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# The formula of life: the maths behind a zebra's stripes, and a leopard's spots

Biology used to be about plants, animals and insects, but five great revolutions have changed the way that scientists think about life: the invention of the microscope, the systematic classification of the planet's living creatures, evolution, the discovery of the gene and the structure of DNA. Now, a sixth is on its way - mathematics.

Maths has played a leading role in the physical sciences for centuries, but in the life sciences it was little more than a bit player, a routine tool for analysing data. However, it is moving towards centre stage, providing new understanding of the complex processes of life.

The ideas involved are varied and novel; they range from pattern formation to chaos theory. They are helping us to understand not just what life is made from, but how it works, on every scale from molecules to the entire planet - and possibly beyond.

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 proteins from which the gene is made. But when attention shifted to what genes do in an organism, the true depth of the problem of life became ever more apparent. Listing the proteins that make up a cat does not tell us everything we want to know about cats.

A creature's genome is fundamental to its form and behaviour, but the information in the genome no more tells us everything about the creature than a list of components tells us how to build furniture from a flat-pack. 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 processes do is mathematics.

The resulting discipline of "biomathematics" is a huge topic, so I am going to limit myself to three examples - animal markings such as spots and stripes, the structure of viruses, and an ecological puzzle called the paradox of the plankton.

First, animal markings. Painters, musicians and writers have long been captivated by the extraordinary beauty of wild creatures. Who could fail to be moved by the power and elegance of a Siberian tiger, the ponderous bulk of an elephant, the haughty poise of a giraffe, or the pop-art stripes of a zebra? Yet each of these animals began life as a single cell, a fusion of sperm and egg. How do you cram an elephant into a cell?

When the paradigm of DNA as information was at its height, the answer seemed simple: you don't. What you cram into an egg is the

information required to make an elephant. An awful lot of molecular information can fit inside a cell. However, an elephant has many more cells in its body than its DNA has bases (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 the animal's DNA. There must be something else going on.

Alan Turing - celebrated for helping to crack the Enigma code during the Second World War and for pinning down the limitations of computers - tackled a special instance of this puzzle: animal markings. In 1952 he suggested that a biochemical process produces something known as a "pre-pattern" in the developing embryo, which is later expressed as the real-life pattern of protein pigments, such as the melanin that gives our skin its colour.

But how does the pre-pattern form? Turing thought it arose through a series of reactions among molecules that he called morphogens: "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 diffuse from cell to cell through the relevant regions of the embryo. It is this process that leads to the creation of the "pre-pattern"; chemical information that tells the cells where to put pigment as they develop, like invisible writing. As the embryo grows, the physical pattern appears.

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 the particular combination of reaction and diffusion in any given animal can create striking patterns: spots, stripes, or more complex markings.

Turing's specific model turned out to be too simple to explain many details of animal markings, but it captured many important features in a simple context and pointed the way to more realistic theories. The developmental biologist Hans Meinhardt has used variants of Turing's equations to study patterns on seashells and work out which types of chemical reaction lead to which kinds of pattern.

The word "pattern", by the way, does not imply regularity. Many seashell patterns are complex and irregular. Some cone shells have what seem 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 public notice by Benoît Mandelbrot in the 1970s. He described fractals as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole".

In 1995 the Japanese scientists Shigeru Kondo and Rihito Asai applied Turing's equations to the beautiful tropical angelfish Pomacanthus imperator, which displays striking yellow and purple stripes. Turing's model made a surprising prediction: the stripes of the angelfish move along its body (unlike those on an adult zebra, for example, which are fixed).

It seemed wildly unlikely, but when Kondo and Asai photographed specimens of the angelfish over periods of several months, they found that the stripes slowly migrated across its surface. Moreover, defects in the pattern of otherwise regular stripes, known as dislocations, broke up and re-formed exactly as Turing's equations predicted. They did this because the pigment proteins leaked from cell to cell, drifting from the fish's tail towards its head. (In animals whose stripes are fixed, this does not happen; but once the size of the animal and other factors are known, the maths can predict whether its markings will move.)

## Disease-bearing footballs

The structure of viruses - a major cause of disease in human beings, animals and plants - can also be explained by maths. Viruses are larger than most biological molecules, but you could pack a million of them into a single bacterium. They outnumber human beings ten septillion (1025) to one. More than 5,000 distinct types are known, and there may be millions more. They consist of genetic material wrapped in a protein coat, and each virus has a definite structure. Most are either icosahedral or helical: shaped like a football or shaped like a spiral staircase.

