Non-local models of cell-cell interactions in development
Bloomfield, Jennifer Martha
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This thesis explores pattern formation caused by cell-cell interactions in animal tissues during development, whether that patterning be natural or experimental. We produce three non-local integrodifferential equation models that each explore a different driver for pattern formation: cell proliferation, cellular differentiation, and cell migration. We consider how the local environment of a cell – or, more specifically, a cell’s neighbours – affects that spatial patterning in each case. In the first part of this thesis we present mathematical modelling of proliferationdriven patterning in mosaic tissues, where mosaic tissues are those composed of two or more genetically distinct cell types. The results of our modelling suggets that small changes in the type of interaction that cells have with their local environment can lead to very different outcomes for the composition of mosaics. We study two variations of a cellular automaton model based on simple rules for renewal, and then do the same for an integrodifferential equation model. The results of the continuous and cellular automata models are qualitatively the same, and we observe that changes in local environment interaction affect the dynamics for both. Furthermore, we demonstrate that the models reproduce some of the patterns seen in actual mosaic tissues. In the next model we consider cellular differentiation, which is the process whereby cells form into their final state. We investigate three different versions of the model, with differentiation being cell autonomous, regulated via a community effect, or weakly dependent on the local cellular environment. We consider the spatial patterns that such different modes of differentiation produce, and investigate the formation of both stripes and spots by the model. We show that pattern formation only occurs when differentiation is regulated by a strong community effect. Finally we present a general model of contact-dependent interactions, and consider the role of both interaction ranges and strengths on patterning formation. Our analysis of the model equations shows that the magnitudes and signs of both of these terms affect whether the system will show instability or stability, as will the initial densities of each cell population. Furthermore, our simulations show that whilst increased ranges lead to increased aggregation widths, and attractive forces lead to stationary aggregations, repulsive forces can cause a range of behaviours, including travelling waves, oscillatory behaviour, and breathing bands.