Math Biology Seminar: Monika Twarogowska

Ectodermal derivatives such as teeth, hair, feathers or scales share similar morphological features and spatial patterning mechanisms. From the mathematical point of view, pioneering works of Alan Turing showed that spatial-temporal self-organization structures can emerge from reaction-diffusion systems. However, recent biological and mathematical studies give evidence that there is a substantial difference in pattern generation between static and growing domains. The latter may contain a key to understanding the problem of sequential patterning in developmental biology.

In this talk we present a macroscopic model of gene expression dynamics in the growing field where molars appear sequentially. Our model mimics the expression of the Edar gene during the formation of signaling centers, from where future teeth originate. We rely on a reaction-diffusion system of an activator-inhibitor type on a dynamically evolving tissue. The key element is not only the tissue growth but also its non-constant properties, which affect the reaction kinetics, depending on the presence of the activator. The purpose of the model is twofold. On one hand it describes a sequential formation of individual spots through Turing instability mechanism. On the other hand, it produces the activator up-regulation waves starting at distal field thanks to reaction functions containing bistable solutions. We present numerical studies of two dynamics on growing domain: under wild conditions and under a mutation regulating the inhibitor concentrations. For a fixed and fully matured domain, we analyze the effect of chemotaxis on the wavelength of Turing patterns and, as a consequence, on the merging of signaling centers that is observed in some biological conditions.