298. Do all spherical viruses have icosahedral symmetry?

This is a guest post by Eric Lewin Altshuler (UMDNJ, Newark) and Antonio Perez-Garrido (U. of Cartagena). Dmitry.

More than half a century ago Crick and Watson (Nature, 177, 473-475 (1956)) had the ingenious insight that viral capsids must be made of multiple units of the same small number of proteins, lest the viral genome be orders of magnitude too large – if coding for each of the hundreds or thousands of capsid proteins separately – to fit inside its capsid. Caspar and Klug (Cold Spring Harbor Symposia On Quantitative Biology 27, 1 (1962)) made a significant advance in appreciating that the structure of a number of viral capsids had icosahedral symmetry. They described capsids by a number (, non-negative integers) having  subunits arranged into an icosadeltahedral lattice. However, determination of the structure of these large capsids is a tour de force of experimentation, and until recently capsid structure was actually determined by fitting relatively low resolution data to an assumption of an icosadeltahedral capsid (see Strauss, “Viruses and Human Disease”, p. 34, Academic Press, San Diego (2002) and refs. therein.) Recently, though high resolution studies have confirmed icosadeltahedral configurations for  1,  3 (Cardone et al., Nature 457, 694-698 (2009)) and  7 (Jiang et al., Nature 439 612-616 (2006) and Gertsman et al., Nature doi:10.1038/nature07686) viruses. So do all spherical viruses have icosahedral symmetry?

The subunits of the viral capsids themselves can often have many components so theory or modeling might seem most difficult. However, because electrostatic interactions between subunits are likely important, it turns out that a more than century old model problem, we have found (arXiv:0902.3566) may shed important light on spherical viral capsid structure and evolution.

Over one hundred years ago J. J.Thomson (Philos. Mag. 7, 237-265 (1904)) asked the question of the minimum energy configuration of  unit point charges on (the surface of) a unit conducting sphere. Much theoretical, numerical and experimental work since then has made considerable progress on Thomson’s problem yielding interesting and nonobvious results: For  charges the global minimum energy configuration is the geometrically symmetric configuration of a tetrahedron. However, for  the minimum energy configuration is not a cube, but rather an anticube – four charges arranged in a square parallel to the equatorial plane in both the Northern and Southern hemispheres, but with the squares rotated by 45 degrees with respect to each other. This configuration has a lower energy than a cube as the rotation of the squares lowers the energy between nearest neighbor charges between the two squares. The case of also illustrates a general phenomenon in Thomson’s problem whereby the most symmetric configuration is not necessarily the configuration of minimum energy.

For (and likely for the most part up to ) attack of Thomson’s problem by multiple numerical and theoretical approaches and methods has likely found the minimum energy configurations. In most cases there are exactly twelve charges with five nearest neighbors – pentamers and the rest of the charges with six nearest neighbors – hexamers. Euler’s theorem for convex polygons – the number of vertices plus faces equals the number of edges plus 2 () – has the result for points on a sphere that there must be at least twelve pentamers with the rest of the charges being hexamers or pentamer/septamer pairs. Only for  N of the form  (where ) is it possible for the twelve pentamers and the entire configuration to have icosahedral symmetry.

Now, for N = 12, 32, 72, 122, 132, 192, 212, 272 and 282 (arXiv:0902.3566, Table 1, Figure 1) the icosahedrally symmetric configuration (an icosadeltahedral configuration) is the best known energyminimum(and the presumed global energy minimum configuration). But for  N = 42, 92, 162, 252 (and then  numbers larger than 282) the icosadeltahedral configuration is not the global energyminimum. Instead configurationswith exactly twelve pentamers, but with the pentamers arranged in  , , and  symmetries are the global energy minima for  N = 42, 92, 162 and 252 respectively (arXiv:0902.3566, Table I, Figure 2). These symmetries of global energy minima hold not only for the  1/r Coulomb potential, but for other representative electrostatic potentials as well (arXiv:0902.3566, Table II).

