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

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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.