Membrane Tension Governs Particle Wrapping-Unwrapping Transitions and Stalling
Membrane wrapping controls nanoparticle uptake during endocytosis, whereas the reverse process of membrane unwrapping accompanies particle expulsion and membrane fusion events. Existing theoretical descriptions typically focus on adhesion and bending energies within the particle membrane contact region and often neglect the deformation energy of the membrane outside the contact zone. This approximation is valid only in the limit of vanishing membrane tension, where the non contact membrane assumes a catenoid like configuration with negligible bending energy. However, at finite tension the deformation of the non contact membrane becomes a dominant energetic contribution. Here we show that this tension dependent non contact energy governs the progression of particle wrapping. By analysing the variation of the total membrane energy with wrapping degree, we uncover a competition between particle adhesion, membrane tension and particle size that determines whether wrapping proceeds, stalls, or reverses into spontaneous unwrapping. This framework reveals a stalling boundary separating regimes of particle uptake and expulsion. To capture the non contact deformation efficiently, we derive a compact phenomenological approximation that accurately reproduces the full numerical solution of the membrane shape. The resulting energetic map provides a unified physical description of particle wrapping and unwrapping, with implications for endocytosis, membrane fusion, and nanoparticle design.