Trajectory Inference approaches for multimodal SDEs with applications in developmental biology

The purpose of the work is to describe the mechanisms of cells differentiation in various biological organisms.
Originally, stem cells are not specialised and share very similar genetic characteristics. At the very first stages of development, these cells start dividing, they increase in number and at the same time become more specialised, starting to commit to a different fate. For example in mammals, after the first pluripotent state, cells can either become trophectoderms -which are cells forming the outer placenta- or they stay in the inner part of the egg and become part of the Inner Cellular Mass (ICM). While the trophectoderms will not change their type and characteristics anymore, in the ICM there is another binary decision that cells make as of becoming Epiblasts or Primitive Endoderm cells, etc... until forming all tissues and apparatuses in the organisms. Their choice of committing to a certain type of cell is driven by the genetic signals, that is, the expression or inhibition of certain genes which guide them in their choice. There is knowledge, in the biology community, of genes which are relevant for some developmental decisions but there might be other unknown genes involved which could help explain and predict the decision-making. There are various experiments to study developmental phenomena in embryos and the data that are collected are the gene expression profile for each cell in the study. Among current techniques there are: FACS: which allows to collect a limited number of genes (5-10) for hundreds of cells. Once the genes are measured, the cells are killed, so we don't track the same cells over time but rather carry out similar repeated experiments and measure the genes at different developmental time points. Single-cell RNA sequencing (scRNAseq): allows to collect hundreds to thousands of genes for a limited number of cells, over time. High-dimensional and noisy. My collaborators have data collected with both techniques and my supervisor has published papers on the proposed modelling approach applied to FACS data. However, the lab is trying to work and publish more using scRNAseq as this is the new cutting-edge technique in the field. There are also a number of published works from other groups both with scRNAseq and FACS data which are available and could be used to apply our techniques. Data challenge: there is currently a lot of interest towards the newest method (scRNAseq)because of the amount of data that can be retrieved. However, having so many genes means there is a lot of noise and confounding when trying to detect what really drives the decision-making. How to distinguish between relevant and irrelevant genes? What dimensionality reduction technique can be employed among several available ones?

Impact from the MathSys CDT will arise from three separate mechanisms, each of which will generate a spectrum of academic, economic and societal impacts.

1) Most prominently, this CDT will create the next generation of quantitative researchers that are trained in the necessary skills and techniques to make substantial impact in academia, industry and government agencies. Creation of skilled researchers with a broad scientific outlook will have a number of beneficiaries. We expect that our students will be in high demand within academia and will be the researcher leaders of tomorrow. In addition, many of our brightest students post-PhD are now moving out of academia to research positions within industry or government agencies; such students are likely to generate substantial financial impact within industry and societal benefits within government agencies. By encouraging strong collaboration with our external partner organisations throughout their training, our PhD students will have a broad insight into the impact that mathematics can bring, and the routes through which academic excellence can be translated into meaningful applied outputs with impact. The assembled team of supervisors has an excellent track-record of supporting and training high calibre PhD students with skills that are in demand both within and outside of academia.

2) More immediate economic and societal benefits will accrue from the direct interaction of our students with external partners that is an integral part of their training. We anticipate that 4-6 students per cohort will undertake a PhD that is co-supervised by one of our external partner organisations; in addition all students during their MSc year will partake in one of several group projects led and supported by one of our external partners. In both cases, research will be focused towards real-world problems that are of current concern to the partners. It is anticipated that through these close interactions our students will develop methodologies and results that will address real-world problems. These new solutions to particular challenging real-world problems from external partners are likely to have substantial industrial, economic or societal benefits as they directly tackle prominent and pressing issues set by those with the greatest knowledge of the real-world challenges. Impact will therefore be generated through direct problem-solving research with a number of the UK's leading organisations.

3) Finally, we envisage that the mathematical techniques that are developed in the context of one real-world problem will have wider benefit to other academic fields. Although the immediate beneficiaries are likely to be other academics who will gain from an increased repertoire of tools and techniques, in the longer term these insights are likely to lead to new applications that feed back into industry, finance and society in general. The transdisciplinary nature of our MathSys CDT will facilitate such interactions, promoting the exchange of ideas between diverse subject areas. We firmly believe that such cross-fertilisation of ideas will be a feature of the MathSys CDT, where students are united by common goals of quantitative understanding and prediction and a common language of mathematics. We therefore expect rapid impact in a variety of applied areas, as novel techniques are introduced.