XIII C: Role of Mathematics behind amazing patterns of spots and stripes in nature.
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]
"The most beautiful gift of nature is that it gives one pleasure to look around and try to comprehend what we see."-------Albert Einstein.
A spotted Hyena.
Spots on the Giraffe.
Strip on the (predator) Tiger.
All the images in this article are from Public Domain.
[To continue]
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]
"The most beautiful gift of nature is that it gives one pleasure to look around and try to comprehend what we see."-------Albert Einstein.
Two spectacular patterns
we find abundantly in the natural world are Spots and Stripes. Let us look at
the leopards, ladybird (a species of beetles with bright colour and dotted
wings), guinea fowls, some woodpeckers, owls, and so many creatures which are
spotted whereas zebras, tigers, western bongo, an antelope with spiral horns,
hyenas, tiger, spider etc. which are striped.
These patterns have many
explanations and of late biology is undergoing a renaissance as scientists
apply mathematical ideas to old theories.
These patterns have an
evolutionary explanation: they have functions which increase the chances that
youngsters of the patterned animals will survive to reproduce. One function of
animal patterns is camouflage, for instance, a leopard, that is inconspicuous to its prey
catches more prey than other hunting animals.
Another functioning is self-defense, for instance, a ladybird is likely to be
attacked by predatory birds that hunt by sight. If it has bold warning colours,
and is also distastefully bitter or poisonous or mimics other distasteful
insects, a young bird may see a warning patterned insect like ladybird and try
to eat it, but it will only do this only once; very soon it will spit out the
bitter insect; the other ladybirds will remain unmolested. The young leopards
and lady birds inheriting genes that somehow create spottedness, survive. But
while these evolutionary and functional arguments explain why these animals
need their patterns, they do not explain how the patterns formed.
Now let us look for an
explanation, how these patterns are formed by application of mathematics to
biology or in other word by “biomathematics”, a new branch of science.
Biology essentially deals with
plants, animals and insects, but five great revolutions have changed the way of
thinking about life. The invention of microscope, the systematic classification
of our planet’s living creatures, evolution, the discovery of the structure of
DNA* and the Gene**. Now, a sixth is on its way ---- mathematics.
Mathematics has played a leading
role in the physical sciences for centuries, but in the Life Sciences it has
little role to play and not used as a routine tool for analysing data. However, it
is moving towards centre stage, providing new understanding of the complex
processes of life.
The ideas of math involved here
are varied and novel; they range from pattern formation to chaos theory. The
chaos theory is actually a branch of mathematics that deals with the complex
systems whose behavior is highly sensitive to slight changes in conditions, so
that small alterations can give rise to strikingly great consequences. These
ideas are helping us to understand not just what life is made from, but how it
works, on every scale from molecules to entire planet--- and possibly beyond.
[*DNA
: The full form is : Deoxyribonucleic acid. It is the
molecule that carries the genetic instructions used in the growth, development,
functioning and reproduction of all known living organisms and many viruses.
**Gene : A gene is the basic physical and
functional unit of heredity. Genes, which are made up of DNA, act as
instructions to make molecules called proteins in humans. Genes vary in sizes
from a few hundred DNA bases to move than two million bases. The Human Genome
Project has estimated that humans have between 20,000 and 25,000 genes. Every
person has two copies of each gene, one inherited from each parent.]
The biggest revolution in modern
biology was the discovery of the molecular structure of DNA, which turned
genetics into a branch of chemistry,
centred on a creature’s genes – sequences of DNA code that specify the protein
from which the gene is made . But when attention shifted what gene do in an
organism, the true depth of the pattern of life aggravate. Listing the proteins
that make up a living being, say a cat, does not tell us everything we want to
know about cats.
A creature’s genome (a complete
set of genes) is fundamental to its form and behavior, but the information in
genome no more tells us everything about the creature than a list of components
tells us how to build furniture from its pieces packed flat in the box. What
matters is how those components are used, the processes that they undergo in a
living creature. And the best tool we possess for finding out what process do
is mathematics. The resulting discipline of biomathematics can play an
important role in explaining animal markings such as spots and stripes.
Painters and writer have long
been captivated by the extraordinary beauty of wild creatures, particularly,
with their spots and stripes. Who fail to be moved by the power and elegance of
a Royal Bengal Tiger, the ponderous bulk of an elephant, the haughty poise of a
giraffe, or artistic stripes of a Zebra? Yes each of these animals began life
as a single cell, a fusion of sperm and egg. But do you squeeze an elephant
into a cell?
When the paradigm of DNA as
information was at its height, the answer seemed simple. The answer is you
can’t squeeze an elephant in to a cell but what you can cram into an egg is the
information required to make an elephant. It is astonishing that lots of molecular
information can fit inside a cell. However an elephant has many more cells than
its DNA constituent units, and they have to be assembled in the right way. An
accurate cell by cell map of an elephant would never fit into its DNA. So there
must be something else going on.
