of the protein distribution vanishes, i.e., $g_{\phi}=0$ and $\xi_{\phi}=0$. Then, one has $b=0$ and the solution of (26) is trivially given by $\Phi(u)=h(u)$. This leads to the protein distribution $$\Phi^0(u)  = \left\{ \begin{array}{lll}
        h(u) & \mbox{for }     & h(u) \geq 0  \\
        0    & \mbox{for }  & h(u) < 0 \\
                \end{array} \right.$$ where the restriction to a finite range in $u$ follows since $\Phi^0(u)$ has to be positive. In fact, for $b=0$, only for $h(u) \geq 0$ the protein distribution is correctly described by the differential equation (26); inspection of the free energy density ${\cal I}$ in the limit $\alpha \rightarrow \infty$ shows that the value of $\Phi(u)$ which minimizes ${\cal I}$ for $h(u)<0$ is given by $\Phi(u)=0$. This failure of the variational methods used in deriving (26) is due to the fact that one requires $\Phi(u)$ to be positive in the limit of very large proteins, $\alpha \rightarrow \infty$.

Hereafter, we choose $c\geq 0$ with no loss of generality, since the problem defined by (26) and (27) is symmetric under a simultaneous inversion of $c$ and $u$ ($c\rightarrow -c$ and $u \rightarrow -u$). Using the asymptotic behavior of $h(u)$, $$h(u)    =\left\{ \begin{array}{lll}
         a-1+c & \mbox{for }     & u 