eMB: Mathematical analyses of multidimensional single cell transcriptional vector fields

细胞可以存在于不同的类型，如肝细胞和神经元。一个细胞如何发育成由不同细胞类型组成的生物体是生物学中的一个基本问题。此外，可以诱导细胞从一种类型（例如，皮肤细胞）转变为另一种类型（例如，神经元或心肌细胞）用于治疗各种疾病，如阿尔茨海默病。单细胞技术和机器学习算法的当前进展导致数据驱动的数学方程的发展，以描述大量基因如何相互调节以调节细胞类型。资助的研究是应用最初在其他领域开发的数学工具来分析这些方程。这种数学分析将为理解基因如何协调调节细胞类型提供机制见解，并指导实验设计，以有效地加速或减缓不同细胞类型之间的转换，这具有很高的生物医学意义。该项目还促进了数学家和细胞生物学家之间的合作，并培养下一代跨学科研究的学生和研究人员。随着单细胞基因组学技术的发展，一个新兴的方向是从数据中学习动力学模型。该研究旨在解决一个新的挑战，即扩展动力学方程下游分析的工具，并将其他背景下开发的工具应用于单细胞数据分析。在数学生物学中，研究人员有工具箱来分析由一组动力学方程描述的生物系统的行为。这些工具，如相平面分析和分叉分析，通常被限制为具有几个自由度或几个选定的参数的系统。因此，需要新的工具来分析具有大量（例如， 20)of degrees度of freedom自由.研究人员将使用最近开发的矢量场的离散图表示以及离散霍奇分解和其他网络分析方法来表征多维矢量场和系统动力学，包括发育和细胞命运转变过程的循环转变。研究人员还将研究非线性动力系统，包括动力学模式分解，Koopman算子分析，伪谱分析和谱子流形分析，并分析吸引子内和吸引子之间的细胞动力学集体动力学模式。该奖项反映了NSF的法定使命，并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而被认为值得支持。

Cells can exist in different types such as liver cells and neurons. It is a fundamental question in biology how a cell can develop into an organism composed of different cell types. Furthermore, one can induce a cell to change from one type (e.g., skin cells) to another type (e.g., neurons or heart muscle cells) for curing various diseases such as Alzheimer’s disease. Current advances of single cell techniques and machine learning algorithms lead to development of data-driven mathematical equations to describe how a large number of genes regulate each other to regulate the cell types. The funded research is to apply mathematical tools originally developed in other fields to analyze these equations. Such mathematical analyses will provide mechanistic insights on understanding how genes coordinately regulate cell types, and guide experimental designs to effectively accelerate or slow down transitions between different cell types, which are of high biomedical significance. The project also fosters collaborations between mathematicians and cell biologists as well as train students and researchers of the next generation in interdisciplinary research. With advances of single cell genomics techniques, an emerging direction is to learn dynamical models from the data. The research aims to address a new challenge to expand the tools for downstream analyses of the dynamical equations and apply tools developed in other contexts to single cell data analyses. In mathematical biology researchers have toolkits to analyze the behaviors of a biological system described by a set of dynamical equations. These tools, such as phase-plane analyses and bifurcation analyses, are typically restricted to either a system with a few degrees of freedom or a few selected parameters. Thus, new tools are needed for analyzing dynamical equations with a large number (e.g., 20) of degrees of freedom. The investigators will use the recently developed discrete graph representation of a vector field together with discrete Hodge decomposition and other network analysis methods for characterizing multi-dimensional vector field and system dynamics including cyclic transitions on developmental and cell fate transition processes. The investigators will also study nonlinear dynamical systems, including dynamical mode decomposition, Koopman operator analysis, pseudospectral analyses, and spectral submanifold analyses, and analyze the collective dynamical modes of cellular dynamics within and between attractors.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.