Structural constraints on the three-dimensional geometry of simple viruses: case studies of a new predictive tool

Abstract

Understanding the fundamental principles of virus architecture is one of the most important challenges in biology and medicine. Crick and Watson were the first to propose that viruses exhibit symmetry in the organization of their protein containers for reasons of genetic economy. Based on this, Caspar and Klug introduced quasi-equivalence theory to predict the relative locations of the coat proteins within these containers and classified virus structure in terms of T-numbers. Here it is shown that quasi-equivalence is part of a wider set of structural constraints on virus structure. These constraints can be formulated using an extension of the underlying symmetry group and this is demonstrated with a number of case studies. This new concept in virus biology provides for the first time predictive information on the structural constraints on coat protein and genome topography, and reveals a previously unrecognized structural interdependence of the shapes and sizes of different viral components. It opens up the possibility of distinguishing the structures of different viruses with the same T-number, suggesting a refined viral structure classification scheme. It can moreover be used as a basis for models of virus function, e.g. to characterize the start and end configurations of a structural transition important for infection.

Keywords: affine-extended symmetry group; genome organization; structural constraints; tiling theory; virus structure.

Figure 1

Quasi-equivalence is part of a wider set of structural constraints on virus architecture. (a) A large number of viruses exhibit icosahedral symmetry in the organization of their protein containers, i.e. they share a set of five-, three- and twofold axes with an icosahedron. Here, four example symmetry axes are shown on a viral protein container. (b) The surface lattice from quasi-equivalence theory that encodes the protein organizations in capsids composed of coat proteins. (c), (d) A packing of overlapping icosahedra generates a partition of the icosahedral face akin to the structure: (c) shows three of the translated and rotated icosahedra as solids; (d) shows the edges of these icosahedra, together with the vertices of all 60 icosahedra. This is an example of the structural constraints implied by our theory.

Figure 2

The geometric principle can be encoded in a tiling. The figure demonstrates the relation of extended symmetry groups with tilings. For simplicity, the principle is demonstrated for the two-dimensional case of tenfold symmetry; the same principle has been applied to icosahedral symmetry in three dimensions and has resulted in the library of point arrays used here (Keef & Twarock, 2009; Wardman, 2012 ▶). (a) A decagon, a geometric representation of tenfold symmetry, superimposed on a Penrose-type tiling such that its corners coincide with vertices of the tiling. (b) Addition of a translation to the rotational symmetries of the decagon (i.e. an affine extension of tenfold rotational symmetry) results in translated copies of the decagon (shown in blue) with corners also coinciding with vertices of the tiling. (c) Subsequent rotations about the tenfold axis at the centre of the original decagon result in ten copies of the translated decagon. (d) Since corner points of the decagon are geometric representations of the (rotational) symmetries, the (artificial) decagonal edges are faded away. An iterative process of translation and rotation leads to the addition of further points. For example, in the second iteration step the red points in (e) are added and more vertices of the tiling are covered.

Figure 3

An illustration of the procedure underlying the best-fit algorithm. Only those points in each array in the library of 569 point arrays are used as input that overlap with the capsid, here shown as a shaded ring structure. Once the best-fit point array has been selected, its remaining points overlapping with the capsid interior must also be descriptors of capsid geometry. They therefore predict additional structural constraints. In our case studies below we show that these predictions provide additional insights into the genome packaging structure that agree well with experimental data.

Figure 4

Structural constraints encode protein topography and genome organization in PaV. (a) The capsid of PaV is organized according to icosahedral symmetry as illustrated by its match to the superimposed icosahedron (red). The magenta points are additional constraints encoded by our theory and correspond to the outermost points of the best-fit array. They match to the tops of the trimeric protein spikes, which is striking given that these are not located on axes of icosahedral symmetry. (b) A cross-sectional view (52 Å thick) of the capsid, showing the locations of the best-fit array points relative to the protein container and its closely associated dsRNA cage (light yellow). (c)–(e) Close-ups of the two trimers bounded by the green rhomb in (a) together with an associated portion of the dodecahedral RNA cage viewed from (c) outside the particle, (d) the side and (e) inside the particle. The point array encodes constraints on the trimeric protein complex (orange and yellow points) and the relative sizes of the capsid and RNA cage. Strikingly, (predictive) green points map on the three-way junctions of this cage, and (predictive) blue points fit snugly into the minor grooves of the A-type RNA duplexes. Since the locations of all points are fixed by extended symmetry with respect to the outermost array points (magenta), this implies that protein topography and RNA organization are correlated by a geometric scaling principle that is encoded by extended icosahedral symmetry. For clarity, array points are shown here and throughout as spheres of 4.5 Å radius, colour coded by their radial positions. Note, the PaV crystal structure is the result of icosahedral symmetry averaging. This procedure does not assume any interdependence of molecules at different radial levels and does not alter the conclusions from the matching to the array described above.

Figure 5

The structural constraints predict a two-shell genome organization in bacteriophage MS2. (a) The outermost points of the best-fit array scale to the N-terminal β-hairpins of the capsid proteins in MS2 (magenta). (b), (c) Central sections through the particle; (b) illustrates the match to the crystal structure in surface representation (Valegård et al., 1990 ▶), with TR stem-loops shown in yellow. As can be seen in the close-up in (d), array points are located at the contact points between stem-loops and protein. This is even more striking given that yellow and orange points are predicted to be located at different radial levels, corresponding to the contacts with the two different types of dimeric building blocks (A/B and C/C) of the capsid. (c) An illustration of the match with the cryo-EM RNA density (Toropova et al., 2008 ▶), shown here as a radially coloured transparent surface. Array points map the inside (maroon points, radius of 38 Å) and outside (blue and mid-blue points, 62 and 65 Å, respectively) surfaces of the inner RNA shell, and also mark the density connecting the inner and outer RNA shells (cyan, 75 Å). Strikingly, magenta and maroon points together define the spatial extent of material in this particle.

Figure 6

Non-quasi-equivalent configurations can also be predicted. The viral capsid of SV40 contains 360 identical coat protein subunits arranged as 72 pentamers, an example of a non-quasi-equivalent capsid organization. (a) The outermost constraints (magenta) are grouped around the 12 clusters of five proteins (pentamers) at the particle fivefold axes of icosahedral symmetry. (b) Cross-sectional view of the capsid, showing the locations of points in the array relative to protein. (c)–(g) Ribbon representations of the two different types of pentamers in the capsid; (c), (d) top and side view of the 12 pentamers at the fivefold axes, viewed from outside the capsid; (e), (f) show the corresponding views for the 60 pentamers off the symmetry axes; (g) shows both pentamer environments simultaneously as situated in the capsid. The new geometric principle of virus architecture distinguishes between the two types of pentamer environments and incorporates this viral geometry that cannot be modelled in quasi-equivalence theory.

Figure 7

(a) The best-fit point array for hepatitis B viewed down a fivefold axis and (b) overlaid on the cryo-EM data of the genomic material, adapted from Wynne et al. (1999 ▶). (c) The best-fit point array for TBSV viewed down a fivefold axis and (d) overlaid on neutron scattering data, adapted from Hopper et al. (1984 ▶), Olson et al. (1983 ▶) and Hogle et al. (1983).