The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis", which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomously from a homogeneous, uniform state.¹² The pattern arises due to Turing instability, which in turn arises due to the interplay between differential diffusion of chemical species and chemical reaction.³ The instability mechanism is surprising because a pure diffusion, such as molecular diffusion, would be expected to have a stabilizing influence on the system (i.e., complete mixing).

Overview

In his paper,¹ Turing examined the behaviour of a system in which two diffusible substances interact with each other, and found that such a system is able to generate a spatially periodic pattern even from a random or almost uniform initial condition.⁴ Prior to the discovery of this instability mechanism arising due to unequal diffusion coefficients of the two substances, diffusional effects were always presumed to have stabilizing influences on the system.

Turing hypothesized that the resulting wavelike patterns are the chemical basis of morphogenesis.⁴ Turing patterning is often found in combination with other patterns: vertebrate limb development is one of the many phenotypes exhibiting Turing patterning overlapped with a complementary pattern (in this case a French flag model).⁵

Before Turing, Yakov Zeldovich in 1944 discovered this instability mechanism in connection with the cellular structures observed in lean hydrogen flames.⁶ Zeldovich explained the cellular structure as a consequence of hydrogen's diffusion coefficient being larger than the thermal diffusion coefficient. In combustion literature, Turing instability is referred to as diffusive–thermal instability.

Concept

The original theory, a reaction–diffusion theory of morphogenesis, has served as an important model in theoretical biology.⁷ 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.⁸

Biological application

A mechanism that has gained increasing attention as a generator of spot- and stripe-like patterns in developmental systems is related to the chemical reaction-diffusion process described by Turing in 1952. This has been schematized in a biological "local autoactivation-lateral inhibition" (LALI) framework by Meinhardt and Gierer.²³ LALI systems, while formally similar to reaction-diffusion systems, are more suitable to biological applications, since they include cases where the activator and inhibitor terms are mediated by cellular "reactors" rather than simple chemical reactions,²⁴ and spatial transport can be mediated by mechanisms in addition to simple diffusion.²⁵ These models can be applied to limb formation and teeth development among other examples.

References

1Turing, Alan (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. S2CID 120437796.
2Stewart, Ian (1998). Life's Other Secret: The New Mathematics of the Living World. London: Allen Lane. pp. 138–140, 142–146, 148, 149, 151, 152. ISBN 0-713-99161-5. OCLC 43126766.
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Biological application

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Key concepts

Overview

Concept

Biological application

See also

References

Further reading