Determining the Pivotal Plane of Fluid Lipid Membranes in Simulations

Abstract

Each leaflet of a curved lipid membrane contains a surface at which the area strain vanishes, the so-called pivotal plane. Its distance z0 from the bilayer's mid-plane arises in numerous membrane structure related contexts, for instance the connection between monolayer and bilayer moduli, stress-profile moments, or area-difference elasticity theories. Experimentally, the pivotal plane is known to lie close to the glycerol backbone of a lipid, but only very few simulations have tried to locate it. Here we propose two precise methods for determining z0, both of which rely on monitoring the lipid imbalance across a curved bilayer. The first method considers the ratio of lipid number between the two leaflets of cylindrical or spherical vesicles and requires lipid flip-flop for equilibration. The second method looks at the leaflet difference across local sections cut out from a buckled membrane and is free of this limitation. Our simulations rely on two different coarse-grained lipid models. First, the generic three-bead solvent-free Cooke model, which is amenable to both methods and gives results for z0 that agree at the percent level. And second, a ten-bead representation of dimyristoylphosphocholine (DMPC) with the explicit solvent MARTINI model, which can only be analyzed with the buckling method. For the latter, the obtained value z0 = 0.850(11) nm lies about 0.4 nm inwards of the glycerol backbone, essentially in the middle of the leaflet, and is hence unexpectedly small. We attribute this to limitations of the coarse-grained description, suggesting that the location of the pivotal plane might be a novel indicator for how well lipid models capture the microscopic origins of curvature elasticity, especially the relative contributions of head and tail regions to the overall curvature modulus.