Turing Patterns in Animal Coats
Junping Shi
Alan Turing (1912-1954)
One of greatest
scientists in 20th
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/
James Murray
(author of books: Mathematical Biology)
Emeritus Professor
University of Washington, Seattle
Oxford University, Oxford
http://www.amath.washington.edu/people/faculty/murray/
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 reactiondiffusion 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.
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
Reaction-diffusion systems
Domain: rectangle
Boundary conditions:
head and tail (no flux),
body side (periodic)
The full reaction-diffusion system:
Solution of the system:
“Theorem 1”: Snakes always have striped (ring)
patterns, but not spotted patterns.
Turing-Murray Theory: snake is the example of b/a is
large.
Snake pictures (stripe patterns)
“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)
Spotted body and striped tail or legs
Cheetah (upper), Okapi (lower)
Tiger (upper), Leopard (lower)
Spotted body and striped tail
Genet (left), Giraffe (right)
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.
(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.
Natural Patterns of cos(kx)
cos(x): Valais goat
(single color: f(x)=1, a lot of examples)
Cos(2x): Galloway belted Cow
cos(2x): Giant Panda
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
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.
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.
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)!
Chlorite-Iodide-Malonic Acid (CIMA) reaction
CIMA reaction
spots
CIMA reaction
stripes
Fish skin
Leopard body
Fingerprint
Zebra stripes
Other related researches
Patterns of sea shells
Patterns of tropical fishes