¿ìè¶ÌÊÓƵ

Animals come in complicated shapes, but these are almost always regular,
not random. They can come in striking colours, and they, too, often form
geometric patterns such as spots, stripes or dapples. And there are numerical
patterns: for example, the human upper arm has a single bone (the humerus),
there are two bones in the forearm, irregular but distinguishable rows of
three followed by four in the wrist, and sets of five for the fingers. Is
this 1-2-3-4-5 sequence a coincidence? Or do mathematical patterns lie behind
the biological ones?

Physics uses mathematical equations to describe the behaviour of space,
time and matter. The flow of a fluid, for example, is governed by a set
of equations which physicists have understood for nearly two centuries.
The regular patterns that form in a moving fluid, such as whirlpools or
ocean waves, are mathematical consequences of these equations, and in this
sense the equations explain the patterns. Are there equations that similarly
explain the shapes and patterns of animals? Is there a tiger equation whose
solutions are stripes?

An orthodox explanation of the form and colouring of a tiger is that
they are completely specified by the DNA of its genome. Various sub-sequences
of its DNA specify the proteins from which the tiger is made and direct
them to where they are used. Some proteins are pigments, and give the tiger
its stripes. So the sequence of DNA bases in the genome might be seen as
the formula for a tiger.

But just as it takes more to build a car than an engineer’s blueprint,
it takes more to build a tiger than just a list of instructions for what
proteins to make and where to put them. The proteins also have to be made,
and put in place. Both these processes are governed by the laws of physics
and chemistry. The remarkable mathematical regularities in the form of living
creatures suggests that these laws may have a major influence on the creatures’
form, rather than just being passive carriers for genetic instructions.

Shape and pattern are two aspects of morphology – the name given to
form in its most general sense. The change of morphology as an organism
develops is called morphogenesis, and DNA is the blueprint for this process.
What we need is a ‘morphogenetic equation’ to describe how the physics and
chemistry interact with the DNA instructions. Such an equation would include
some features of the organism’s biology, chemistry and physics, but to be
useful it would have to simplify those features. It would be a model of
the broad sweep of development, not a description of every tiny biological
detail.

What might such an equation look like? In 1917, in On Growth and Form,
D’Arcy Thompson explained the shape of a jellyfish by an analogy with drops
of gelatin falling through water. Implicitly, he was modelling jellyfish
development by using the equations of fluid dynamics. His ideas have an
immediate appeal to anybody interested in patterns, but they tend to fall
down when it comes to the biological details. Nonetheless, he had an important
point: it is not surprising that animal and plant development should follow
geometric rules, because we live in a geometric universe. You don’t need
instructions in DNA to tell water to behave like a liquid, or salt crystals
to be composed of cubes. Does the natural geometric structure of the world
have implications for morphogenesis?

In 1952 the role of geometry in the process was picked up by the mathematician
and computing pioneer Alan Turing. He argued that a system of chemical
substances reacting together and diffusing through tissue could explain
the formation of pat-terns. Specifically, he devised a set of ‘reaction-diffusion’
equa-tions to describe the distribution of chemicals within the tissue.
When Turing solved these equations he found that patterns form spontaneously
when the homogeneous state – in which concentrations of the chemicals are
identical everywhere – becomes unstable. Any slight lack of uniformity in
the distribution of chemicals will spread rapidly. You might expect this
to produce a random patchwork of chemicals, but the patches are linked by
diffusion, so they arrange themselves into coherent spatial patterns resembling
spots, stripes and other geometric textures.

A mechanical analogy illustrates how these ‘Turing patterns’ form. Cut
out a thin strip of plastic and place it on its long edge, like a fence,
between two books about a centimetre apart. Grasp both ends of the strip
between your fingers and gently push them towards each other, and the strip
will change from its uniform, unbent state, and buckle into a surprisingly
regular series of waves. The local buckling under pressure is like the instability
caused by a chemical reaction; the analogue of diffusion is the transmission
of elastic forces through the material. The result is a regular pattern
of waves. Turing found the same thing, but with chemical waves, not elastic
ones.

