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Modelling virus selfassembly pathways: Avoiding dynamics using geometric constraint decomposition
 J. Comp. Biol
, 2006
"... We develop a 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 selfassembly occuring in nature and possesses many features that are desirable whi ..."
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We develop a 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 selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. The model uses static geometric and tensegrity 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 properties are carefully defined and computed using computational algebra and geometry, specifically stateofart concepts in geometric constraint decomposition. The model is analyzable and refinable and avoids expensive dynamics. We show that it has a provably tractable and accurate computational simulation and that its predictions are roughly consistent with known information about viral shell assembly. Justifications for mathematical and biochemical assumptions are provided, and comparisons are drawn with other virus assembly models. A method for more conclusive experimental validation involving specific viruses is sketched. Overall, the paper indicates a strong and direct, mutually beneficial interplay between (a) the concepts underlying macromolecular assembly; and (b) a wide variety of established as well as novel concepts from combinatorial and computational algebra, geometry and algebraic complexity. Key words: selfassembly, geometric constraints nanoscience, viral pathway, tractable modeling. 1.
Deletion of Highly Conserved ArginineRich RNA Binding Motif in Cowpea Chlorotic Mottle Virus Capsid Protein Results in Virion Structural Alterations and RNA Packaging Constraints
, 2004
"... The Nproximal region of cowpea chlorotic mottle virus (CCMV) capsid protein (CP) contains an argininerich RNA binding motif (ARM) that is also found in the CPs of other members of Bromoviridae and in other RNA binding proteins such as the Tat and Rev proteins of human immunodeficiency virus. To ass ..."
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The Nproximal region of cowpea chlorotic mottle virus (CCMV) capsid protein (CP) contains an argininerich RNA binding motif (ARM) that is also found in the CPs of other members of Bromoviridae and in other RNA binding proteins such as the Tat and Rev proteins of human immunodeficiency virus. To assess the critical role played by this motif during encapsidation, a variant of CCMV RNA3 (C3) precisely lacking the ARM region (C3/�919) of its CP gene was constructed. The biology and the competence of the matured CP derived in vivo from C3/�919 to assemble and package progeny RNA was examined in whole plants. Image analysis and computerassisted threedimensional reconstruction of wildtype and mutant virions revealed that the CP subunits bearing the engineered deletion assembled into polymorphic virions with altered surface topology. Northern blot analysis of virion RNA from mutant progeny demonstrated that the engineered mutation downregulated packaging of all four viral RNAs; however, the packaging effect was more pronounced on genomic RNA1 and RNA2 than genomic RNA3 and its CP mRNA. In vitro assembly assays with mutant CP subunits and RNA transcripts demonstrated that the mutant CP is inherently not defective in packaging genomic RNA1 (53%) and RNA2 (54%), but their incorporation into virions was competitively inhibited by the presence of other viral RNAs. Northern blot analysis of RNA encapsidation in vivo of two distinct bromovirus RNA3 chimeras, constructed by exchanging CPs having the �919 deletion, demonstrated that the role of the
COMBINATORIAL DECOMPOSITION, GENERIC INDEPENDENCE AND ALGEBRAIC COMPLEXITY OF GEOMETRIC CONSTRAINTS SYSTEMS: APPLICATIONS IN BIOLOGY AND ENGINEERING
, 2006
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Tree Orbits under Permutation Group Action: Algorithm, Enumeration and Application to Viral Assembly
, 2009
"... 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 ca ..."
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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 Gorbits 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.
SelfAssembly of Brome Mosaic Virus Capsids: Insights from Shorter TimeScale Experiments†
, 2008
"... An amended kinetic model for the selfassembly of empty capsids of brome mosaic virus is proposed. The model has been modified to account for a new feature in the assembly kinetics revealed by timecourse light scattering experiments at higher temporal resolution than previously attempted. To be abl ..."
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An amended kinetic model for the selfassembly of empty capsids of brome mosaic virus is proposed. The model has been modified to account for a new feature in the assembly kinetics revealed by timecourse light scattering experiments at higher temporal resolution than previously attempted. To be able to simulate the sharp takeoff from the initial lag phase to the growth phase in the kinetic curves, a monomer activation step was proposed.
Modeling Virus SelfAssembly Pathways Using Computational Algebra and Geometry
 APPLICATIONS OF COMPUTER ALGEBRA (ACA2004)
, 2004
"... 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 selfassembly occuring in nature and possesses many features that are d ..."
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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 selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly 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
COUNTING AND ENUMERATION OF SELFASSEMBLY PATHWAYS FOR SYMMETRIC MACROMOLECULAR STRUCTURES
"... 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 selfassembly occuring in nature and ..."
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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 selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly 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.
THE INFLUENCE OF SYMMETRY ON THE PROBABILITY OF ASSEMBLY PATHWAYS FOR ICOSAHEDRAL VIRAL SHELLS
"... 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. Besid ..."
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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.
Role of Surface Charge Density in NanoparticleTemplated Assembly of Bromovirus Protein Cages
"... n the past few years increased evidence was provided that established symmetric protein cages as a common architectural paradigm of the subcellular world. Intracellularly, protein cages may serve as microreactors, concentrators, or vehicles for metabolites.1 For example, clathrin cages are involv ..."
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n the past few years increased evidence was provided that established symmetric protein cages as a common architectural paradigm of the subcellular world. Intracellularly, protein cages may serve as microreactors, concentrators, or vehicles for metabolites.1 For example, clathrin cages are involved in endocytosis,2 carboxysomes in CO2 fixation,3 and ferritin cages in iron metabolism.4 In all these cases, the cages shield their cargo from the influence of external conditions and provide a controlled microenvironment, which is chemically welldefined by virtue of their regular nature. Protein cages also represent