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

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.

The main potential economic and societal impacts are in medical and health areas. Understanding the way in which cells assess the signals they perceive from their external and internal environments and the way they use this to change their state is absolutely critical to working out how to improve disease outcomes in patients. For example, cancer occurs when this decision-making goes wrong and mutated cells which should have committed suicide instead decide to proliferate. On a different note, if we can understand how this decision-making provides the mechanisms by which the body develops from a single cell and renews itself throughout life, we can hope to be be able to replace damaged tissues and help the body regenerate itself, potentially curing or easing the suffering of those afflicted by disorders like heart disease, Alzheimers, Parkinsons, diabetes, spinal cord injury and cancer. This is regenerative medicine.

Mathematical research is needed here for two main reasons. Firstly, cell fate decisions are made by networks of interacting genes and proteins reacting to and extra- and intra-cellular signals and that network determines a complex stochastic dynamical system that can only be properly understood with mathematics. Secondly, recent technological developments have led to potentially very powerful ways of obtaining data about how individual cells are working. However, this produces huge amounts of data and we need new statistical methods in order to extract the information from all the noise in this data. To develop the ways to achieve this we need more new mathematics.

This project introduces a new way to mathematically model the decision-making process that can deal with the difficulties inherent in the incompleteness of biological understand and the complexity of the underlying system. It uses advanced mathematics (singularity theory, dynamical systems) and statistics (rare-event stochastic simulation) to do this.

We expect that the mathematical methodologies that we will develop will have use in other areas of economic and societal impact. For example, the project will provide a new method of data dimension reduction, a key theme in data science where there is a need for methods that allow one to find the structure in high-dimensional noisy data. In the biological applications the dynamical systems that determine the temporal evolution of the data are defined by the interactions of genes and proteins. Although in other applications they will be due to completely different processes, they nevertheless will very often have the same mathematical properties (technically called Morse-Smale after two mathematicians who did fundamental work on them), and if this is the case then the same approach can be used to analyse them. Thus the project will produce a new methodology for model-led data analysis.