Geometric and Topological Modeling and Computation of Biomolecular Structure, Function, and Dynamics

21世纪生物科学的一个主要特征是其从定性和描述性向定量和分析性的转变。在过去的几十年里，对自组织生物分子系统（如阿尔茨海默病中的病毒、分子马达和蛋白质）的实验探索一直是科学发现和创新的主要推动力。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。阻碍当前生物分子系统定量理解的根本挑战是它们的巨大复杂性和过多的自由度。大多数生物过程发生在水里，水占人体细胞总量的65- 90%。人类蛋白质平均有大约5500个原子，加上周围的水分子，大约有10万个自由度。亚细胞细胞器和多蛋白复合物的维数显著增加。目前，利用全原子模型对人体细胞中的分子马达和/或病毒进行实时结构优化、动态模拟和功能预测是非常困难的。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学。由于异常庞大的数据集，该项目解决了自组织生物分子系统在结构、功能和动力学方面的重大挑战。这些挑战是通过引入新的数学模型和先进的计算方法来处理过大的生物分子数据集来解决的。该提案为海量数据分析、降维、计算数学和数学建模等重要领域提供了创新方法。该项目通过一系列几何和拓扑方法解决了上述挑战。首先，提出了一个多尺度框架，通过宏观连续描述水生环境和微观离散描述生物分子来降低维数和自由度。此外，引入基于持续稳定流形的自适应粗粒度方法，进一步降低超大生物分子系统的维数。引入总自由能泛函，使宏观表面张力和微观势相互作用相等。利用曲面的微分几何理论来描述宏观和微观领域之间的界面。势驱动的几何流被构造为最小化总自由能泛函。此外，还引入了进化拓扑和总曲率来分析生物分子的拓扑-功能关系。利用Frenet框架表征生物分子系统动力学数据中的局部几何和相关稳定流形。提出了一种提取稳定流形的机器学习算法。最后，引入微扰策略探讨了稳定流形的持久性，为粗粒度模型的可靠性提供了保证。除了有希望和广泛的初步结果说明了这种方法的力量之外，还提出了广泛的验证和应用，以确保所提出的方法为生物分子结构优化，功能预测和动态模拟提供强大而强大的工具。该项目由数学科学部和分子与细胞生物科学部共同资助。

A major feature of the biological science in the 21st century is its transition from qualitative and descriptive to quantitative and analytical. 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. Fundamental challenges that hinder the current quantitative understanding of biomolecular systems are their tremendous complexity and excessively large number of degrees of freedom. Most biological processes occur in water, which constitutes 65-90 percent human cell mass. An average human protein has about 5500 atoms, which, together with its surrounding water molecules, involve about 100,000 degrees of freedom. The dimensionality increases dramatically for subcellular organelles and multiprotein complexes. The real-time structure optimization, dynamic simulation, and function prediction 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. This project 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 mathematical models, together with advanced computational methods to deal with excessively large biomolecular data sets. This proposal offers innovative approaches to an important area in massive data analysis, dimensionality reduction, computational mathematics and mathematical modeling.The project addresses the aforementioned challenges by a number of geometric and topological approaches. First, a multiscale framework is proposed to reduce the dimensionality and number of degrees of freedom by a macroscopic continuum description of the aquatic environment, and a microscopic discrete description of biomolecules. Additionally, adaptive coarse-grained approach based on persistently stable manifolds is introduced to further reduce the dimensionality of excessively large biomolecular systems. 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. Furthermore, evolutionary topology and total curvature are introduced to analyze the topology-function relationship 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. Finally, perturbation 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 the proposed methodology yields robust and powerful tools for biomolecular structure optimization, function prediction and dynamical simulation.This project is funded by the Division of Mathematical Sciences with cofounding from the Division of Molecular and Cellular Biosciences.