Understanding their structure could suggest new cures for disease. The main mathematical tool is geometry, but with a twist: it is calculated in more than three dimensions.

The classical geometry of Euclid is formulated in two dimensions (the plane) or three (space). The shapes of viruses lead to something less familiar: geometry in six dimensions. I am not suggesting that viruses come from the sixth dimension - that might make a good title for a sci-fi movie, but little more. However, the mathematics of six dimensions turns out to be a good way to understand viruses in three dimensions, because the complicated three-dimensional shapes observed in viruses turn out to be "shadows" or slices of simpler, six-dimensional shapes.

The climax of Euclid's classic geometry text, the Elements, identifies five regular solids: cube, tetrahedron, octahedron, dodecahedron and icosahedron. The names, cube aside, refer to the number of faces: four, six, eight, 12 and 20, respectively. The cube has square faces, the dodecahedron has pentagonal ones, and the other three are made from equilateral triangles.

Euclid's elegant icosahedron, devoid of practical application for more than 2,000 years, turned out to be just the right shape for making a virus. The big question was: why?

Part of the answer is energy. Virus coats are typically constructed from many copies of a single protein molecule, a different one for each virus. A collection of such molecules has the least energy - which nature finds desirable - if it is as close as possible to being a sphere.

Virus coats can't form exact spheres - try to fit a hundred tennis balls together to make a smooth sphere - but they do the best they can. Among Euclid's solids, the icosahedron is closest to a sphere. An icosahedron with its corners cut off is even more spherical, hence its use

for most footballs used in international games.

The protein coats of icosahedral viruses are made from 20 triangles, and each triangle is an array of protein units, like the balls at the start of a game of pool or snooker. In 1962 the biologists Donald Caspar and Aaron Klug realised they had seen arrangements like this before, in the work of the architect Buckminster Fuller.

Fuller is renowned for the "geodesic dome", a roughly spherical enclosure made by fitting a large number of triangular panels together (think of the Eden Project dome in Cornwall, although this simplifies the structure by using hexagons and pentagons). Caspar and Klug discovered that most viruses have a similar geometry to Fuller's domes, which form structures known as pseudo-icosahedra - "pseudo" because each of the 20 triangular faces is subdivided into further triangles.

Their geometry predicts very specific numbers of protein units, such as 32, 42, 72, 92, 162, 252 and 362, forming corners on the surface. The theory agreed very well with real viruses - for example, infectious canine hepatitis has 362 units, and the human wart virus has 72. In both cases, the units are arranged like a geodesic dome. However, there are exceptions to the Caspar-Klug theory, such as simian virus 40, which can cause tumours in apes and human beings.

Just over ten years ago, the German-born mathematician Reidun Twarock was pondering this problem. Her answer was to develop a more general theory of the geometry of viruses based on the symmetries of the icosahedron. Unlike with the geometry of Euclid, however, she used shapes in six dimensions, not three.

This is not (quite) as complicated as it sounds, as "dimensions" here broadly means "variables in the equation". Imagine the solar system: if you want to plot the position of the earth, you need to know where it is in space (three dimensions) and how fast it is moving through space (another three). If you then wanted to plot the position of the sun, you would need another six. For the moon, another six.

As such, the mathematical laws that govern motion refer to a space with 18 dimensions. The actual configuration of the bodies at any instant lies in ordinary three-dimensional space, and is a kind of "shadow" of the 18-dimensional description.

We are quite used to squashing down dimensions in this way from three to two. Imagine drawing a tree on paper, for example. Moving from six or 18 dimensions down to three uses the same idea, just with more variables.

Twarock used this idea to imagine 3D virus structures as shadows of simpler structures in higher dimensions. For example, if you stack a large number of cubes together like a 3D chessboard, and then slice through the stack in the right way, you get an elegant tiling pattern with both triangles and hexagons (see above). The original stack uses just one shape, the cube, but two different shapes appear in the slice.

Or, to put it another way, imagine looking at a cat, first from the front, and then from the side. In two dimensions, these shapes are very different. But look at a living, breathing, three-dimensional cat, and the shapes of its different parts make sense.

Twarock used a similar trick for the protein units of icosahedral viruses. An icosahedron is highly symmetrical, in a technical sense - there are 120 ways to rotate or reflect an icosahedron so that it occupies its original space. If the arrangement of units comes from a pattern in a higher-dimensional space, this new pattern should also have the same 120 symmetries. There is a well-developed branch of mathematics, called group theory, which tackles this kind of question. It led to a specific list of patterns in a space of six dimensions.

