DTP–05/09 A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Assembly models are developed for viral capsids built from protein building blocks that can assume different local bonding structures in the capsid. This situation occurs, for example, for viruses in the family of Papovaviridae, which are linked to cancer and are hence of particular interest for the health sector. More specifically, the viral capsids of the (pseudo-) T = 7 particles in this family consist of pentamers that exhibit two different types of bonding structures. While this scenario cannot be described mathematically in terms of Caspar-Klug Theory (Caspar and Klug 1962), it can be modelled via tiling theory (Twarock 2004). The latter is used to encode the local bonding environment of the building blocks in a combinatorial structure, called the assembly tree, which is a basic ingredient in the derivation of assembly models for Papovaviridae along the lines of the equilibrium approach of Zlotnick (Zlotnick 1994). A phase space formalism is introduced to characterize the changes in the assembly pathways and intermediates triggered by the variations in the association energies characterizing the bonds between the building blocks in the capsid. Furthermore, the assembly pathways and concentrations of the statistically dominant assembly intermediates are determined. The example of Simian Virus 40 is discussed in detail.

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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. 1.

We consider the problem of explicitly enumerating and counting the assembly pathways by which an icosahedral viral shell forms from identical constituent protein monomers. This poorly understood assembly process is a remarkable example of symmetric macromolecular self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly at the nanoscale. We use the new model of���that employs a static geometric constraint graph to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The model was developed to answer focused questions about the structural properties of the most probable types of successful assembly pathways. Specifically, the model reduces the study of pathway types and their probabilities to the study of the orbits of the automorphism group of the underlying geometric constraint graph, acting on the set of pathways. Since these are highly symmetric polyhedral graphs, it seems a viable approach to explicitly enumerate these orbits and count their sizes. The contribution of this paper is to isolate and simplify the core combinatorial questions, list related work and indicate the advantages of an explicit enumerative approach. 1.

We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly at the nanoscale. The model uses static geometric constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If G is a finite group acting on a finite set X, then there is a natural induced action of G on the set TX of trees whose leaves are bijectively labeled by the elements of X. If G acts simply on X, then |X |: = |Xn | = n · |G|, where n is the number of G-orbits in X. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of G on TXn, for every n, and (2) a simple algorithm to find the stabilizer of a tree τ ∈ TX in G that runs in linear time and does not need memory in addition to its input tree.

Affine Toda field theories related to non-crystallographic Coxeter groups Affine Toda field theories related to Coxeter groups of non-crystallographic type

Affine Toda field theories related to non-crystallographic Coxeter groups Affine Toda field theories related to Coxeter groups of non-crystallographic type

Previous studies have reported the production of malformed virus-like-particles (VLP) in recombinant host systems. Here we computationally investigate the case of a large triple-layered rotavirus VLP (RLP). In vitro assembly, disassembly and reassembly data provides strong evidence of microscopic reversibility of RLP assembly. Light scattering experimental data also evidences a slow and reversible assembly untypical of kinetic traps, thus further strengthening the fidelity of a thermodynamically controlled assembly. In silico analysis further reveals that under favourable conditions particles distribution is dominated by structural subunits and completely built icosahedra, while other intermediates are present only at residual concentrations. Except for harshly unfavourable conditions, assembly yield is maximised when proteins are provided in the same VLP protein mass composition. The assembly yield decreases abruptly due to thermodynamic equilibrium when the VLP protein mass composition is not obeyed. The latter effect is more pronounced the higher the Gibbs free energy of subunit association is and the more complex the particle is. Overall this study shows that the correct formation of complex multi-layered VLPs is restricted to a narrow range of association energies and protein concentrations, thus the choice of the host system is critical for successful assembly. Likewise, the dynamic control of intracellular protein

SV40 is a small, non enveloped DNA virus with an icosahedral capsid of 45 nm. The outer shell is composed of pentamers of the major capsid protein, VP1, linked via their flexible carboxy-terminal arms. Its morphogenesis occurs by assembly of capsomers around the viral minichromosome. However the steps leading to the formation of mature virus are poorly understood. Intermediates of the assembly reaction could not be isolated from cells infected with wt SV40. Here we have used recombinant VP1 produced in insect cells for in vitro assembly studies around supercoiled heterologous plasmid DNA carrying a reporter gene. This strategy yields infective nanoparticles, affording a simple quantitative transduction assay. We show that VP1 assembles under physiological conditions into uniform nanoparticles of the same shape, size and CsCl density as the wild type virus. The stoichiometry is one DNA molecule per capsid. VP1 deleted in the C-arm, which is unable to assemble but can bind DNA, was inactive indicating genuine assembly rather than non-specific DNA-binding. The reaction requires host enzymatic activities, consistent with the participation of chaperones, as recently shown. Our results demonstrate dramatic cooperativity of VP1, with a Hill coefficient of,6. These findings suggest that assembly may be a concerted reaction. We propose that concerted assembly is facilitated by simultaneous binding of multiple capsomers to a single DNA molecule, as we have recently reported, thus increasing their local concentration. Emerging principles of SV40 assembly may help understanding assembly of other complex systems. In addition, the SV40-based nanoparticles described here are potential gene therapy vectors that combine efficient gene delivery with safety and flexibility.