Virus from the Fourth Dimension - Mathematics of Life

Mathematics of Life (2011)

Chapter 10. Virus from the Fourth Dimension

Geometry became a well-developed and powerful branch of mathematics through the work of ancient Greek philosophers and mathematicians. The most famous of the ancient Greek geometers, though not the most talented, was Euclid of Alexandria.1 He put geometry on a systematic basis in his Elements, a logical development of geometry written around 300 BC. It became the most successful textbook ever written, with thousands of editions since it first saw print in Venice in 1482.

The climax of the Elements is the classification and construction of the five regular solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. The names, cube aside, refer to the number of faces: 4, 6, 8, 12 and 20, respectively.2 The cube has square faces, the dodecahedron has pentagonal ones and the other three are made from equilateral triangles (see Figure 24).

In most areas of science, discoveries made 2,300 years ago are no longer terribly relevant – although Archimedes’ principle about floating bodies and his law of the lever still have their uses, and they are about that old.3 But mathematics builds new discoveries on top of previous ones, and once something has been proved true it remains that way. So it tends to hang around. New standards of rigour come into being, definitions are made more watertight, new interpretations are introduced, and topics that were once flavour of the month may sink into obscurity, but fundamental mathematical ideas are pretty much permanent.

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Fig 24 The regular solids. Left to right: Tetrahedron, cube, octahedron, dodecahedron, icosahedron.

The icosahedron is a case in point. It has always played a role in pure mathematics: in 1908 the French mathematician Charles Hermite discovered how to use the geometry of the icosahedron to solve algebraic equations of the fifth degree. But until the twentieth century there were no significant applications of the icosahedron to the real world, because it did not seem to arise in nature. Since 1923 it has appeared in the design of objects: the engineer Walther Bauersfeld used it as the basis of the first planetarium projector, and the architect Buckminster Fuller reinvented the idea and used it to design geodesic domes. It also underlies the geometry of the modern football, which is an elaboration on the icosahedron: you truncate it – cut off the corners – to make it more rounded (of course). The same elaboration captures the structure of buckminsterfullerene, a recently discovered molecule consisting of 60 atoms of carbon and nothing else, which form a roughly spherical cage (see Figure 25).

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Fig 25 Left: Icosahedron. Middle: Truncated icosahedron and football. Right: Buckminsterfullerene.

When electron microscopy and X-ray diffraction methods got going, however, Euclid’s 20-faced solid became a regular feature of biology. The context was viruses – diminutive structures too small to be seen in an optical microscope, but visible in a more powerful electron microscope. Viruses are a major cause of diseases in humans, animals and plants; the Latin word ‘virus’ means ‘poison’. Viruses are a bit larger than most biological molecules, but a typical virus is about one-hundredth the size of a typical bacterium. Since volume varies as the cube of length, you could pack a million (100×100×100) viruses inside a single bacterium if no space were wasted. There are about 5×1030 bacteria on Earth, but the number of viruses is about ten times that. Neither of these figures is particularly accurate, and both may be underestimates, but they give a general feel for the numbers. Viruses outnumber humans 1022 to one.

Bacteria definitely qualify as life, because they can reproduce using their own genetic processes. Viruses may or may not qualify: they have genes – DNA or RNA sequences – but they cannot reproduce using just their own genes. Instead, they reproduce explosively by subverting the reproductive biochemical machinery of a bacterium, much as a document can reproduce through the intermediary of a photocopier. (Actually, a few viruses can replicate unaided, but these are exceptional.) Some biologists argue that the definition of life should be extended to include viruses.

Over 5,000 distinct types of virus have been found since Martinus Beijerinck made the first discovery, of the tobacco mosaic virus, in 1898, and indirect evidence tells us that there must be millions more. Most viruses have two main components: genes formed from DNA or RNA, enclosed within a protein coat known as a capsid. The capsid is typically formed from identical protein units, known as capsomers. Some viruses also possess an extra layer of lipids (fat) for protection when outside a cell.