A good example of real-world Turing patterns is the Belousov-Zhabotinskii
(BZ) reaction. If a particular cocktail of chemicals is thoroughly mixed
together and placed in a shallow dish, they at first form a uniform blue
layer, which suddenly turns reddish-brown. After a few minutes, for no apparent
reason, a few tiny blue spots appear. They grow, and their centres turn
red. As the blue rings move outwards, and the red centre spot expands with
them, new blue spots emerge at their centres. Soon the dish is filled with
concentric rings of red and blue, all slowly growing and colliding with
each other. Until this reaction was documented by B. P. Belousov in 1958
and by A. M. Zhabotinskii in 1963, most chemists didn’t believe that cyclic
chemical oscillations were possible, because they produced an ordered pattern
from what seemed to them to be an utterly disordered state in which there
is no pattern. Biologists, being accustomed to the cyclical chemical processes
that occur in living cells, had less trouble accepting the BZ reaction.
And mathematicians took to it like ducks to water, seeing it as a key example
of ‘nonlinear’ dynamics.

Turing saw that similar chemical waves in the early stage of an organism’s
development might act as ‘prepatterns’, laying down a kind of template or
scaffolding for its subsequent development. If pigments are deposited according
to the peaks and troughs of parallel waves, you get stripes. More complex
waves produce spots, and so on. Form can be controlled in a similar way,
by growing bumps and dents according to the chemical prepattern. Turing
showed in particular that a circular ring of cells can form wavy patterns
of several kinds. He compared these to the hydra, a tiny creature. It has
a bulging, cylindrical body with tentacles at one end. The tentacles are
regularly spaced, like the peaks and troughs of Turing’s chemical waves.

This all looked promising, and biologists turned to other examples
where Turing patterns appeared to be involved in development. John Maynard
Smith and K. C. Sondhi showed that hairs on fruit flies occur in a variety
of Turing patterns whose genetic variants are also Turing patterns. It is
hard to explain this if patterns are arbitrary consequences of DNA codes.
Why should natural selection prefer Turing patterns?

However, this initial finding was not repeated for systems such as feather
development. When feathers are grown from tissue at different temperatures,
the changes in their patterns do not fit Turing’s equations. Another problem
is that in many cases what appear in adults to be Turing patterns must have
been laid down when the organism’s shape was quite different. For example,
the ‘eyes’ on a peacock’s tail look like BZ rings; but any chemical pattern
that caused them would have been formed when the budding feathers were tiny
cylinders. The pre-pattern required inside these cylinders is nothing like
a BZ ring.

Trouble with Turing

Moreover, chemists were having trouble creating the static chemical
patterns required by Turing’s theories: the best they could achieve were
mobile patterns such as those of the BZ reaction. The essence of Turing’s
approach was that mathematics drives the chemistry, which in turn drives
the biology. Unfortunately, the chemistry wasn’t working.

By the 1970s, most biologists had become bored with finding Turing patterns
that turned out to be false alarms, and had moved on. Instead, they concentrated
on the DNA code and how it is expressed. Lewis Wolpert of Middlesex Hospital
Medical School developed the theory of ‘positional information’, in which
each cell is equipped with the equivalent of a map to tell it where in the
organism it is, and a book of instructions to tell it which genes need to
be switched on once it has found out from the map where it is. The map is
provided by changes in chemical concentrations or ‘chemical gradients’.
The book is the DNA code in the nucleus of every cell of a given organism.

Mathematicians, however, realised that all systems like Turing’s – ‘mechano-chemical’
equations describing the interaction of chemical changes and tissue growth
– would produce the same general range of patterns. The specifics of particular
equations were unimportant, so they were not too worried by the failure
of Turing’s equations. What mattered to them were the common features of
the whole class of equations, which were the key to the problem of pattern
formation. This viewpoint led to a general principle of pattern formation
called symmetry-breaking.

