\rightarrow - \infty\\
         a-1-c& \mbox{for }     & u \rightarrow +\infty\\
                \end{array} \right.$$ the following classification emerges: (i) For $a\leq 1-c$, the protein distribution vanishes both for positive and negative values of $u$ at a sufficiently large but finite distance from the interface (which is located at $u=0$); one actually obtains a nonvanishing, [*localized*]{} distribution of proteins provided that $h(u)>0$ for some range of $u$, but this cannot be seen from the bulk behavior; (ii) for $1-c<a \leq 1+c$, the distribution is [*semi-localized*]{} and vanishes only for sufficiently large positive values of $u$ and stays finite as $u \rightarrow - \infty$, and (iii) for $a>1+c$ the distribution is [*delocalized*]{} and stays finite in both limits $u \rightarrow \pm \infty$. These three regimes are in accord with the phase diagram obtained in Sect. III and Fig. 1 for finite $\alpha$ and in the $\alpha \rightarrow \infty$ limit.

An additional observation can be made for $c \leq 2$, where $h(u)$ has one maximum located at $$u_{max}=-\tanh^{-1}(c/2)$$ with a height $$h(u_{max})=a+c^2/4$$ (in the limit $c \rightarrow 2$ one obtains $u_{max} \simeq \log(2-c)/2$). Consequently, for $c \leq 2$, the line defined by $a=-c^2/4$ marks the border between a fourth regime where the protein distribution vanishes 