As early as 1956, it was noticed that the majority of viruses are either icosahedral or helical: shaped like a football or shaped like a spiral staircase. Some have a more complex structure: for example, enterobacteriophage T4 has an icosahedral head, a helical stalk and a hexagonal baseplate from which fibres extend. It looks like a lunar landing molecule (see Figure 26). But the main form observed is Euclid’s elegant icosahedron, which, devoid of practical application for more than two thousand years, turns out to be just right for making a virus (see Figure 27).

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Fig 26 Enterobacteriophage T4.

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Fig 27 Icosahedral structure of two viruses. Left: Foot-and-mouth disease virus differs from its mirror image and has 60 symmetries. Right: Herpes simplex virus is mirror-symmetric and has 120 symmetries.

There is a hand-waving explanation based on minimising energy, which goes like this. Virus coats are typically constructed from many copies of a roughly spherical protein molecule. A collection of such molecules has the least energy – something that nature finds desirable – if it is as close as possible to being a sphere. Soap bubbles are spheres, because this is the shape that has the least surface area, hence the least energy of surface tension, while enclosing a given volume. Virus coats can’t form exact spheres, because the component protein units cause bumps. (Try to fit a hundred tennis balls together to make a smooth sphere.) So they do the best they can. Of all Euclid’s solids, the icosahedron is closest to a sphere. The truncated icosahedron is even more spherical, hence its use for footballs (and as a bonus, the panels become even rounder when the ball is inflated). So evolution and FIFA (the Fédération Internationale de Football Association) independently came up with the same shape, for the same reason. Until the 2010 World Cup, that is, when the balls were made in a different way – and everyone complained.

At the heart of the icosahedron, indeed of all five regular solids, is the concept of symmetry. Since the early 1800s, mathematicians have developed a profound theory of symmetry, with applications throughout the sciences, known as group theory. I could spend a book discussing how group theory came about (and I have done4). The key point is that a symmetry of an object is not a thing, but a transformation, whose application leaves the object looking exactly the same.

Put baldly, that sounds like gobbledegook: a symmetry transforms the object but leaves it looking exactly the same. Yes, and there are little green men on the Moon but they’re invisible so nobody can ever detect them ... Right. Actually, the statement makes sense if properly construed. A transformation is a way to rearrange things or move them around. In this case, the relevant transformations are rigid motions, especially rotations and reflections. Now, think of a square, which intuitively has quite a lot of symmetry. For example, all four corners are the same shape. One way to capture this feature is to rotate the square about its centre through a right angle. The result is an identical square, in an identical orientation. If you shut your eyes while the square was being rotated, and it had no markings on it, then when you opened them again you wouldn’t notice that anything had happened. So the transformation ‘rotate through a right angle’ is a symmetry of the square. In all, there are eight such symmetries: leave the square alone; rotate it through one, two or three right angles; reflect it in either diagonal; reflect it in either line joining the midpoints of opposite sides.

These transformations have a pleasant kind of ‘closure’: perform any two of them in turn, and the result is one of the same eight symmetries. They are said to form a group. The same is true for the symmetries of any other object. A circle is much more symmetric than a square, and has infinitely many symmetries: rotate through any angle, reflect in any diameter. But again, any two symmetries performed in turn have the same effect as a single symmetry. Rotate by 14° and then rotate by 53°: the result is the same as a single rotation through 67°, which is 14°+53°.

Euclid’s regular solids have rich and beautiful symmetry groups. The tetrahedron has 24 symmetries, the cube and octahedron have 48, and the dodecahedron and icosahedron have a massive 120. The symmetry properties of these solids are what make them so prevalent in modern pure mathematics. Notice that different solids here, such as the dodecahedron and icosahedron, can have the same number of symmetries. There is a good reason: if you draw a dot in the middle of each face of an icosahedron, you get the 20 vertices of a dodecahedron. Similarly, if you draw a dot in the middle of each face of a dodecahedron, you get the 12 vertices of an icosahedron. Suitable geometric relations between shapes can give them the same symmetry groups.

