Spots and Stripes - Mathematics of Life

Mathematics of Life (2011)

Chapter 13. Spots and Stripes

Painters, poets and writers have long been captivated by the extraordinary beauty of animals in the wild. Who could fail to be moved by the power and elegance of a Siberian tiger, the ponderous enormity of an elephant, the haughty pose of a giraffe or the popart 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. Since that is in molecular form, an awful lot of information can be confined to the interior of a single cell.

If you do the sums, however, it’s clearly not that straightforward. An elephant has many more cells in its body than its DNA has bases. The cells come in many different types. They have to be fitted together in the right way ... Have you ever considered how complicated an accurate, cell-by-cell map of an elephant would be? Let alone putting in the complex organelles and cytoskeleton inside each cell.

Part of the answer is the background of physical and chemical laws (see p. 192). These laws operate automatically; in fact, you can’t stop them doing so. A tiger’s DNA doesn’t have to contain information about how chemical bonds fit together, how cells that are sticky adhere to one another, how electrical impulses pass along its neurons. All these things are implicit in the laws of nature.

However, that’s only part of the answer. It’s not enough to assign anything you don’t find in DNA to the action of physical laws, because this fails to answer any of the big questions. In particular: how does DNA regulate the complex processes of chemistry, to turn that egg – seething as it may be with potential – into a gigantic striped cat?

This is where mathematics can play its part in the great scientific enterprise. We don’t, yet, have accurate mathematical models for all the processes that convert egg into tiger, but we do have models that provide insight into various stages of that process. Mathematics can help us understand simple features of a growing embryo, such as gastrulation, where a spherical mass of tiny cells turns inwards on itself, the first stage in giving the eventual animal an inside and an outside. There are many other applications of mathematics to biological development, but I want to skip a few stages to look at the most obvious features of many animals: their markings.

The mathematician who opened up this area was an Englishman, Alan Turing. Turing is famous for his wartime work at Bletchley Park breaking the Enigma code, for developing the Turing test for artificial intelligence, and for establishing the undecidability of the halting problem for Turing machines. (That is, there is no systematic way to determine whether any given computer program will terminate with an answer, or go on for ever – for example, by repeating the same instructions over and over again indefinitely.) From these activities it might appear that Turing was a pioneer in computer science and cryptography, which is true, and that he specialised in these areas – which is false. Another mathematical interest of his was the markings on animals. Spots, stripes, dappled patterns ...

For half a century, mathematical biologists have built on Turing’s ideas. His specific model, and the biological theory of pattern-formation that motivated it, turns out to be too simple to explain many details of animal markings, but it captures many important features in a simple context, and points the way to models that are biologically realistic (Figure 53, see over).

It all started in the early 1950s, when Turing became puzzled about the geometry of animal form and markings: the stripes on tigers and zebras, the spots on leopards, the dappled patches on Friesian cows. Although these patterns do not display the exact regularity that people often expect from mathematics, they have a distinct mathematical ‘feel’. We now know that the mathematics of pattern formation can produce irregular patterns as well as regular ones, so the irregularities are not evidence that mathematical models of animal patterns are wrong.

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Fig 53 Left: Boxfish. Right: Calculated Turing pattern.

Turing presented his theory of pattern formation in a celebrated paper entitled ‘The chemical basis of morphogenesis’, published in 1952.1 He modelled the formation of animal markings as a process that laid down a ‘pre-pattern’ in the developing embryo. As the embryo grew, this pre-pattern became expressed as a pattern of protein pigments. He therefore concentrated on modelling the pre-pattern. His model has two main ingredients: reaction and diffusion. Turing imagined some system of chemicals, which he called morphogens, ‘form-generators’. At any given point on the part of the embryo that eventually becomes the skin – in effect, the embryo’s surface – these morphogens react together to create other chemical molecules. The chemicals and their reaction products can also diffuse, moving across the skin in any direction.

Chemical reactions require nonlinear equations, ones in which – for example – twice the input does not yield twice the output. Diffusion can sensibly be modelled by simpler linear equations: twice as much of some molecule, diffusing from some given location, leads to twice as much everywhere. The most important result to emerge from Turing’s ‘reaction – diffusion’ equations is that local nonlinearity plus global diffusion creates striking and often complex patterns (see Figure 54). Many similar equations can produce patterns, not just the specific ones proposed by Turing. What distinguishes them is which patterns occur in which circumstances.

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Fig 54 Above: Regular Turing patterns: spots and stripes. Below: Less regular Turing patterns.

