Mathematical modelling of pattern formation in developmental biology
Hunt, Gordon S.
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The transformation from a single cell to the adult form is one of the remarkable wonders of nature. However, the fundamental mechanisms and interactions involved in this metamorphic change still remain elusive. Due to the complexity of the process, researchers have attempted to exploit simpler systems and, in particular, have focussed on the emergence of varied and spectacular patterns in nature. A number of mathematical models have been proposed to study this problem with one of the most well studied and prominent being the novel concept provided by A.M. Turing in 1952. Turing's simple yet elegant idea consisted of a system of interacting chemicals that reacted and di used such that, under certain conditions, spatial patterns can arise from near homogeneity. However, the implicit assumption that cells respond to respective chemical levels, di erentiating accordingly, is an oversimpli cation and may not capture the true extent of the biology. Here, we propose mathematical models that explicitly introduce cell dynamics into pattern formation mechanisms. The models presented are formulated based on Turing's classical mechanism and are used to gain insight into the signi cance and impact that cells may have in biological phenomena. The rst part of this work considers cell di erentiation and incorporates two conceptually di erent cell commitment processes: asymmetric precursor di erentiation and precursor speci cation. A variety of possible feedback mechanisms are considered with the results of direct activator upregulation suggesting a relaxation of the two species Turing Instability requirement of long range inhibition, short range activation. Moreover, the results also suggest that the type of feedback mechanism should be considered to explain observed biological results. In a separate model, cell signalling is investigated using a discrete mathematical model that is derived from Turing's classical continuous framework. Within this, two types of cell signalling are considered, namely autocrine and juxtacrine signalling, with both showing the attainability of a variety of wavelength patterns that are illustrated and explainable through individual cell activity levels of receptor, ligand and inhibitor. Together with the full system, a reduced two species system is investigated that permits a direct comparison to the classical activator-inhibitor model and the results produce pattern formation in systems considering both one and two di usible species together with an autocrine and/or juxtacrine signalling mechanism. Formulating the model in this way shows a greater applicability to biology with fundamental cell signalling and the interactions involved in Turing type patterning described using clear and concise variables.