  Biomembranes play a central role in various phenomena like locomotion of
cells, cell-cell interactions, packaging of nutrients, and in maintaining
organelle morphology and functionality. During these processes, the membranes
undergo significant morphological changes through deformation, scission, and
fusion. Modeling the underlying mechanics of such morphological changes has
traditionally relied on reduced order axisymmetric representations of membrane
geometry and deformation. Axisymmetric representations, while robust and
extensively deployed, suffer from their inability to model symmetry breaking
deformations and structural bifurcations. To address this limitation, a 3D
computational mechanics framework for high fidelity modeling of biomembrane
deformation is presented. The proposed framework brings together Kirchhoff-Love
thin-shell kinematics, Helfrich-energy based mechanics, and state-of-the-art
numerical techniques for modeling deformation of surface geometries. Lipid
bilayers are represented as spline-based surfaces immersed in a 3D space; this
enables modeling of a wide spectrum of membrane geometries, boundary
conditions, and deformations that are physically admissible in a 3D space. The
mathematical basis of the framework and its numerical machinery are presented,
and their utility is demonstrated by modeling 3 classical, yet non-trivial,
membrane problems: formation of tubular shapes and their lateral constriction,
Piezo1-induced membrane footprint generation and gating response, and the
budding of membranes by protein coats during endocytosis. For each problem, the
full 3D membrane deformation is captured, potential symmetry-breaking
deformation paths identified, and various case studies of boundary and load
conditions are presented. Using the endocytic vesicle budding as a case study,
we also present a "phase diagram" for its symmetric and broken-symmetry states.
