The interplay of proteins and curvature of lipid bilayers is well-known to regulate cell morphology and a variety of cellular functions, such as traf.cking 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 .ssion. For example, the membrane remodeling during endocytosis is a consequence of the interplay between the elastic membrane and concerted actions of highly specialized membrane proteins that can both sense and create membrane curvature. In case of the clathrin-mediated endocytosis more than 40 different proteins are involved, many of which are only transiently recruited to the plasma membrane.
A major difficulty in computer simulations of these fully coupled processes lies in the vastly different scales involved that make fully atomistic models intractible, but also exclude pure continuum modeling, because some (but not all) proteins may occur in low copy numbers or involve genuinely molecular effects, such as conformational changes or structural changes of the membrane (fusion, pore formation). As a consequence, mathematical modeling and associated numerical algorithms have to bridge spatial scales from 5 nm (size of proteins, membrane thickness) to 1000 nm (vesicles, functional membrane domains) and copy numbers ranging from about 10 to more than 1000 particles.
The goal of the project is to derive a hierarchy of discrete-continuum models of protein-membrane interactions and corresponding numerical algorithms, providing a seamless transition from molecular to macroscopic scales. In particular, we aim at derivations and multiscale extensions of existing purely phenomenological macroscopic models.
As a root model, we will select a continuum-mechanical description of membrane interacting with rigid particles of finite size. This kind of hybrid approach is well-established among physicists and biochemists. We will start with the mathematical and numerical analysis of hybrid models in the Monge gauge that is suited to studying the case of small deformations. In particular, we will investigate the transition from .nite-size to point-like particle descriptions. Utilizing numerical and analytical upscaling techniques, we will also derive multiscale hybrid models incorporating discrete and continuous representations of different types of particles with strongly differing copy numbers. In the next step, we plan to give up the Monge gauge and consider the fully nonlinear case involving the coupling of discrete particles and pdes on surfaces. A longterm goal is to study time-dependent processes.
Gräser, C. and Kies, T. (2018) Discretization error estimates for penalty formulations of a linearized Canham-Helfrich-type energy. IMA Journal of Numerical Analysis, drx071 . ISSN 0272-4979
Djurdjevac, A. and Elliott, C.M. and Kornhuber, R. and Ranner, Th. (2017) Evolving surface finite element methods for random advection-diffusion equations. arXiv:1702.07290 . pp. 1-25. (Submitted)
Elliott, C.M. and Gräser, C. and Hobbs, G. and Kornhuber, R. and Wolf, M.W. (2016) A Variational Approach to Particles in Lipid Membranes. Archive for Rational Mechanics and Analysis, 222 (2). pp. 1011-1075. ISSN Print: 0003-9527; Online:1432-0673
Djurdjevac, A. (2015) Advection-diffusion equations with random coefficients on evolving hypersurfaces. Interfaces and Free Boundaries . (Submitted)