Quantifying the Energetics of Protein-Mediated Stabilization of Membrane Deformation

Membrane curvature plays a central role in a wide range of biological processes, yet a quantitative and transferable description of its underlying energetics remains challenging. In this work, we develop a "reaction coordinate" within the Umbrella-Sampling framework to compute the free-energy cost associated with local membrane bending in both coarse-grained and atomistic simulations. The methodology is systematically validated across diverse lipid environments, including pure bilayers with varying tail unsaturation, tail length, and headgroup chemistry, where it reproduces established trends in membrane-mechanical rigidity. Extension to mixed lipid systems further captures composition-dependent modulation of bending energetics, including cholesterol-induced responses. Finally, we apply the framework to membrane-active proteins with distinct curvature-generation mechanisms and demonstrate that the method successfully captures the protein modulation of the free-energy cost of local membrane deformation. Furthermore, the method is transferable to atomistic membrane systems while maintaining reasonable computational efficiency, providing a unified approach for quantifying membrane bending energetics across lipids, proteins, and simulation scales.