Form, symmetry and packing of biomacromolecules. II. Serotypes of human rhinovirus

Viral Phrenology

We introduce Viral Phrenology, a new scheme for understanding the genomic composition of spherical viruses based on the locations of their structural protrusions. We used icosahedral point arrays to classify 135 distinct viral capsids collected from over 600 capsids available in the VIPERdb. Using gauge points of point arrays, we found 149 unique structural protrusions. We then show how to use the locations of these protrusions to determine the genetic composition of the virus. We then show that ssDNA, dsDNA, dsRNA and ssRNA viruses use different arrangements for distributing their protrusions. We also found that Triangulation number is also partially dependent on the structural protrusions. This analysis begins to tie together Baltimore Classification and Triangulation number using point arrays.

Unveiling the Hidden Rules of Spherical Viruses Using Point Arrays

Since its introduction, the Triangulation number has been the most successful and ubiquitous scheme for classifying spherical viruses. However, despite its many successes, it fails to describe the relative angular orientations of proteins, as well as their radial mass distribution within the capsid. It also fails to provide any critical insight into sites of stability, modifications or possible mutations. We show how classifying spherical viruses using icosahedral point arrays, introduced by Keef and Twarock, unveils new geometric rules and constraints for understanding virus stability and key locations for exterior and interior modifications. We present a modified fitness measure which classifies viruses in an unambiguous and rigorous manner, irrespective of local surface chemistry, steric hinderance, solvent accessibility or Triangulation number. We then use these point arrays to explain the immutable surface loops of bacteriophage MS2, the relative reactivity of surface lysine residues in CPMV and the non-quasi-equivalent flexibility of the HBV dimers. We then explain how point arrays can be used as a predictive tool for site-directed modifications of capsids. This success builds on our previous work showing that viruses place their protruding features along the great circles of the asymmetric unit, demonstrating that viruses indeed adhere to these geometric constraints.

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

The architecture of infinite structures with non-crystallographic symmetries can be modelled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic group G is embedded into the point group P of a higher-dimensional lattice and the chains of all G-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensional G-invariant subspace consists of nested point sets with G-symmetry at each radial level. The number of different radial levels is bounded by the index of G in the subgroup of P. In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart.

Approximation of virus structure by icosahedral tilings

Viruses are remarkable examples of order at the nanoscale, exhibiting protein containers that in the vast majority of cases are organized with icosahedral symmetry. Janner used lattice theory to provide blueprints for the organization of material in viruses. An alternative approach is provided here in terms of icosahedral tilings, motivated by the fact that icosahedral symmetry is non-crystallographic in three dimensions. In particular, a numerical procedure is developed to approximate the capsid of icosahedral viruses by icosahedral tilesviaprojection of high-dimensional tiles based on the cut-and-project scheme for the construction of three-dimensional quasicrystals. The goodness of fit of our approximation is assessed using techniques related to the theory of polygonal approximation of curves. The approach is applied to a number of viral capsids and it is shown that detailed features of the capsid surface can indeed be satisfactorily described by icosahedral tilings. This work complements previous studies in which the geometry of the capsid is described by point sets generated as orbits of extensions of the icosahedral group, as such point sets are by construction related to the vertex sets of icosahedral tilings. The approximations of virus geometry derived here can serve as coarse-grained models of viral capsids as a basis for the study of virus assembly and structural transitions of viral capsids, and also provide a new perspective on the design of protein containers for nanotechnology applications.

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

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.