Hans Meinhardt, at the Max Planck Institute for Developmental Biology in Tübingen, has made extensive studies of many variants of Turing’s equations. In his elegant book The Algorithmic Beauty of Seashells he examines many different kinds of chemical mechanism, showing that particular types of reaction lead to particular kinds of pattern. For example, some of the chemicals inhibit the production of others, some activate the production of others. Combinations of inhibitors and activators can cause chemical oscillations, resulting in regular patterns of stripes or spots. Meinhardt’s theoretical patterns match those found on real shells.

Use of the word ‘pattern’ does not imply regularity. Many striking patterns on seashells are complex and irregular. Some cone shells have what seem to be random collections of triangles of various sizes (Figure 55, see over). Mathematically, patterns of this kind can arise from Turing-like equations; they are fractals, a complex kind of geometric structure popularised by the French-American mathematician Benoît Mandelbrot working at Yale in the 1960s. Fractals are closely associated with dynamical chaos – irregular behaviour in a deterministic mathematical system. So the cone shell combines mathematical features of order and chaos in one pattern.

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Fig 55 Cone shell pattern.

The Scottish mathematician James Murray of the Universities of Washington and Oxford has applied Turing’s ideas, suitably modified and extended, to the markings on big cats, giraffes, zebras and related animals.2 Here the two classic patterns are stripes (tiger, zebra) and spots (cheetah, leopard). Both patterns are created by wave-like structures in the chemistry. Long, parallel waves produce stripes. A second system of waves, at an angle to the first, can cause the stripes to break up into series of spots. Mathematically, stripes turn into spots when the pattern of parallel waves becomes unstable. Pursuing this led Murray to an interesting theorem: a spotted animal can have a striped tail, but a striped animal cannot have a spotted tail. The smaller diameter of the tail leaves less room for stripes to become unstable, whereas this instability is more likely on the larger-diameter body.

In 1995 the Japanese scientists Shigeru Kondo (Kyoto University Centre for Molecular Biology and Genetics) and Rihito Asai (Kyoto University Seto Marine Biological Laboratory) used Turing’s equations to make a startling discovery about the colourful tropical angelfish Pomacanthus imperator. Along two-thirds of its body run parallel stripes of yellow and purple. Stripes are an archetypal Turing pattern, but there is an apparent technical difficulty. In this particular case the mathematics of Turing patterns makes a surprising prediction: the stripes of the angelfish have to move. Stable steady patterns are not consistent with the mathematical formalism.

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Fig 56 Moving stripes on angelfish. Top: Angelfish whose stripes have a Y-shaped branch (circled). Middle: Observations at intervals of 6 weeks show that the branch moves. Bottom: Theoretical movement according to Turing’s model.

Moving stripes? It seemed bizarre. However, Kondo and Asai decided to keep an open mind. Maybe angelfish stripes do move. To find out, they photographed specimens of the angelfish over periods of several months. They found that the stripes do, in fact, migrate across its surface (see Figure 56).3 As they move, certain defects in the pattern of otherwise regular stripes, known as dislocations, break up and reform exactly as Turing’s equations predict.

Pattern formation is predicted by a variety of mathematical models, many of which give rise to the same catalogue of possible patterns – those that occur in nature as stripes in ocean waves, on tigers and on angelfish, for instance. There ought to be some deeper, general reason for these similarities – indeed, for the patterns themselves. There is, and it’s called symmetry breaking.

In normal parlance, something is symmetric if it has an elegant, balanced form. More specifically, we talk of an object having bilateral or mirror symmetry if it looks the same as its reflection in a mirror. The human body comes close to this, although a person and their mirror image differ in minor details, and the arrangement of internal organs can differ markedly.

There are more complicated types of symmetry; my favourite example is the starfish. The most common starfish found around the British Isles has five arms, each arm more or less identical to the others, all of them arranged in something close to a five-pointed star. A starfish does have mirror symmetry, but the most obvious symmetry is not a reflection, but a rotation: each arm is separated from the next by 72°, one-fifth of a turn.

Symmetry is not about parts of some shape being similar to one another: it is about the effect of some transformation on the whole shape. Does the shape appear to stay the same when reflected? If so, it has mirror symmetry. Does it appear to stay the same when rotated? Then it has rotational symmetry. As we’ve seen, this idea has become a formal mathematical theory: group theory, so named because any two symmetries combine to yield another one. The set of all symmetries of an object is its symmetry group.

Not only can the solutions of mathematical equations have symmetries; the equations themselves can have symmetries. The algebraic formula x+y is symmetric in the two numbers x and y: if you swap them, you get the same sum. More generally, an equation is symmetric if applying some transformation to the symbols yields the same equation. By the middle of the last century’s first decade, Albert Einstein had already noted the importance of this type of symmetry in the laws of physics, the basic equations for space, time, energy and matter. He insisted that these laws must be the same at every point of space and at every instant of time.