The architecture of the viral capsid is important biologically: in particular, it helps with the analysis of images of the virus (such as those obtained by X-ray crystallography) and the construction of models of how the virus assembles. The icosahedral structure of viruses does not just determine the overall shape: it is inherent in the arrangement of the protein units. Until recently the main theoretical description of capsid architecture was the one derived in 1962 by the American and British biophysicists Donald Caspar and Aaron Klug.5

Icosahedral virus coats are made from triangular arrays of capsomers, fitted together like the faces of an icosahedron. Each triangle is made from row upon row of capsomers, arranged like the balls at the start of a game of snooker or pool. Looking more closely still, that’s not quite the full story: the rows of capsomers can be skewed, so that some rows run into the edge of the triangle and warp across it into the next triangle. Nature obviously has no trouble making such shapes, but they are slightly odd mathematically, and the first step in understanding the patterns is to work out their mathematical status and find their common features.

Like pool balls, most capsomers are surrounded by six others (hexamers). However, some are surrounded by only five others (pentamers). It turns out that this condition is forced by geometry. If we represent the virus capsid as a polyhedron by placing vertices at the capsomer and joining adjacent capsomers by edges, then hexamers are vertices lying on 6 edges, and pentamers lie on 5 edges. This fact alone imposes mathematical conditions on the possible number of capsomers. Leonhard Euler, one of the all-time mathematical greats, discovered a formula relating the number of faces, edges and vertices of a solid. Namely, for any polyhedron topologically equivalent to a sphere,

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where F is the number of faces, E the number of edges and V the number of vertices. For example, a cube has F=6, E=12, V=8, and 6-12+8=2. This general fact is known as Euler’s formula for polyhedra. A simple calculation using the formula shows that in any virus coating composed purely of hexamers and pentamers, there must be exactly 12 pentamers. The method doesn’t specify where they occur, but it proves that they have to be present.

Caspar and Klug followed up these topological clues. They looked first at helical viruses, and then went on to consider icosahedral ones. Here the basic mathematical problem is to pack identical units together to form shapes that are close to spherical, bearing in mind that the relation between each unit and those adjacent to it is likely to be restricted by the available chemical bonds. The simplest case is when there is only one such relationship; geometrically, this means that each unit is surrounded by exactly the same configuration of adjacent units. This in turn implies a high degree of symmetry, which for brevity I’ll call ‘perfect symmetry’, and immediately suggests considering the regular solids. Among these, the icosahedron is the most plausible candidate: of all the regular solids, it forms the best approximation to a sphere. In addition, electron microscope images of several viruses appeared to be icosahedral, although Caspar and Klug note that this ‘does not necessarily mean that the symmetry down to the molecular level is icosahedral’.

With the requirement of perfect symmetry, an icosahedral arrangement can clearly accommodate 12 or 20 units: 12 if they are situated at the corners, 20 if they are placed at the centres of the faces. The largest number of units that can fit together so that each unit has the same immediate neighbourhood is 60 (see Figure 28). This increases to 120 if mirror images are considered identical, but biological molecules tend to have a specific ‘handedness’, so this is unlikely. Again, the symmetry is icosahedral.

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Fig 28 Fitting 60 identical units together so that each has the same relation to its neighbours.

It follows that the number of units in a virus with perfect symmetry must be 12, 20 or 60. However, no viruses known to Caspar and Klug employed these numbers, and most of them had more than 60 units. Indeed, none of them had a multiple of 60 units, which might be realised by relaxing the requirement of perfect symmetry a little. The most likely way out is to relax the requirement even further, and Caspar and Klug found inspiration from an unusual source: the architect Buckminster Fuller. Fuller had a liking for geometric forms, and the geodesic dome is one of his more famous ideas, a roughly spherical enclosure made by fitting a large number of triangular panels together. Such a dome featured as a pavilion for the 1964 New York World’s Fair, and hemispherical versions can be found at the Eden Project in Cornwall.

