Mathematical Models for Biological Pattern Formation

The formation of patterns in developing biological systems involves the spatiotemporal co-ordination of growth, cell-tell signalling, tissue movement, gene expression, and cell differentiation. The interactions of these complete processes are generally non-linear, and thus mathematical modelling and analysis are needed to provide the framework in which to compute the outcome of different hypotheses on modes of interaction and to make experimentally testable predictions. This collection contains papers exploring several aspects of the hierarchy of processes occurring during pattern formation. A number of papers address the modelling of cell movement and deformation, with application to pattern formation within a collection of cells in response to external signalling cues. The results are considered in the context of pattern generation in Dictyostelium discoideum and bacterial colonies. A number of models at the macroscopic level explore the possible mechanisms underlying spatiotemporal pattern generation in early development, focussing on primitive streak, somitogenesis, vertebrate limb development, and pigmentation patterning. The latter two applications consider in detail the effects of growth on patterning. The potential of models to generate more complex patterns is considered, and models involving different modes of cell-cell signalling are investigated. Pattern selection is analysed in the context of chemical Turing patterns, which serve as a paradigm for morphogenesis, and a model for vegetation patterns is presented.