It may seem surprising that solutions can have less symmetry than the equations they solve. Working in Paris a few years before Einstein, Pierre Curie suggested that this should not be possible because of a simple physical principle: symmetric causes produce equally symmetric effects. Curie’s principle seems to rule out a change of symmetry from equations (‘causes’) to solutions (‘effects’). But if the universe itself had that much symmetry, looking exactly the same in every location and at every instant, it would be uniform and unchanging, like a universe composed entirely of cosmic custard. So there must be some escape clause. There is: it’s symmetry breaking, and its consequences are both beautiful and far-reaching.

How can symmetry break? More precisely, how can a solution of an equation have less symmetry than the equation itself? If an equation is symmetric, then some transformation of the symbols yields the same equation. So applying that transformation to a solution also yields a solution.

However, it need not be the same solution. This is the loophole that makes symmetry breaking possible.

In our algebraic example, the sum x+y of two numbers is symmetric in x and y. If we swap x and y, it becomes y+x; formally, this looks different, but it always yields the same answer. That statement remains true for specific choices of x and y, such as 17 and 36. With these choices 17+36=36+17. But this does not imply that 17 and 36 remain the same if we swap them. On the contrary, 17 turns into 36, which is different. Of course, 36 correspondingly turns into 17 – but again, those two numbers are different. So the solution x=36, y=17 is different from the solution x=17, y=36. Curie’s principle is correct as it stands for equations that have only one solution. But when there are many solutions, which turns out to be very common, the principle as originally stated fails. Instead, all we can assert is that whenever we have a solution, we can obtain other solutions by applying symmetry transformations.

We’ve seen that symmetry is very common in mathematical models of the natural world, so symmetry breaking should also be common. And so it is. In fact, it provides a very general mechanism for the formation of nature’s patterns. Those patterns are the explicit realisations, in specific physical systems, of the abstract symmetries that are implicit in the laws that describe those systems.

Multiple solutions open the door to symmetry breaking. What shoves the mathematics through that door is instability.

On the left of Figure 57 (see over) is a satellite photo of part of the Rub’ al-Khali desert in Saudi Arabia, often known by its English name, the Empty Quarter. The stripes are huge sand dunes. Although there are irregularities, the stripes are pretty much parallel to one another and equally spaced. The pattern is caused by strong and steady trade winds, which in this case blow in the same direction as the stripes: accordingly, these are known as longitudinal dunes. Striped dunes can also occur when the wind blows at right angles to the stripes: the result looks much the same, but the different mode of formation requires a different name: transverse dunes. The right-hand photo shows transverse dunes on Mars.

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Fig 57 Left: Longitudinal dunes in the Empty Quarter (width is roughly 40 km). Right: Transverse dunes on Mars.

There are many other dune patterns, including wonderful crescent shapes and stars, but striped dunes are the simplest. Now, it might seem that the strong patterns in the dunes must reflect equally strong patterns in the way the wind blows, but longitudinal and transverse dunes typically form when the wind is steady. In fact, the steadier the wind, the more regular the dune pattern.

Of course once the dunes have formed, they affect the wind in their locality, but you don’t need a striped wind to create striped dunes.

Why not?

To avoid confusion, let me focus on transverse dunes. Longitudinal dunes have a similar, perhaps simpler, explanation, but the transverse ones will serve my purpose better later on.

Imagine a perfectly flat desert, over which a steady wind blows, at the same speed and in the same direction everywhere. No such desert can exist in reality, of course, but this ideal case homes in on the essence of the puzzle: how a uniform wind can lead to a striped desert. The key is symmetry, and how it breaks. My idealised desert is very symmetric. In fact, the only departure from the symmetry of a mathematical plane is the existence of a preferred direction, that of the wind. So the transformations that preserve the system include no rotations, and the only reflectional symmetries occur for mirrors aligned with the wind direction. However, I can slide the entire desert north, south, east or west, and the system – and therefore also its mathematical representation – will look exactly the same.

If the behaviour of the sand in response to the wind were as symmetric as the system itself, there would be no patterns. The state of the sand, and in particular the height of the desert surface, would be identical at every point. So the sand would stay flat, and the symmetry of the system would not break.

If we inject just one element of realism, however, this picture changes dramatically. Sand is not smooth: it comes in tiny grains. Those grains poke above the surface in places, and there are gaps between them. The surface departs from perfect planarity by a very tiny amount. And those departures are pretty much random. Such a system has no symmetry at all. However you transform the desert, the sand grains will not repeat the exact same pattern.

