As long as they do not infect us, viruses are nothing but fascinating. One aspect of this fascination is based on the shapes of viruses. Looking at electron microscopy images of viruses or at computer models based on X-ray crystallography, my impression is that mother nature has copied from a geometry book. This impression is echoed by what you find in books and papers on virology. Why do biologists think about possible polyhedra with icosahedral symmetry? Why is it that many viruses have the shape of such polyhedra? For some years, it seemed like biologists had a very accurate theory of the construction of such regular viruses. Advances in imaging have left them less confident but with an even higher appreciation of the formation of biological shapes.
« Assembly of virus particles is a highly ordered process which requires the bringing together of various particle components, including the genome nucleic acid, in a controlled sequence of events. »
— Dimmock, Easton, Leppard: Introduction to Modern Virology
In the beginning, there were Watson and Crick. Not only was their work on DNA necessary for the modern understanding of viruses but they also inspired a highly successful theory on the shape of regular viruses by their paper on the Structure of Small Viruses from which we quote most of the conclusion:
« We assume that the basic structural requirement for a small virus is the provision of a shell of protein to protect its highly specific packet of ribonucleic acid. This shell is necessarily rather large, and the virus, when in the cell, finds it easier to control the production of a large number of identical small protein molecules rather than that of one or two very large molecules to act as its shell. These small protein molecules then aggregate around the ribonucleic acid in a regular manner, which they can only do in a limited number of ways if they are to use the same packing arrangement repeatedly. Hence small viruses are either rods or spheres. The number of sub-units in a rod-shaped virus is probably unrestricted, but for a spherical virus the number is likely to be a multiple of 12. Every small virus will contain symmetry elements and in favourable cases these can be discovered experimentally.
We believe that this hypothesis is likely to apply (in this form or a simple variant of it) to all small viruses which have a fixed size and shape. »
— Crick and Watson: Structure of Small Viruses. Nature 177: 473–475
The general idea here is that in order to be a successful vicious little beast of a virus, it is a good idea to have a small efficient genome protected by a protein shell — the virus capsid — which also helps you getting into those cells you would like to convince to reproduce you. Now if the size of your genome is small, you do not have space for describing a complex shell. Moreover, the larger the genome, the larger that shell has to be. Therefore, it is beneficial to construct a shell from small identical pieces.
There are two kinds of structures observed for viruses constructed from small parts. First, the parts can form a helix leading to tubular viruses. Second, the parts can be distributed in a spherical shape, leading to polyhedral viruses.
Based on these ideas and on their studies of the structure of various viruses, Caspar and Klug developed their quasi-equivalence theory of icosahedral virus construction. It is detailed in their seminal paper Physical Principles in the Construction of Regular Viruses.
If the parts constituting the capsid of a polyhedral viruses were to be completely equivalent in their arrangement, the virus should show the symmetry of a point group, leaving the centre of the virus fixed. However, experimental data shows that there are capsids formed from more than 60 chemically identical proteins whereas no object with more than 60 identical shapes can exist under point group symmetry. Hence the need for a theory of quasi-equivalence based on small deviations from exact symmetry.
It is really interesting to look at the original paper of Caspar and Klug and see how they have illustrated their idea. Lacking powerful computer graphics tools, all the models of capsid subunits distributed over polyhedra that are depicted in the paper have actually been built by hand — some of them from over 100 parts.
We are quite lucky that modern computer graphics and advances in imaging techniques which allow to deduct the structure of virus capsids pretty exact allow us to generate wonderful libraries of virus structures. One such library is ViperDB. Here is an impression of its contents:
Images from: http://viperdb.scripps.edu/info_page.php?VDB=3gzt
Caspar’s and Klug’s theory is wonderful and still important enough to be found in pretty much any textbook on viruses. However, not all icosahedral viruses can be explained in this way.
In his chapter Form, Function, and Functioning in the book Shaping Space, George Fleck cites Caspar with the words:
« The theories we have formulated in the past have given a very good explanation of why icosahedral viruses are icosahedral. Now we don’t know! »
Clearly, nature tends to surprise us by being ever more flexible than we expect, but this does not change the impression that it is having a good study of its mathematics books.