Mathematical modelling of mammalian pigmentation patterns: Stochastic modelling of melanoblast neural crest cells.

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Embryogenesis defines the early stages of embryonic development. Many such developments are attained via afamily of cells called the neural crest cells. Neural crest cells play a vital role in many biological developments in theearly stages of the growing embryo, e.g. formation of bones, cartilage & pigmentation of hair & skin.Epidermal pigmentation is a product of melanogenesis which is achieved via melanocytes. Melanoblasts, the earlyprecursors of melanocytes, are the pigment-producing cells responsible for producing melanin. Melanoblastsoriginate in the trunk region of the neural crest from which they delaminate & migrate dorsoventrally along theirmigratory pathway to colonise the developing epidermis. They then differentiate into melanocytes & start producingmelanin. Survival of melanoblasts is dependent on signalling between the receptor Kit & its ligand Kitl. Mutations inthe Kit gene can alter the signalling mechanism causing melanoblasts to behave erroneously which ultimately leadsto incomplete colonisation of the epidermis. Recent research suggests that the melanoblasts of Kit mutant miceexhibit longer cell-cycle times, leaving parts of the epidermis deprived of these pigment-producing cells. The resultingneurocristopathy is called piebaldism. A piebald mouse shows a white spot on the belly. Erroneous behaviour ofother neural crest cells leads to more serious conditions such as Neurofibromatosis & Hirschsprung's disease.Larger mammals also show piebaldism, e.g. patches of unpigmented skin in cows.My project will concern modelling melanoblast behaviour using mathematical models. So far, using experimentallyparameterised stochastic agent-based models I have been able to replicate the white belly spots in mice on agrowing domain. We also showed that varying the parameter values our model can produce patch-like patterns oflarger mammals such as those seen in some cow species. We would now like to expand on this work along the following key linesPattern formation in larger mammals: Although the on-lattice agent-based model could seemingly successfullyreplicate patches in cows, our work lacks mathematical analysis to accompany these results. We will be developing amore mathematically rigorous continuum model to accompany the simulation algorithm which will quantify the patchformation in cows. It has been observed that cow patches have sharper & well-defined boundaries in contrast tothe belly spots in mice. We will develop hybrid deterministic-continuum models which exploit Turing's theory ofdiffusion-driven pattern formation in order to investigate this phenomenon.Localisation of mesenchyme dermal cells to hair follicles: Mesenchymal dermal cells play a key role in hair folliclemorphogenesis. Responding to certain signalling pathways the mesenchymal cells aggregate & form a periodicpattern of dermal condensates. The locations of these condensates mark the positions of future hair follicles. In thisstrand of the project, we are interested in the mechanisms which drive this periodic pattern. We will use simulationmodels to understand this patterning.Realistic cell-cycle time distribution: Much of the stochastic modelling during this PhD will employ the GillespieStochastic Simulation Algorithm. The Gillespie algorithm assumes that cell-cycle times are exponentially distributed& exhibit the memoryless property. However, it is well-known that cells-cycle times are not drawn from thisdistribution in reality. It has been shown previously that cell-cycle times of certain cells in mice are more accuratelymodelled using an Erlang distribution. We will be developing models to realistically model cell-cycle time data formelanoblasts. Evidence from existing work suggests correlations between cell cycle times of daughter cells & theirmore distant relatives. We would like to expand on the existing models to incorporate the effects of correlationbetween generations of