arXiv subjects

R. Twarock

Publications and source records attributed to R. Twarock.

13 recordsLinked to original sources

arXiv2008

DNA duplex cage structures with icosahedral symmetry

A construction method for duplex cage structures with icosahedral sym- metry made out of single-stranded DNA molecules is presented and applied to an icosidodecahedral cage. It is shown via a mixture of analytic and computer techniques that there exist realisations of this graph in terms of two circular DNA molecules. These blueprints for the organisation of a cage structure with a noncrystallographic symmetry may assist in the design of containers made from DNA for applications in nanotechnology.

q-bio.BM↗

arXiv2007

Dynamical Implications of Viral Tiling Theory

The Caspar-Klug classification of viruses whose protein shell, called viral capsid, exhibits icosahedral symmetry, has recently been extended to incorporate viruses whose capsid proteins are exclusively organised in pentamers. The approach, named `Viral Tiling Theory', is inspired by the theory of quasicrystals, where aperiodic Penrose tilings enjoy 5-fold and 10-fold local symmetries. This paper analyzes the extent to which this classification approach informs dynamical properties of the viral capsids, in particular the pattern of Raman active modes of vibrations, which can be observed experimentally.

q-bio.BM↗

arXiv2006

A Note on Genome Organisation in RNA Viruses with Icosahedral Symmetry

The structural organisation of the viral genome within its protein container, called the viral capsid, is an important aspect of virus architecture. Many single-stranded (ss) RNA viruses organise a significant part of their genome in a dodecahedral cage as a RNA duplex structure that mirrors the symmetry of the capsid. Bruinsma and Rudnick have suggested a model for the structural organisation of the RNA in these cages. It is the purpose of this paper to further develop their approach based on results from the areas of graph theory and DNA network engineering. We start by suggesting a scenario for pariacoto virus, a representative of this class of viruses, that is energetically more favorable than those derived previously. We then show that it is a representative of a whole family of cage structures that abide to the same construction principle, and then derive the energetically optimal configuration for a second family of cage structures along similar lines. Finally, we give reasons for the conjecture that these two families are more likely to occur in nature than other scenarios.

q-bio.BM↗

arXiv2005

A new series of polyhedra as blueprints for viral capsids in the family of Papovaviridae

In a seminal paper Caspar and Klug established a theory that provides a family of polyhedra as blueprints for the structural organisation of viral capsids. In particular, they encode the locations of the proteins in the shells that encapsulate, and hence provide protection for, the viral genome. Despite of its huge success and numerous applications in virology experimental results have provided evidence for the fact that the theory is too restrictive to describe all known viruses. Especially, the family of Papovaviridae, which contains cancer-causing viruses, falls out of the scope of this theory. In a recent paper we have shown that certain members of the family of Papovaviridae can be described via tilings. In this paper, we develop a comprehensive mathematical framework for the derivation of all surface structures of viral particles in this family. We show that this formalism fixes the structure and relative sizes of all particles collectively so that there exists only one scaling factor that relates the sizes of all particles with their biological counterparts. The series of polyhedra derived here complements the Caspar-Klug family of polyhedra. It is the first mathematical result that provides a common organisational principle for different types of viral particles in the family of Papovaviridae and paves the way for an understanding of Papovaviridae polymorphism. Moreover, it provides crucial input for the construction of assembly models.

q-bio.BM↗

arXiv2005

Classification of capped tubular viral particles in the family of Papovaviridae

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Viral capsids are usually spherical, and for a significant number of viruses exhibit overall icosahedral symmetry. The corresponding surface lattices, that encode the locations of the capsid proteins and intersubunit bonds, can be modelled by Viral Tiling Theory. It has been shown in vitro that under a variation of the experimental boundary conditions, such as the pH value and salt concentration, tubular particles may appear instead of, or in addition to, spherical ones. In order to develop models that describe the simultaneous assembly of both spherical and tubular variants, and hence study the possibility of triggering tubular malformations as a means of interference with the replication mechanism, Viral Tiling Theory has to be extended to include tubular lattices with end caps. This is done here for the case of Papovaviridae, which play a distinguished role from the viral structural point of view as they correspond to all pentamer lattices, i.e. lattices formed from clusters of five protein subunits throughout. These results pave the way for a generalisation of recently developed assembly models.

q-bio.BM↗

arXiv2005

Master equation approach to the assembly of viral capsids

The distribution of inequivalent geometries occurring during self-assembly of the major capsid protein in thermodynamic equilibrium is determined based on a master equation approach. These results are implemented to characterize the assembly of SV40 virus and to obtain information on the putative pathways controlling the progressive build-up of the SV40 capsid. The experimental testability of the predictions is assessed and an analysis of the geometries of the assembly intermediates on the dominant pathways is used to identify targets for antiviral drug design.

q-bio.BM↗

arXiv2005

Assembly Models for Papovaviridae based on Tiling Theory

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.

q-bio.BM↗

arXiv2002

Quantum Mechanics with Difference Operators

A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a "natural" choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a "new" framework.

quant-ph↗

arXiv2002

Affine extension of noncrystallographic Coxeter groups and quasicrystals

Unique affine extensions $H^{\aff}_2$, $H^{\aff}_3$ and $H^{\aff}_4$ are determined for the noncrystallographic Coxeter groups $H_2$, $H_3$ and $H_4$. They are used for the construction of new mathematical models for quasicrystal fragments with 10-fold symmetry. The case of $H^{\aff}_2$ corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project scheme we obtain by construction finite point sets, which grow with a model specific growth parameter.

math.GR↗

arXiv2002

New group structures for Carbon onions and Carbon nanotubes via affine extensions of non-crystallographic Coxeter groups

We present results underlining the conjecture that affine extensions for non-crystallographic Coxeter groups are suitable mathematical objects for the description of the symmetries of Carbon onions and Carbon nanotubes. It is the hope that these considerations will shed new light on open questions concerning structure, stability and formation of these fullerenes.

physics.chem-ph↗

arXiv2002

Construction of Diffusion Algebras

In cond-mat/0103603 Diffusion algebras have been introduced in the context of one-dimensional stochastic processes with exclusion in statistical mechanics. While this reference is focused on the needs of the physicist reader and thus states results without proofs and focuses on the discussion of lower-dimensional examples, it is the purpose of this paper to present a construction formalism for Diffusion algebras and to use the latter to prove the results in that reference.

math.QA↗

arXiv1996

Representations of $U_h(su(N))$ derived from quantum flag manifolds

A relationship between quantum flag and Grassmann manifolds is revealed. This enables a formal diagonalization of quantum positive matrices. The requirement that this diagonalization defines a homomorphism leads to a left \Uh -- module structure on the algebra generated by quantum antiholomorphic coordinate functions living on the flag manifold. The module is defined by prescribing the action on the unit and then extending it to all polynomials using a quantum version of Leibniz rule. Leibniz rule is shown to be induced by the dressing transformation. For discrete values of parameters occuring in the diagonalization one can extract finite-dimensional irreducible representations of \Uh as cyclic submodules.

q-alg↗

arXiv1995

The Rigged Hilbert Space Formulation of Quantum Mechanics and its Implications for Irreversibility

Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.

quant-ph↗