The architecture of Platonic polyhedral links

Hu, Guang; Zhai, Xiaofang; Lu, Dan; W, Qiu

doi:10.1007/s10910-008-9487-z

Cited by 36 publications

(37 citation statements)

References 33 publications

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“…The edges in this structure show two crossings, giving rise to one full twist of every edge. This example belongs to the class of T 2 k polyhedral links [18], where k denotes the number of full-twists along each edge. In the present example k = 1.…”

Section: Methodsmentioning

confidence: 99%

A New Euler's Formula for DNA Polyhedra

Ceulemans

2011

PLoS ONE

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DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus.

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Section: Methodsmentioning

confidence: 99%

A New Euler's Formula for DNA Polyhedra

Ceulemans

2011

PLoS ONE

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“…In [34], an analogous pattern, called the '3-branched curves and m-twisted double-lines covering' pattern, was also introduced in which the three single rings at a vertex are not locked. This pattern can be dealt with by replacing the vertex color set in Model 1 by C v = C * v = {0}, in which 0 is achiral.…”

Section: P Gmentioning

confidence: 99%

“…This enumeration model is expected to have various applications in counting three dimensional chemical compounds or other three dimensional objects. In particular, we apply this technique to the enumeration problem of polyhedral links which have received special attention from biochemists, mathematical chemists and mathematicians over the past two decades [29][30][31][32][33][34][35].…”

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confidence: 99%

Enumerating the total colorings of a polyhedron and application to polyhedral links

2012

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In this paper we consider the enumeration problem of a particular three-dimensional molecular or chemical compound system which has a polyhedral frame where the vertices, edges and faces represent 'units' such as atoms, bonds, ligands, polymers, or other objects of chemical interests. In this system, chirality is also taken into account. This enumeration problem is mathematically modeled as the 'total coloring' enumeration problem of a polyhedron: i.e., the number of ways to color all the vertices, edges and faces of the polyhedron by using three or more corresponding color sets, in which some colors may be chiral. We establish a general formula for this enumeration problem by extending the fundamental version of Plya's enumeration theorem. In particular, we apply this technique to the enumeration problem of polyhedral links which have received special attention from biochemists, mathematical chemists and mathematicians over the past two decades.NSFC [10831001

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“…In [28], the authors introduced an uni-variable polynomial from the adjacency matrix of a link diagram. The point symmetry group was used to detect the chirality in [23] and the writhe was applied to detect the topological chirality in [24,29]. The HOMFLY polynomial [30,31] is more powerful, but difficult to compute.…”

Section: Introductionmentioning

confidence: 99%

“…See [29,[32][33][34][35][36]. The tool we shall use is the celebrated Jones polynomial [37] for oriented links and its counterpart for unoriented links.…”

Section: Introductionmentioning

confidence: 99%

Topological chirality of a type of DNA and protein polyhedral links

Cheng

2015

J Math Chem

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Polyhedral links, interlinked and interlocked architectures, have been proposed for the description and characterization of DNA and protein polyhedra. In this paper, we study the topological chirality of a type of DNA polyhedral links constructed by the strategy of "n-point stars" and a type of protein polyhedral links constructed by "three-cross curves and untwisted double-line" covering. Furthermore, we prove that links corresponding to bipartite plane graphs have antiparallel orientations, and under these orientations, their writhes are not zero. As a result, the type of double crossover DNA polyhedral links are topologically chiral. We also prove that the unoriented link corresponding to a connected, even, bipartite plane graph always has self-writhe 0. Using the Jones polynomial for unoriented links we derive two simple criteria for chirality of unoriented alternating links with self-writhe 0. By applying this criterion we show that 3-regular protein polyhedral links are also topologically chiral. Topological chirality always implies chemical chirality, hence the corresponding DNA and protein polyhedra are all chemically chiral. Our chiral criteria may be used to detect the topological chirality of more complicated DNA and protein polyhedral links that may be synthesized by chemists and biologists in the future.

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The architecture of Platonic polyhedral links

Cited by 36 publications

References 33 publications

A New Euler's Formula for DNA Polyhedra

A New Euler's Formula for DNA Polyhedra

Enumerating the total colorings of a polyhedron and application to polyhedral links

Topological chirality of a type of DNA and protein polyhedral links

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