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

Abstract

Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present report 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 values 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 for the first time 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 quasi-equivalence by the introduction of subunitsubunit 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.

Geometric considerations in virus capsid size specificity, auxiliary requirements, and buckling

“…3A represents a canonical capsid subunit (described in ref. 7) with its interaction types that give rise to all possible capsid sizes, and Fig. 3B represents a pentamer-hexamer cluster present in T Ͼ 1 capsids.…”

“…We studied dihedral angles within X-ray structures of all natural capsids unambiguously denotable as canonical capsids (capsids that are representable by monohedral tilings) (7). The stringency of these qualities is crucial to the dihedral angle comparisons, and so only a portion of those capsids deemed as ''canonical'' in ref.…”

A minimal representation of the self-assembly of virus capsids

“…Generalizations of the CK rules properly account for the geometry of some exceptional icosahedral capsids [3][4][5][6] and other elongated virus capsids that share coordination numbers with the icosahedral ones [7][8][9] . Recent work [10][11][12] provides important further development about the effect of the geometrical and topological constraints on the capsid structure.…”

Curvature Concentrations on the HIV-1 Capsid

“…The concept of the discrete Gaussian curvature on a polyhedral surface is based on the triangulation of such a surface. Triangulation, in this case, is equivalent to the idea of tiling, see [23,24], in which the tiles or subsections within one polygon are related by the theory of quasi-equivalence. For convenience, we assume each tile is equivalent, resulting in similar triangles within each hexagon or pentagon along the polyhedral surface.…”

“…The Euler characteristic of a closed polyhedral surface is given as χ M = V − E + F, regardless of how the surface is bent. Any closed convex polyhedral surface has an Euler characteristic χ M = 2, see [23,32]. This characteristic is independent of the choice of subsections, triangles, or tiles, since it is assumed each polygon is a planar object.…”