Budding and domain shape transformations in mixed lipid films and bilayer membranes

We study the stability and shapes of domains with spontaneous curvature in fluid films and membranes, embedded in a surrounding membrane with zero spontaneous curvature. These domains can result from the inclusion of an impurity in a fluid membrane or from phase separation within the membrane. We show that for small but finite line and surface tensions and for finite spontaneous curvatures, an equilibrium phase of protruding circular domains is obtained at low impurity concentrations. At higher concentrations, we predict a transition from circular domains, or caplets, to stripes. In both cases, we calculate the shapes of these domains within the Monge representation for the membrane shape. With increasing line tension, we show numerically that there is a budding transformation from stable protruding circular domains to spherical buds. We calculate the full phase diagram and demonstrate two triple points of, respectively, bud-flat-caplet and flat-stripe-caplet coexistence.

Figure 1

(a) Flat symmetric phase-separated membrane and (b) curved, asymmetric phase-separated membrane.

Figure 2

(Color online) Schematic representation of phase-separated circular (top) and stripe (middle) domains and a bud (bottom) of the $B$ component, with induced deformations.

Figure 3

A sketch of the stripe and caplet morphologies. (a) shows a linear array of $B$-phase stripes of width $2x_0$ (black) in a matrix of phase $A$ (grey). (b) shows a hexagonal array of $B$-phase circular caplets of radius $r_0$ (black) in a matrix of phase $A$ (grey). The dashed lines in each figure show the appropriate Wigner-Seitz cell boundaries, while the membrane displacement profile of an appropriate unit cell cross section is shown above each figure.

Figure 4

Reduced domain size as a function of $\tau$ in the dilute limit $(\varphi \to 0,\widehat{\mu }\to 0)$. The solid lines denote the stable phase. Caplets are stable for $\tau \lesssim 0.2115$, and stripes are stable for $0.2115\lesssim \tau <0.25$. For $\tau >0.25$ the stable state is a macroscopically phase separated flat phase.

Figure 5

Chemical potential as a function of generalized tension for different frame tensions $\widehat{\sigma }$. Note the extrapolation to small $\widehat{\mu }$ with increasing $\tau$.

Figure 6

Phase diagram as a function of $B$-phase area fraction $\varphi$ and generalized reduced tension $\tau$, for different constant frame tensions $\widehat{\sigma }$ ($\widehat{\lambda }$ was varied to scan $\tau$). The region between upper and lower pairs of lines are two-phase regions of coexisting stripes and caplets. The inset shows convergence for large $\widehat{\sigma }$ near the small $\tau$ phase boundary. An equilibrium flat phase is predicted for $\tau >0.25$. Scans for different $\widehat{\sigma }$ converge to the results of Ref. 32, which corresponds to the limit $\widehat{\sigma }\to \infty$.

Figure 7

Phase boundaries ${\varphi }_{\text{caplet}}$ and ${\varphi }_{\text{stripe}}$ and stripe and caplet slopes $dh∕dx$ for fixed $\tau =0.001$ and varying $\widehat{\sigma }$.

Figure 8

Scaled domain sizes for caplet and stripe phases along the phase coexistence line, for both the concentrated and dilute limit calculations. For each geometry the two dimensions (stripe half-period ${\xi }_0$ and minority domain half width ${\Xi }_0$, and caplet inner radius ${\rho }_0$, and Wigner-Seitz radius $\overline{R}$, all scaled by $\widehat{\sigma }$) are shown. The variation in domain sizes with tension $\widehat{\sigma }$ can be seen to be very small.

Figure 9

Data collapse of the free energy per unit area (in units of $\kappa c_0^2$), domain size (units of $c_0$), and boundary slope as a function of $\tau$ for both stripes and caplets, for various values of the line tension $\widehat{\lambda }$.

Figure 10

The calculated phase diagram, indicating bud, caplet, and stripe phases, as well as the flat phase corresponding to macroscopic phase separation. This has been obtained by the numerical procedure outlined in Sec. 5. The solid lines indicate thermodynamic phase boundaries, while the dashed line indicates the limit of metastability of caplets. As discussed in the text, we expect that buds are metastable throughout the phase diagram, within the approximations used here. The thin region shown is that of the stripe phase. This vanishes at a triple point near $(\widehat{\sigma },\widehat{\lambda })=(1.0,0.25)$. The bud, caplet, and flat phases meet at the other triple point along $\widehat{\sigma }=0.5$. Upper and right-hand axes are dimensionless, while lower and left-hand axes are in physical units for typical values of the bending energy and spontaneous curvature, $\kappa =100\mathrm{pN}\mathrm{nm}$, $c_0^{-1}=100\mathrm{nm}$.