Mixed lipid bilayers with locally varying spontaneous curvature and bending

Abstract

A model of lipid bilayers made of a mixture of two lipids with different average compositions on both leaflets, is developed. A Landau hamiltonian describing the lipid-lipid interactions on each leaflet, with two lipidic fields $\psi_1$ and $\psi_2$, is coupled to a Helfrich one, accounting for the membrane elasticity, via both a local spontaneous curvature, which varies as $C_0+C_1(\psi_1-\psi_2)/2$, and a bending modulus equal to $\kappa_0+\kappa_1(\psi_1+\psi_2)/2$. This model allows us to define curved patches as membrane domains where the asymmetry in composition, $\psi_1-\psi_2$, is large, and thick and stiff patches where $\psi_1+\psi_2$ is large. These thick patches are good candidates for being lipidic rafts, as observed in cell membranes, which are composed primarily of saturated lipids forming a liquid-ordered domain and are known to be thick and flat nano-domains. The lipid-lipid structure factors and correlation functions are computed for globally spherical membranes and planar ones. Phase diagrams are established, within a Gaussian approximation, showing the occurrence of two types of Structure Disordered phases, with correlations between either curved or thick patches, and an Ordered phase, corresponding to the divergence of the structure factor at a finite wave vector. The varying bending modulus plays a central role for curved membranes, where the driving force $\kappa_1C_0^2$ is balanced by the line tension, to form raft domains of size ranging from 10 to 100~nm. For planar membranes, raft domains emerge via the cross-correlation with curved domains. A global picture emerges from curvature-induced mechanisms, described in the literature for planar membranes, to coupled curvature- and bending-induced mechanisms in curved membranes forming a closed vesicle