You can’t make a geodesic dome from equilateral triangles arranged six to a vertex, because they would form a flat plane. Fuller, following various predecessors, realised that triangles that are nearly equilateral can fit together to make a dome. Such arrangements do not have perfect symmetry; instead, triangles have two different kinds of neighbourhood. Consistency with Euler’s formula demands that some must be arranged so that five of them fit round a vertex, while the rest fit six to a vertex. Caspar and Klug realised that although adjacent units are generally held together by the same arrangement of chemical bonds, these bonds can be bent by a small amount, so that the bond angles can be slightly different for units that are not symmetrically related. Experiments performed by the Nobel Prize-winning chemist Linus Pauling suggest that bond angles can be changed by about 5° from their average values, which allows some flexibility.

Caspar and Klug were led to an unorthodox range of solids called pseudo-icosahedra, familiar to expert geometers but not to most mathematicians. They are solids that resemble icosahedra but are less regular. They can be constructed from a tiling of the plane (see p. 154) by equilateral triangles. First, choose two numbers a and b (see Figure 29). Starting from a vertex, move a units to the right and b units at 120° to this direction to get a second vertex; then locate the third vertex to form a large equilateral triangle containing many vertices of the original tiling. Twenty of these triangles can then be fitted together to form an icosahedral polyhedron with 10(a2+ab+b2)+2 vertices, of which 12 are pentamers and the rest hexamers. The pentamers always lie on the axes of icosahedral symmetry (the ‘corners’). Examples of pseudo-icosahedral architecture are given in Table 8.

The Caspar – Klug theory applies to many different icosahedral viruses, but there are exceptions. Forty years ago, Nicholas Wrigley noticed that some icosahedral viruses do not have this pseudo icosahedral architecture. Instead, they can be described by so-called Goldberg polyhedra, which are hexagonal packings on the surface of an icosahedron.6 However, even these structures are insufficient to classify the arrangements of capsomers in icosahedral viruses: for example, in 1991 Robert Liddington and colleagues pointed out that polyoma virus has many more pentamers than the 12 found in pseudo-icosahedra and Goldberg polyhedra.7 So some more general mathematical description was needed.

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Fig 29 Construction of basic triangle of a pseudo-icosahedron.

Table 8 Numerology of pseudo-icosahedral viruses.

{a, b}

No. of capsomers

Virus

{1, 1}

32

Turnip yellow mosaic

{2, 0}

42

Bacteriophage ΦR

{2, 1}

72

Rabbit papilloma

{1, 2}

72

Human wart

{3, 0}

92

Reo

{4, 0}

162

Herpes, chickenpox

{5, 0}

252

Adenovirus type 12

{6, 0}

362

Infectious canine hepatitis

By now, biologists’ minds were on other things, but mathematicians were still puzzling over the exceptions. Around the year 2000 the German-born mathematician Reidun Twarock and her research team at the University of York developed a more general theory of the geometry of viruses based on symmetry principles closely analogous to the group theory of the icosahedron.8

There was only one difference: now the geometry took place in four dimensions, not three.

The fourth dimension ...

It sounds like something from a science-fiction story, a hidden realm adjacent to our everyday world in which all manner of strange creatures lurk ... Indeed, this is how the concept is portrayed in H.G. Wells’s The Time Machine of 1895, in which the Time Traveller is taken to the far future of humanity, now separated into the languid Eloi and the grotesque Morlocks. But Wells based his novel on a topic from real science that was becoming popular at the time. He credited the idea to ‘student discussions in the laboratories and debating society of the Royal College of Science in the [eighteen-]eighties.’ As the story opens, the Time Traveller invokes the fourth dimension to explain why such a machine is possible:

There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives ... But some philosophical people have been asking why three dimensions particularly – why not another direction at right angles to the other three? – and have even tried to construct a Four-Dimensional geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago.

