Fig. 2: Theoretical and computational modeling.
From: Dynamic mechanochemical feedback between curved membranes and BAR protein self-organization
a Schematic diagram of a BAR domain interacting with a lipid membrane. Protein elastic energy depends on surface curvature and protein orientation. For cylindrical surface, curvature is maximal (dark green) and minimal (light green) along perpendicular directions. b–e Landscape of free-energy density per unit area Fprot according to our mean-field density functional theory (Section 4.1 of Supplementary Note 1) depending on protein coverage Φ, nematic alignment S, and the shape and size of the underlying membrane (sphere or cylinder as illustrated on top of each plot, where we have generated microscopic realizations of molecular organization consistent with coverage and orientational order of the mean-field theory using a Monte Carlo algorithm). Red dots denote states of equilibrium alignments S for a given protein coverage \(\phi\), i.e., minimizers of the free energy along vertical profiles, depicting the transition from isotropic (i) to nematic phase (ii–iii). The white region in the energy landscape is forbidden due to steric protein interactions. b, c Discontinuous transitions for protein alignment on isotropically curved membranes. d, e Continuous transitions for anisotropically curved membrane. The intrinsic protein radius of curvature is \(\frac{1}{\begin{array}{c}\bar{C}\\ \,\end{array}}=15\) nm (see Supplementary Note 1 for other model parameters). f Free-energy density profiles for spheres and cylinders of different sizes along the equilibrium paths. The chemical potential of proteins is the slope of these curves. All points marked with red circles have the same chemical potential at the tangent points µb and hence are in chemical equilibrium. g Membrane protrusions obtained by lateral compression of an adhered membrane patch of radius R0 interacting with a substrate with a potential U(z) and for various amounts of enclosed volume V0, see Supplementary Note 1. h Schematic of reshaping dynamics involving membrane relaxation, and protein binding, diffusion, and ordering.
