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

DNA packaging in the bacteriophage φ29 involves a molecular motor. It is proposed that dsDNA is packaged through a channel in a connector located at the 5-fold vertex of a preformed prolate icosahedral capsid. The packaging motor also consists of virally-encoded RNA molecules (pRNA) coupled to ATPases. Data obtained from studies using surface plasmon resonance, fluorescence quenching and circular dichroism are presented to demonstrate the importance of the N-termini of the connector protein subunits in pRNA interaction and in conformational change. Based on our findings, we propose a model of DNA packaging based on connector conformational change.

Arron C. Tolley

Nicola J. Stonehouse

Conformational Changes in the Connector Protein Complex of the Bacteriophage φ29 DNA Packaging Motor

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

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?

This paper motivates and sets up the mathematical framework for a new program of investigation: to isolate and clarify the precise influence of symmetry on the probability space of assembly pathways that successfully lead to icosahedral viral shells. Several tractable open questions are posed. Besides its virology motivation, the topic is of independent mathematical interest for studying constructions of symmetric polyhedra. Preliminary results are presented: a natural, structural classification of subsets of facets of T = 1 polyhedra, based on their stabilizing subgroups of the icosahedral group; and a theorem that uses symmetry to formalize why increasing depth increases the numeracy (and hence probability) of an assembly pathway type (or symmetry class) for a T = 1 viral shell.

Miklós Bóna

Meera Sitharam

The Influence of Symmetry on the Probability of Assembly Pathways for Icosahedral Viral Shells

As computational and mathematical studies become increasingly central to studies of complicated reaction systems, it will become ever more important to identify the assumptions our models must make and determine when those assumptions are valid. Here, we examine that question with respect to viral capsid assembly by studying the ‘pathway complexity’ of model capsid assembly systems, which we informally define as the number of reaction pathways and intermediates one must consider to accurately describe a given system. We use two model types for this study: ordinary differential equation models, which allow us to precisely and deterministically compare the accuracy of capsid models under different degrees of simplification, and stochastic discrete event simulations, which allow us to sample use of reaction intermediates across a wide parameter space allowing for an extremely large number of possible reaction pathways. The models provide complementary information in support of a common conclusion that the ability of simple pathway models to adequately explain capsid assembly kinetics varies considerably across the space of biologically meaningful assembly parameters. These studies provide grounds for caution regarding our ability to reliably represent real systems with simple models and to extrapolate results from one set of assembly conditions to another. In addition, the analysis tools developed for this study are likely to have broader use in the analysis and efficient simulation of large reaction systems.

Navodit Misra

Daniel Lees

Tiequan Zhang

Russell Schwartz

Pathway Complexity of Model Virus Capsid Assembly Systems

The Simian virus 40 (SV40) capsid is a T = 7d icosahedral lattice ∼45 nm in diameter surrounding the ∼5 kb circular minichromosome. The outer shell is composed of 360 monomers of the major capsid protein VP1, tightly bound in 72 pentamers. VP1 is a jellyroll β-barrel, with extending N- and C-terminal arms. The N-terminal arms bind DNA and face the interior of the capsid. The flexible C-arms tie together the 72 pentamers in three distinct kinds of interactions, thus facilitating the formation of a T = 7 icosahedron from identical pentameric building blocks. Assembly in vivo was shown to occur by addition of capsomers around the DNA. We apply a combination of biochemical and genetic approaches to study SV40 assembly. Our in vivo and in vitro studies suggest the following model: one or two capsomers bind at a high affinity to ses, the viral DNA encapsidation signal, forming the nucleation centre for assembly. Next, multiple capsomers attach concomitantly, at lower affinity, around the minichromosome. This increases their local concentration facilitating rapid, cooperative assembly reaction. Formation of the icosahedron proceeds either by gradual addition of single pentamers to the growing shell or by concerted assembly of pentamer clusters.

Ariella Oppenheim

O. Ben-nun-Shaul

S. Mukherjee

M. Abd-El-Latif

SV40 Assembly In Vivo and In Vitro

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