It is a matter of historical record that Newcomb, one of the most eminent American mathematicians of his day, published on the topic of four-dimensional space from 1877, and he spoke about it to the New York Mathematical Society in 1893. Four dimensions (and more) were important research topics in mathematics and physics.

Some Victorian theologians saw ‘the’ fourth dimension as a convenient location for God, contiguous with our universe at every point, yet outside it, and affording the Deity a complete view of the entire universe in a single glance. But then the hyperspace theologians decided that the fifth dimension would be even better, the sixth better still ... and that nothing short of the infinitieth dimension would serve for an omnipotent, omnipresent Deity. At much the same time, spiritualists found the fourth dimension equally suitable as a home for the spirits of the dead; believers in ghosts had a similar view, with the obvious change in the identity of the entities that inhabited this hidden dimension; various cults and pseudoscientific bodies found effective ways to rationalise their own beliefs by throwing in a few references to the fourth dimension; and a few outright crooks used topological trickery to ‘prove’ that they had access to the fourth dimension and con people into giving them money.

It was the mathematicians’ fault, really – they had set this particular ball rolling, and physicists picked it up and ran with it. Then popular culture, unrestrained by the need to stay close to reasonable speculation, pushed the idea to the limits – much as the more recent media image of cloning was populating the planet with exact copies of human beings at a time when biologists were failing to duplicate a cell. With Dolly the Sheep, fact began to catch up with fiction, but cloned humans do not yet exist.

The fourth dimension has fared much better, and if anything, fact is now ahead of fiction – except when it comes to time travel. The concept has been entirely respectable in mathematics and physics for over a century, and scientists now routinely employ mathematical concepts of any number of dimensions – four, ten, a hundred, a million. Even infinity. The imagery of multidimensional spaces has spread to biology and economics, together with the associated mathematical techniques. The idea may sound outlandish, but it is actually very natural. Its main relevance to this chapter, though, is a more direct application of four-dimensional geometry to, of all things, viruses. Strange though it may seem, the fourth dimension and its higher cousins are providing important insights into how viruses pack their protein units together.

When mathematicians start talking about familiar terms in ways that make no sense, it usually turns out that they have either appropriated the word and given it a totally different meaning, or they have extended the usual meaning to a wider context. A group is not merely a collection of similar objects, a ring has nothing to do with jewellery or even with circles, and you won’t find sheep grazing in a mathematical field. Terms like ‘dimension’, ‘space’ and ‘geometry’ fall into the second category, and are easier to misunderstand because the new meaning is not so obviously different from the old one.

The unwritten rule for extending the meaning of a word is that it should retain its original meaning in its original context. It is all right to introduce a new concept of ‘space’, for example, as long as the familiar spaces of Euclid’s geometry – the plane and, well, space – are still included. Provided this convention is obeyed, you won’t get confused by applying the new meaning in the old context. Though you might get confused in the new context if you assume that specific features of the old one still hold good – such as ‘space’ being something that humans can or do live in.

‘Dimension’ is a case in point. The plane has two dimensions, our familiar notion of space has three. Any extension of the word to other ‘spaces’ should preserve those facts. However, the traditional definition – the number of independent directions – is not sacred. It’s not even sacred in the traditional contexts: you can change the definition as long as the answers are still two and three respectively.

Traditionally, we sometimes speak of individual ‘dimensions’. Length is a dimension, so is width, so is height. Some care is needed, because the same word can also mean ‘size’: the longest dimension (in that sense) of a box is whichever of these three is biggest, but actually the longest ‘dimension’ is the diagonal, which is bigger than any of them. Mathematics and science have settled on a more general notion of dimension in which we can safely say that some space has, say, ten dimensions, without saying what these dimensions are. The emphasis has shifted: the space has dimension (or dimensionality) 10. We don’t define things called dimensions and count them to show there are ten of them. That said, lists of ten things do exist – but we don’t call the things dimensions. We call them coordinates.

