Using catastrophes, dynamics & data analysis to uncover how differentiating cells make decisions

景观模型由参数化的势函数族和黎曼度量组成。与此相关的动力系统由相应的梯度矢量场给出。任何只有静止点吸引子的莫尔斯-小动力系统和任何允许过滤的系统都允许这种表示，除非在吸引子和排斥子的小邻域中。这种景观模型在发育生物学中引起了极大的兴趣，因为它们与Waddington著名的表观遗传景观相对应，但也可以与相关遗传系统的网络模型紧密相关。当用来模拟细胞的动力学时，景观的参数对应于细胞接收到的信号。这可能是由于细胞环境中的形成因子或来自其他细胞的信号。当这些信号被改变时，环境就会发生变化，这就会导致分叉，从而破坏控制细胞状态的吸引子，从而导致细胞状态的变化。这就是细胞分化，细胞可以改变它们的细胞类型和规格。例如，干细胞以这种方式分化，最终为体内所有组织类型提供细胞。脊椎动物躯干的形成提供了一个重要的例子，说明发育组织中细胞命运的决定是如何由信号控制的基因调控网络做出的。我们的生物学合作者一直在研究其中的一部分，即利用这种多维单细胞数据研究小鼠胚胎干细胞向前神经或神经-中胚层祖细胞分化的时间过程。这些实验和相关的数学分析表明，这个系统的基础是一个高度非平凡的景观，其复杂性远远超过任何已发表的。这将是一个关键的探索性系统，我们将使用它来开发我们的想法，我们将与Briscoe和Warmflash实验室密切合作。然而，重要的是要强调，本提案的目的是强烈关注发展数学思想和工具，而不仅仅是嵌入到特定的生物学项目中。另一方面，获得最先进的数据是非常重要的。它确保了生物学上的相关性，并使用真实数据，而不是模拟数据，这提出了真正的数学挑战。越来越多的强大的生物工具可以用来研究这些过程，但是所产生的数据的数量和复杂性的增加，以及这些过程是由复杂系统执行的这一事实意味着需要新的数学工具来帮助理解正在发生的事情。特别是，生物学家现在可以在一次实验中测量成千上万个细胞中每个细胞中多个分子的数量。该项目的主要目的是增加我们对景观模型的理解，并将其与最先进的统计技术相结合，提供新的工具来分析这些数据，并利用这些数据来探索一些重要生物系统中细胞分化和细胞决策的机制。该项目涉及与生物实验室在数据和生物学理念方面的深度合作。这将是数据科学的一个很好的例子，因为它涉及信息学（生物信息学），统计学，数学（分析，几何和概率），hp计算和科学（生物学）。它提供了一种新的数据降维方法，这是数据科学的一个关键主题。

A landscape model consists of a parameterised family of potential functions together with a Riemannian metric. The dynamical system associated with this is given by the corresponding gradient vectorfield. Any Morse-Smale dynamical system with only rest point attractors and any system that admits a filtration admits such a representation except in a small neighbourhood of attractors and repellers. Such landscape models are of great interest in Developmental Biology because they correspond to Waddington's famous epigenetic landscapes but can also be rigorously associated with network models of the relevant genetic systems.When used to model the dynamics of a cell the parameters of the landscape correspond to signals being received by the cell. These can be due to morphogens in the cell's environment or signals coming from other cells. When these signal are altered, the landscape changes and this can cause bifurcations which destroy the attractor governing a cell's state and this can lead to a change in the cell's state. This is cellular differentiation, the way by which cell can change their cell type and specification. For example, stem cells differentiate in this way eventually to provide cells for all the tissue types in the body.The formation of the vertebrate trunk provides an important example of how cell fate decisions in developing tissues are made by signal controlled gene regulatory networks. Our biological collaborators have been studying part of this, namely the time course of differentiation of mouse embryonic stem cells to anterior neural or neural-mesodermal progenitors using such multidimensional single cell data. These experiments and the associated mathematical analysis has suggested that underlying this system is a highly non-trivial landscape of a complexity significantly greater than any published. This will be a key exploratory system that we will use to develop our ideas and we will work closely with the Briscoe and Warmflash labs to do this. However, it is important to stress that the purpose of this proposal is to focus strongly on developing mathematical ideas and tools and not just to be embedded in a particular biological project. On the other hand, access to state-of-the art data is very important. It ensures biological relevance and work with real data, rather than simulated data, raises real mathematical challenges.More and more powerful biological tools are becoming available to study such processes but the increasing amount and complexity of the data produced and the fact that the processes are carried out by complex systems means that new mathematical tools are need to help understand what is going on. In particular, biologists can now measure the numbers of multiple molecules in each of tens of thousands of cells in a single experiment.The key aim of this project is to increase our understanding of landscape models and combine this with state-of-the-art statistical techniques to provide new tools to analyse such data and to use it to probe the mechanisms of cellular differentiation and cellular decision-making in some important biological systems.The project involves deep collaboration with biological labs both in terms of data and biological ideas. It will be an excellent example of data science since it involves informatics (bioinformatics), statistics, mathematics (analysis, geometry & probability), hp computing and science (biology). It provides a new method of date dimension reduction a key theme in data science.