An object, pattern, or mathematical system is said to have a particular
symmetry if, after being transformed in a particular way, it looks exactly
the same as it did to begin with. For example, if you rotate a square by
90 degrees it ends up looking exactly the same as it did to start with.
That transformation – rotation through 90 degrees – is a symmetry of the
square. But if you rotate it through 45 degrees it ends up in a quite different
orientation, so rotation through 45 degrees is not a symmetry of a square.
A circle, however, has more symmetries: it is symmetric under rotation through
any angle, and under reflection in any diameter.

The simplest way for a symmetric system to behave is to retain its symmetry:
for a spherical cell, for example, to remain spherical. However, systems
can – and usually do – break their symmetry, changing state from a symmetric
one to a less symmetric one. For example, a spherical cell can buckle, forming
a dent at one end. Just such a change, triggered by tiny instabilities,
occurs in the early development of a frog embryo. The plastic strip mentioned
earlier does something similar. Before buckling, the strip has the symmetry
of translation: it looks exactly the same if it is translated (or slid)
along its length. But a buckled strip, with its series of waves, has fewer
symmetries. It looks the same only when translated by a whole number of
wavelengths. The waves – that is, the patterns – break some of the symmetries
of the unbuckled strip.

Symmetry-breaking explains the apparent production of order from disorder
in the BZ reaction. The homogeneous state has more symmetry than the patterned
one: it looks the same no matter how it is transformed geometrically. This
symmetry is a kind of order, but we tend not to see it as such. We are more
impressed by discrete repetitions; we see the stripes of the tiger but not
the cloudless sky behind it. So instead of a breaking of symmetry – which
is what actually occurs – we see an apparently paradoxical increase of pattern.

An extensive theory of this kind of emergence of pattern has been pioneered
by Stuart Kauffman and collected together in his recent book The Origins
of Order. Here he criticises the orthodox approach, with its emphasis on
ever more detailed aspects of DNA chemistry. He argues: ‘Both the prepattern
con-cept and the latter-day theory of positional information rely on a conceptual
separation between positional-information assess-ment and the subsequent
interpretation of that information by the cell, with no provision whatsoever
for the obvious possibility that the very interpretation made by the cell
might feed back and modify the information. Yet . . . such phenomena are
common, not rare.’

Turing’s ideas are coming back into vogue, but in a more subtle guise.
Brian Goodwin, a biologist at the Open University, has studied ‘mechano-chemical’
models of development which incorporate not only the reaction and diffusion
of chemicals, but the response to these of the shape of the tissues in which
they lie. In these models, as a creature develops, its older parts become
relatively fixed in form. If the creature begins with circular symmetry,
then the parts of the creature which form first also have circular symmetry.
Eventually the symmetry breaks, say via one of Turing’s waves, and then
the creature develops a number of equally spaced bulges that grow into branches,
tentacles or petals. So the symmetry of the creature changes as you cast
your eyes along the tissue from old to new – just like a hydra.

Goodwin’s equations successfully describe one key step in the morphogenesis
of Acetabularia, a single-celled marine alga. The creature begins as a spherical
egg, which puts out a root-like structure and a stalk. The stalk grows,
and produces a ring of small hairs, called a whorl. The tip continues growing
from the centre of the whorl, producing more whorls; then it develops a
capped structure. Goodwin’s equations relate the formation of the whorl
to a broken symmetry in calcium distribution within the organism. This affects
local growth rates and hence its shape.

More realistic biology

Philip Maini, a biomathematician at the University of Oxford, has generalised
Turing’s scheme to include more realistic biology, writing down equations
for a complex hierarchy of processes in which patterning mechanisms at one
level regulate those at higher levels. In a two-tier model of this type,
an existing prepattern modifies a subsequent reaction-diffusion process.
One application is the development of the skeleton in vertebrate limbs –
like the 1-2-3-4-5 pattern of bones in the human arm. The figure on the
next page shows a chemical prepattern generated by Maini’s equations: it
has the same basic structure, a reliable and stable sequence of 1, 2, 3,
. . . elements.

