Results 1 
8 of
8
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 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
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.
COMBINATORIAL DECOMPOSITION, GENERIC INDEPENDENCE AND ALGEBRAIC COMPLEXITY OF GEOMETRIC CONSTRAINTS SYSTEMS: APPLICATIONS IN BIOLOGY AND ENGINEERING
, 2006
"... ..."
(Show Context)
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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.
–The Virus Assembly Model: Pathways and Effort
"... 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 ..."
Abstract
 Add to MetaCart
(Show Context)
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.
Modeling Autonomous Supramolecular Assembly
"... Abstract Supramolecular assembly is often a remarkably robust, rapid and spontaneous process, starting from a small number of monomeric types. Although, the process occurs widely in nature and is increasingly important in healthcare and engineering, it is poorly understood. Icosahedral viral shell ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract Supramolecular assembly is often a remarkably robust, rapid and spontaneous process, starting from a small number of monomeric types. Although, the process occurs widely in nature and is increasingly important in healthcare and engineering, it is poorly understood. Icosahedral viral shell assembly is one such outstanding example. We sketch the experimental roadblocks that necessitate mathematical and computational modeling of assembly, and list the types of experimental data available for model validation, thereby defining the models ’ input and output, and framing the scope of model predictions. We isolate the various factors, specifically configurational and combinatorial entropy that influence spontaneous supramolecular assembly, pinpointing the modeling challenges and motivating the use of multiscale models. We then survey existing modeling paradigms for the modeling different scales, emphasizing the newest models and paradigms developed by the author’s group, geared towards not only predicting, but also intuitively explaining, analyzing and engineering assembly processes. The models leverage geometric and algebraic characteristics unique to molecular assembly (as opposed to folding), and permit provable performance guarantees together with some level of forward and backward analysis as well as a desired level of precision and refinability of prediction. 1