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Modelling Pigmentation Patterns


Animals demonstrate a staggering range in pigmentation colour and pattern. Different skin markings have developed for a variety of reasons; for example, the characteristic spots of the leopard, right, allow the animal to be effectively camouflaged, whereas the bright colours of posion arrow frogs provide a warning against potential predators. Certain animals, such as chameleons or flounders, have the ability to rapidly change colour and blend in with different backgrounds. Such changes are termed physiological colour changes and are controlled by response to nerve or hormonal signals. Other colour changes occur on a far slower time scale, either via increased deposition of pigment or via increased numbers of pigment cell, and are termed morphological colour changes.

Pigmentation in the Koran Angelfish

A striking example of the latter occurs in species of marine angelfish, such as the Koran angelfish ( Pomacanthus semicirculatus) right. As the fish matures from juvenile to adult, the pigmentation changes. Young fish display a sequence of white curved stripes on a dark background. As the fish grows and doubles in length, the number of these stripes also doubles, with new stripes inserting between the older stripes. A further change takes place from juvenile to adult. The juvenile pattern fades, and the adult pattern consisting of small white spots on a gray background appears. Related species demonstrate similar juvenile stages, yet the adult pattern can be very different.

Modelling fish pigmentation patterns

The first attempt to model this process was made by Kondo and Asai (1995), who considered a Turing mechanism on a growing domain. Such mechanisms, originally proposed by the British mathematician Alan Turing, consist of a system of reacting and diffusing chemicals which interact in a manner to produce spatially varying patterns. By incorporating a growing domain, Kondo and Asai demonstrated how a doubling sequence could emerge, reminiscent of the stripe doubling behaviour of juvenile semicirculatus

Growth in Two-Dimensions

The one-dimensional model proposed by Kondo and Asai demonstrates the process by which the number of "pigment stripes" double as the domain doubles in length. To investigate whether this same behaviour is found in a more realistic geometry, we have performed detailed numerical simulations (K.J. Painter, H.G. Othmer and P.K. Maini, 1999) on a two-dimensional domain. The movie on the right demonstrates a stripe doubling sequence in two dimensions: (length scale indicated by x-axis). Thus a robust sequence of stripe doubling, as seen on the body of the angelfish can be generated by the reaction diffusion model.

Under certain conditions, however, the robustness of the stripe doubling sequence fails. A simulation where we have considered a slightly larger initial domain size is shown on the right. Although early behaviour indicates the same stripe doubling behaviour, this eventually breaks down to form a pattern of convoluted stripes. Such "labyrinthian" patterns are commonly observed in nature; for example, on species of fish species such as the wrasse.

Incorporation of cell movement and realistic body growth

In addition to exploration into the effect of the two-dimensional geometry, we have further explored the process of pigmentation in Pomacanthus semicirculatus by incorporating the effect of pigment cell movement and a more realistic reflection of body growth and domain shape. The movie on the right demonstrates how the hypothesis of a chemotactic-type mechanism by which pigment cells organise themselves in the skin can account for further details of the pigmentation process. Stripes insert in a gradual manner in the early growth and, as the growth of the fish slows, the pattern reorganises into a spot pattern. This is consistent with the patterning described above.

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