Mathematicians did not introduce spaces of many dimensions for fun, or to impress people. They did so because they needed them. By the end of the nineteenth century, a variety of developments, motivated by everything from pure geometry to celestial mechanics, all seemed to point towards the same new idea. At much the same time, physicists started to realise that many key discoveries made more sense if they were formalised within a ‘space-time’ of four dimensions: the three traditional dimensions of space, plus an extra one of time. But time was not the fourth dimension: just one possibility.

To cut a long story short: the dimension of a space is the number of independent coordinates needed to specify the things that belong to it. Spaces with many dimensions provide a convenient way to describe systems in which many distinct variables can be set to whatever values we desire. The ‘space’ of all such choices has a natural structure – a direct generalisation of the familiar mathematics of two and three dimensions. In particular, we can specify what it means for two ‘points’ in such a space to be close together: corresponding variables should have values that are close together.

Moreover, the ‘points’ need not actually be points. The plane is a set of points, but it can also be thought of as, say, a collection of ellipses. The set of all ellipses in the plane is an interesting mathematical object in its own right. How can you specify an ellipse? Let’s do it in Euclidean geometry, where the pictures are more familiar. You need to know:

• where the centre of the ellipse is (2 numbers),

• how long it is (1 number),

• how wide it is (1 number), and

• at what angle it is tilted (1 number).

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Fig 30 Five numbers are required to determine an ellipse in the plane.

So in all, it takes five numbers to specify an ellipse (see Figure 30).

The ‘space’ of ellipses is five-dimensional. And it is a space, in the sense that if you change the numbers representing some ellipse by small amounts, you get an ellipse that is ‘nearby’. It looks very similar. And the smaller the change, the more similar it looks.

From one point of view, the plane is two-dimensional. From another, it is five-dimensional. But it’s the same plane either way, so it makes no sense to maintain that a two-dimensional space exists but a five-dimensional one doesn’t. Here, they are two aspects of the same thing. Aside from familiarity and tradition, there is no good mathematical reason to prefer the set of points to the set of ellipses. Which viewpoint is best depends on what questions you’re asking. It was for this kind of reason that mathematicians came not only to tolerate spaces of higher dimensions, but to feel lost without them.

This simple idea turned out to be so useful that it rapidly invaded physics. Today’s particle physics, for instance, cannot even be set up properly without using spaces with pretty much any number of dimensions. Engineers got in on the act, too. If you are trying to calculate the stresses and strains in a grid of 100 metal girders, then you have 100 forces to work with. Since you don’t know what they are until you do the sums, you are conceptually looking at lists of 100 arbitrary numbers and trying to select the correct one. That is, you are seeking a point in a space of 100 dimensions.

Engineers find that terminology off-putting, and prefer the more physical concept of ‘degrees of freedom’. How many different things can vary independently? But it’s the same idea. Thinking of all possible configurations of some complicated system as a ‘point’ in a ‘space’ of all potentially possible configurations gives such a vivid image, and such a conceptual boost, that it has pervaded every branch of science – and beyond.

A case in point is DNA-space. A sequence of (for simplicity) ten DNA bases permits four choices (A, C, G, T) at each separate location. So ‘DNA-space’, the ensemble (as physicists say) or set (as mathematicians say) of all possible such sequences can be thought of as a ‘space’ of ten dimensions, with each individual dimension taking four possible values: A, C, G or T. Replace ‘ten’ by any other number, such as a million, and the same holds good.

This space has a natural geometry. Two sequences are close to each other if they differ in a small number of locations. For instance, AAAAAAAAAA is very close to AAAAACAAAA, a bit further from AAAAACTAAA, further still from AAGAACTAAA, and so on. The ‘distance’ between two sequences is the number of bases at which they differ. This notion resembles distance in the two-dimensional plane or ordinary three-dimensional space in many respects, although it differs in others. If you are interested in the genetic basis of evolution, whose simplest manifestation is ‘point mutations’ at which one base changes, this notion of distance is ideal: it equals the minimum number of mutations that could lead from one sequence to another.

