Reaction & Diffusion system Xinyao Wang 2018.11.27

Outline: I: Patterns in nature II:Define reaction - diffusion system III: Gray-Scott III:Simulation IV: Applications

Patterns in nature

More patterns

Pattern formation The science of pattern formation deals with visible, orderly outcomes of self-organization and the common principles behind similar patterns in nature. Examples of pattern formation can be found in Biology, Chemistry, Physics and Mathematics. It can also be simulated by using computer graphics techniques. Reaction-diffusion model can be used in biological systems and widely used in chemistry systems. .

Alan Turing’s contribution Alan Turing was not a biologist, but a mathematician and the founder of computer science. Well-known for cracking the Nazis’ Enigma Code during World War II, Turing sought to crack another kind of code – how animals could develop from chemical substrates. He believed development could be reduced to mathematical axioms and physical laws. His landmark paper, “The Chemical Basis of Morphogenesis,” was published in 1952, two years before his death. In it he theorized a system of two different interacting molecules, called morphogens, which could establish chemical gradients through a “reaction-diffusion system.” The central idea behind the theory is that two homogeneously distributed substances within a certain space, one “locally activated” and the other capable of “long-range inhibition,” can produce novel shapes and patterns. The results of these substance interactions are dependent on just four variables– the rate of production, the rate of degradation, the rate of diffusion and the strength of their activating/inhibiting interactions.

What is reaction diffusion system? A model which correspond to change in space and time of the concentration of one or more chemical substance 1: local chemical reactions in which the substances are transformed into each other 2: the substances spread out over a surface in space

Six stable states toward which the two-factor RD system can converge. The elements selected by Turing were a theoretical pair of interacting molecules diffused in a continuous field. In his mathematical analysis, Turing revealed that such a system yields six potential steady states, depending on the dynamics of reaction term and wavelength of the pattern The finding of this type of wave is the major achievement of Turing’s analyses, and these are usually referred to as Turing patterns. A Turing pattern is a kind of nonlinear wave that is maintained by the dynamic equilibrium of the system. Its wavelength is determined by interactions between molecules and their rates of diffusion. Such patterns arise autonomously, independent of any preexisting positional information.

Two-dimensional patterns generated by the Turing model. These patterns were made by an identical equation with slightly different parameter values. These simulations were calculated by the software provided as supporting online material.

Reproduction of biological patterns created by modified RD mechanisms. The intricate involutions of seashells (5), the exquisite patterning of feathers (12), and the breathtakingly diverse variety of vertebrate skin patterns (13) have all been modeled within the framework of the Turing model

Gray Scott model Partial differential equations

Possible patterns? The colors show the concentration of U at each locus. Red labels the maximum concentration and Blue labels the minimum concentration. The patterns that you see are stable -- no significant change will occur, regardless of how long we run the process. Each pattern starts off from the same slightly randomized initial condition. The differences among the patterns is due to either the differences of the grids or the differences in the initial condition.

F=0.018, k=0.047

Increase diffusion F=0.025, k=0.047

Increase reaction F=0.025, k=0.0547

Computational simulation https://vimeo.com/170073061

Various simulation: https://pmneila.github.io/jsexp/grayscott/

Compatibility of RD and Gradient Models In the classic gradient model, the fixed source of the morphogen at a specific position provides positional information. Although the assumption of a morphogen source appears to be different from the assumptions of the RD model, it can be introduced into the RD model quite naturally as a “boundary condition.” In other words, the classic morphogen gradient model can be thought of as the specific case of the RD model in which the reaction term is removed. In many simulation studies, such boundary conditions are used to make the pattern more realistic

Turing Patterns in Vertebrate Skin Movement of zebrafish stripes and the interaction network among the pigment cells. The pigment pattern of zebrafish is composed of black pigment cells (melanophores) and yellow pigment cells (xanthophores). The pattern is made by the mutual interaction between these cells. (A) Melanophores in the two black stripes were ablated by laser, and the process of recovery was recorded. (B) Results of simulation by the Turing model.

References: https://pmneila.github.io/jsexp/grayscott/ https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system http://math.bu.edu/people/heijster/PRESENTATIONS/HEIJSTER_RPI2011.pdf http://www.karlsims.com/rd.html https://groups.csail.mit.edu/mac/projects/amorphous/GrayScott/ https://phylogenous.wordpress.com/2010/12/01/alan-turings-reaction-diffusion-model-simplification-of-the-complex/ http://science.sciencemag.org/content/329/5999/1616 https://mrob.com/pub/comp/xmorphia/index.html https://www.sciencedirect.com/science/article/pii/S0169534708000281