Topological and Geometric Modeling and Computation of Structures and Functions in Single-Cell Omics Data

不同的细胞相互作用以维持生物组织的功能。最近的单细胞技术以前所未有的分辨率和规模描绘组织，例如，数千个单个细胞中数千个基因的表达水平。从这些数据中提取生物学见解依赖于结构表示，例如如何描述细胞之间的相似性以及数据呈现的全局形状。虽然已经开发了许多方法来执行各种分析任务，但表示数据结构的初始步骤尚未得到充分研究。本项目将开发新的拓扑和几何方法，一种描述形状的形式语言，以研究和表征单细胞数据的结构。结构表征将与细胞功能联系起来，以揭示结构-功能关系。这些方法将被整合到现有的单细胞数据分析工具中，以提高生物学结论和预测的可靠性和鲁棒性。这些工具的应用将有助于识别携带关键功能的细胞以及这些细胞的特性。这些方法将作为公开提供的开放源码软件包加以实施。该研究将促进生物学家和数学家之间的跨学科合作，并对推进单细胞数据中的结构-功能关系感兴趣。该项目还将为学生和代表性不足的群体提供高等数学和现代生物数据分析的培训。许多单细胞数据分析工具依赖于降维的结构表示，观察结果可能对所使用的低维表示敏感。因此，需要对结构表示进行系统的探索，以控制下游分析结果的可靠性和可解释性。将开发基于应用拓扑学和几何学的方法，通过扫描广泛的尺度和参数，从高维单细胞组学数据中提取低维结构特征。将开发适应单细胞组学数据分析应用的方法，例如局部拓扑指纹和拓扑引导的最佳运输。将组装具有量化结构之间差异的明确定义的度量的单细胞数据集的结构表示的图谱，以提供表示单细胞组学数据的结构的系统方式。将在各种应用案例中介绍和评估在该结构图谱上应用下游分析工具的普遍适用的管道。系统的结构分析方法将与机器学习相结合，进一步解决两个重要问题：在单细胞数据集中建立结构-功能关系，例如基于数据集中的局部结构识别过渡细胞，以及基于拓扑和几何特征的单细胞多组学数据集的整合，特别是对于没有共享特征的数据集。高效，稳定，准确的数值方法和算法将开发这些数学问题的生物应用的动机。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Different cells interact to maintain the functions of biological tissues. Recent single-cell technologies profile a tissue with unprecedented resolution and scale, for example, expression levels of thousands of genes in thousands of individual cells. Extracting biological insights from this data relies on structural representations, such as how to describe similarities between cells and what global shape the data presents. While numerous methods have been developed to perform various analysis tasks, this initial step of representing the structure of data is understudied. This project will develop new topological and geometric methods, a formal language of describing shapes, to investigate and characterize the structure of single-cell data. The structural characterizations will be linked to cell functions to reveal structure-function relationships. These methods will be integrated into the large collection of existing analysis tools for single-cell data to improve the reliability and robustness of the biological conclusions and predictions. Application of these tools will help to identify cells carrying critical functions and the properties of these cells. The methods will be implemented as publicly available open-source software packages. The research will promote interdisciplinary collaborations between biologists and mathematicians with an interest in advancing the structure-function relationship in single-cell data. This project will also provide training for students and underrepresented groups at the interface of advanced mathematics and modern biological data analysis.Numerous single-cell data analysis tools rely on structural representations with reduced dimensions, and the observations could be sensitive to the low-dimensional representation used. A systematic exploration of structural representations is thus needed to control the reliability and interpretability of downstream analysis results. Methods based on applied topology and geometry will be developed to extract low-dimensional structural characteristics from the high-dimensional single-cell omics data by scanning a wide range of scales and parameters. Methods will be developed to adapt to the application of single-cell omics data analysis, for example, local topological fingerprints and topology-guided optimal transport. An atlas of structural representations for a single-cell dataset with well-defined metrics quantifying the difference between structures will be assembled to provide a systematic way of representing the structures of single-cell omics data. A generally applicable pipeline of applying downstream analysis tools upon this structure atlas will be introduced and evaluated in various application cases. The systematic structural analysis method will be combined with machine learning to further address two important questions: establishment of structure-function relationships in single-cell datasets, such as identifying transition cells based on their local structures in the dataset, and integration of single-cell multi-omics datasets based on topological and geometric characterizations, especially for datasets without shared features. Efficient, stable, and accurate numerical methods and algorithms will be developed for these mathematical questions motivated by biological applications. The tools will be implemented to be easily usable by both computational and experimental scientists.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.