IAN STEWART
Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?
In this opening verse of William Blake’s “The Tyger” from his Songs of Experience of 1794, the poet is using “symmetry” as an artistic metaphor, referring to the great cat’s awe-inspiring beauty and terrible form. But the tiger’s form and markings are also governed by symmetry in a more mathematical sense. In 1997, when delivering one of the Royal Institution’s televised Christmas Lectures, I took advantage of this connection to bring a live tiger into the lecture theatre. I have never managed to create quite the same focus from the audience in any lecture since then.
Taking the term literally, the only symmetry in a tiger is an approximate bilateral symmetry, something that it shares with innumerable other living creatures, humans among them. A tiger viewed in a mirror continues to look like a tiger. But the markings on the tiger are the visible evidence of a biological process of pattern formation that is closely connected to mathematical symmetries. Nowhere is this more evident than in one of the cat’s most prominent, and most geometric, features: its elegant, cylindrical tail. A series of parallel circular stripes, running round the tail, has continuous rotational symmetries, and (if extended to an infinitely long tail) discrete translational symmetries as well.
The extent to which these symmetries are real is a standard modeling issue. A real tiger’s tail is furry, not a mathematical surface, and its symmetries are not exact. Nevertheless, our understanding of the cat’s markings needs to explain why they have these approximate symmetries, not just explain them away by observing that they are imperfect. For the past few decades, biology has been so focused on the genetic revolution, and the molecules whose activities determine much of the form and behavior of living organisms, that it has to some extent lost sight of the organisms themselves. But now that focus is starting to change, and an old, somewhat discredited theory is being revived as a consequence.
Sixty years ago, when Francis Crick and James Watson first worked out the molecular structure of DNA (with vital input from Rosalind Franklin and Maurice Wilkins), many biologists hoped that most of the important features of a living organism could be deduced from its DNA. But, as the molecular biology revolution matured, it became clear that instead of being some fixed blueprint, DNA is more like a list of ingredients for a recipe. An awful lot depends on precisely how those ingredients are combined and cooked.
The best tool for discovering what a process does if you know its ingredients and how they interact is mathematics. So a few mavericks tried to understand the growth and form of living creatures by using mathematical techniques. Unfortunately, their ideas were overshadowed by the flood of results appearing in molecular biology, and the new ideas looked old-fashioned in comparison, so they weren’t taken seriously by mainstream biologists. Recent new results suggest that this reaction was unwise.
The story starts in 1952, a year before Crick and Watson’s epic discovery, when the mathematician and computing pioneer Alan Turing published a theory of animal markings in a paper with the title “The Chemical Basis of Morphogenesis.” Turing is famous for his wartime code-breaking activities at Bletchley Park, the Turing test for artificial intelligence, and the undecidability of the halting problem for Turing machines. But he also worked on number theory and the markings on animals.
We are all familiar with the stripes on tigers and zebras, the spots on leopards, and the dappled patches on some breeds of cow. These patterns seldom display the exact regularity that people often expect from mathematics, but nevertheless they have a distinctly mathematical “feel.” Turing modeled 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 modeling the pre-pattern.
Turing’s model has two main ingredients: reaction and diffusion. He imagined some system of chemicals, which he called morphogens. 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. These reactions can be modeled by ordinary differential equations. However, the skin also has a spatial structure, and that is where diffusion comes into play. The chemicals and their reaction products can also diffuse, moving across the skin in any direction. Turing wrote down partial differential equations for processes that combined these two features. We now call them reaction–diffusion equations, or Turing equations.
The most important result to emerge from Turing’s model is that the reaction–diffusion process can create striking and often complex patterns. Moreover, many of these patterns are symmetric. A symmetry of a mathematical object or system is a way to transform it so that the end looks exactly the same as it started. The striped pattern on a tiger’s tail, for example, has rotational and translational symmetries. It looks the same if the cylindrical tail is rotated through any angle about its axis, and it looks the same if the tail (here we assume that it is infinitely long in both directions) is translated through integer multiples of the distance between neighboring stripes at right angles to the stripes.
Turing’s theory fell out of favor because it did not specify enough biological details—for example, what the morphogens actually are. Its literal interpretation also failed to predict what would happen in various experiments; for example, if the embryo developed at a different temperature. It was also realized that many different equations can produce such patterns, not just the specific ones proposed by Turing. So the occurrence of patterns like those seen in animals does not of itself confirm Turing’s proposed mechanism for animal markings. Mathematically, there is a large class of equations that give the same general catalog of possible patterns. What distinguishes them is the details: which patterns occur in which circumstances. Biologists tended to see this result as an obstacle: How can you decide which equations are realistic? Mathematicians saw it as an opportunity: Let’s try to find out.
