Thursday 15 Jan 2015: SDAC seminar : Localised patterns and cellular level morphogenesis
Prof Alan Champneys - Bristol
In this talk I shall describe recent and ongoing work inspired by problems in plant biology in which the shape of a cell
undergoes rapid change. The mechanism is believed to involve the dynamics of structural G-proteins known as ROPs that come in in active and inactive forms. The active forms are bound to the cell membrane and so diffuse more slowly. Canonical problems involve the formation of hairs in plant roots and the interdigitation of leaf pavement cells. In both cases, it has been observed that patches of active ROPs provide the precursor to sites of cell wall softening and active growth. Mathematically, the problem lends itself to reaction diffusion models and Turing instabilities. But, in the presence of source and loss terms it is found that the Turing bifurcations are subcritical. Instead of wavy patterns, localised patterns are observed, which can be explained using the so-called theory of homoclinic snaking. The case of plant root hairs is worked out in detail in both one and two dimensions using semi-strong asymptotic analysis, in which the spatial location of the hair is controlled by a gradient of the plant hormone auxin.