Scale transitions in models for multiphase biomembranes

• Applicants: Professor Dr. Patrick Dondl; Professor Dr. Matthias Röger
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 553092739

Biological membranes contain a variety of different components. The formation and maintenance of different phases is essential for many biological functions. Curvature-dependent shape energies such as the Willmore or Canham-Helfrich energies have led to significant advances in understanding the organizational principles of single- and multi-phase membranes. The mathematical analysis of such energies remains an active area of research in the calculus of variations, geometric measure theory and differential geometry, with challenging open questions, particularly in the multiphase case. Macroscopic shape energies are largely phenomenological and are based on a number of assumptions that cannot be justified within the theories. In particular, such descriptions are only well justified as long as the membrane curvature assumes values in the order of magnitude of the inverse membrane thickness. On the other hand, experiments show that this assumption does not have to be fulfilled, for example in the presence of several phases or during vesiculation processes. The aim of this proposal is a systematic and mathematically sound derivation of multiphase energies from models on finer scales and a rigorous mathematical analysis of the resulting limit energies. To do this, we consider models on a molecular or "mesoscopic" scale and derive asymptotic reductions using Gamma-convergence methods. In particular, we want to identify the scaling of different energy contributions (e.g. stretching, bending, phase separation, kinks) with characteristic parameters such as the membrane thickness. Such a characterization of the scaling of different energy contributions is particularly important when several small parameters are present, such as a small bending stiffness or a large strain. The overall goal of the project is to develop suitable mathematical methods that allow to characterize which specifics of fine-scale models induce a certain behavior of the resulting macroscopic reductions.

• DFG Programme: Research Grants