Elasticity of Lipid Membrane Leaflets: Determining Pivotal Plane and Tilt Modulus in Computer Simulations

This thesis introduces a novel protocol to measure two elastic quantities of lipid membranes, the pivotal plane of a single leaflet and the lipid tilt modulus. The key setup relies on the simulation of a buckled membrane, a configuration that had been previously proposed to measure a membrane’s bending rigidity. Since this method solely depends on the geometrical analysis of the membrane shape, and can hence even be applied to existing simulation trajectories of buckled membranes, it can provide two additional physical quantities at no extra simulation cost. Due to the high precision of its result, many technical subtleties are considered throughout the derivation, pertaining for instance to higher order corrections due to Poisson ratio effects. This calculation, as shown in this thesis, can be used to provide both a reference for other study as well as deeper insight into the nature of lipid models, especially the limitation of the coarse-grained ones. After introducing the subject in the first chapter, the second chapter is mainly intended as a concise review of some key technical details for readers not familiar with the basic concepts in membrane and lipid structure, membrane elasticity theory and molecular dynamics simulations of membranes. Once the reader is prepared with the basic background knowledge, the third chapter dives into the first major topic of determining the pivotal plane position. This is a foundational step in the construction of well-known energy functionals for membranes (such as the Helfrich Hamiltonian), which describe the membrane as a continuous surface with no thickness. Among all the possible choices of a dividing surface, the pivotal plane, which is the location where the area strain vanishes, is one of the most common ones. We provide two measurement methods, both of which are based on monitoring the number imbalance across two opposing lipid layers. The first method is based on simple geometric configurations: cylinders and spheres. However, the process of lipid number equilibration needs a high flip-flop rate of the lipids between the two layers, which is difficult to achieve in simulation for many lipid models. The second method relies on a buckled membrane. This solves the equilibration problem, but requires more sophisticated lipid counting. The generic Cooke three bead model is amenable for both methods, and so it can serve as an excellent test case for the more complicated buckling approach. Agreement between the two methods hence shows that the latter one is indeed reliable. Following this proof of principle, two membrane models of higher resolution, MARTINI DMPC and Berger DMPC are investigated, and the positions of their pivotal planes are determined. The value obtained for MARTINI, z0 = 0.850(11)nm lies about 0.4 nm below the glycerol backbone, which is unexpectedly small. In contrast, the united atom Berger model’s value, z0 = 1.3225(44)nm, is much closer to the glycerol backbone, where most people believe the pivotal plane is. We attribute this discrepancy to a limitation of the coarse graining description, which makes the tail part of MARTINI DMPC less responsive to curvature changes. The last chapter illustrates that an extension of the analysis process for the buckle, with which the pivotal plane was determined, can be used to measure the tilt modulus, which describes the energetic cost for tilting a lipid’s orientation away from the local normal vector. The same three lipid models are investigated. For the Cooke model, a bilayer tilt modulus of 29 ± 9 pN/nm is found, while for MARTINI DMPC, 115 ± 6 pN/nm and for Berger DMPC, 39 ± 2 pN/nm are measured. The value found for the Berger model agrees extremely well with both previous simulation results and the most recent experimental measurement.