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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 repre- sentation 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 ca- nonical 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 struc- tural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in under- standing 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.

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