III: Medium: Geometric and topological approaches to biomolecular structure and dynamics

自组织生物分子系统的实验探索，如病毒，分子马达和阿尔茨海默病中的蛋白质，在过去几十年中一直是科学发现和创新的主导动力。不幸的是，对生物分子结构、功能和动力学的定量理解严重滞后于实验进展的步伐。人体中的平均蛋白质约有5500个原子，加上周围的水分子，涉及约10万个自由度。对于复杂的生物过程和生物分子系统，维数急剧增加。目前，全原子模型难以解决人体细胞中分子马达和病毒的真实的时间结构优化、动力学模拟和数据分析等问题。一个关键的问题是如何减少自由度的数量，同时保留复杂生物系统的基本物理。拟议的研究可能是变革性的。作为第一个基于微分几何的生物分子系统多尺度/多分辨率建模方法，它将在未来的研究中为其他大数据系统的多尺度建模开辟一个新的方向，并促进类似的方法。此外，新的持续稳定流形策略可以应用到其他领域，如图像处理，计算机辅助设计和流体力学。此外，所提出的新的耦合方程将导致新的研究课题的几何，拓扑，偏微分方程分析和数学生物学。最后，我们的新理论框架被直接集成到流行的软件包中，以确保整个数学，计算机科学和生物学领域的研究人员广泛使用。拟议的研究具有坚实的教育组成部分。该项目将支持学生和初级研究人员在数学建模、数据分析和算法开发方面的培训。计划从拟议的研究中加强课程，作为PI教学研究实践的延续。特别课程开发，推广计划和年度研讨会旨在进一步扩大教育和社会影响。拟议的研究解决了由于异常庞大的数据集而导致的自组织生物分子系统的结构，功能和动力学方面的巨大挑战。这些挑战是通过引入一个新的微分几何为基础的多尺度模型，连同一个多分辨率的粗粒度的方法，在分子动力学数据的基础上持续稳定的流形。这一建议为海量数据管理、降维、计算数学和数学建模等重要领域提供了创新的新方法。该项目使用了一些几何和拓扑方法来解决缩放问题。 首先，多学科团队将使用多尺度框架，通过对水环境的宏观连续描述和对生物分子的微观离散描述来降低维度和自由度。为了进一步降低过大的生物分子系统的维数，他们引入了一种基于分子动力学数据中持续稳定流形的多分辨率粗粒度方法。引入总自由能泛函，使宏观表面张力和微观势相互作用处于同等地位。利用曲面微分几何理论描述了宏观域和微观域之间的界面。势驱动的几何流构造，以最小化总自由能泛函。利用欧拉特征线和全曲率分析了生物分子的拓扑结构和相应的功能。Frenet标架被用来描述生物分子系统动力学数据中的局部几何和相关的稳定流形。提出了机器学习算法来提取稳定流形。在最后一步中，引入了一种策略来探索稳定流形的持久性，为粗粒度模型的可靠性提供了保证。除了有前途的和广泛的初步结果说明这种方法的力量，广泛的验证和应用已经提出，以确保这种方法产生强大的和强大的工具，生物分子结构优化和动态模拟。

Experimental exploration of self-organizing biomolecular systems, such as viruses, molecular motors and proteins in Alzheimer's disease, has been a dominating driving force in scientific discovery and innovation in the past few decades. Unfortunately, quantitative understanding of biomolecular structure, function, and dynamics severely lags behind the pace of the experimental progress. An average protein in human body has about 5500 atoms, which, together with its surrounding water molecules, involve about 100,000 degrees of freedom. The dimensionality increases dramatically for complex biological processes and biomolecular systems. The real time structure optimization, dynamic simulation, and data analysis of molecular motors and/or viruses in human cells are intractable with full-atom models at present. A crucial question is how to reduce the number of degrees of freedom, while retaining the fundamental physics in complex biological systems. The proposed research may be transformative. As the first differential geometry based multiscale/ multiresolution approach to biomolecular systems, it will open a new direction and foster similar approaches in multiscale modeling of other large data systems in future research. Additionally, new persistently stable manifold strategy can be applied to other fields, such as image processing, computer aided design, and fluid mechanics. Furthermore, the proposed new coupled equations will lead to new research topics in geometry, topology, PDE analysis and mathematical biology. Finally, our new theoretical framework is directly integrated into popular software packages to ensure extensive usage by the community of researchers throughout mathematics, computer science and biology. The proposed research has a solid educational component. The project will support the training of student and junior researchers in mathematical modeling, data analysis and algorithm development. The enhancement of curricula from the proposed research is planned as a continuation of PIs teaching-research practice. Special curriculum development, outreach program and annual workshops are designed to further broaden educational and societal impacts. The proposed research addresses grand challenges in the structure, function and dynamics of self-organizing biomolecular systems due to exceptionally massive data sets. These challenges are tackled through the introduction of a new differential geometry based multiscale model, together with a multiresolution coarse grained method based on persistently stable manifolds in molecular dynamics data. This proposal offers innovative new approaches to an important area in massive data management, dimensionality reduction, computational mathematics and mathematical modeling. This project uses a number of geometric and topological approaches to address the scaling issues.. First, the multidisciplinary team will use multiscale framework which reduces the dimensionality and number of degrees of freedom by a macroscopic continuum description of the aquatic environment, and a microscopic discrete description of biomolecules. To further reduce the dimensionality of excessively large biomolecular systems, they introduce a multiresolution coarse-grained approach based on persistently stable manifolds in molecular dynamics data. A total free energy functional is introduced to bring the macroscopic surface tension and microscopic potential interactions on an equal footing. The differential geometry theory of surfaces is utilized to describe the interface between macroscopic and microscopic domains. Potential driven geometric flows are constructed to minimize the total free energy functional. Euler characteristic and total curvature are employed to analyze the topology and corresponding function of biomolecules. Frenet frames are utilized to characterize the local geometry and associated stable manifolds in dynamical data of biomolecular systems. Machine learning algorithms are proposed to extract stable manifolds. In the last step, a strategy is introduced to explore the persistence of stable manifolds, which provides the assurance for the reliability of the coarse grained model. In addition to promising and extensive preliminary results illustrating the power of this approach, extensive validation and application have been proposed to ensure that this methodology yields robust and powerful tools for biomolecular structure optimization and dynamical simulation.