The interplay of curvature and particles diffusing in biological membranes is responsible for organizing and shaping the membrane and
gives rise to a variety of cellular functions. Hybrid models combining a continuum representation of the membrane with discrete, highly coarse grained descriptions of particles have a long history in physics, while mathematical analysis is still in its infancy.
We present a hierarchy of variational formulations of existing hybrid models, where the coupling of particles and membrane is formulated in terms of linear constraints to the minimization of the Canham–Helfrich energy of the membrane.
Utilizing concepts from shape calculus, we derive a numerically feasible representation of the derivative of the minimal Canham–Helfrich energy for given particle positions with respect to the particle positions. This representation is applied in numerical investigations of the clustering behaviour of BAR domains and paves the way to Langevin dynamics of particles in membranes.
This is joint work with C.M. Elliott, C. Gräser and T. Kies