to three distinct phase regions and is shown in Fig. 1 for the values $\alpha=10$, $50$, and $200$. Anticipating the definitions (28) and (30), we present the results in terms of the rescaled variables $a \equiv \mu/(3J)$ and $c \equiv \lambda /\sqrt{3J/2}$. The results obtained for $\alpha=\infty$ are denoted by solid lines. In the region denoted “no proteins” both protein concentrations $\phi_1$ and $\phi_2$ are very small (exponentially in $-\alpha$); in the region “semi-localized” only one concentration is small while the other is finite (distinguished by the criterion described above), and in the region “delocalized” both phases have finite protein concentrations.

In the next section we will calculate the protein profile explicitly and, in addition, obtain a “localized” phase. This phase cannot be distinguished from the “no protein” phase by just looking at the bulk free energy. In fact, in this phase there is a finite protein concentration only at a finite distance from the boundary between the LE and LC regions. As one can see from Fig. 1, the phase boundary for $\alpha=50$ (long dashes) is already fairly close to the asymptotic boundary ($\alpha \rightarrow 
\infty$, solid line), so that neglecting the protein entropy is already a good approximation 