The interplay of proteins and curvature of lipid bilayers is well-known to regulate cell morphology and a variety of cellular functions, such as trafficking or signal detection. Here, interplay not only means that proteins can induce curvature by shaping and remodeling the membrane, but also that the membrane curvature plays an active role in creating functional membrane domains and organizing membrane proteins, including their conformation dynamics. Moreover, microscopic causes, such as hydrophobic mismatch of proteins and amphiphilic lipids, may have macroscopic effects, such as budding or fission.

Coarse grained molecular models explicitly describe proteins as well as lipid membrane constituents as discrete particles. Given the large-scale characteristic (both in space and time) of these fully coupled processes, corresponding molecular dynamics simulations are often prohibitive. At the macroscopic end of existing modelling concepts are fully continuum models, describing the membrane as a surface governed by Canham--Helfrich-type energy functionals and embedded particles by densities. By construction, these models are limited to very large particle counts. Against this background, so-called hybrid models, treating proteins as discrete particles coupled to a continuum description of the lipid membrane, have become a long-standing active field in physics and biology. At the heart of hybrid models is the representation of membrane-mediated particle interaction by suitable coupling conditions.

This project aims at extending well-established concepts of particle dynamics from Euclidean space to particles in and interacting with membranes by combining basic concepts from coarse grained molecular dynamics and statistical mechanics with analysis and numerical analysis of geometric and stochastic partial differential equations. A major goal is to derive, analyse, and numerically approximate Langevin dynamics of discrete particles in fluctuating membranes. This novel class of dynamical hybrid models could be directly linked to atomistic and coarse grained molecular dynamics and would directly lead to efficient computational methods for macroscopic quantities in terms of free energies, thus bridging the complete range of scales from molecular to macroscopic descriptions in a mathematically consistent way. In the long term perspective these results might also lead to a better understanding of existing fully continuum models.