Biologists have found the concept of DNA-space, or sequence space, very informative. It matches a similar idea used in computer science to describe digital messages in information theory. Biologists are not alone. Economists view the prices of a million goods as a point (or ‘vector’) in a hypothetical space of a million dimensions, and the mathematical processes they use, such as optimisation methods, are explicitly motivated by this image. Astronomers, discovering they need 18 numbers to describe the state of the Earth – Moon – Sun system, work in an 18-dimensional mathematical space. The geometry of this 18-dimensional space tells them a lot about how such a three-body system behaves.

The propensity of viruses to undergo genetic mutations at the drop of a hat is informed by this kind of use of multidimensional spaces. It formalises the use of similarities in DNA sequences to infer past evolutionary changes, and it classifies viruses into ‘strains’, different variants that arise by mutation or exchange of genetic material. These changes are important in medicine, because vaccines that work for one strain may be ineffective for a different strain.

You probably recall the swine flu pandemic of 2009, when a flurry of deaths in Mexico announced the arrival of a new strain of influenza virus. Sequencing revealed that this strain, known as H1N1, had evolved by combining genetic material from four previous strains of flu. Some time earlier, three strains had combined in this way: one occurred in pigs, one in birds and one in humans. This new strain went largely unnoticed until it combined with another pig flu virus. At that point, the World Health Organization took charge and declared the virus to be a global pandemic. Governments worldwide rushed to order suitable supplies of vaccine, tailored to the new strain. In the event, H1N1 proved less dangerous than feared; by August 2010 only 15,000 people had died from the virus – far less than the millions trumpeted by the tabloid press, and less than the numbers typically killed by ordinary seasonal flu – and the response has since been criticised as overkill. But H1N1 was unusual: it had worse effects on the young than on older people who had previously been exposed to a related strain and built up immunity. It is unclear whether the authorities overreacted, or whether most of us got lucky.

Sequence space uses multidimensional geometry as a descriptive framework, and similar ideas could be used without the associated geometric language. Twarock’s work on viruses involves a deeper use of multidimensional geometry: employing detailed theorems about the intricate geometry of spaces with four or more dimensions to understand the structure of viruses.

In a series of papers, especially one published in 2004, Twarock developed a more general version of the Caspar – Klug theory, applicable to the polyoma virus and other exceptions.9 This approach, known as viral tiling theory, allows the capsomers to be arranged in more general ways than the ‘pool ball’ hexagonal lattice. In particular, the pentamers need not lie at the vertices of the underlying icosahedron. Viral tiling theory is not straightforward, because regular pentagons do not tile the plane – if you try to fit them together to cover the plane, they either overlap or leave gaps – and crystallographic lattices in two and three dimensions cannot have fivefold rotational symmetry.

A key insight arose indirectly through the discovery of quasicrystal patterns, such as the famous Penrose tilings (see Figure 31). In patterns of this kind, the tiles fit together according to specific mathematical rules, but they do not form a lattice, that is, a pattern that repeats the same arrangement over and over again like a wallpaper pattern. A Penrose tiling covers the infinite plane without gaps or overlaps, using two types of tile. The resulting patterns incorporate the fivefold symmetry of the regular pentagon. Though these are not in fact lattice patterns, they can be understood using a mathematical trick: they are representations in 2D and 3D spaces of suitable parts of lattice patterns in spaces of higher dimension.

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Fig 31 A Penrose tiling.

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Fig 32 Non-lattice pattern in two dimensions.

This idea can be understood through an analogy: constructing non-lattice patterns in 2D from lattice patterns in 3D. The lattice patterns in 2D are based on tiling by parallelograms: certain shapes of parallelogram lead to patterns with more symmetry than the rest, notably the square and hexagonal lattices. Figure 32 shows a tiling of the plane (2D) by two different shapes of tile: equilateral triangle and hexagon. This is not a lattice pattern, which among other things would use a single shape of tile, but it is very symmetric and regular nonetheless.

