Turing Pattern in a Reaction-Diffusion System

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In the last few decades, developmental biologists have extensively used the reaction-diffusion model to explain pattern formation in living organisms. The original model was proposed by Alan Turing in 1952 [1]. The model is based on the idea that pattern formation results from two fundamental mechanisms: (1) coupled catalytic and autocatalytic reactions in a space element between two chemical species, an activator and an inhibitor, and (2) transfer of the interacting species to and from the neighboring space elements through a diffusional transport mechanism. Under appropriate reaction and diffusion conditions, a periodic pattern is formed from an initially homogeneous spatial distribution of activator and inhibitor [2, 3]. Examples of pattern formation can be found in biology, chemistry (the famous Belousov–Zhabotinskii reaction), physics, and mathematics [4, 5].

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To illustrate the mechanism of pattern formation, consider the hypothetical activator-inhibitor reaction sequence:

(R1)

(R2)

(R3)

(R4)

(R5)

and .

From the expression of the reaction rate of , , the species inhibits the activator production, hence its name: the larger its concentration, the lower the production rate of .

Introducing the variables and , the governing equations of the reaction-diffusion system in 2D can be written in the following nondimensional form:

,

,

with Set . For the formation of spatial patterns, the diffusion rates of activator and inhibitor should be very different: set the diffusion coefficients to be and .

For the numerical solution of the system of ODEs, assume (1) periodic boundary conditions as well as (2) the initial conditions:

,

,

where and are numbers taken randomly in the interval .

The Chebyshev orthogonal collocation method applied with nodes in both spatial directions transforms the system of two coupled nonlinear PDEs into a system of 392 nonlinear coupled first-order ordinary differential equations. This system of ODEs is solved using the built-in Mathematica command NDSolve. The snapshots show the results for the 2D inhibitor concentration distribution at times and at . It is interesting to note, in particular, the emergence of Turing patterns similar to leopard spots in the concentration distribution at .

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Contributed by: Housam Binous and Ahmed Bellagi (August 2015)
Open content licensed under CC BY-NC-SA


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Details

References

[1] A. M. Turing, "The Chemical Basis of Morphogenesis," Philosophical Transactions of the Royal Society, 237(641), 1952 pp. 37–72.

[2] H. Meinhardt, The Algorithmic Beauty of Sea Shells, New York: Springer-Verlag, 1995.

[3] T. Miura and P. K. Maini, "Periodic Pattern Formation in Reaction-Diffusion Systems: An Introduction for Numerical Simulation," Anatomical Science International, 79(3), 2004 pp. 112–123.

[4] J. D. Murray, Mathematical Biology: I. An Introduction, 3rd ed., New York: Springer, 2002.

[5] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed., New York: Springer, 2003.



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