}(E_{PL} + E_{PV})$$ $$\lambda \equiv 
-\frac{1}{2}(E_{LL} - E_{VV}- E_{PL} + E_{PV})$$ $$\mu \equiv \mu_P+
\frac{1}{2}(E_{LL} + E_{VV}+ E_{LV}- E_{PL}
- E_{PV})-\log 2$$ The constant $\log 2$ appears in the definition of $\mu$ in order to render the simplified expression (13) in a simpler form.

The above expression for ${\cal G}$ is studied in Sect. III for different values of the various parameters and the corresponding bulk 