Turing Patterns

The study of pattern formation in nonequilibrium reaction-diffusion systems began with the theoretical analysis of Turing structures2,3, which are stationary, spatially periodic patterns resulting from the interplay between pure diffusion and nonlinear reaction kinetics. Turing1 suggested that such structures could play a role in morphogenesis, and his point of view has gradually become prominent The first experimental observation of Turing patterns, however, occurred nearly 40 years after Turing's work, in a chemical reaction-diffusion system.4 Later, Kondo and collaborators5-7 showed that skin patterns in various small fish also develop according to the Turing mechanism.

Models for Turing patterns are two basic categories distinguished by Gierer and Meinhardt [8]: activator-substrate, and activator-inhibitor.

  1. Turing, A.M."The Chemical Basis of Morphogenesis.pdf" Philos. Trans. R. Soc. London B 1952, 237, 37.
  2. Meinhardt, H. Models of Biological Pattern Formation; Academic: London, 1982.
  3. Murray, J. D. Mathematical Biology; Springer: Berlin, 1989.
  4. V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Phys. Rev. Lett.,64, 2953 (1990); Q. Ouyang and H.L. Swinney, Nature 352, 610 (1991).
  5. Kondo, S; Asai, R. A Reaction-Diffusion Wave on the Skin of the Marine Angelfish Pomacanthus,Nature 1995, 376, 765.
  6. Asai, R.; Taguchi, E.; Kume,Y; Saito, M. Kondo Mechanisms of Development 1999, 89, 87.
  7. Kondo, S. The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin, Genes to Cells 2002, 7, 535.
  8. A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30-39, (1972).

Classical Turing patterns are stationary. We are interested in oscillatory Turing patterns which are caused from interplay between Turing and Hopf instabilities. There are two distinct ways:

  • When the Turing instability is stronger than the Hopf, Turing patterns grow first, then they lose stability through Hopf bifurcation. You can find this routine in Phys. Rev. Lett. 90(17), 178303 (2003)
  • The other way happens when the Hopf instability if stronger than Turing, and a bulk oscillation occurs first, then, the limit-cycle solution turns unstable toward spatial perturbation and develop Turing structure through period-doubling bifurcation. You can find this routine in Phys. Rev. Lett. 92, 198303 (2004)

Turing patterns typically consist of hexagonal arrays of spots or labyrinthine arrangements of stripes with a single characteristic wavelength. We found superlattice Turing patterns, such as "black-eye", "white-eye", "twinkling-eye", ...

In two-layer systems, the arrangement of the Turing patterns can be symmetric, asymmetric or antiphase. You can find them in Phys. Rev. E 69, 026211 (2004).

Turing instability (Diffusive instability, diffusion-driven instability)

Homogeneous steady state solution is stable to spatially homogeneous perturbations, but unstable to some inhomogeneous perturbations.

Turing's original model:

Turing52:

Activator-inhibitor and substrate-depleted model:

Based on the Jacobian matrix. The model is called activator-inhibitor if the matrix is and substrate-depleted if the matrix is .

The beginning of studies on diffusive instability is traditionally associated with the work of Turing (1952), who studied stratification of an active medium caused by differences in diffusivities. However, qualitative arguments that provided an understanding of the mechanism of this instability were already suggested by Zel’dovich as early as 1944. [see "Physics of reaction waves" Rev. Mod. Phys. 71, 1173–1211 (1999)]

Pattern selection

 A two-dimensional Turing pattern can be spot- or stripe-like. Spot patterns can be hexagonal (H0, or H\pi=honeycomb), or square lattices; the stripes can be parallel stripe, zigzag-stripe, or labyrinthine stripes. Pattern selection is all about, of them, what kind of symmetry will be selected.
  1. Hexagons and stripes
  2. Squares and stripes
  3. Secondary Instabilities
Turing+wave | Two-layer Turing | Interacting Turings(t2) | Superlattice Turing | Oscillatory Turing | Square Turing

Alan TuringAlan Mathison Turing (Born: 23 June 1912 in London, England, Died: 7 June 1954 in Wilmslow, Cheshire, England)

Alan Mathison Turing was one of the great pioneers of the computer field. He inspired the now common terms of "The Turing Machine" and "Turing's Test."

In 1952 he published the first part of his theoretical study of morphogenesis, the development of pattern and form in living organisms.

Turing died of potassium cyanide poisoning while conducting electrolysis experiments.



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