Counting the tiger’s stripes
S Ananthanarayanan
| 06 January, 2016
(Photo: SNS)
Evolutionary biologists have
explanations for why animals have patterns on their coats or even birds on
their wings, but the mechanism by which the patterns actually come about has
not been understood. The legendary Alan Turing created a mathematical framework
of how chemical interactions could lead to the periodic expression of growth
factors, but it took many decades before the possible substances involved could
be identified.
The patterns on the coats or animals
are a familiar sight and clearly help them blend with the surroundings — for
concealment, for safety or for greater stealth in the hunt. It is not difficult
to understand that the chance appearance of features like the stripes on the
tiger or the zebra or the spots on the leopard would have conferred a survival
advantage, and genes that made for these features soon became the rule for the
species.
But genes lead to particular
proteins and tissue types, so how do colours get distributed? How, in fact,
does tissue grow in specific shapes and how do patterns, or even limbs, grow in
specific directions?
In a celebrated 1951-52 paper,
Turing considered a simplified case of just a few cells and how chemical agents
within the cells may interact and diffuse from one cell to another. He
considered, first, a case of just two cells and, then, a case of a ring of
cells, where agents that he called morphogens, or generators of form, would
interact and then diffuse from cell to cell at different speeds, reinforcing or
cancelling their effects at different places. He simplified the case to be
studied so that the mathematics of the way mechanical or electrical
disturbances spread could be applied to the progress of morphogens and he found
that the effects of the chemical agents manifested in waves, with crests and
troughs, just like waves in a vibrating string or electromagnetic waves.
The theory, when applied to the
animal called the hydra, a simple, tubular creature that has a ring of
tentacles at one end, led naturally to the regularly spaced concentration
of morphogens in the ring that forms an end of the hydra, to give rise to the
ring of tentacles. Although without speaking of what the morphogens may be,
Turing’s theory of “reaction and diffusion” was able to explain the black and
white patches on Friesian cows, which he was interested in. The theory was
carried forward by others and the stripes that appear on the back of the tiger
are now understood as a pattern of pigmentation laid down by periodic waves of
diffusion of chemicals in the animal’s embryo.
More complex patterns appear by
considering more sets of waves. Just as one set of waves can lead to a stripe
pattern, like the waves at the seaside, two sets of waves that cross each other
would lead to a distribution of points or peaks, leading to a spotted pattern,
like the coat of the leopard. The wave pattern is seen to be affected by
different factors that include the dimensions of the area in which they occur,
which determine the “boundary conditions”. This, in turn, leads to more
conclusions about the spots or stripes on the tails of the great cats. The
tiger has stripes on its body and tail and the leopard has spots on the body
and the tail.
The cheetah has spots on the body
and stripes on the tail, but no animal has spots on the tail and stripes on the
body. The reason seems to be the space available for two sets of waves,
which lead to spots. If these could exist in the tail, for a spotted one, then
they can certainly exist in the body, and the body must be spotted, rather than
striped!
Recent studies of the ridges on the
roof of the mouth of the common mouse have now identified a hormone called
Fibroblast Growth Factor and a protein called Sonic Hedgehog as the chemicals
involved. Other studies of the shapes of the cactus plant and the role of a
hormone called auxin, by Allen Newel at the University of Arizona, have found
that three sets of waves could be the reason for cells to grow in particular
shapes. Tom W Hiscock and Sean G Megason at Harvard University note that there
can be other drivers, besides the Turing process, of periodic phenomena, like
cell-based mechanisms or mechanical properties, like elasticity, of a system.
They have just published in the journal Cell Systems an integration of the
different models in the form of a combined mathematical formulation to find
bases for more features. Why the stripes of the tiger are vertical, while those
of the zebra can be both ways, for example, or and even why the fingers grow
along the length of the arm and not any other way.
The work of Newel, in fact, connects
the waves of morphogens with some fascinating numbers that are found to appear
in the plant kingdom. The Fibonacci series is a list of numbers where each is
formed by adding the previous two, like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,
55, 89, 144… The interesting property of this series is that ratio of each pair
of successive numbers gets closer and closer to 1.61803398874… the so called
Golden Ratio, which appears in geometry, aesthetics, music, art and
architecture. The ratio of successive numbers in the Fibonacci series, in fact,
gets closer and closer to an infinitely long, non-repeating decimal number that
actually cannot be expressed exactly as a ratio of two integers (and is hence
known as an “irrational number”).
Series: 0 1 1 2 3 5 8 13 21 34
89 Ratio: 0 1 2 1.5 1.666 1.6 1.625 1.615 1.619 1.618
Fibonacci numbers are also found to appear everywhere in nature. One typical
instance is the placement of leaves along the stalk of a plant. The angular
separation of the leaves, as one goes up the stalk, is found to form a Fibonacci
series. The fact that the ratio of successive numbers in the series tends to be
an irrational number has an important consequence — that no leaf in the plant
will ever find itself exactly above or below another leaf. This property
accurately optimises the function of the leaves to receive the best of sunlight
or rainfall, to make for the energy efficiency of natural systems to have
evolved to be at the maximum that is possible.
While natural selection can explain
how plants, and even snail shells, seashells, and a host of natural shapes have
selected the most efficient and economical dimensions, the question always
arises of how do species ensure that they grow according to these dimensions.
The work of Newel and colleagues provides an illuminating lead — they find that
the waves of the three morphogens involved in the cactus plant are connected by
the Golden Ratio; the number of wave crests of the third wave is actually the
sum of the numbers in the first two waves!
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