Computational topology and geometry for systems biology

拟议的项目重点是创建新的数学工具，使用拓扑学、几何学和机器学习来分析生物学中的复杂数据集。在拓扑数据分析中心(TDA)取得成功的基础上，这一新倡议旨在建立和加强与德国萨克森州研究人员的合作，特别是与马克斯·普朗克科学数学研究所(MPI-MIS)和马克斯·普朗克分子细胞生物学和遗传学研究所(MPI-CBG)的合作。这些研究所与德累斯顿/莱比锡可扩展数据分析和人工智能中心(ScaDS.AI)和德累斯顿系统生物学中心(CSBD)密切合作。这些机构处于计算几何、机器学习和系统生物学前沿研究的前沿。该项目的主要目标是通过集成数据科学技术、代数和几何方法以及拓扑学来推进拓扑数据分析(TDA)。通过与MPI-CBG的实验者和模型师密切合作，该项目旨在推动TDA的边界并将其应用于生物系统，在现实世界应用程序和方法进步之间创造一个迭代循环。这个合作项目寻求发现生物数据中的形状和结构，最终导致对分子生物学的开创性见解。生物数据集通常是复杂、噪声和高维的。传统的方法，如聚类或回归，在捕捉数据的复杂形状方面存在局限性，并且无法识别更高阶的结构。TDA通过描述和量化多尺度系统的固有形状或结构，为理解多尺度系统提供了一种独特的方法。虽然TDA已经在医学上证明了它的有效性，包括在肿瘤免疫相互作用和血管网络中的应用--甚至导致了乳腺癌新亚型的发现--但这项提议的重点是扩大拓扑数据分析(TDA)的领域，以处理(空间)系统生物学中遇到的生物数据集。在TDA中扩展数学将提供一个通用的工具包，该工具包可以处理具有多个参数的各种数据。通过与马克斯·普朗克分子细胞生物学和遗传学研究所的实验者和模型师的密切合作，该项目将获得不同的生物数据集，使该团队能够推动TDA的理论、计算和实践边界。该计划的核心重点是将TDA扩展到其他数学和数据科学技术领域。这种多学科方法将在实际应用和方法进步之间建立一个迭代循环。通过与应用代数几何、微分几何和AI/TDA接口方面的领先研究人员的合作，该项目旨在开发新的理论框架、案例研究和软件。这些资源将展示拓扑和几何工具对数据分析的直接适用性。总而言之，该方案将有助于扩大英国的拓扑数据分析(TDA)社区，并为今后参与更大规模的项目铺平道路。拟议的研究项目旨在开发分析空间和时间多参数生物数据集的创新数学方法。通过利用拓扑学、几何学和机器学习的力量，该项目寻求解开机械论的洞察力，揭示生物系统中的结构，并彻底改变我们对生物学的理解。与国际研究中心的合作将最大限度地发挥影响。

The proposed project focuses on creating novel mathematical tools to analyse complex datasets in biology using topology, geometry, and machine learning. Building upon the success of the Centre for Topological Data Analysis (TDA), this new initiative aims to establish and strengthen collaborations with researchers in Saxony, Germany, specifically at the Max Planck Institute for Mathematics in the Sciences (MPI-MiS) and the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG). These institutes are closely associated with the Center for Scalable Data Analytics and Artificial Intelligence in Dresden/Leipzig (ScaDS.AI) and the Centre for Systems Biology Dresden (CSBD). These institutions are at the forefront of cutting-edge research in computational geometry, machine learning, and systems biology. The project's main objective is to advance topological data analysis (TDA) through the integration of data science techniques, algebraic and geometric methods, and topology. By working closely with experimentalists and modellers at MPI-CBG, the project aims to push the boundaries of TDA and apply it to biological systems, creating an iterative cycle between real-world applications and methodological advancements. This collaborative programme seeks to uncover shapes and structures within biological data, ultimately leading to groundbreaking insights in molecular biology.Biological datasets are often complex, noisy, and high-dimensional. Traditional methods, such as clustering or regression, have limitations when it comes to capturing the intricate shape of the data and cannot identify higher-order structures. TDA offers a unique approach to understanding multiscale systems by characterising and quantifying their inherent shape or structure. While TDA has already demonstrated its effectiveness in medicine, including applications in tumour-immune interactions and vascular networks--even led to the discovery of new subtypes of breast cancer-- the focus of this proposal is to expand the field of topological data analysis (TDA) to handle biological datasets encountered in (spatial) systems biology. Extending the mathematics in TDA will provide a versatile toolkit that can handle a wide range of data with multiple parameters. Through close collaboration with experimentalists and modellers at the Max Planck Institute of Molecular Cell Biology and Genetics, the project will have access to diverse biological datasets, enabling the team to push the theoretical, computational, and practical boundaries of TDA.The core focus of this programme is the expansion of TDA with other areas of mathematics and data science techniques. This multidisciplinary approach will create an iterative cycle between practical applications and methodological advancements. Through collaborations with leading researchers in applied algebraic geometry, differential geometry, and the AI/TDA interface, the project aims to develop new theoretical frameworks, case studies, and software. These resources will demonstrate the immediate applicability of topological and geometric tools for data analysis.In summary, the programme will contribute to the expansion of the UK topological data analysis (TDA) community and pave the way for future involvement in larger-scale projects. The proposed research project aims to develop innovative mathematical approaches for analysing spatial and temporal multi-parameter biological datasets. By harnessing the power of topology, geometry, and machine learning, the project seeks to unlock mechanistic insights and reveal structures within biological systems and revolutionise our understanding of biology. The collaboration with international research centres will maximise impact.