The difference is tiny, but what actually happens in a (fairly) flat desert subjected to (fairly) constant winds is huge and dramatic. Great mounds of sand appear, many thousands of times the size of the sand grains that cause the departure from exact symmetry. And, very commonly, the dunes have large-scale patterns: transverse dunes, for example, are arranged in regularly spaced parallel stripes like waves on a beach. Ocean waves move, and so do transverse dunes: they move at right angles to the stripes. But they do so very slowly, as sand is blown off the crest of each dune and is deposited somewhere ahead of the crest.

Parallel rows of dunes have quite a lot of symmetry. They form a striped pattern in the plane, and this can be slid sideways along the direction of the stripes. It is also unchanged if it is slid perpendicular to the stripes, through a distance that is any integer multiple of the distance between adjacent stripes – the actual spacing, twice that, three times, and so on.

This is remarkable. The symmetry of the typical pattern of behaviour resembles neither that of the perfect idealised model, with complete translational symmetry, nor the small but total asymmetry of real sand grains. Instead, it lies somewhere in between. And it turns out that the idealised model can reproduce exactly that kind of pattern, if it is tweaked very slightly to mimic the random but tiny deviations from perfection introduced by the granular nature of sand. In such a model, the sand remains very close to a perfect plane provided the wind speed is sufficiently low – low enough not to disturb any sand grains. But for higher wind speeds this undisturbed state becomes unstable. Any tiny imperfection, however small, grows. If a grain of sand pokes up slightly more than its neighbours, the wind picks it up and blows it somewhere else. The resulting hole creates a bigger difference in height, and the grains on either side become more exposed and also get blown away. The hole grows, and the displaced sand piles up.

However, the sand does not pile up at random. Instead, feedback between the shape of the desert surface and the movement of the wind homes in on a stable pattern: waves of sand and waves of wind. In the right range of wind speeds, that pattern is transverse dunes.

Where has the symmetry of the system gone?

That could be a silly question. Symmetry isn’t a physical substance that can’t be annihilated without creating something else. Symmetry is a concept, a property. But in this case it’s not a silly question, because the missing symmetry has gone somewhere. It exists as unrealised potential. That pattern of stripes could have formed in any position. Its actual position was a consequence of the first grain of sand starting to move. Potentially, that grain of sand might have been anywhere in the desert. If so, the entire process would have taken place some distance away – so the crests of the potential dunes would have formed somewhere in the troughs of the actual ones. The symmetry of the system is not so much broken as shared among the entire set of possible solutions.

A model that predicts the formation of transverse dunes cannot specify the exact location of their crests and troughs. If we moved the entire pattern forward ten metres, it would also satisfy the equations of the model. In fact, if we waited long enough the wind would actually move the pattern into that position. At any given moment, the location of the crests and troughs depends on the past history of the dunes.

There are profound mathematical techniques for calculating stabilities and working out which patterns appear when fully symmetric states become unstable. They are very technical, but in general terms they suggest that nature prefers not to break symmetry much more than it has to. Typical patterns arising in a symmetric system through spontaneous symmetry breaking tend to have a lot of symmetry rather than just a little. This statement can be made precise, in any particular instance: if taken too literally it is false, but on the whole not by much.4 When it is false, the mathematics tells us what to expect instead.

What about animal form? Form and pattern are two aspects of the same thing: morphology. Both form and pattern seem to be set up in embryos through a chemical pre-pattern induced by a morphogen. The pre-pattern sits there until the organism reaches an appropriate stage of development, at which point the varying chemical concentrations in the pre-pattern trigger either the formation of pigment proteins, creating visible patterns, or cellular changes, creating form.

There are disagreements about the precise mechanism that sets up the pre-patterns and about the precise mechanism that turns pre-patterns into visible patterns or form. Many of the chemical changes involved clearly have a genetic component – particular genes ‘switch on’ simultaneously in blocks of cells, stimulating the production of some pigment, or causing the cells to modify their mechanical or chemical properties. Pre-patterns alone cannot explain morphology: their interaction with genes presumably can.

Meinhardt has applied Turing equations, together with simple genetic ‘switching’ ideas, to the formation of somites in developing vertebrate embryos.5 Somites are equally spaced pairs of blocks of differentiated tissue which eventually form the basis of the backbone. They come into being, one pair at a time, starting from the head end of the animal. On general mathematical grounds, however, Meinhardt was led to a counter-intuitive mathematical prediction. The diffusing chemical waves that trigger the formation of somites should originate not from the front of the animal, but from the back.

