pattern The original theory, a
reaction–diffusion theory of morphogenesis, has served as an important model in
theoretical biology. 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. Turing proposed a model wherein two homogeneously distributed substances (P and S) interact to produce stable patterns during morphogenesis. These patterns represent regional differences in the concentrations of the two substances. Their interactions would produce an ordered structure out of random chaos. In Turing's model, substance P promotes the production of more substance P as well as substance S. However, substance S inhibits the production of substance P; if S diffuses more readily than P, sharp waves of concentration differences will be generated for substance P. An important feature of Turing's model is that particular wavelengths in the substances' distribution will be amplified while other wavelengths will be suppressed. In dye-doped
liquid crystals, a 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). The systems have very different physical mechanisms on the chemical reactions and diffusive process, but on a phenomenological level, both have the same ingredients. 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. 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. This mode of pattern formation may act in tandem with classical reaction-diffusion systems, or independently to generate patterns in biological development. Turing patterns may also be responsible for the formation of human
fingerprints. As well as in biological organisms, Turing patterns occur in other natural systems – for example, the wind patterns formed in sand, the atomic-scale repetitive ripples that can form during growth of bismuth crystals, and the uneven distribution of matter in
galactic disc. Although Turing's ideas on morphogenesis and Turing patterns remained dormant for many years, they are now inspirational for much research in
mathematical biology. It is a major theory in
developmental biology; the importance of the Turing model is obvious, as it provides an answer to the fundamental question of morphogenesis: "how is spatial information generated in organisms?". The result is a pattern of gene activity that changes as the shape of the tooth changes, and vice versa. Under this model, the large differences between mouse and vole molars can be generated by small changes in the binding constants and diffusion rates of the BMP and Shh proteins. A small increase in the diffusion rate of BMP4 and a stronger binding constant of its inhibitor is sufficient to change the vole pattern of tooth growth into that of the mouse. Experiments with the sprouting of
chia seeds planted in trays have confirmed Turing's mathematical model. ==Classic example: radiolarian shells==