Mini-Workshop: Mathematics of Biological Membranes

Abstract

From the early 1970's on, the understanding of the variety of equilibrium shapes of cell membranes has been a very active field in biophysics. Relaxation dynamics, phase separation on membranes, the coupling to inner/outer flows, and the contribution of domains on membranes to the biological function of cells are still a major focus of research.

The mathematical analysis of the shape and evolution of cell membranes is challenging. Over the last few years we have seen an increasing number of contributions from modelling, analysis and numerics. This workshop reports on recent progress made by the different communities – biophysics, numerics, analysis – and aims at a more complete picture of the relevant questions and adequate techniques.

Variational approaches for the description of equilibrium shapes of membranes are based on phenomenological bending energies, such as proposed by Canham [4] and Helfrich [7]. The Willmore functional is the most important reduction of the Helfrich energy, and understanding its properties is key in the analysis. Recent progress on the Willmore functional and its $L^2$-gradient flow by Kuwert–Schätzle [10, 9, 11] and Rivière [12] [Rivière], such as a new formulation of the Euler–Lagrange equation, provide promising techniques to investigate (constrained) minimization problems for different bending energies. (Here and below, names between square brackets refer to workshop presentations).

In many minimization or evolution problems approximations by phase-field models are convenient, since they can deal with topological changes of membranes and phases on the membrane respectively [2, 16, 8]. The connection between the Willmore energy and its phase field approximations are meanwhile better understood [13]. Emphasis in talks has been on thermodynamically consistent phase-field models [Misbah] as well as on their capability to recover singularities in the membrane shapes [Stinner,Rätz].

To find equilibrium shapes a $L^2$ gradient flow of an appropriate bending energy is the most practical approach. A discussion of more physical models for the relaxation towards an equilibrium has started. One choice is to take the flow of the lipid molecules on the membranes into account [Arroyo]. Other effects have to be considered as well, such as the orientation of molecules in ordered phases [Dolzmann].

Much progress in the numerical simulation of the Willmore flow, of partial differential equations on surfaces, and the coupling of reaction-diffusion processes or inner flows on evovling surfaces with their motion laws has been made [6, 5, 1]. For example, new weak formulations of the gradient flow for the Willmore functional avoid the evaluation of higher derivatives [Dziuk], and the inclusion of tangential motion in the numerical scheme leads to a convenient redistribution of the mesh [Garcke]. It has become feasible to compare equilibrium shapes and phase transition diagrams that have been determined either experimentally or by minimization within some restricted classes of shapes (for instance rotational symmetry) [14, 3] [Svetina].

Cell membranes consist of many different lipid molecules (and other components, such as rafts) and phase separation processes and evolution of phase boundaries are important for the function of a biological cell [Ben Amar]. Variational models account for different physical properties of the distinct phases (for instance different bending or spontaneous curvature constants) and a line tension energy at the boundary between the phases. First simulations of equilibrium configurations and relaxations have been obtained and are becoming more and more efficient [16, 15] [Stinner, Rätz].

The participants of this workshop came from 8 different countries and their expertise was evenly spread over the areas biophysics, analysis, numerics and modeling. On the first day there were three introductory talks on modeling, analysis and numerics respecvtively. On the following days there were more specialized talks, two in the morning and one in the afternoon. The schedule was kept flexible and allowed to have room for extensive and lively discussions which arose after virtually every talk. This was very much appreciated by all participants and was repeatedly reported as a significant advantage of the concept of a miniworkshop in comparison to the regular Oberwolfach meetings.