Why? Imagine ocean waves carrying floating debris up onto a beach, as the tide is going out. One wave reaches up to the top of the beach and deposits a strip of driftwood and seaweed. The next wave doesn’t reach quite as far, thanks to the falling tide, so it leaves a line of debris further down the beach. Step by step, the waves travelling up the beach create a series of stripes of debris that accumulates down the beach. That way, the beach between the waves and the debris remains pristine, so the waves can deposit more debris using exactly the same process. The existing debris doesn’t get in the way.

For somites, the waves are waves of concentration of some morphogen, and the debris that is deposited is a series of genetic ‘switches’ that change the state of the relevant cells. Again, the somites that have already formed would interfere with incoming waves unless the waves came from the back. Meinhardt made this prediction over fifteen years ago, as a consequence of Turing’s mathematical scheme. The new-found ability to make genetic switching visible has shown that he was basically correct.

Despite these remarkable achievements, Turing’s equations for animal markings have unsurprisingly – given the pioneering nature of his work – not been a complete success: they often fail to predict experimental details, such as what happens when you grow organisms at different temperatures. Turing was the first to attempt this kind of modelling, and he kept the model as simple as he dared: at that time solutions had to be calculated by hand. His theory has spun off many more sophisticated modern descendants, each of which attempts to address such deficiencies.

Whatever the details, though, the spatio-temporal patterns of activity of the genes are taken, virtually unchanged, from Turing’s mathematical pattern-book. So it looks as if DNA guides morphogenesis along certain lines, but what then happens is heavily dependent on the laws of physics and chemistry too, hence upon context.

We know a lot about how DNA makes proteins, but in comparison we know very little about how those proteins are marshalled together to create an organism. The problem of biological development is one of the biggest scientific challenges we face. How do organisms regulate their own growth patterns? What defines an animal’s body plan? How is its form transferred from the DNA drawing-board to the developmental assembly line? The answers will involve chemistry, biology, physics and mathematics. And they will be nowhere near as straightforward as just obeying a list of genetic instructions.

The availability of powerful computers provides an alternative to classical continuum models like Turing’s. Instead of approximating animal tissue by an infinitely divisible region of space, we can model the tissue cell by cell. We can study how cells affect their neighbours, how their internal dynamics and genetic regulatory systems conspire with the external world to determine their fate.

At the start of this chapter I mentioned gastrulation, the stage in embryonic development at which the growing mass of cells pretty much turns itself inside out. This process looks mathematical: it typically starts when a circular arc forms on the surface of a hollow sphere of cells; this becomes the lip of the inwardly folding portion of the surface. Many people have built mathematical models of gastrulation, but biologists know that the entire process is regulated by a few genes, and what they’d like to know is how the genetics interacts with the geometry.

In the 1990s, working at the Artificial Intelligence Laboratory in Zurich, Peter Eggenberger Hotz devised a series of mathematical models that incorporated the role of genes.6 A typical strategy is to start with a grid of cells, represented as adjacent spheres, and write down a list of dynamical rules for how the genes present in these cells interact with those in neighbouring cells. A gene is activated (or inhibited) only when the total concentration of incoming signalling molecules exceeds a threshold level. The cell then responds to the activity of this gene, either by sending out its own signalling molecule, making a cell adhesion molecule that connects it to a nearby cell, or by equipping that cell with a receptor that can respond to an incoming signalling molecule. Additionally, the cell may respond by dividing or by dying.

The model is then simulated on a computer, following the dynamical rules and seeing what transpires. Depending on the choices made for the dynamics, the collection of cells may grow and develop in interesting ways. The shape that the mass of cells takes up is calculated using a fairly realistic set of equations for a collection of objects that interact through stickiness and elasticity, two key properties of a real cell.

As the mass of simulated cells develops, genetic signals cause the cells to change position in space. These changes in turn affect the activity of the genes and the signals they produce. The feedback loop between genes and form leads to the eventual shape of the mass of cells. The model can be used for many purposes, for example to explore the effect of a morphogen on morphology. One version of the model mimics gastrulation in a hollow sphere of cells.

Because the entire model exists in a computer, it is possible to investigate features of the structure that are difficult or impossible to observe in a real developing embryo, such as the concentrations of signalling molecules at various locations. This is a major advantage of all models. The corresponding disadvantage is that they are not the real system. As Eggenberger remarks, ‘Putting evolutionary techniques on firm ground, where the mechanisms can be understood, is itself a major reason to investigate the potential of such systems.’