HOW THE LEOPARD GOT ITS SPOTS

How the leopard got its spots?

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**MATHEMATICS & BIOLOGY**

Alan Turing is a widely-known name, usually because of his role in code-breaking (specifically the Enigma code)
Did you know he was also a pioneer of the maths theories which determine what markings, if any, animals have?

Why are the coats of certain types of animals such as the leopard spotted, whereas the coats of others are striped (tiger, zebra). Why are the spots of the giraffe much bigger than and different from those of the leopard? Why do certain animals, such as the mouse and elephant not have a pattern? Why do some animals, such as the cheetah, jaguar, leopard and genet, have spotted bodies and striped tails, but there are no animals with striped bodies and spotted tails?

All of these questions now have a mathematical answer; they are the consequence of mathematical dynamics.
. The model in question describes the way in which two different chemical products react and are propagated on the skin : one which colors the skin, and one which does not color it.
More precisely, one which stimulates the production of melanin (coloring the skin); and one which inhibits this production.

Turing discovered two separate formulae & functions for the behaviour of such chemicals: combined, they form grid-like patterns of shapes.
Whether spots, stripes, or other patterns, these are determined & limited by one factor: size.
(Size here is two-dimentional, not three; as the surface area of a solid object is in only two dimensions.)

The same basic equation explains all of the patterns!

The tiger and the leopard have different patterns, although their bodies are similar in shape, because the formation of the patterns would not be produced at the same stage of growth of the embryo:
in the first instance, the embryo would be still small; in the other, it would be much bigger.

More precisely, the equations show that no pattern is formed if the embryo is very small; that a striped pattern is formed if the embryo is a little bigger; a spotted pattern if it is bigger yet; and again no pattern at all if it is too big.
This is why the mouse and the elephant do not have a pattern.

Regarding comparable surfaces: the shape/size of the surface makes a difference within the same animal.
Thus, if a surface is large enough to permit the formation of spots; and one area (e.g. the tail) has a long, cylindrical form; then the spots are transformed into stripes.

In this way, a unique system of differential equations seems to govern all the coat patterns that one finds in nature. The same type of equations also explain the patterns of the wings of butterflies, & certain colored patterns of exotic fish.

However, the processes of chemical diffusion which we have just mentioned (called a diffusion-reaction mechanism) have not yet been directly observed on the skin of animals: although certain indirect evidence seems to confirm their presence.
The chemical substances in question would be actually found in the epidermis or just below, and it is very difficult to detect them experimentally.
For the moment, then, this model remains a model, (although considerable circumstantial evidence seems to confirm it).

The history of this development of knowledge began in the early 50’s with two quite separate strands.
There was theoretical work by Alan Turing on Morphogenesis, not long before he died in 1954.
There was the very important experimental work of a Russian biophysicist, Boris Belousov.
Much of this was either unpublished or unrecognised for many years, & the theory of reaction-diffusion lay dormant until around 1968 when a number of strands came together.
A Russian biochemist named Anatoly Zhabotinsky had been improving on Belousov’s experimental work; and Western scientists learnt of it at a symposium in Prague in 1968.
Simultaneously, the famous Russian chemists Ilya Prigogine and Rene Lefevre, following up Turing’s work, formulated and analysed a model for a chemical self-organising reaction, which fitted in with some of Prigogine’s earlier work. This earlier work showed that the spontaneous creation of order is not forbidden by the Second Law of Thermodynamics, and later played a significant role in Prigogine’s 1977 Nobel Prize for Chemistry.
Finally, the English mathematical biologist James Murray brought it all together in 1988-89.

Further reading can be found at the following link:
http://plus.maths.org/content/how-leopard-got-its-spots

(The article was submitted to us by our reader ‘Aileen Tree’)
lifepotter.com feels copiously thankful to Aileen for her moral and intellectual support.

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