Form, symmetry and packing of biomacromolecules. V. Shells with boundaries at anti-nodes of resonant vibrations in icosahedral RNA viruses

The RNA viruses cowpea chlorotic mottle, satellite tobacco mosaic, pariacoto and MS2, already considered in part IV of this series of papers [Janner, A. (2011a), Acta Cryst. A67, 517-520], are investigated further, with the aim to arrive at a possible physical basis for their structural properties. The shell structure of the filled capsid is analyzed in terms of successive spherical boundaries of the sets of icosahedral equivalent chains. By inversion in the sphere enclosing the capsid, the internal boundaries are transformed into external ones, which are more easily visualized. This graphical procedure reveals the presence of regularly spaced shells with boundaries fitting with anti-nodal surfaces of the virus considered as an elastic resonator. The centers of gravity of the various chains occur in the nodal regions of eigenvibrations with wavelength λ = R(0)/K(0), where R(0) is the radius of the virus and K(0) takes one of the values 12, 6, 4, 3, depending on the mode. The resonator model is consistent with practically all spherical shell boundaries, whereas deviations are observed for the icosahedral axial modes, which apparently play a secondary role with respect to the spherical ones. Both the spherical and the axial anti-nodal surfaces fit very well with the packed structure of the viruses in the crystal which, accordingly, is expected to have eigenfrequencies related to those of the virus. These results open the way to a better understanding of the possibility of breaking the capsid using resonant forced oscillations excited, for example, by an applied elastic shock or by irradiation with femtosecond laser pulses, as already realised by K.-T. Tsen and co-workers. An alternative `plywood' model connected to the extreme elastic properties of the capsid is also considered.

Form, symmetry and packing of biomacromolecules. III. Antigenic, receptor and contact binding sites in picornaviruses

The relation between serotype differentiation and crystallographic symmetry, revealed by the contact fingerprint diagrams investigated in Part II [Janner (2010). Acta Cryst. A66, 312-326] for the human rhinovirus, is extended to the Picornaviridae family. The approach, outlined in Part I [Janner (2010). Acta Cryst. A66, 301-311] and Part II for biomacromolecules packed in a crystal and based on concepts such as packing lattice, kissing points and crystal-packing parameters, can directly be applied to the picornaviruses. In particular, the contact fingerprint diagrams of 20 different virus strains have been derived. In these cases, as for the rhinovirus, these diagrams are serotype/strain specific, justifying the name fingerprint. The molecular basis for the serotype variability, and the associated conservation requirements, is usually analysed by considering antigenic sites, where capsid residues bind with antibodies, and receptor sites, where other residues bind with molecular receptors of the host cell membrane. Both the antigenic variation and the receptor conservation allow repeated infection of the host cells of the given animals. The graphical description of these sites is usually done by footprints and roadmap diagrams, mapping properties of the capsid surface and using the icosahedral symmetry of the capsid. The alternative fingerprint diagrammatic description, based on the crystal symmetry, adopted in Part II for the contact sites, where a capsid is bound to the next one in the crystal packing, is extended to the antigenic and receptor binding sites. Again, the antigenic/receptor fingerprints are specific, at least for the nine picornaviruses investigated so far, despite the more than a factor of ten larger coarse graining with respect to the corresponding footprint and roadmap diagrams. The latter are based on a grid spacing of about 2 Å, whereas the spacing implied by the packing-lattice approximation adopted in fingerprints varies typically from 20 to 50 Å. The fingerprint diagrams are accordingly simpler (because approximated), but nevertheless still serotype specific, despite the complex character of the interactions involved.

Form, symmetry and packing of biomacromolecules. I. Concepts and tutorial examples

The aim of this paper is to relate morphological properties of single biomacromolecules based on molecular enclosing forms indexed by an appropriate form lattice to the symmetry of the crystal where the molecules are periodically packed. Similar to the way in which the 'molécule intégrante' of Haüy permitted a molecular interpretation of the law of rational indices of crystal growth forms, alternative molecular enclosing forms, indexed by a so-called packing lattice, allow one to bridge the gap between form and crystal lattices. In this first part, selected tutorial examples illustrate the validity of the approach and the crystallographic compatibility between molecular and crystal structures. In particular, integral molecular lattices are shown to imply the observed axial ratios between crystal lattice parameters, leading sometimes to surprising results, like a cubic crystal lattice with a unit cell having a trigonal molecular filling with hexagonal enclosing form.