A similar model recently devised by R. Dillon and H. G. Othmer of the
University of Utah is consistent with what has been found in experiments
on the effects of grafting developing tissue. Growth occurs in a particular
set of cells, the ‘progress zone’, and is maintained by a chemical factor
produced either within the progress zone or in a specialised group of cells
on its boundary, called the ‘zone of polarising activity’, or ZPA. If cells
from this zone are transplanted to the front edge of the developing limb,
parts of the skeleton tend to double up.

A ‘one-tier’ model in-volving only a single reaction-diffusion scheme,
as in Turing’s original work, gives incorrect predictions for these experiments.
For example, it predicts that a transplant from the ZPA carried out at one
particular stage of development will lead either to elimination of the humerus,
its duplication, or the formation of a mirror-image duplicate. But in 1987
Wolpert and A. Hornbruch showed experimentally that the humerus is never
eliminated and only occasionally duplicated. When it is, the duplicate can
be either a normal humerus or a mirror image. The one-tier theory also predicts
that multiple ZPA grafts should create fused or very thick digits, and this
is not observed either. The two-tier model fits the experiments much better.

Kauffman has devised yet other types of mathematical model for aspects
of morphogenesis. One of the simplest of his models is a line of cells,
each equipped with a rudimentary internal dynamic governing cell division.
Cells ‘communicate’ with each other using microhormones – chemical products
of their genes. Computer simulations show that patterns form spontaneously
in such systems. What we see here is a combination of the dynamic ideas
of Turing with the controlling role of DNA. It opens up an enormous range
of new mathematical research.

Ironically, one of the early problems with Turing’s scheme has also
now been overcome: chemists Vincent Castets, Etienne Dulos, Jacques Boissonade
and Patrick de Kepper of the University of Bordeaux have produced complex,
but static Turing patterns. Harry Swinney and colleagues at the University
of Texas at Austin have found that a simpler way to do this is to slow the
system down by using a gel instead of a solution. In retrospect it is obvious
that living tissue resembles a gel much more closely than a liquid. They
have been able to create static but irregular patterns with a very ‘organic’
look to them. John Pearson at Los Alamos National Laboratory has performed
num-erical simulations of reaction-diffusion equations representing a reaction
taking place inside a vessel fed with chemicals from an outside source,
in which he observed a vast range of patterns, some regular, some chaotic.
Which pattern occurs depends upon just two parameters: the rate at which
the chemicals react, and the rate at which they are fed in.

Early mathematical equations for development of organisms were too
far removed from real, detailed biology to provide accurate models. The
current emphasis on DNA goes too far the other way: it explains the production
of proteins, but it does not adequately explain how they are assembled to
form an organism, or – crucially – why nature so often prefers mathematical
patterns of Turing’s symmetry-breaking variety.

To see the difference between the two approaches, and how both fall
short of reality, imagine a vehicle (corresponding to a developing organism)
driving through a landscape (representing all the possible forms that the
organism might take, with valleys corresponding to common forms and peaks
to highly unlikely ones). In models like Turing’s, once you have set the
vehicle rolling, it has to follow the contours of the landscape. It can’t
suddenly decide to change direction and head uphill if the ‘natural’ dynamic
is to go straight on. But in the conventional view of the role of DNA, any
destination is possible, given the right instructions, and no particular
destination is preferred.

The true picture, however, must combine genetic ‘switching’ instructions
and free-running mechano-chemical dynamics. An organism cannot take up
any form at all: its morphology is constrained by its dynamics – the laws
of physics and chemistry – as well as by its DNA instructions. But where
several different lines of development are consistent with the dynamical
laws, the DNA instructions can make arbitrary choices between them. The
new mathematical models are finally beginning to put these two aspects of
development together. It is not DNA alone, or dynamics alone, that controls
development. It is both, interacting with each other.

We still can’t yet write down a tiger equation. But we’re getting a
much clearer idea of what one would look like.

Jack Cohen is a reproductive biologist and author of The Privileged
Ape.

Ian Stewart’s latest book Fearful Symmetry: is God a Geometer? is published
by Penguin.