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

Abstract—This paper is the result of analyzing the growth pattern of the first 500 prime numbers. The growth rate was found to be related to two characteristics – two parallel threads and the almost viral impact of the multiples of 6 on these two threads. When corresponding cause and effect of these impacts are overlaid, a double-helix structure is the result. This concept and modeling approach is submitted with the hope of providing a model that helps connect the research benefits of bioinformatics and the potential impact of the prime number structure of the Riemann Hypothesis when applied by professionals in their scientific computing fields. Index Terms—Double-helix, gap, growth, prime numbers. I.

Hints of beauty in social cognition: Broken symmetries in mental dynamics

We propose a new modeling framework inspired by chemical reaction processes. Our approach consists in defining the processes and the interactions within the system in term of reactions. Such a definition can be applied to many systems, ranging from biochemical systems to swarm robotics. In particular, we aim at exploiting the toolbox developed in the context of hybrid system modeling and simulation. The concept of extended self-assembly is the following: given a set of passive building blocks A, B, C, and D, how to obtain, with a maximal yield, the products X, Y, and Z using a set of N active transporters? What is the smallest set of reactions leading to these products? More importantly, how shall we design the building blocks and their transporters in order to fit this set of reactions? The reaction set may also involve intermediate products and be influenced by external factors. We draw inspiration from the DNA

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.