Turing pattern

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Three examples of Turing patterns
Six states generating a Turing pattern
A Turing bifurcation pattern
An example of a natural Turing pattern on a giant pufferfish

The concept of a Turing pattern (often referred to in the plural as Turing patterns) was introduced by the English mathematician Alan Turing in a 1952 paper entitled The Chemical Basis of Morphogenesis.[1] This foundational paper describes the way in which patterns in nature such as stripes and spots can arise naturally out of a homogeneous, uniform state.


The original theory, a reaction–diffusion theory of morphogenesis, has served as an important model in theoretical biology.[2] Reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. Patterns such as fronts, hexagons, spirals, stripes and dissipative solitons are found as solutions of Turing-like reaction–diffusion equations.[3] The parameters will depend on the physical system that is under consideration. In the context of fish skin pigmentation, the associated equation is a three field reaction–diffusion where the linear parameters are associated with pigmentation cell concentration and the diffusion parameters are not the same for all fields.[4] In dye-doped liquid crystals, photoisomerization process in the liquid crystal matrix is described as a reaction–diffusion equation of two fields (liquid crystal order parameter and concentration of cis-isomer of the azo-dye).[5] Both systems have very different physical mechanisms on the chemical reactions and diffusive process, but on a phenomenological level, both have the same ingredients.

It is a major theory in developmental biology; for example, there was a theoretical study on the potential of VEGFC to form Turing patterns to regulate lymphangiogenesis in the zebrafish embryo.[6]

Turing-like patterns have also been demonstrated to arise in developing organisms without the classical requirement of diffusible morphogens. Studies in chick and mouse embryonic development suggest that the patterns of feather and hair-follicle precursors can be formed without a morphogen pre-pattern, and instead are generated through self-aggregation of mesenchymal cells underlying the skin.[7][8] In these cases, a uniform population of cells can form regularly patterned aggregates that depend on the mechanical properties of the cells themselves and the rigidity of the surrounding extra-cellular environment. Regular patterns of cell aggregates of this sort were originally proposed in a theoretical model formulated by George Oster, which postulated that alterations in cellular motility and stiffness could give rise to different self-emergent patterns from a uniform field of cells.[9] This mode of pattern formation may act in tandem with classical reaction-diffusion systems, or independently to generate patterns in biological development.

As well as in biological organisms, Turing patterns occur in other natural systems – for example, the wind patterns formed in sand. Although Turing's ideas on morphogenesis and Turing patterns remained dormant for many years, they are now inspirational for much research in mathematical biology.[10]

Turing patterns can also be created in nonlinear optics as demonstrated by the Lugiato–Lefever equation.

See also[edit]


  1. ^ Turing, A. M. (1952). "The Chemical Basis of Morphogenesis" (PDF). Philosophical Transactions of the Royal Society of London B. 237 (641): 37–72. Bibcode:1952RSPTB.237...37T. doi:10.1098/rstb.1952.0012. JSTOR 92463.
  2. ^ Harrison, L. G. (1993). Kinetic Theory of Living Pattern. Cambridge University Press.
  3. ^ Kondo, S.; Miura, T. (23 September 2010). "Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation". Science. 329 (5999): 1616–1620. doi:10.1126/science.1179047. PMID 20929839.
  4. ^ Nakamasu, A.; Takahashi, G.; Kanbe, A.; Kondo, S. (11 May 2009). "Interactions between zebrafish pigment cells responsible for the generation of Turing patterns". Proceedings of the National Academy of Sciences. 106 (21): 8429–8434. doi:10.1073/pnas.0808622106. PMC 2689028. PMID 19433782.
  5. ^ Andrade-Silva, Ignacio; Bortolozzo, Umberto; Clerc, Marcel G.; González-Cortés, Gregorio; Residori, Stefania; Wilson, Mario (27 August 2018). "Spontaneous light-induced Turing patterns in a dye-doped twisted nematic layer". Scientific Reports. 8 (1): 12867. doi:10.1038/s41598-018-31206-x. PMC 6110868. PMID 30150701.
  6. ^ Roose, Tiina; Wertheim, Kenneth Y. (3 January 2019). "Can VEGFC Form Turing Patterns in the Zebrafish Embryo?". Bulletin of Mathematical Biology. 81 (4): 1201–1237. doi:10.1007/s11538-018-00560-2. ISSN 1522-9602. PMC 6397306. PMID 30607882.
  7. ^ Glover, James D.; Wells, Kirsty L.; Matthäus, Franziska; Painter, Kevin J.; Ho, William; Riddell, Jon; Johansson, Jeanette A.; Ford, Matthew J.; Jahoda, Colin A. B.; Klika, Vaclav; Mort, Richard L. (2017-07). "Hierarchical patterning modes orchestrate hair follicle morphogenesis". PLoS biology. 15 (7): e2002117. doi:10.1371/journal.pbio.2002117. ISSN 1545-7885. PMC 5507405. PMID 28700594. Check date values in: |date= (help)
  8. ^ Shyer, Amy E.; Rodrigues, Alan R.; Schroeder, Grant G.; Kassianidou, Elena; Kumar, Sanjay; Harland, Richard M. (08 25, 2017). "Emergent cellular self-organization and mechanosensation initiate follicle pattern in the avian skin". Science. 357 (6353): 811–815. doi:10.1126/science.aai7868. ISSN 1095-9203. PMC 5605277. PMID 28705989. Check date values in: |date= (help)
  9. ^ Oster, G. F.; Murray, J. D.; Harris, A. K. (1983-12). "Mechanical aspects of mesenchymal morphogenesis". Journal of Embryology and Experimental Morphology. 78: 83–125. ISSN 0022-0752. PMID 6663234. Check date values in: |date= (help)
  10. ^ Woolley, T. E., Baker, R. E. Maini, P. K., Chapter 34, Turing's theory of morphogenesis. In Copeland, B. Jack; Bowen, Jonathan P.; Wilson, Robin; Sprevak, Mark (2017). The Turing Guide. Oxford University Press. ISBN 978-0198747826.