The secret of animal markings
was first cracked by Alan Turing (1912 – 1954), a pioneer British Computer
Scientist, mathematician, logician, cryptanalyst and theoretical biologist. He
was celebrated for helping to crack the Enigma code* during the second world
war (1st Sept. 1939 to 2nd Sept.1945) and for pinning down
the limitations of computers. In 1952 he suggested that a biochemical process
produces something known as “pre-pattern” in developing embryo, which is later
expressed as the real-life pattern of protein pigments, such as the melanin
(pigment that gives human skin, hair, eyes their colour. Dark Skinned people
have more melanin in their skin than light skinned people have).
*[Enigma
code: During Second World War, Germans
believed that its secret codes for radio messages were indecipherable to the
enemy, Allies. But the meticulous work of code breaker based at Bletchley Park,
England, cracked the secret of German wartime communication and played such an
important role in the final defeat of Germans that the then British Prime
Minister told to the king George VI, that “It was thank to ULTRA that we won
the war”, ULTRA being the code name at information obtained from such high
level German sources.
The Enigma story began
in 1920s when the German using an ‘Enigma’ machine, invented by the German
engineer Arthur Scherbius at the end of World War-I (July 28, 1911 to 11th
November,1918), developed for business market began to communicate in
unintelligible coded messages (ULTRA),]
But how does the pre-pattern
form? Turing thought it arose through a series of reactions among molecules
that he called morphogen which may be termed as “form – generators”. At each
point on the part of the embryo that eventually becomes the skin, morphogens react
together to create other molecules.
Simultaneously, these molecules
and their reaction products also defuse from cell to cell through the relevant
regions of the embryo. It is this
process that leads to the creation of pre-patterning chemical information that
tells the cells where to put pigment as they develop, like invisible writing.
As embryo grows, the physical pattern appears.
There are evidences that
mathematics has a role to play here. The way this process unfolds can be
formulated as a system of mathematical equations. The most important result to
emerge from Turing’s equations is that to emerge combination of reaction and
diffusion in any particular animal can create striking patterns: spots,
stripes, or more complex markings.
Turing’s specific model is to be
too simple to explain many details of animal markings, but it has many
important features in a simple context and lead to more realistic theories.
The word “pattern” does not
imply regularity and should follow mathematical rules. Many sea shell patterns
are complex and irregular. These shells because of their bright colours, rich
variety of shades and designs attracted us most and collecting them from sea
shore is a great hobby for many of us. Cone shells or cone snails have more
than 600 species and found abundantly along the sea shore. But be careful in
collecting them, for some are very deadly. Now look at their patterns : they
are almost of conical shape and that seems to be random collections of triangles
of various sizes, yet it turns out that patterns of this kind are common in
Turing – like equations. In fact, they are fractals, a complex type of
geometric structure brought to our notice by Benoit Mandelbrot in 1970s, which
we have found earlier in fractal patterns.
Here we see some spectacular pictures of dots and stripes on
some animal bodies:
dots on a plain grey Guinea Fowl [Pic. No.
16a], beautiful spots on a Cheetah [Pic. No.16b], Spotted Hyenas [Pic. No.
16c], Spotted Deer [Pic. No. 16d], Spots on majestic Giraffe [Pic. No. 16e],
the bright beautiful stripes on the predator Tiger and the prey Zebra [Pic.
No.16f(i) and(ii) respectively], the dazzling stripes on fishes [Pic. No. 16g].
Pic.No16a.
Pic.No16a.
Dots on a grey Guinea Fowl.
pic.No16b.
Spots on a Cheetah.
pic.No16c.
Pic.No16d.
A spotted Deer.
pic.No16e.
Pic.No16f(i)
Pic.No16f(ii).
Stripes on the (prey) Zebra.
Pic.No16g.
Dazzling stripes on a fish.
We have seen many beautiful
stripes on canvas of the skins of many animals but do not know that these
stripes are sometimes not stationary and may move over bodies of some of them.
Turing’s equations contained an astonishing prediction of it.
In 1995, the Japanese Scientists
Shigeru Kondo and Rihito Asai applied Turing’s equations to the beautiful
tropical angelfish Pomacanthus, a genus of it, which displays striking yellow
and purple stripes. Turing’s model made a surprising prediction the stripes of
the angelfish move along its body, which is not the case for an adult Zebra
whose stripes are fixed.
Though it seemed unlikely, yet
they observed and photographed specimens of the angelfish for a period of
several months and found the stripes slowly migrated across its surface.
Moreover, defects in the pattern of otherwise regular stripes broke up and
reformed exactly as Turing’s equations predicted. The cause behind this
phenomenon of drifting of stripes is that the pigment proteins leaked from
cell, drifting from fish’s tail towards its head. This does not happen in
animal whose stripes are fixed. Maths can predict whether its markings will
move, if the size of the animals and other factors are known.
Reference: Internet All the images in this article are from Public Domain.
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