Interfacial Mechanics of Cell Membranes: Stochastic Exterior Calculus Approaches for Curved Fluid Lipid-Protein Bilayers

数学方法在生物科学中发挥着越来越重要的作用。这包括开发新的模型和理论，以提供对生物过程的见解，开发新的分析工具和软件，用于实验设计和实验数据分析。在细胞生物学中，脂质双层膜可以被认为是一种有效的二维材料，由脂质、蛋白质和其他小分子的异质混合物组成。这些分子的个体和集体动力学被微调，以执行复杂的细胞过程，从信号传导到突触传递到细胞器形状的调节。细胞膜有效的二维流体弹性特性产生了影响分子相互作用和动力学的界面现象和复杂的几何形状，这与它们的体积三维对应物非常不同。为了更深入地了解弯曲的蛋白质-脂质双分子层中的细胞过程，需要新的方法来结合几何所起的重要作用。本研究开发了新的数学方法和软件，通过实验室实验和膜蛋白系统的计算模拟来处理复杂的几何形状。该项目的教育活动将进一步加强对本科生和研究生在生物科学新的定量方法方面的培训。该项目的研究将开发新的数学方法，以捕获由主动集体运动和被动热波动引起的弯曲膜内蛋白质的扩散运输，曲面的流体力学和粘弹性变形，以及包含随机动力学的包涵体之间的离散非均质相互作用。数学方法将扩展到连续介质力学和粗粒度分子描述，以便深入了解双层膜内包裹体的涌现力学和动力学。为了进行计算模拟，将设计一类基于外部微积分的一般数值方法来解决与求解弯曲流形上的微分方程相关的挑战。该项目的研究为研究膜蛋白系统提供了新一代的理论模型和分析工具。这项工作的社会影响来自于对细胞生物学中膜蛋白系统的理解的基本贡献，这可能会对膜相关技术的相关过程有深入的了解，例如通过渗透给药或膜干扰物等抗菌剂的设计。

Mathematical approaches are playing an increasingly important role in the biological sciences. This ranges from the development of new models and theories that provide insights into biological processes, to the development of new analytic tools and software that are useful in experimental design and in the analysis of experimental data. In cell biology, lipid bilayer membranes can be thought of as effectively two-dimensional materials comprising of a heterogeneous mixture of lipids, proteins, and other small molecules. The individual and collective dynamics of these molecules are fine-tuned to carry out complex cellular processes ranging from signaling to synaptic transmission to the regulation of shapes of organelles. The effective two-dimensional fluid-elastic nature of cell membranes yields interfacial phenomena and complicated geometric shapes effecting both molecular interactions and dynamics that can be very distinct from their bulk three dimensional counterparts. To gain a deeper understanding of cellular processes in curved protein-lipid bilayers, there is a need for new approaches for incorporating the important roles played by geometry. This research develops new mathematical approaches and software for handling complicated geometries in investigations conducted through laboratory experiments and computational simulations of membrane-protein systems. The educational activities of the project will further enhance the training of both undergraduate and graduate students in new quantitative approaches to the biological sciences. The research of this project will develop new mathematical approaches to capture diffusive transport of proteins within curved membranes arising from both active collective motions and passive thermal fluctuations, hydrodynamics and viscoelastic deformations of curved surfaces, and discrete heterogeneous interactions between inclusions incorporating stochastic kinetics. Mathematical approaches will be extended for both continuum mechanics and coarse-grained molecular descriptions so as to provide insights into the emergent mechanics and dynamics of inclusions within bilayer membranes. To perform computational simulations, a general class of numerical methods based on exterior calculus will be designed to address the challenges associated with solving differential equations on curved manifolds. The research of this project advances the next generation of theoretical models and analysis tools for investigating membrane-protein systems. Societal impacts of the work derive from fundamental contributions to the understanding of membrane-protein systems in cell biology, which may give insights into processes relevant to membrane-related technologies such as drug delivery through permeation or design of antimicrobials such as membrane disruptors.