Turing-like pattern formation in discrete models

A crucial step in the early development of multicellular organisms involves the establishment of spatial patterns of gene expression. These spatial patterns later dictate the varying fates of proliferating cells and allow for robust development of complex biological systems such as organs. How exactly this spatial patterning arises in biological systems is still an unresolved question. One major contender for a model of biological pattern formation was developed by Alan Turing, which showed that heterogeneous structure could emerge on a global level from the synergetic effect of local processes, which individually have no pattern forming ability. Turing's theory argues that biological function emerges from the integration of processes rather than being assignable to a single process. Although Turing's theory was developed more than 65 years ago the debate over what mechanisms actually underpin biological pattern formation remains an active and growing area of research. The reason for this is partly due to the main pitfall of Turing's model: reaction systems that show pattern forming ability tend to only have a small Turing regime. That is, the system can only produce patterns within a small restricted area of parameter space. This sized Turing regime seems to be in stark contrast to what is observed in practice, as the early development of multicellular organisms proves to be a robust process, insensitive to stochastic perturbations and variance.
The project focuses on developing novel frameworks that are capable of coping with the computationally demanding task of exploring the design space of pattern forming systems. In addition, these frameworks allow for the abstraction of pattern forming mechanisms and facilitate the identification of common features. Current methods used to identify systems that produce Turing instabilities, and which are capable of pattern formation, are mostly done within the continuous setting and do not scale well due to being computationally costly. We therefore adopt discrete modelling approaches for two beneficial reasons. First, to gain the computational advantage and numerical stability inherent in a discretised spatio-temporal system. This allows the computational time to be significantly reduced in comparison to continuous models of systems with equivalent complexity and non-linearity. This advantage enables us to preform large exhaustive explorations of the design space for Turing patterns. Second, a further benefit of a discrete approach is that by stripping the model down to a discretised skeleton framework of pattern formation we do away with unnecessary components, and this reduction allows the key features of pattern producing mechanism to be more readily identified. Additional questions naturally arise from the ability to efficiently simulate spatio-temporal systems. For example, how often do Turing instabilities truly relate to pattern formation in silico? Are there more reliable predictors than Turing instabilities for predicting in silico pattern formation? Do the same reaction systems identified to produce patterns within the continuous also do so in the discrete? The project confronts these questions and attempts to provide answers in addition to developing a framework package within the Julia language to aid future studies of large-scale design space analysis. In recent years, an increasing number of studies combining both experimental and theoretical work has sought to reveal how Turing models could underlie a variety of morphogenetic processes. These include important processes such as lung branching and limb development. It has been noted in current literature that there is a need for theoretical models capable of providing more extensive searches of Turing design space and model frameworks that allow for larger Turing systems to be analysed. This project seeks to deliver this and in so doing it provides an advance in developmental biology and regenerative medicine.