Applications of Turing Pattern Formation, from Geometric Visual Hallucinations to Eukaryotic Chemotaxis

In 1952 Alan Turing proposed a mechanism for the development of spatial patterns, such as animal coat patterns, from spatially homogeneous initial conditions, such as a putative uniform embryo. Many systems have invited analogous analyses, from segmentation of the Drosophila embryo to Meinhardt's model for establishing direction in eukaryotic chemotaxis. The two essential elements underlying the Turing mechanism, a short-range activator and a long-range inhibitor, have not always been easily identified as the biology underlying pattern formation becomes better understood. In this talk I will explore two systems in which the biological details gave new insights into the possibilities of pattern formation. In the cerebral cortex, the local connectivity of nervous tissue gives an effective long-range inhibitory and short-range excitatory interaction that can lead to the creation of spontaneous patterned activity in the cortical sheet. In the visual cortex this spontaneous activity gives rise to a distinct set of geometric visual hallucinations. Careful analysis of the geometry of cortical connectivity allows classification of the observed patterns in terms of a particular class of subgroups of the Euclidean motions of the plane. As a second example, I will return to the problem of eukaryotic chemotaxis first addressed (incorrectly, from a biological perspective) by Meinhardt. The problem is for an unbiased cell to respond to a weak chemical gradient signal in the surrounding medium, identifying the direction of the gradient and rearranging its internal chemistry to prepare to crawl up the gradient (chemotaxis). Using data from mutant screens of cells with anomalous chemotaxis, we have identified a rapidly diffusing intracellular inhibitory molecule that facilitates sharpening of the directional response. In addition to amplifying the weak spatial gradient signal, this variant of the Turing mechanism also exploits timing characteristics of the extracellular signal. (The Mathematical Biology Seminar on Thursday 1/16 will discuss biochemical networks in more detail.)

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).