Why aren’t icosadeltahedral configurations the global minima for  N = 42, 92 and 162 ((2, 0), (3, 0) and 4, 0))? For these  N (see Figure 1) one sees that the vertices of the pentamers (known as disclinations in the language of elasticity and continuum mechanics) are lined up point to point. If the pentamers could be arranged with a different energy the cost of the disclinations being lined up and relatively closer to each other could be removed, as long as the new configuration itself does not impose an energy cost even higher. Conversely, for  N = 32, 72 ( N = 12 is a uniquely special case) one sees (Figure 1) that the pentamers are rotated with respect to each other to reduce strain of aligned disclinations. In general it seems that in seeking global energy minima for N<200 Nature uses the general strategies of moving and rotating pentamers. In some cases either because of pure geometrical constraints or just to minimize the energy, occasionally a pentamer/heptamer defect pair (dislocation defect in the language of elasticity) is needed to achieve a global energy minimum. As  N grows larger – or so – important papers by Dodgson and Moore (Dodgson J., Phys. A 29, 2499 – 2508 (1996) and Dodgson and Moore, Phys. Rev. B 55, 3816 – 3831 (1997)) showed that the energy strain of the pentamers which is necessitated by the topology of a sphere but which distort the pure hexagonal lattice that would be the energy minimum on a flat sheet, is such that to lower the energy pentamer/heptamer defect pairs are needed between all of the pentamers.

Fig. 1. Icosadeltahedral configurations.

Further, as  grows the number of local energy minima,  (1) was found to grow exponentially (Erber and Hockney, Phys. Rev. Lett. 74, 1482 (1995)):

.

Thus, if Nature is using an energy minimization strategy to find the configuration of alignment of molecules in a viral capsid it would seem that as  N grows it will become increasingly difficult if not impossible to find the global energy minimum configuration. Furthermore, the number and relative depth and breadth of good local minima could also be a constraint on kinetic strategies that Nature may use to find the ultimate configuration. In arXiv:0902.3566, Table I we show the number of times we found the various local minima in runs where we started the charges from 5000 random configurations and then used standard conjugate gradient methods to go to a local minimum. We see that for  N = 12 , 32 and 72 the minimum energy configuration and overwhelmingly the most common is in fact the icosadeltahedral configuration. For  N = 42 , 92 and 162 however, the icosadeltahedral configuration is not only not the minimum energy configuration, it is not reached from random configurations ever. Similarly, we found that for others interaction potentials again the icosahedral configurations for  N = 42 , 92 and 162 are not global minima and virtually never occur in the simulations (arXiv:0902.3566, Table II).

Fig. 2. Icosadeltahedral configuration b) Global minima, c) Icosadeltahedral and global minima overlap.

Fig. 3. Models of an icosahedron (left) and one with D_5h symmetry (right) that resembles the global minimum for N=42 .

For  N = 42 ( T4 ) icosahedral and  symmetries look quite similar (Figure 2 and Figure 3). If we only take into account the balls’ positions in Fig. 3 we can pass from the left model to the right one just by rotating an hemisphere by an angle of . Based on these electrostatic model potentials we would predict that , and virus would have icosadeltahedral configurations as recently found experimentally. viruses may have an icosadeltahedral configuration. We believe that high resolution studies of viruses will show a configuration in particular and not an icosadeltahedral configuration. We predict that , and viruses will not be found to have an icosadeltahedral configuration though we do not have a clear prediction for the structure of these viruses. Conversely, if a , , or virus were found to have an icosadeltahedral configuration given the essentially vanishing possibility of this from energetic considerations or statistical considerations based on electrostatic potentials, it would indicate a mechanical rule of assembly to be discovered that is of expontentially good precision. It is still also a mystery why nature seems to so prominently use capsids with numbers of subunits to the exclusion of other numbers of subunits. The energetics and statistics are so favorable for  N = 12, 32 and 72 (, , ) that protein subunits consistent with these configurationsmust have emerged. The same factors would suggest a  configuration for  N = 42. A key question could be understanding the evolution of  capsids. In general, geometry and topology seem to be important constraints that need to be considered in viral evolution or even possible treatments for viral.

In summary, we have shown that theory and simulations for a broad and representative range of electrostatic potentials for interaction of capsid subunits predict an icosaheltahedrally symmetric configuration for , , and viruses (and maybe for viruses). However, the prediction for , , and  viruses Is for a capside of lower symmetry, in particular  for  viruses. If, conversely, icosadeltahedral structures are found in high resolution studies, then any model of capsid assembly would have to account for such a structure in the face of energy and vast statistical disadvantages.

The other mystery – potentially linked? – is to uncover the principle of why Nature has chosen to make capsids only with  numbers of subunits, whereas geometry and topology do not exclude other  N .  N = 12 has a relatively lower energy per subunit than other  N near 12, so this one seems to have arose first. In terms of understanding the evolution of viruses one would guess that  viruses somehow arose as a variation of  viruses.

The next challenge is to understand how  viruses evolved. This understanding of viral capsid structure and evolution is not purely a theoretical exercise, as such understanding could lead to the development of new therapeutic agents.