It turns out that we can construct the same tiling pattern from a lattice, provided we move into 3D space. The lattice required is the cubic lattice, the simplest of all 3D lattices. It is like a 3D chessboard with cubes in place of the traditional squares. The most obvious feature of a cube is its square faces, but the pattern we’re considering doesn’t involve squares: it is made from equilateral triangles and hexagons. Nonetheless, it is concealed within the cubic lattice pattern. In fact, if you slice this 3D pattern using a plane that is tilted to run through the midpoints of three adjacent edges of one of the constituent cubes, you get exactly the required 2D pattern.

Here, we obtain the more complex lower-dimensional pattern by taking a slice through a simpler 3D pattern. Another strategy is also available: to project parts of the 3D pattern onto a suitable plane, much as a movie image is projected from the film onto the screen. Better still, both sections and projections can be used. The details don’t greatly matter: the mathematical advantage is that we can understand complex patterns in terms of simpler ones. The complexity of Penrose tilings, for example, can be simplified in this way. The price we pay is having to extend the pattern into a higher-dimensional space, but as far as mathematicians are concerned, this is harmless, easy and standard. It may sound a big deal to anyone not used to this trick, especially when the spaces involved are 4D or 5D, but algebraically it makes good sense, and that justifies using it, as well as providing a way to do the sums.

Twarock wondered whether it was possible to use this trick on icosahedral viruses, which are non-lattice patterns in 3D space. That means going to 4D space, at the very least – perhaps more. The powerful mathematics of symmetry groups can be brought to bear, and it shows that the smallest dimension of a lattice with icosahedral symmetry is not 4, but 6.

Although 6D spaces may seem irrelevant to biology, it is worth remembering that the ingredients of a mathematical description often lack direct physical counterparts. For example, the motion of the Earth and Moon around the Sun is most naturally represented as a dynamical system in 18D space, with 3 position and 3 velocity coordinates for each body, even though any configuration of the bodies lies in ordinary 3D space. So the geometry of lattices in ‘unphysical’ spaces of high dimension should be considered as a useful technique for determining 3D non-lattice patterns with special features, not as a literal description of a physical process.

The icosahedral group belongs to an important class of symmetry groups known as Coxeter groups after the geometer H.S. M. (‘Donald’) Coxeter,10 which are higher-dimensional analogues of a kaleidoscope. Working with Tom Keef, Twarock has applied this class of groups to the structure of icosahedral viruses in terms of a 6D lattice with icosahedral symmetry known as D6. By adapting the methods from Penrose tilings, they have constructed a class of possible virus structures defined as projections from the 6D lattice D6 into 3D space.

The projection of the entire D6 lattice would fill space densely with points, but the same issue arises for Penrose tilings, and the answer is to consider only a subset of D6, a slice with non-zero thickness, so to speak, and to project only that part of D6. This ‘cut and project’ technique yields all the pseudo-icosahedra, but also additional structures with more than 12 pentamers. In particular the structures of polyoma virus, Simian virus 40, and bacteriophage HK97 are now accounted for.

The mathematical techniques employed here range from well-established ideas in group theory and crystallography, through more modern contributions such as Coxeter groups, to recent innovations inspired by Penrose tilings. The resulting structures have definite biological interest. 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. Viral tiling theory goes beyond the Caspar – Klug approach by allowing different types of bonding among capsomers, so it represents the actual molecular configurations more closely. It also opens up new ways of thinking about tubular malformations, where the virus assembles into a tube rather than an approximate sphere. For example, if changes in the chemical environment of an assembling virus can cause it to form into a tube (a non-infectious form) rather than an infectious icosahedron – which seems plausible – then it might be possible to interfere with virus replication.

Other applications include cross-linking structures, such as exist in bacteriophage HK97, in which additional chemical bonds occur between adjacent capsomers; physical properties of capsids, which could suggest new ways to destroy the virus; and the way the genome is packaged within the virus. Research into this intriguing area continues. But what is known to date fully justifies the view that abstract geometry in higher dimensions can tell us a lot of useful things about real viruses in three dimensions.