Geometric evolution towards the understanding of biomembranes

生物膜是多种不同类型的脂类和蛋白质组分的混合物，它们的相对含量和组成在不同的功能区域之间有所不同。对脂膜的强烈兴趣源于假设膜中的脂相分离与基本的细胞生物学过程的耦合，如膜信号和运输[1]。具有不同曲率的亚区可能具有精确的生物学特性[2]，因此了解脂类成分如何动态地影响膜的形态是至关重要的。脂质成分的变化被认为有助于或对抗一侧的膜曲率，但也可能通过集中在他们喜欢的另一侧的曲率域来对曲率作出反应。最近在实验中观察到了巨型脂质体中强烈的曲率变化，其中不同的脂类根据其化学性质分离并导致芽的形成[3，4]。分子动力学[5]和蒙特卡罗模拟[6]也显示了脂膜中相分离和形状动力学的强耦合。然而，这种原子模拟在可获得的长度和时间尺度上是有限的。由于曲率是决定膜性质的关键成分之一，在连续统框架内模拟进化似乎是很自然的。发挥作用的不同长度尺度进一步证明了这一点。膜的厚度在纳米范围内，而生物膜的典型尺寸在微米范围内。这种长度尺度的分离允许生物膜被描述为弹性表面[7]，这是我们治疗的基础。在这样的连续体描述中，可以通过通过萌发这些结构域来降低与结构域边界相关的线能量的可能性来理解观察到的多组分脂双层中的萌发[8]，这是整体相分离过程中不存在的附加自由度。然而，到目前为止，在连续介质水平上对多组分生物膜的动态模拟仅限于小变形或特殊形状[9，10，11]，这是由于控制方程中的高阶非线性项描述了演化表面上的相分离和磁区形成。我们建议在脂类双层生物膜的数学模型中研究膜结构、结构域形成和形状变形之间的相互作用的动力学，以克服这一局限性。一个热力学上一致的模型将被导出，它在数学上导致了一个在演化表面上的更高阶的演化方程。对于这类问题，我们将考虑各种数值方法，包括联合前沿跟踪和相场模型，联合水平集和相场模型，以及结合表面相分离来考虑表面演化和相分离的完全相场模型。所有方法都将使用自适应有限元和多层技术。此外，并行化还将允许在合理的时间内在3D中求解高度非线性的系统，并回答有关长时间行为的问题。

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.