Equilibrium shapes of tubular lipid membranes

Tubular vesicles represent abundant structural motifs which are observed both in experiments and in nature. We analyse them within the theory of bending elasticity and determine the equilibrium solutions at fixed volume, surface area, and segment length without imposing any specific symmetry or periodicity. We identify four different non-periodic equilibrium shapes. Depending on the precise value of the constraints or the corresponding Lagrange multipliers, these four shapes include: (i) snake-like and (ii) helical structures, (iii) tubes with a spherical body, and (iv) tubes with a discoidal body. However different in the details, all of the shapes have the same general cylindrical morphology which is either globally modulated or is a superposition of an additional structural motif and the cylinder. These results point to a great significance of the circular cylindrical shape and offer a comprehensive and general analysis of the shape of tubular vesicles.