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

Polyomaviridae assemble in vitro into different aggregates depending on experimental conditions. We use an energy landscape approach using empirical energy calculations to quantify how the formation of these different aggregates depends on pH, the presence of bound calcium ions and disulfide linkages. Computations are carried out for SV40, a member of the Polyomaviridae family and are based on the binding free energy landscape of three distinct trimers of pentamers that correspond to the different bonding configurations between the capsid proteins observed in its crystal structure. Our computational analysis shows that the energetics of one of these environments is pivotal for the polymorphic assembly behaviour of SV40, whilst the binding energy landscapes of the other two environments are broadly funnel-shaped and thus contribute little to the formation of particles other than virus-like particles (VLP). We have quantified how the existence of bound calcium ions in the absence of disulfide linkages enhances the binding free energies of all three environments and hence, favours the assembly of VLPs. Moreover, estimation of the relative binding free energies of the three environments at pH 5 and pH 8 reveals that they are destabilized at pH 5 relative to pH 8. The extent of this destabilization is dependent on the presence of disulfide linkages and bound calcium ions and accounts for the experimentally observed polymorphic behaviour of VP1 proteins at pH 5. Interestingly, concurrent existence of bound calcium ions and disulfide linkages is found to be destabilizing and thus may disrupt the assembly of VLPs at pH 8.

Karim M. ElSawy

Leo S. D. Caves

Reidun Twarock

Polyomaviridae Assembly Polymorphism from an Energy Landscape Perspective

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

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

We present a top-down approach to the study of the dynamics of icosahedral virus capsids, in which each protein is approximated by a point mass. Although this represents a rather crude coarse-graining, we argue that it highlights several generic features of vibrational spectra which have been overlooked so far. We furthermore discuss the consequences of approximate inversion symmetry as well as the role played by viral tiling theory in the study of virus capsid vibrations.

Kasper Peeters

Anne Taormina

Dynamics of Icosahedral Viruses: What Does Viral Tiling Theory Teach Us?

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

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

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

Polyhedra with icosahedral symmetry and vertices labelled by rational indices of points of a six-dimensional lattice left invariant by the icosahedral group allow a morphological characterization of icosahedral viruses which includes the Caspar&#x2013;Klug classification as a special case. Scaling transformations relating the indexed polyhedron enclosing the surface with the one delimiting the central cavity lead to models of viral capsids observed in nature. Similar scaling relations can be obtained from projected images in three dimensions of higher-dimensional crystallographic point groups having the icosahedral group as a subgroup. This crystallographic approach can be extended to axial-symmetric clusters of coat proteins around icosahedral axes of the capsid. One then gets enclosing forms with vertices at points of lattices left invariant by the corresponding point group and having additional crystallographic properties also observed in natural crystals, but not explained by the known crystallographic laws.

Aloysio Janner

Exploring the Interplay between Virology and Molecular Crystallography

The series of ordered mutations that cause a specific cell to become cancerous is modeled so that the fraction of a risk population (e.g. White men) that has developed a specific cancer (e.g. melanoma) at any age can be calculated. The saturated model constructed and solved here is isomorphic to the physical model describing an ordered chain of radioactive nuclei decays with the exception that it allows for the possibility that a fraction of a risk population may be immune to developing a specific cancer.The simplest model developed here depends on only three independent parameters: the number of ordered mutations necessary for a cell to become cancerous, the fraction of the risk population that is immune to developing a specific cancer and the average time between mutations (a time defined as the mutation lifetime). The values of these independent parameters are determined by fitting the model's cancer incidence function to the cancer incidence data.This model was applied to five widely different cancers: melanoma, pancreatic cancer, female breast cancer, non-Hodgkin lymphoma and prostate cancer. The modeling predicts that all White males in the USA are vulnerable to developing melanoma, five-ordered mutations are required to develop it and the mutation lifetime is 48.3 years. By contrast, the modeling predicts that 80.7&#x25; of White females in the USA are immune to developing melanoma, three-ordered mutations are required to develop it and the mutation lifetime is 78.9 years. Remarkably, it was also found that about 70&#x25; of females are immune to developing breast cancer and about 70&#x25; of males are immune to developing prostate cancer, predictions that fit in with the experimental evidence of cancer immunosurveillance and immunoediting.Clearly, different risk populations can develop the same cancer through different pathways. Delineating the mechanism underlying the prevalence of immunity to specific cancers in specific risk populations should become a research priority. Finding ways of blocking or repairing cellular mutations and/or destroying mutated, potentially cancerous cells would prevent cancers from developing altogether and eliminate a major cause of mortality.

Ivan Kramer

Evidence that Natural Immunity to Breast Cancer and Prostate Cancer Exists in the Majority of Their Risk Populations Is Predicted by a Novel, Inherently Saturated, Ordered Mutation Model

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Computational and Mathematical Methods in Medicine