Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence protect the viral genome, it has been recognized that icosahedral symmetry is crucial for the structural organization of viruses. In particular, icosahedral symmetry has been invoked in order to predict the surface structures of viral capsids in terms of tessellations or tilings that schematically encode the locations of the protein subunits in the capsids. Whilst this approach is capable of predicting the relative locations of the proteins in the capsids, a prediction on the relative sizes of different virus particles in a family cannot be made. Moreover, information on the full 3D structure of viral particles, including the tertiary structures of the capsid proteins and the organization of the viral genome within the capsid are inaccessible with their approach. We develop here a mathematical framework based on affine extensions of the icosahedral group that allows us to address these issues. In particular, we show that the relative radii of viruses in the family of Polyomaviridae and the material boundaries in simple RNA viruses can be determined with our approach. The results complement Caspar and Klug's theory of quasi-equivalence and provide details on virus structure that have not been accessible with previous methods, implying that icosahedral symmetry is more important for virus architecture than previously appreciated.

T. Keef

R. Twarock

New Insights into Viral Architecture via Affine Extended Symmetry Groups

Virus assembly, utilizing a limited number of viral coat protein (CP or VP) building blocks, is an excellent example of a directed macromolecular interaction occurring in nature. Two basic principles govern the assembly of spherical (icosahedral) viruses: (i) Genetic Economy – the encapsidated genome encodes a single or few CPs that assemble a protective shell (the viral capsid); and (ii) specificity – the CPs have to fold to recognize each other and form exact CP–CP interfacial interactions during the assembly pathway. Using a variety of biophysical techniques, including X-ray crystallography and cryo-electron microscopy, combined with homology model building, biochemistry and molecular biology, the nature of the interactions between protein–protein subunits and protein–nucleic acid that facilitate viral capsid assembly have been studied. This review discusses both the similarities and differences that have been elucidated for ssDNA Microviridae, Geminiviridae, and Parvoviridae virus families. The Microviridae represent a family of bacteriophages that utilize several CPs and scaffold proteins to assemble a T = 1 icosahedral capsid, the Geminiviridae plant viruses assemble a unique twinned quasi-isometric (geminate) pseudo T = 1 virion assembled from a single CP and the Parvoviridae represent animal viruses whose T = 1 capsids are formed from the common overlapping region of two to four VPs that have unique N-terminal extensions. A survey of the three-dimensional (3D) data available for these viruses shows that they utilize structural commonalities, facilitated by disparate CP/VP amino acid sequences for the successful assembly of mature infectious virions.

Antonette Bennett

Robert McKenna

Mavis Agbandje-McKenna

A Comparative Analysis of the Structural Architecture of ssDNA Viruses

Whereas many viruses have capsids of uniquely defined sizes that observe icosahedral symmetry, retrovirus capsids are highly polymorphic. Nevertheless, they may also be described as polyhedral foldings of a fullerene lattice on which the capsid protein (CA) is arrayed. Lacking the high order of symmetry that facilitates the reconstruction of icosahedral capsids from cryo-electron micrographs, the 3D structures of individual retrovirus capsids may be determined by cryo-electron tomography, albeit at lower resolution. Here, we describe computational and graphical methods used to construct polyhedral models that match in size and shape, capsids of Rous sarcoma virus (RSV) observed within intact virions. The capsids fall into several shape classes, including tubes, ‘lozenges’ and ‘coffins’. The extent to which a capsid departs from icosahedral symmetry reflects the irregularity of the distribution of pentamers, which are always 12 in number for a closed polyhedral capsid. The number of geometrically distinct polyhedra grows rapidly with increasing quotas of hexamers, and ranks in the millions for particles in the size range of RSV capsids, which typically have 150–300 hexamers. Unlike the CAs of icosahedral viruses that assume a minimal number of quasi-equivalent conformations equal to the triangulation number (T), retroviral CAs exhibit a near-continuum of quasi-equivalent conformations – a property that may be attributed to the flexible hinge linking the N- and C-terminal domains.

J. Bernard Heymann

Carmen Butan

Dennis C. Winkler

Rebecca C. Craven

Alasdair C. Steven

Irregular and Semi-Regular Polyhedral Models for Rous Sarcoma Virus Cores

HIV-1 integrase is the viral enzyme responsible for the insertion of the viral DNA into the host cell chromosome. This process occurs through two distinct biochemical reactions: the 3&#x2032;-processing of the viral DNA and the transesterification reaction. Because experimental structural information on the reaction intermediate is not available, several molecular models have been developed. Unfortunately, none of the models of the enzyme&#x2013;substrate complex is fully consistent with the available molecular biological data. We have constructed a new theoretical model based on mutagenesis experiments and cross-linking data, using a relatively accurate computational setup. The structural features of the model along with its limitations are discussed here.

C. Fenollar-Ferrer

V. Carnevale

S. Raugei

P. Carloni

HIV-1 Integrase&#x2013;DNA Interactions Investigated by Molecular Modelling

A number of idealized models have been proposed to explain the long range organization of the DNA in bacteriophages. However, none of these models can account for the distributions of complex knots found when examining DNA extracted from bacteriophage P4 capsids. Furthermore, these models do not consider possible chirality biases in the arrangement of the DNA molecule inside the capsid. In this paper, we address these two issues by proposing a randomized version of one of the most popular models: the coaxially spooled model. We present analytical and numerical results for the properties of the random polygons (knots) generated using this model. We show that such model can easily generate complex knotted conformations and although it accounts for some chirality of the organization of the DNA molecules inside bacteriophage capsids does not fully explain the experimental data.

