Cross-disciplinary investigation of pattern formation in Zebrafish using spatially extended mathematical models with volume exclusion

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Considerable theoretical and experimental investigations-have prompted hypotheses that a Turing-type reaction-diffusion mechanism may be responsible for the striped pigment pattern of zebrafish. However, recent experimental studies suggest that the mechanism uses cell-cell interactions, rather than the classic pre-pattern- generating pair of interacting, diffusible ligands.Indeed, Yamanaka and Kondo (2014) have suggested that the asymmetric interactions and differential movement rates of xanthophores and melanophores - leading to escape-pursuit behaviour - can induce pattern formation. Recently a third cell type - iridophores - have also been suggested as critically important for zebrafish pattern formation. Finally, we have identified Agouti Signalling Peptide (ASIP) as a further patterning component, likely acting in parallel to the stripe-forming mechanism.Mathematical modelling has played an important role in the elucidation of zebrafish striping mechanisms. The framework typically employed to represent this phenomenon utilises deterministic continuum partial differential equations (PDEs). These two-species PDE-based models fail to capture the underlying biology appropriately for four main reasons: (i) they do not account for the fluctuations inherent in biological systems; (ii) finite cell size effects are neglected; the crucial role of (iii) iridophores and (iv) ASIP signalling are ignored.The central goal of this project is to generate a framework to investigate the effects of stochasticity and finite cell size upon pattern formation models which include iridophores and ASIP signalling, and to apply it to the study of zebrafish skin patterning.I will construct an on-lattice exclusion-process model in which cells interact with neighbours, and move to neighbouring lattice sites. A detailed investigation of the relationship between key length scales in the model, the system size and compartment size, and their impact on the patterns formed, will be carried out. Subsequently hypotheses on pigment cell interactions will be explicitly encoded in this framework to explore the potential of the model to replicate the patterns on both wild-type and mutant fish.These theoretical studies will alternate with experimental investigations. Guided by the modelling outputs, I will generate data to test the models efficacy. Data will be obtained from the literature and guided experimental studies for which our ongoing collaboration with Rotllant and Cerda-Reverter will be invaluable. Experimental studies will involve direct quantitative measurements of important properties of the pigment cell pattern, e.g. perhaps including dynamic cell-cell distances and density; pair correlation functions; gene expression (e.g. using qRT-PCR on skin samples); and RT-PCR comparing skin samples at different stages.