Determining the bending modulus of a lipid membrane by simulating buckling.

The force needed to buckle a thin elastic surface is proportional to its bending rigidity. This fact suggests using a buckling setup to measure the bending modulus of lipid membranes. Extending the work of Noguchi [Phys. Rev. E 83, 061919 (2011)], we systematically derive highly accurate analytical expressions for the forces along and perpendicular to the buckle, and we elucidate some of their counterintuitive properties using the framework of a surface stress tensor. Furthermore, we estimate the corrections to buckling forces due to thermal fluctuations and find them significant only for stresses along the ridges. We then apply this buckling protocol to four different lipid membrane models, which widely differ in their level of resolution and the treatment of solvent, and show that in all cases buckling is a reliable and accurate means for measuring their rigidity. Finally, we show that monitoring both stresses and energies during a simulation offers additional insights into the thermodynamics of curvature elasticity and permits one to predict the bending rigidity for a range of temperatures around the actual simulation temperature.