J. Arsuaga

Y. Diao

DNA Knotting in Spooling Like Conformations in Bacteriophages

A classification of icosahedral viral capsids is proposed. We show how the self-organization of capsids during their formation implies a definite composition of their elementary building blocks. The exact number of hexamers with three different admissible symmetries is related to capsids&#39; sizes, labelled by their T-numbers. Simple rules determining these numbers for each value of T are deduced and certain consequences concerning the probabilities of mutations and evolution of viruses are discussed.

Richard Kerner

Classification and Evolutionary Trends of Icosahedral Viral Capsids

The mathematic modelling of icosahedral virus assembly has drawn increasing interest because of the symmetric geometry of the outer shell structures. Many models involve equilibrium expressions of subunit binding, with reversible subunit additions forming various intermediate structures. The underlying assumption is that a final lowest energy state drives the equilibrium toward assembly. In their simplest forms, these models have explained why high subunit protein concentrations and strong subunit association constants can result in kinetic traps forming off pathway partial and aberrant structures. However, the cell biology of virus assembly is exceedingly complex. The biochemistry and biology of polyoma and papillomavirus assembly described here illustrates many of these specific issues. Variables include the use of cellular ‘chaperone’ proteins as mediators of assembly fidelity, the coupling of assembly to encapsidation of a specific nucleic acid genome, the use of cellular structures as ‘workbenches’ upon which assembly occurs, and the underlying problem of making a capsid structure that is metastable and capable of rapid disassembly upon infection. Although formidable to model, incorporating these considerations could advance the relevance of mathematical models of virus assembly to the real world.

Robert L. Garcea

Biologic Constraints on Modelling Virus Assembly

There is now a considerable literature on computer simulations of DNA packaging inside bacteriophage capsids. While most studies have reached a semiquantitative or qualitative agreement with single molecule packaging and ejection studies, several quantitative answers are to date still lacking, needing either more accurate measurements or more realistic or difficult simulations. Here, I briefly review the outstanding questions in this field and report some new numerical results on DNA packaging inside the phi29 phage, modelled either as a capped sphero-cylinder or as a sphere with the same internal volume. These simulations include electrostatics and a realistic genome length, and contribute to seriously questioning the inverse spool model, which arises from a purely continuum mechanics view of the problem, and is still commonly adopted to describe the shape of the packaged genome.

D. Marenduzzo

Computer Simulations of DNA Packing inside Bacteriophages: Elasticity, Electrostatics and Entropy

Using a combination of biochemistry, mass spectrometry, NMR spectroscopy and cryo-electron microscopy (cryo-EM), we have been able to show that quasi-equivalent conformer switching in the coat protein (CP) of an RNA bacteriophage (MS2) is controlled by a sequence-specific RNA–protein interaction. The RNA component of this complex is an RNA stem-loop encompassing just 19 nts from the phage genomic RNA, which is 3569 nts in length. This binding results in the conversion of a CP dimer from a symmetrical conformation to an asymmetric one. Only when both symmetrical and asymmetrical dimers are present in solution is assembly of the T = 3 phage capsid efficient. This implies that the conformers, we have characterized by NMR correspond to the two distinct quasi-equivalent conformers seen in the 3D structure of the virion. An icosahedrally-averaged single particle cryo-EM reconstruction of the wild-type phage (to ∼9 Å resolution) has revealed icosahedrally ordered density encompassing up to 90% of the single-stranded RNA genome. The RNA is seen with a novel arrangement of two concentric shells, with connections between them along the 5-fold symmetry axes. RNA in the outer shell interacts with each of the 90 CP dimers in the T = 3 capsid and although the density is icosahedrally averaged, there appears to be a different average contact at the different quasi-equivalent protein dimers: precisely the result that would be expected if protein conformer switching is RNA-mediated throughout the assembly pathway. This unprecedented RNA structure provides new constraints for models of viral assembly and we describe experiments aimed at probing these. Together, these results suggest that viral genomic RNA folding is an important factor in efficient assembly, and further suggest that RNAs that could sequester viral CPs but not fold appropriately could act as potent inhibitors of viral assembly.

Ottar Rolfsson

Katerina Toropova

Victoria Morton

Simona Francese

Gabriella Basnak

Gary S. Thompson

Stephen W. Homans

Alison E. Ashcroft

Nicola J. Stonehouse

Neil A. Ranson

Peter G. Stockley

RNA Packing Specificity and Folding during Assembly of the Bacteriophage MS2

The bacteriophages of the Cystoviridae family package their single stranded RNA genomic precursors into empty capsid (procapsids) using a hexameric packaging ATPase motor (P4). This molecular motor shares sequence and structural similarity with RecA-like hexameric helicases. A concerted structural, mutational and kinetic analysis helped to define the mechanical reaction coordinate, i.e. the conformational changes associated with RNA translocation. The results also allowed us to propose a possible scheme of coupling between ATP hydrolysis and translocation which requires the cooperative action of three consecutive subunits. Here, we first test this model by preparing hexamers with defined proportions of wild type and mutant subunits and measuring their activity. Then, we develop a stochastic kinetic model which accounts for the catalytic cooperativity of the P4 hexamer. Finally, we use the available structural information to construct a quantum-chemical model of the chemical reaction coordinate and obtain a detailed description of the electron density changes during ATP hydrolysis. The model explains the results of the mutational analyses and yields new insights into the role of several conserved residues within the ATP binding pocket. These hypotheses will guide future experimental work.

Jelena Telenius

Anders E. Wallin

Michal Straka

Hongbo Zhang

Erika J. Mancini

Roman Tuma

RNA Packaging Motor: From Structure to Quantum Mechanical Modelling and Sequential-Stochastic Mechanism

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