for moderately large macromolecules.

Euler-Lagrange equations
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In this section we calculate the protein concentration profile based on the free energy expression (15). Minimization of the line free energy $\gamma$ (14) leads to the Euler-Lagrange equations (denoting $\d \phi/\d x$ by $\phi^\prime$, etc.) $$\frac{\partial {\cal I}}{\partial \eta} - \frac{\d}{\d x} 
\frac{\partial {\cal I}}{\partial \eta^\prime}=0$$ $$\frac{\partial {\cal I}}{\partial \phi} - \frac{\d}{\d x} 
\frac{\partial {\cal I}}{\partial \phi^\prime}=0$$ Using the full free energy of mixing (7), one obtains two coupled second-order and non-linear differential equations of the form $$-\mu_{\eta}-(2J+1) \eta+\lambda \phi +
\frac{1}{2} \log \left( \frac{1+\eta-\phi}{1-\eta-\phi} \right) =
g_{\eta}\frac{\d^2 \eta}{\d x^2}$$ $$-\mu -\log 2 + 2L\phi +\lambda \eta +\frac{1}{2}
\log \left( \frac{4(1-\phi)^2}{(1+\eta-\phi)(1-\eta-\phi)} \right)+
\frac{1}{\alpha} \log \left( \frac{\phi}{1-\phi} \right) =
g_{\phi}\frac{\d^2 \phi}{\d x^2}$$ For the actual calculation of concentration profiles, we will use the simplified free energy expression (13), leading to the more compact expressions $$-\mu_{\eta}-2J\eta+\frac{1}{3}\eta^3 +\lambda \phi +\eta \phi=
g_{\eta}\frac{\d^2 \eta}{\d x^2}$$ $$-\mu + 2L\phi +\lambda \eta +\frac{1}{2}\eta^2 =
g_{\phi}\frac{\d^2 \phi}{\d x^2}$$ These are the same equations that were considered by Halperin and Pincus in the context of polymer adsorption at liquid-liquid interfaces[@Pincus].

Instead of solving (21)-(22) numerically, we recall that for large values of $L$ we can treat the protein area fraction 