Geometric evolution towards the understanding of biomembranes

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Biological membranes are a mixture of many different types of lipids and protein components, and their relative amount and composition differ between functionally distinct domains. The strongly increasing interest in lipid membranes results from the hypothesized coupling of lipid phase segregation in the membrane to fundamental cell biological processes, such as membrane signaling and trafficing [1]. Sub-domains of distinct curvature may have precise biological properties [2], thus an understanding how lipid components can dynamically influence to membrane morphology is of utmost importance. Changes in lipid composition are assumed to assist or antagonize the membrane curvature on one side, but also might respond to the curvature by concentrating in domains of curvature that they prefer on the other side. Strong curvature variations have recently been observed experimentally in giant liposomes, where different lipids segregate according to their chemical properties and lead to the formation of buds [3, 4]. The strong coupling of phase separation and shape dynamics in lipid membranes has also been shown numerically by molecular dynamics [5] and Monte Carlo simulations [6]. Such atomistic simulations however are limited in the accessible length and time scales. With the curvature as one of the crucial ingredients to determine properties of membranes it seems natural to model the evolution within a continuum framework. This is further justified by the different length scales which come into play. The thickness of the membrane is in the nm-range, while a typical size of a biomembrane is in the µm-range. This length scale separation allows the biomembrane to be described as an elastic surface [7], which is the basis for our treatment. Within such a continuum description the observed budding in multicomponent lipid bilayers can be understood, by the possibility to reduce the line energy associated with the domain boundaries by budding these domains [8], an additional degree of freedom which is not present for phase separation processes in the bulk. A dynamic simulation of multicomponent biomembranes on a continuum level however is until now limited to small deformations or special shapes [9, 10, 11], which is due to the high-order nonlinear terms in the governing equations to describe the phase separation and domain formation on evolving surfaces. We propose to study the dynamics of the interactions between membrane structure, domain formation and shape deformation within a mathematical model for lipid bilayer biomembranes which will overcome this limitations. A thermodynamically consistent model will be derived, which mathematically leads to a higher order evolution equation on an evolving surface. We will consider various numerical approaches for such problems, including combined front-tracking and phase-field models, combined level-set and phasefield models and fully phase-field model to consider the evolution of the surface combined with the phase-separation on the surface. All approaches will use adaptive finite elements and multilevel techniques. Parallelization furthermore will allow to solve the highly nonlinear system in 3d in a reasonable amount of time and to answer questions concerning the long time behavior.