Equilibrium microphase separation in the two-leaflet model of lipid membranes

Because of the coupling between local lipid composition and the thickness of the membrane, microphase separation in two-component lipid membranes can take place; such effects may underlie the formation of equilibrium nanoscale rafts. Using a kinetic description, this phenomenon is analytically and numerically investigated. The phase diagram is constructed through the stability analysis for linearized kinetic equations, and conditions for microphase separation are discussed. Simulations of the full kinetic model reveal the development of equilibrium membrane nanostructures with various morphologies from the initial uniform state.

Figure 1

(Top) Opposite spontaneous curvatures in two membrane leaflets. (Bottom) Local height and thickness variables that describe the shape of a two-leaflet membrane.

Figure 2

The phase diagram in the plane $\left(\chi ,\alpha \right)$ for $\kappa =20,c_0=0.2$, and $\gamma =1$. Region I: no phase separation; region II: microphase separation; region III: coexistence of micro- and macrophase separation; region IV: macroscopic phase separation (spinodal decomposition). (Inset) The wave number of the critical mode of the Turing bifurcation as a function of parameter $\alpha$.

Figure 3

Increments of growth $\omega$ as functions of the wave number $q$ at different values of parameter $\chi$ for (a) $\alpha =1.5$ and (b) $\alpha =1.7$. The choices of $\chi$ correspond to the vertical dashed lines in Fig. 2. (Inset) Enlargement of panel (b) for small wave numbers. Other parameters are $\kappa =20,c_0=0.2$, and $\gamma =1$.

Figure 4

(a–c) Examples of equilibrium patterns for membrane composition $\varphi$: (a) complete phase segregation (region IV, $\alpha =1.75,\chi =35,\overline{\varphi }=0$); (b) lamellae (region II, $\alpha =1.5,\chi =15,\overline{\varphi }=0$); (c) spots $\alpha =1.5,\chi =15,\overline{\varphi }=0.15$). Panel (d) shows profiles of composition order parameter (thick) and thickness variation (thin) in a one-dimensional simulation for $\alpha =1.5,\chi =15,\overline{\varphi }=0$. Other parameters are the same as in Fig. 1. The linear size of the system is 100 dimensionless length units corresponding to 200 nm.

Figure 5

Four subsequent snapshots (from left to right, ending at $t=5\times {10}^4{t}_u$) showing evolution of the local membrane composition inside region III $\left(\alpha =1.7,\chi =27,\overline{\varphi }=0\right)$. Other parameters and the size of the system are the same as in Fig. 4.