Demixing of homogeneous binary lipid membranes induced by protein inclusions

We study a model of a lipid bilayer membrane described by two order parameters: the chemical composition described using the Gaussian model and the spatial configuration described with the elastic deformation model of a membrane with a finite thickness or, equivalently, for an adherent membrane. We assume and explain on physical grounds the linear coupling between the two order parameters. Using the exact solution, we calculate the correlation functions and order parameter profiles. We also study the domains that form around inclusions on the membrane. We propose and compare six distinct ways to quantify the size of such domains. Despite its simplicity, the model has many interesting features like the Fisher-Widom line and two distinct critical regions.

Figure 1

Schematic plot of the two possible applications of the model. Different types of lipids are denoted with blue and yellow. (a) The membrane is flat, but its thickness is not constant and described by the order parameter $\overline{h}\left(\mathit{r}\right)$. Due to hydrophobic mismatch of the anchor (difference between the equilibrium thickness of the membrane and size of the hydrophobic region on the anchor) the thickness of the membrane is alerted around the anchor, which modifies local composition of the lipids. (b) The membrane has a constant thickness, but it is not flat, and its height over the reference plane is described by the order parameter $\overline{h}\left(\mathit{r}\right)$. In this case the anchor (illustrated in the middle of the picture) is locally fixing the height of the membrane, and, as a result, the membrane is bent. The external potential that acts differently on different types of lipids changes the composition of the membrane around the anchor.

Figure 2

Plot of the three zones in the space of parameters defined by the properties of roots of the polynomial $W(z;\tau ,\kappa {\mu }^2)$; see Eq. (11). The behavior of the correlation functions for large $\rho$ is different in each zone. The blue dots denote the values of parameters used in Fig. 3, and the black square marks the point where all three zones meet $({\omega }^{*},{\tau }^{*})=(8/\left(3\sqrt{3}\right),1/\left(6\sqrt{3}\right))$.

Figure 3

Example plots of the correlation functions ${\mathfrak{C}}_{hh}$, ${\mathfrak{C}}_{h\varphi }$, and ${\mathfrak{C}}_{\varphi \varphi }$, and their amplitudes in different zones as functions of the distance $\rho$. (a) $\kappa =10$, $\tau =0.02$, and $\mu =0.5$ (zone I), (b) $\kappa =10$, $\tau =0.15$, and $\mu =0.2$ (zone II), (c) $\kappa =10$, $\tau =0.30$, and $\mu =0.5$ (zone III). The values of parameters used here have been marked with blue dots in Fig. 2. In each panel the plot on the left presents the correlation functions, and the plot on the right their amplitudes [see Eq. (16)]. The behavior of the amplitudes is different in each zone.

Figure 4

Correlation length $\xi$ as a function of $\tau$ for different values of $\omega$. (a) For $\omega <{\omega }^{*}$ the minimum gets deeper upon increasing $\omega$; (b) for $\omega >{\omega }^{*}$ the value in the minimum increases upon increasing $\omega$ and, eventually, disappears for $\omega ={\omega }^{†}$. The dashed lines denote the asymptotic behavior of the correlation length (which is independent of $\omega$): ${\left(2\tau \right)}^{-1/2}$ for small $\tau$, and $\sqrt{2}$ for large $\tau$. The kinks, visible for all the curves, are associated with crossing the border between two different zones: in (a) from zone II for small $\tau$ to zone III for large $\tau$, and in (b) from zone I to zone III. Crossing from zone I to zone II is not associated with any nonanalyticity of $\xi$, and for the selected values of $\omega$ it is located beyond the range of the plot.

Figure 5

Plots of the effective radii ${\mathfrak{R}}_{\mathrm{\Gamma }}$, ${\mathfrak{R}}_V$, ${\mathfrak{R}}_{\text{infl}}^{\varphi }$, ${\mathfrak{R}}_{\text{infl}}^h$, ${\mathfrak{R}}_{1/2}^{\varphi }$, and ${\mathfrak{R}}_{1/2}^h$ describing the size of the domain induced by the defect. (a), (b) Plot of six radii as functions of $\tau$ for $\kappa =10$ and $\mu =0.4$. (c) Plot of the radii as a function of $\mu$ for fixed $\kappa =10$ and $\tau =0.15$. In all panels, the dashed lines denote the asymptotic behavior for $\tau$, $\mu \to 0$, $\infty$, and the dots for $\tau =0$ and $\mu =0$ denote the finite limits of the radii; cf. Table 2.

Figure 6

Schematic plot presenting how the three zones are changed by the surface tension $\eta$ of the membrane. (a) $\vartheta =\eta /{\left(\kappa \gamma \right)}^{1/2}=0$, (b) $\vartheta =0.2$, (c) $\vartheta =0.5$, (d) $\vartheta =1.0$, (e) $\vartheta =1.5$, and (f) $\vartheta =2$. On every plot the black square marks the point where all three zones meet. The color code used to denote the zones is the same in every panel.

Figure 7

Parameters $t_1$, $t_2$, $t_3$, and $a$ that describe the complex roots of the polynomial $W(z;\omega ,\tau )$ as a function of $\tau$ for fixed (a) $\omega =1.0$, and (b) $\omega =1.9$. The zones on the plane of parameters have been marked on top of the graphs, and gray vertical lines denote the borders between zones (see Fig. 2).