Coalescent models for developmental biology and the spatio-temporal dynamics of growing tissues.

Development is a process that needs to tightly coordinated in both space and time. Cell tracking and lineage tracing have become important experimental techniques in developmental biology and allow us to map the fate of cells and their progeny in both space and time. A generic feature of developing (as well as homeostatic) tissues that these analyses have revealed is that relatively few cells give rise to the bulk of the cells in a tissue; the lineages of most cells come to an end fairly quickly. This has spurned the interest also of computational and theoretical biologists/physicists who have developed a range of modelling - perhaps most notably are the agent-based modelling (ABM) -- approaches. These can become computationally prohibitively expensive but seem to capture some of the features observed in experiments. Here we develop a complementary perspective that allows us to understand the dynamics leading to the formation of a tissue (or colony of cells). Borrowing from the rich population genetics literature we develop genealogical models of tissue development that trace the ancestry of cells in a tissue back to their most recent common ancestors. We apply this approach to tissues that grow under confined conditions -- as would, for example, be appropriate for the neural crest -- and unbounded growth -- illustrative of the behaviour of 2D tumours or bacterial colonies. The classical coalescent model from population genetics is readily adapted to capture tissue genealogies for different models of tissue growth and development. We show that simple but universal scaling relationships allow us to establish relationships between the coalescent and different fractal growth models that have been extensively studied in many different contexts, including developmental biology. Using our genealogical perspective we are able to study the statistical properties of the processes that give rise to tissues of cells, without the need for large-scale simulations.