Determining the Lipid Tilt Modulus by Simulating Membrane Buckles

Lipid tilt affects the energy of membrane deformations on scales comparable to the membrane's thickness. The surface divergence of the tilt field creates a local spontaneous curvature, while tilt itself is quadratically penalized with a strength given by the tilt modulus. Traditionally, this modulus is determined by measuring the power spectrum of lipid orientation fluctuations. Here we present a novel approach which does not rely on fluctuations but instead exploits the fact that curvature gradients induce a tilt field. Its implementation extends a technique previously developed by us for localizing the position of the pivotal plane in buckling simulations, which quantifies the lipid imbalance across segments cut out from a complete buckle. Lipid tilt affects this count in a predictable way, and the signal can be quantified well enough to back out the tilt modulus at no additional cost and with about 5% precision for not too coarse models. We apply our technique to three lipid models of very different resolution: the highly coarse grained Cooke model, and two versions of DMPC, using both the (less highly coarse grained) MARTINI and the (united atom) Berger force field. For Cooke, we find an effective bilayer tilt modulus of 29 +/- 9 pN/nm, and for the less generic DMPC lipid, we find 115 +/- pN/nm for MARTINI and 39 pN/nm for Berger, both in reasonable agreement with existing studies for these models. We also show that the position of the pivotal plane for Berger DMPC lies just below the glycerol backbone unlike for MARTINI DMPC, where this plane lies closer to the middle of the lipid.