Local and non-local mathematical modelling of signalling during embryonic development

Embryonic development requires cells to communicate as they arrange into the adult organs and tissues. The ability of cells to sense their environment, respond to signals and self-organise is of crucial importance. Patterns of cells adopting distinct states of differentiation arise in early development, as a result of cell signalling. Furthermore, cells interact with each other in order to form aggregations or rearrange themselves via cell-cell adhesion. The distance over which cells can detect their surroundings plays an important role to the form of patterns to be developed, as well as the time necessary for developmental processes to complete. Cells achieve long range communication through the use of extensions such as filopodia. In this work we formulate and analyse various mathematical models incorporating long-range signalling. We first consider a spatially discrete model for juxtacrine signalling extended to include filopodial action. We show that a wide variety of patterns can arise through this mechanism, including single isolated cells within a large region or contiguous blocks of cells selected for a specific fate. Cell-cell adhesion modelling is addressed in this work. We propose a variety of discrete models from which continuous models are derived. We examine the models’ potential to describe cell-cell adhesion and the associated phenomena such as cell aggregation. By extending these models to consider long range cell interactions we were able to demonstrate their ability to reproduce biologically relevant patterns. Finally, we consider an application of cell adhesion modelling by attempting to reproduce a specific developmental event, the formation of sympathetic ganglia.