Stochastic Turing Patterns

Pattern formation is a phenomenon based on the interaction of different components, possibly under the influence of their surroundings. Alan Turing, a cryptographer and a pioneer in computer science, developed algorithms to describe complex patterns using simple inputs and random fluctuation. In 1952, he proposed that the interaction between two biochemical substances with different diffusion rates have the capacity to generate biological patterns. In his mathematical framework, there is one activating protein (activator) that activates both itself and an inhibitory protein (inhibitor), which only inhibits the activator. He detected that a stable homogeneous pattern could become unstable if the inhibitor diffuses more rapidly than the activator. The interplay between the concentrations of these substances forms a pattern whose spatiotemporal evolution is governed by coupled reaction-diffusion systems (activator-inhibitor model). By his equation he could generate a pattern of regularly-spaced spots, fingerprints, or only simple the stripes of a zebra. This phenomenon is called diffusion-driven instability Turing instability. Thus, the most fundamental phenomenon in pattern-forming activator-inhibitor systems is that a slight deviation from spatial homogeneity has vital positive feedback leading to increase further. The presence of nonlinearities in the local dynamics, for example, due to the inhibitor concentration, saturates the Turing instability into a stable and spatially inhomogeneous pattern. Usually, one models these equations in a deterministic framework. The deterministic model, i.e., the macroscopic system of equations, is derived from the microscopic behavior studying the limit behavior. From the microscopic perspective, one interprets the movements of the molecules as a result of microscopic irregular movement. Taking the limit and passing from the microscopic to the macroscopic equation, one neglects the fluctuations around the mean value. In addition, biological systems are frequently subject to noisy environments, inputs, and signalling. These stochastic perturbations are crucial when considering the ability of such models to reproduce results consistently. In our project we investigate the impact of the randomness to systems generating Turing patterns