Periodic table of virus capsids: implications for natural selection and design

Abstract

Background: For survival, most natural viruses depend upon the existence of spherical capsids: protective shells of various sizes composed of protein subunits. So far, general evolutionary pressures shaping capsid design have remained elusive, even though an understanding of such properties may help in rationally impeding the virus life cycle and designing efficient nano-assemblies.

Principal findings: This report uncovers an unprecedented and species-independent evolutionary pressure on virus capsids, based on the the notion that the simplest capsid designs (or those capsids with the lowest "hexamer complexity", C(h)) are the fittest, which was shown to be true for all available virus capsids. The theories result in a physically meaningful periodic table of virus capsids that uncovers strong and overarching evolutionary pressures, while also offering geometric explanations to other capsid properties (rigidity, pleomorphy, auxiliary requirements, etc.) that were previously considered to be unrelatable properties of the individual virus.

Significance: Apart from describing a universal rule for virus capsid evolution, our work (especially the periodic table) provides a language with which highly diverse virus capsids, unified only by geometry, may be described and related to each other. Finally, the available virus structure databases and other published data reiterate the predicted geometry-derived rules, reinforcing the role of geometry in the natural selection and design of virus capsids.

Figure 1. Capsids are scalable.

(A) Spherical capsids of various sizes are composed of 12 pentamers (represented as darkened pentagons) and a variable number of hexamers. (B) Quasi-equivalence posits that one may produce a pentamer from a hexamer by removing one subunit and its environment (the shaded triangular region) and joining the unpaired interfaces. This operation imposes pentameric dihedral angle values (“endo angles”) onto its neighboring hexameric angles , which, if unchallenged, propagate through the hexamers (depicted by arrows) in what we call endo angle propagation.

Figure 2. The three virus capsid classes.

All canonical capsids (made up of trapezoidal subunits) may be built from a single type of pentamer and a repertoire of distinct hexamer shapes (colored distinctly only once in each capsid; also described in Fig. S2 in File S1). The hexamer shape is described by the number of endo angles it displays. Endo angles are depicted as bold lines within a “face” in its isolated (right) and capsid environment (left) for the first three capsid sizes in each class (excepting ).

Figure 3. Periodic discrimination of spherical capsids.

(A) As predicted by the inverse rule, capsids with high hexamer complexity are under-represented in nature as evident in the observed versus unbiased capsid abundances ( of families that display capids of specific ). (B) is not conveniently correlated with capsid size () or class (symbols). (C) However, trends in are easily discerned from the periodic table, where, in each period (row), , class number and increase (or remain the same), while trends in other capsid properties such as rigidity may also be deciphered.

Figure 4. Spherical capsid phase diagram.

We describe two specific capsid sizes that remain to be elucidated ( and ; the diagram arbitrarily assumes that ). describes the limit of the geometric domain, beyond which our geometric assumptions and predictions may not hold. We expect that all capsid sizes greater than will be exclusively described by continuum elasticity. We also expect that, beyond (i.e., in the purely continuum domain), the Föppl-von Karman number () , that dictates spherical vs. icosahedral morphology will depend primarily on , and so there will be a capsid size () that demarcates the allowance for spherical and icosahedral morphologies in the purely continuum regime (the sigmoidal curve represents the dependence of and hence morphology on ). These assumptions consolidate all observed instances of spherical capsid morphology.