Mathematics

How organ size is controlled during development has been a subject of scientific study for centuries, but the growth control mechanisms are still poorly understood. The Drosophila wing imaginal disc has widely been used as a model system to study the regulation of growth. Growth control in the Drosophila wing disc involves various local signals, including signaling pathways, mechanical signals, etc. We developed a model of the Hippo pathway, which is the core regulatory pathway that mediates cell proliferation and apoptosis in Drosophila and mammalian cells, and contains a core kinase mechanism that affects control of the cell cycle and growth. We investigated the regulatory role of two upstream components Fat and Ds on the downstream mediator Yki of the pathway, and provide explanations to some of the seemingly contradictory experimental results. We found that a number of non-intuitive experimental results can be explained by subtle changes in the balances between inputs to the Hippo pathway. Since signal transduction and growth control pathways are highly con-served across species and directly involved in tumor growth, much of what is learned about Drosophila will have relevance to growth control in mammalian systems. Our recent work on morphogen transport in the wing disc will also be discussed.

The first passage time (FPT) of a diffusive searcher to a target determines the timescale of many physical, chemical, and biological processes. While most studies focus on the FPT of a given single searcher, another important quantity in some scenarios is the FPT of the first searcher to find a target from a large group of searchers. This fastest FPT is called an extreme FPT and can be orders of magnitude faster than the FPT of a given single searcher. In this talk, we will explain recent results in extreme FPT theory and give special attention to the case of extreme FPTs to small targets. Time permitting, we will also explain results on extreme FPTs of subdiffusion modeled by a fractional time derivative and superdiffusion modeled by a fractional Laplacian.

Collective cell movement occurs throughout biology and medicine and there are many common features shared across different areas. I will review work we have carried out over the past few years on (i) systematically deriving a PDE model for tumour angiogenesis from a discrete formulation and comparing this model with the classical, phenomenological snail-trail model; (ii) agent-based models for cranial neural crest cell migration in a collabo-ration with experimental biologists that has revealed a number of new biological insights.

For cells to divide, they must undergo mitosis: the process of spatially organizing their copied DNA (chromosomes) to precise locations in the cell. This procedure is carried out by stochastic components that manage to accomplish the task with astonishing speed and accuracy. New advances from our collaborators in the New York Dept of Health provide 3D spatial trajectories of every chromosome in a cell during mitosis. Can these trajectories tell us anything about the mechanisms driving them? The structure and context of this cutting-edge data makes utilizing classical tools from data science or particle tracking challenging. I will discuss my progress with Alex Mogilner on developing analysis for this data and mathematical modeling of emergent phenomena.