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Turing Patterns in Animal Coats Junping Shi

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Alan Turing (1912-1954) One of greatest scientists in 20 th century Designer of Turing machine (a theoretical computer) in 1930’s Breaking of U-boat Enigma, saving battle of the Atlantic Initiate nonlinear theory of biological growth http://www.turing.org.uk/

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James Murray (author of books: Mathematical Biology) Emeritus Professor University of Washington, Seattle Oxford University, Oxford http://www.amath.washington.edu/people/faculty/murray/

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Murray’s theory Murray suggests that a single mechanism could be responsible for generating all of the common patterns observed. This mechanism is based on a reaction- diffusion system of the morphogen prepatterns, and the subsequent differentiation of the cells to produce melanin simply reflects the spatial patterns of morphogen concentration. Melanin: pigment that affects skin, eye, and hair color in humans and other mammals. Morphogen: Any of various chemicals in embryonic tissue that influence the movement and organization of cells during morphogenesis by forming a concentration gradient.

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Murray’s Theory (Cont.) The development of color pattern on the skin of mammal occurs towards the end of embryogenesis, but it may reflect an underlying pre-pattern that is laid down much earlier. (For zebra, the pre-pattern is formed around 21-35 days, and the whole gestation period is about 360 days.) To create the color patterns, certain genetically determined cells, called melanoblasts, migrate over the surface of the embryo and become specialized pigment cell, called melanocytes. Hair color comes from the melanocytes generating melanin, within the hair follicle, which then pass into the hair. From experiments, it is generally agreed that whether or not a melanocyte produces melanin depends on the presence of a chemical, which we do not know. Embryo of zebra fish

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Reaction-diffusion systems Domain: rectangle Boundary conditions: head and tail (no flux), body side (periodic)

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The full reaction-diffusion system: Solution of the system:

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“Theorem 1”: Snakes always have striped (ring) patterns, but not spotted patterns. Turing-Murray Theory: snake is the example of b/a is large.

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Snake pictures (stripe patterns)

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“Theorem 2”: There is no animal with striped body and spotted tail, but there is animal with spotted body and striped tail. Turing-Murray theory: The body is always wider than the tail. The same reaction-diffusion mechanism should be responsible for the patterns on both body and tail. Then if the body is striped, and the parameters are similar for tail and body, then the tail must also be striped since the narrower geometry is easier to produce strips. Examples: zebra, tiger (striped body and tail), leopard (spotted body and tail), genet, cheetah (spotted body and striped tail)

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Spotted body and striped tail or legs Cheetah (upper), Okapi (lower)Tiger (upper), Leopard (lower)

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Spotted body and striped tail Genet (left), Giraffe (right)

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Tail patterns of big cats Domain: tapering cylinder, with the width becoming narrower at the end. We still use no-flux boundary condition at the head and tail parts, and periodic boundary condition on the side. Predicted patterns: spots on the wider part, and strips on the tail part; all spots; or all strips.

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(a) (b) (c) Numerical simulations (d) Cheetah tail markings (e) Jaguar tail markings (f) Genet tail markings (g) Leopard tail markings Leopard : the spots almost reach the tip of tail, the pre-natal leopard tail is sharply tapered and relatively short; There are same number of “rings” on the pre-natal and post-natal tails; the sharply tapered shape allow the existence of spots on top part of tail; larger spots are further down the tail, and the spots near the body are relatively small. Genet: uniformly striped pattern; the genet embryo tail has a remarkably uniform diameter which is relatively thin.

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Natural Patterns of cos(kx) cos(x): Valais goat ( single color: f(x)=1, a lot of examples )

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Cos(2x): Galloway belted Cow

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cos(2x): Giant Panda

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Effect of scale on pattern very small domain: lambda is small, there is no spatial pattern, and the constant is stable. (small animals are uniform in color: squirrel, sheep, small dogs) medium size domain: lambda is not too large nor too small, and there are many spatial patterns. (zebra, big cats, giraffe) large domain: lambda is large, and there are patterns but the structure of the pattern is very fine. (elephant, bear) (a)small black cat (b) valais goat (c)giant panda (d)cow (e)giraffe (f) ? (g) elephant

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Criticisms on Murray’s work (book review by Leah Edelstein-Keshet) Adorable leopards have graced all three editions of Murray’s books. After all, it was his seminal work in the 1980s that led to the intriguing idea that the spots on those leopard skins were created by Turing-type reaction-diffusion systems [4, 5]. (In other words, he “earned his stripes”—or should we say spots—in this work.) Now, nearly two decades later, one has to wonder whether such systems exist per se in biological pattern formation. (After all, where are those interacting chemicals, with their just-so rates of diffusion and just-so kinetics? If molecular biology has not succeeded in uncovering such precisely tuned activator-inhibitor systems in embryonic development, why should we believe these “just-so” stories?) Let me hasten to add that this skepticism does not mean that I would skip over Turing’s work—far from it. Teaching the basics of Turing pattern formation is still as illuminating as ever, imparting insight into non-intuitive and interesting PDE phenomena. It further provides a relatively accessible “playground” for would-be applied mathematicians to practice their skills, both analytic and numerical. And it provides a clever and instructive model for how patterns might form.

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But given the decades of missing evidence, we should inject a note of caution against taking this mechanism too literally. Such temperance was only hinted at, but not fully realized, in Chapter 3: “The considerable circumstantial evidence that comes from comparing patterns generated by the model mechanism with specific animal pattern features is encouraging. The fact that many general and specific features of mammalian coat patterns can be explained by this simple theory does not, of course, mean that it is correct, but so far, they have not been explained satisfactorily by any other general theory” (p. 154). I would have preferred a stronger statement, to the effect that these models form a classical and historical foundation of mathematical biology, but that current and future generations should aspire to build higher.

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Waves in the Belousov-Zhabotinsky reaction Boris P. Belousov (Soviet Union, 1951,left) Anatol M. Zhabotinsky (Soviet Union, 1961,right) Chemical reactions can be oscillatory (periodic)!

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Chlorite-Iodide-Malonic Acid (CIMA) reaction CIMA reaction stripes Fingerprint Zebra stripes CIMA reaction spots Fish skin Leopard body

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Other related researches Patterns of sea shells Patterns of tropical fishes

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生物数学中的斑图生成 Pattern Formations in Math Biology Junping Shi 史峻平 Department of Mathematics College of William and Mary Williamsburg, VA 23187,USA 中国哈尔滨师范大学数学学院.

生物数学中的斑图生成 Pattern Formations in Math Biology Junping Shi 史峻平 Department of Mathematics College of William and Mary Williamsburg, VA 23187,USA 中国哈尔滨师范大学数学学院.

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