The resulting theory improves on that of Caspar and Klug. It accounts for the exceptional structures of simian virus 40 and others. There are potential medical implications, too. One way to attack a virus is to interfere with its assembly process, and the geometry of the fully assembled virus provides clues about potential weak points in this process.

Moreover, icosahedral viruses sometimes change shape and form tubes, which are not infectious. A treatment that changed the shape from icosahedron to tube might interfere with virus replication and prevent disease. If scientists could "reprogram" a virus to churn out protein units that made it tubular, it might render it harmless.

## Paradox of the plankton

Our final example comes from the uppermost layers of the oceans. The waters there teem with plankton, organisms ranging from microscopic creatures to small jellyfish. Many are the larvae of much larger adults. They all occupy the same kind of habitat and compete for much the same resources.

Yet something isn't right here. The principle of competitive exclusion, introduced in 1932 by the Russian biologist Georgii Gause, states that the number of species in any environment should be no more than the number of available "niches", or ways to make a living. If two species try to compete for the same niche, then natural selection implies that one of them should win. This is the paradox of the plankton: the niches are few, yet the diversity is enormous - many thousands of species. The solution to the paradox comes from chaos theory.

Classical dynamics - based on Newton's laws of motion - focus on steady states, where nothing changes as time passes, and periodic states, where the same sequence of events repeats over and over again. A rock is in a steady state if it is not moving and we ignore erosion. The cycle of the seasons is periodic, with a period of one year.

In the 1960s, however, mathematicians realised that the conventional view had completely missed another, more puzzling kind of behaviour: chaos. This is behaviour so irregular that it may appear random, but it is not.

It might seem that such outlandish behaviour has no place in nature, but chaos is entirely natural. It arises whenever the dynamics of a system mix everything up, much as kneading dough mixes the ingredients. It seems outlandish only if you are looking for solutions that can be expressed by neat, tidy formulas. Those are rare, and nature doesn't need them.

The mathematical model that justifies Gause's principle assumes the populations do not fluctuate over time. But this takes the "balance of nature" metaphor too seriously. Ecosystems must be stable, but a stable system need not remain in exactly the same state for ever, just as a stable economy is not one in which everyone has exactly the same amount of money as they did yesterday. A population is stable if fluctuations remain within fairly tight limits. It is not necessary that there be no fluctuations at all.

Chaos theory solves our oceanic puzzle; it allows for erratic fluctuations, but places limits on their size. Chaotic fluctuations let different species utilise the same resources, but at different times. They still avoid direct competition, but they don't do it by one of them winning and killing off all the others. They do it by taking turns to gain access to the same resource. This is how chaos solves the paradox of the plankton.

## Spherical cows

There is an old joke about a farmer who hires a group of mathematicians to help him improve his milk yield. When they present him with their report, the opening sentence reads: "Consider a spherical cow." This story exposes a misunderstanding about mathematical models. They don't have to be an exact representation of reality to be useful. A spherical cow is useless if you want it to give birth to a calf, but it might be a sensible approximation if you're wondering about the spread of a skin disease.

Part of the art of biomathematics is the selection of useful models. Another part is taking the biology seriously and not missing something crucial. But sometimes it is also necessary to try out a new idea in a simplified setting and see where that leads.

There is another old joke, about a drunk searching under a lamp post for his keys. "Did you drop them here?" "No, but this is the only place where there's enough light to look." The original context, in Computer Power and Human Reason by Joseph Weizenbaum, was an analogy with science, and his point was the exact opposite of the usual interpretation of the joke. In science, you have to search under the lamp post, or you'll never find anything. Even if the keys are somewhere along the road in the gutter, you might find a torch under the lamp post. Then you can search further afield.

I would be surprised if mathematics ever came to dominate biological thinking in the way it does physics, but it is rapidly becoming an essential part of the discipline: 21st-century biology makes use of mathematics in ways that no one would have dreamed of at the start of the 20th. By the time we get to the 22nd, mathematics and biology will have changed each other beyond all recognition, just as mathematics and physics did in the 19th and 20th centuries. Science is changing from a collection of villages to a worldwide community. Welcome to the global ecosystem of tomorrow's science.

*Ian Stewart is a professor of maths at the University of Warwick and the author of "Mathematics of Life" (Profile Books, £20)*

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**This article first appeared in the 25 April 2011 issue of the New Statesman, Easter special