as a small parameter. As a zeroth-order approximation, we neglect the terms depending on $\phi$ in (21) and obtain as a solution the lipid order parameter profile $\eta_0(x)$ in the absence of proteins. This profile is then inserted into (22), yielding the protein profile $\phi(x)$. The validity of this approach, namely solving the equation (21) while neglecting the coupling between $\eta$ and $\phi$ and inserting the solution into equation (22), is critically examined in Appendix B. There, it is found that this approximation indeed corresponds to the first term in an expansion, in which the protein concentration functions as the expansion parameter and which therefore is valid for small protein concentrations.

To proceed, setting $\phi=0$ in (21) leads to $$\eta_0(x)=\eta_{\infty} \tanh (x/\xi_{\eta})$$ with the definitions $$\eta_{\infty} \equiv \sqrt{6J}$$ $$\xi_{\eta} \equiv \sqrt{g_{\eta}/J}$$ This is the solution of the usual 4th order Ginzburg-Landau free energy expansion and is strictly valid here only for the pure lipid. The lipid order parameter varies between $+\eta_\infty$ for $x\rightarrow \infty$ and $-\eta_\infty$ for $x\rightarrow -\infty$, and its width is characterized by the correlation length $\xi_\eta$. The chemical potential $\mu_{\eta}$ is zero in the approximation employed above. The origin is chosen as the symmetric point between the 