Abstract

Cellular processes such as endocytosis, exocytosis, and organelle biogenesis fundamentally rely on the precise control of membrane curvature. This study investigates the **thermodynamics of a lipid membrane with curvature** to elucidate the underlying physical principles that govern these critical biological events. Employing a combination of theoretical modeling, specifically a modified Helfrich-type Hamiltonian, and coarse-grained molecular dynamics simulations, we quantified the free energy cost associated with various degrees of membrane deformation. Our primary objective was to determine how key thermodynamic parameters, including the bending modulus ($K_c$) and spontaneous curvature ($C_0$), are influenced by local curvature and lipid composition. The key finding is the demonstration of a significant, non-linear coupling between the local curvature and the effective bending modulus, suggesting that the membrane's mechanical resistance is not constant but dynamically adapts to the degree of deformation. Furthermore, the introduction of specific lipid species, such as lysophospholipids, was shown to dramatically lower the energy barrier for high-curvature states. These results provide a crucial quantitative framework for understanding how cells regulate membrane shape and stability, offering new insights into the biophysics of membrane remodeling and the function of curvature-sensing proteins.