Mathematical Modelling of Molecular to Tissue Scale Patterning

The patterning of multicellular tissues is an essential process in embryonic development involving dynamic, iterative interactions between components from the molecular to the tissue scale. Mathematical modelling can help disentangle this complexity by integrating disparate experimental observations in a systematic conceptual framework. To that end, this thesis presents a suite of reaction-diffusion models of molecular to tissue scale patterning, ranging from generic, tractable models amenable to mathematical analysis to more detailed models that account for known biochemical interactions. At the molecular scale, we explore the phenomenon of protein complex clustering by modelling transmembrane homodimer formation. We identify conditions in which this model supports spatial patterns, corresponding to clusters, as a Turing or wave-pinning instability. Moving up a length scale, motivated by the `core' planar polarity pathway in the fly wing, we model transmembrane heterodimer complex formation in a one-dimensional line of cells. We explore how different forms of molecular feedback interactions amplify initial asymmetries to drive this system to a spatial pattern (corresponding to planar polarisation). We then extend this model into two spatial dimensions and explore the existence and stability of different forms of patterning, including planar polarisation. Finally, we consider a two-dimensional model of the Fat-Dachsous planar polarity pathway, exploring the relative contributions of local feedback interactions and tissue-scale signalling cues in generating observed levels of planar polarisation in this system. Together, the contributions in this thesis advance the use of mathematical modelling to help understand patterning processes in developmental biology.