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。这包括对生物过程提供洞察的新模型和新理论的开发，到对实验设计和实验数据分析有用的新分析工具和软件的开发。在细胞生物学中，脂质双层膜可以被认为是由脂类、蛋白质和其他小分子的非均相混合物组成的有效的二维材料。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性质产生了界面现象和复杂的几何形状，影响分子相互作用和动力学，这可能与它们的主体三维对应物非常不同。为了更深入地了解弯曲的蛋白质-脂双层中的细胞过程，需要新的方法来结合几何学所扮演的重要角色。这项研究开发了新的数学方法和软件，用于处理通过实验室实验和膜-蛋白质系统的计算模拟进行的调查中的复杂几何图形。该项目的教育活动将进一步加强本科生和研究生在生物科学新的量化方法方面的培训。该项目的研究将开发新的数学方法来捕捉蛋白质在弯曲膜内的扩散传输，这些传输是由主动集体运动和被动热涨落、曲面的流体动力学和粘弹性变形以及包含随机动力学的包裹体之间的离散非均相相互作用引起的。数学方法将扩展到连续介质力学和粗粒度分子描述，以提供对双层膜中夹杂物的紧急力学和动力学的见解。为了进行计算模拟，将设计一类基于外部微积分的通用数值方法，以解决与求解曲线流形上的微分方程相关的挑战。该项目的研究为研究膜-蛋白质系统提供了新一代理论模型和分析工具。这项工作的社会影响源于对理解细胞生物学中的膜-蛋白质系统的基本贡献，这可能使人们深入了解与膜相关技术相关的过程，如通过渗透给药或设计抗菌剂，如膜干扰剂。

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.