Thinking along such lines, Jim Murray has developed more general versions of Turing’s model and has applied them to the markings on many animals, including big cats, giraffes, and zebras. Here the iconic patterns are stripes (tiger, zebra) and spots (cheetah, leopard). Both patterns are created by wavelike structures in the chemistry. Long, parallel waves, like waves breaking on a seashore, 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 pattern-making led Murray to an interesting theorem: A spotted animal can have a striped tail, but a striped animal cannot have a spotted tail.
FIGURE 1. Stripes and spots in reaction–diffusion equations on a cylindrical domain. Source: http://www-rohan.sdsu.edu/˜rcarrete/teaching/M-596_patt/lectures/lectures.html.
Hans Meinhardt has studied many variants of Turing’s equations, with particular emphasis on the markings on seashells. His elegant book The Algorithmic Beauty of Seashells studies many different kinds of chemical mechanism, showing that particular types of reactions lead to particular kinds of patterns. For example, some of the reactants inhibit the production of others, and some reactants 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 compare well with those found on real shells.
FIGURE 2. Cone shell pattern. Source: Courtesy of Shutterstock.com/Kasia
Stripes and spots can be obtained by solving Turing’s equations numerically, and their existence can be inferred by analyzing the equations directly, using methods from bifurcation theory, which tells us what happens if a uniform pattern becomes unstable. But why are we seeing patterns at all? It is here that symmetry enters the picture. Turing himself noticed that the tendency of Turing equations to create patterns, rather than bland uniformity, can be explained using the mathematics of symmetry-breaking. In suitably mathematical regions, such as a cylinder or a plane, Turing equations are symmetric. If a solution is transformed by a symmetry operation, it remains a solution (though usually a different one). The uniform solution—corresponding to something like a lion, the same color everywhere—possesses the full set of symmetries. Stripes and spots do not, but they retain quite a lot of them.
Symmetry-breaking is an important process in pattern formation, which at first sight conflicts with a basic principle about symmetries in physical systems stated by the physicist Pierre Curie: Symmetric causes produce equally symmetric effects. This principle, if taken too literally, suggests that Turing’s equations are incapable of generating patterns. In his 1952 paper, Turing includes a discussion of this point, under the heading “The Breakdown of Symmetry and Homogeneity.” He writes,
There appears superficially to be a difficulty [with the theory] … An embryo in its spherical blastula stage has spherical symmetry … A system which has spherical symmetry, and whose state is changing because of chemical reactions and diffusion, will remain spherical for ever … It certainly cannot result in an organism such as a horse, which is not spherically symmetrical.
However, he goes on to point out that
There is a fallacy in this argument … The system may reach a state of instability in which … irregularities … tend to grow. If this happens a new and stable equilibrium is usually reached…. For example, if a rod is hanging from a point a little above its centre of gravity it will be in stable equilibrium. If, however, a mouse climbs up the rod, the equilibrium eventually becomes unstable and the rod starts to swing.
To drive the point home, he then provides an analogous example in a reaction–diffusion system, showing that instability of the uniform state leads to stable patterns with some—though not all—of the symmetries of the equations.
In fact, it is the reduced list of symmetries that makes the patterns visible. The tiger’s stripes are visibly separated from each other because the translational symmetries are discrete. You can slide the pattern through an integer multiple of the distance between successive stripes—an integer number of wavelengths of the underlying chemical pattern.
Turing put his finger on the fundamental reason for symmetry-breaking: A fully symmetric state of the system may be unstable. Tiny perturbations can destroy the symmetry. If so, this pattern does not occur in practice. Curie’s principle is not violated because the perturbations are not symmetric, but the principle is misleading because it is easy to forget about the perturbations (because they are very small). It turns out that in a symmetric system of equations, instability of the fully symmetric state usually leads to stable patterns instead.
For example, if the domain of the equations is a circle, typical broken-symmetry patterns are waves, like sine curves, with a wavelength that is the circumference of the circle divided by an integer. On a rectangular or cylindrical domain, symmetry-breaking can lead to plane waves, corresponding to stripes, and superpositions of two plane waves, corresponding to spots.
