liquid condensed phase ($x>0$) and the liquid expanded phase ($x<0$). Defining a rescaled length $u\equiv x/\xi_{\eta}$ and a rescaled protein density $\Phi(x) \equiv 4L \phi(x)/(\eta_{\infty})^2$, the second differential equation (22) is reduced to $$\Phi(u)-h(u)=b^2 \frac{\d^2 \Phi(u)}{\d u^2}$$ with the inhomogeneous term $h(u)$ given by $$h(u) \equiv a- \tanh^2(u) -c \tanh(u)$$ The remaining rescaled parameters are $$a \equiv \frac{2 \mu}{(\eta_{\infty})^2}
=\frac{\mu}{3J}$$ $$b^2 \equiv \frac{J g_{\phi}}{2L g_{\eta}}=
\frac{\xi_\phi^2}{ \xi_\eta^2}$$ $$c \equiv \frac{2 \lambda }{\eta_{\infty}}=
\frac{\lambda}{\sqrt{3J/2}}$$ The parameter $a \sim \mu$ is the rescaled chemical potential, $b$ is the relative stiffness of the lipid concentration profile compared to the protein concentration profile, and $c \sim \lambda$ measures the preference of the proteins for the dense ($c<0$) or dilute ($c>0$) lipid domains. The correlation length of the protein distribution is defined by $ \xi_\phi \equiv \sqrt{g_{\phi}/2L}$.

The general solution of the second order differential equation (26) can be written as $$\Phi(u)=A \sinh(u/b) +B \cosh(u/b)+a -\Phi_1(u)/b -c \Phi_2(u)/b$$ where the functions $\Phi_1(u)$ and $\Phi_2(u)$ are given in Appendix A. The constants $A$ and $B$ have to be determined in accord with the boundary conditions.

Protein distribution
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Solution for the case $b=0$
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It is instructive to treat first the limiting case where the stiffness 