Euler-Helfrich problem for open vesicles bounded by elastic filaments

We develop a theoretical framework to elucidate the mechanics and morphogenesis of open vesicles coupled to boundary filaments, capturing their geometric, energetic, and mechanical interplay. Using a spherical harmonics parametrization and constrained energy minimization, we systematically explore equilibrium morphologies as functions of filament length, stiffness, and topology, as well as membrane spontaneous curvature, uncovering a rich spectrum of shapes and transitions. For open vesicles bounded by flexible closed filaments, increasing filament length or spontaneous curvature drives transitions from cuplike to stomatocyte and budded morphologies, both axisymmetric and nonaxisymmetric, with continuous and discontinuous regimes mapped in a phase diagram. For open vesicles partially bounded by open filaments, the coupled filament-membrane system exhibits more complex morphological evolution, including free edge stretching or contraction, localized filament bending, and opening expansion with increasing filament length. The derived force and moment balances at the vesicle edge connect local geometric quantities, such as filament and membrane curvatures, to filament internal forces and membrane tension, providing a direct mechanical interpretation of shape equilibria. Our results reveal how filament geometry, topology, and bending stiffness govern membrane shape transformations and establish a general mechanical framework for coupled filament-membrane systems, such as DNA-vesicle complexes and other filament-mediated membrane structures.