Most biologists found Turing’s ideas unsatisfactory. In particular, his model did not specify what the supposed morphogens were. Biologists came to prefer a different approach to the growth and form of the embryo, known as positional information. Here an animal’s body is thought of as a kind of map, and its DNA acts as an instruction book. The cells of the developing organism look at the map to find out where they are, and then at the book to find out what they should do when they are in that location. Coordinates on the map are supplied by chemical gradients: For example, a chemical might be highly concentrated near the back of the animal and gradually fade away toward the front. By “measuring” the concentration, a cell can work out where it is.
FIGURE 3. Patterns formed from one or more plane waves.
The main difference in viewpoint was that the mathematicians saw biological development as a continuous process in which the animal grew organically by following general rules controlled by specific inputs from the genes, whereas to the biologists it was more like making a model out of chemical Lego bricks by following a plan laid out in the DNA genetic instruction book.
Important evidence supporting the theory of positional information came from transplant experiments, in which tissue in a growing embryo is moved to a new location. For example, the wing bud of a chick embryo starts to develop a kind of striped pattern that later becomes the bones of the wing, and a mouse embryo starts to develop a similar pattern that eventually becomes the digits that make up its paws. The experimental results were consistent with the theory of positional information and were widely interpreted as confirming it.
Despite the apparent success of positional information, some mathematicians, engineers, physicists, and computer scientists were not convinced that a chemical gradient could provide accurate positions in a robust manner. It is now starting to look as though they were right. The experiments that seemed to confirm the positional information theory turn out to have been a little too simple to reveal some effects that are not consistent with it.
In December 2012, a team of researchers led by Rushikesh Sheth at the University of Cantabria in Spain carried out more complex transplant experiments involving a larger number of digits. They showed that a particular set of genes affects the number of digits that a growing mouse embryo develops. Strikingly, as the effect of these genes decreases, the mouse grows more digits than usual—like a human with six or seven fingers instead of five. This and other results are incompatible with the theory of positional information and chemical gradients, but they make complete sense in terms of Turing equations.
Other groups have discovered additional examples supporting Turing’s model and have identified the specific genes and morphogens involved. In 2006, Stefanie Stick and co-workers reported experiments showing that the spacing of hair follicles (from which hairs sprout) in mice is controlled by a biochemical signaling system called WNT and proteins in the DKK family that inhibit WNT.
In 2012, a group at King’s College London (Jeremy Green from the Department of Craniofacial Development at King’s Dental Institute) showed that ridge patterns inside a mouse’s mouth are controlled by a Turing process. The team identified the pair of morphogens working together to influence where each ridge is formed. They are FGF (fibroblast growth factor) and Shh (Sonic Hedgehog, so called because fruit flies that lack the corresponding gene look spiky).
These results are likely to be followed by many others that provide genetic underpinning for Turing’s model or more sophisticated variants. They show that genetic information alone cannot explain the structural changes that occur in a developing embryo. More flexible mathematical processes are also important. On the other hand, mathematical models become genuinely useful only when they are combined with detailed and specific information about which genes are active in a given process. Neither molecular biology nor mathematics alone can explain the markings on animals, or how a developing organism changes as it grows. It will take both, working together, to frame the fearful symmetry of the tyger.
References
Andrew D. Economou, Atsushi Ohazama, Thantrira Porntaveetus, Paul T. Sharpe, Shigeru Kondo, M. Albert Basson, Amel Gritli-Linde, Martyn T. Cobourne, and Jeremy B. A. Green, “Periodic stripe formation by a Turing mechanism operating at growth zones in the mammalian palate.” Nature Genetics 44 (2012) 348–351; doi: 10.1038/ng.1090.
Rushikesh Sheth, Luciano Marcon, M. Félix Bastida, Marisa Junco, Laura Quintana, Randall Dahn, Marie Kmita, James Sharpe, and Maria A. Ros, “Hox genes regulate digit patterning by controlling the wavelength of a Turing-type mechanism.” Science 338 (2012) 6113, 1476–1480; doi: 10.1126/science.1226804; available at http://www.sciencemag.org/content/338/6113/1476.full?sid=dc2a12e8-c1c7-4f2a-bc4e-97962850caa3.
Alan Turing, “The chemical basis of morphogenesis.” Phil. Trans. R. Soc. London B 237 (1952) 37–72.