Tilable nature of virus capsids and the role of topological constraints in natural capsid design

Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present paper we ask to what extent these structures can be viewed as mathematically tilable objects using a single two-dimensional tile. We find that spherical viruses from a large number of families—eight out of the twelve studied—qualitatively possess properties that allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit. This we did by characterizing the extent to which individual spherical capsids display subunit-subunit (1) holes, (2) overlaps, and (3) gross structural variability. All capsids with $T$ numbers greater than 1 from the Protein Data Bank, with homogeneous protein composition, were used in the study. These monohedral tilings, called canonical capsids due to their platonic (mathematical) form, offer a mathematical segue into the structural and dynamical understanding of not one, but a large number of virus capsids. From our data, it appears as though one may only break the long-standing rules of quasiequivalence by the introduction of subunit-subunit structural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in understanding structural aspects of such assemblies, we used graph theory and discrete geometry to enumerate the types of shapes that the tiles (and hence the subunits) must possess. We show that topology restricts the shape of the face to a limited number of five-sided prototiles, one of which is the “bisected trapezoid” that is a platonic representation of the most ubiquitous capsid subunit shape seen in nature (the trapezoidal jelly-roll motif). This motif is found in a majority of seemingly unrelated virus families that share little to no host, size, or amino acid sequence similarity. This suggests that topological constraints may exhibit dominant roles in the natural design of biological assemblies, while having little effect on amino acid sequence similarity.

Figure 1

(Color) Analysis of the tilability of a capsid. (a) The three-dimensional all-atom model is projected onto a sphere of average shell radius (colored to represent individual subunits) upon which a net, or dot matrix, is cast (visible in the subunit cleared area). These dots are used in calculating the (areawise) percentage of holes and overlaps in the capsid shell. (This projected capsid was derived from PDB ID 1smv). (b) The presence of either holes (right) or subunit-subunit overlaps (middle) will result in the inability to represent these structures as well-behaved two-dimensional tilings (left).

Figure 2

There are a large number of spherical capsids, highlighted by the dashed rectangles, that possess one of the three requirements of monohedral tilability.

Figure 3

(a)–(c) Dependence of one of three requirements for monohedral tilability plotted versus each of the others using family-averaged values. The solid lines are added to emphasize the trends. (d) An example of a capsid belonging to the Leviviridae family (PDB ID 1mst), which violates all of the requirements of monohedral tilability, as indicated by excessive overlap of black and white subunits and large holes at five- and pseudosixfold symmetry axes. (Subunit coloring: $A,B$, black; $C$, white.)

Figure 4

Abstract representation of a canonical capsid subunit, represented here in subunit form along with the types of subunit-subunit interactions (a) and in capsomere (hexamer and pentamer) arrangements that are formed from type II interactions (b),(c). The final capsid is composed of 12 pentamers and $10\left(T-1\right)$ hexamers that interact with each other via type I interactions [1]. For our studies, in order to not constrain the number of subunit vertices (and hence edges), we define the curved edge as one with a potentially unlimited number of vertices (with a lower bound of 2).

Figure 5

Pentagonal type 5 tilings. Type 5 tiles [e.g., (a) and (b)] are the only known five-sided tiles that possess six-valent vertices. The interacting edges of the prototile in both (a) and (b) have been marked with one, two, and three dashes. Such edges interact with identically labeled edges of neighboring tiles to form interfaces within dimers, trimers, and hexamers, respectively. Interestingly, a version of this tile (b) resembles a very commonly seen subunit shape found in nature depicted as a space-filled subunit (c) and a subunit within its capsid environment (d) (structure used: 1vak of the sobemoviridae family).

Figure 6

Relationship between mirror symmetry elements and the number of edges $\left(\sigma \right)$ per hexavalent tile. This image illustrates trigonal, tetragonal, and pentagonal tiles in (a) the hexavalent form and (b) as a single tile (along with the two-, three-, and sixfold symmetry relationships denoted). The dashed lines in the figure indicate possible mirror planes cutting the plane of the paper perpendicularly. It is immediately evident that the hexameric cluster for only $\sigma =5$ has no possible mirror symmetries, which is crucial in the chiral-centric biological world.