A New Euler's Formula for DNA Polyhedra

Hu, Guang; W, Qiu; Ceulemans, Arnout

doi:10.1371/journal.pone.0026308

Cited by 18 publications

(8 citation statements)

References 25 publications

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“…A theoretical framework for characterizing DNA cages is a polyhedral link which turns the double-stranded DNA molecule into a computable topological model [24]. In fact, a polyhedral link is an interlinked and interlocked architecture obtained from a polyhedral graph G, by using branchedjunctions and twisted lines to replace the vertices and edges [14]. To fully understand this study, we have to explain some basic concepts used in our previous work.…”

Section: Dna Polyhedral Linksmentioning

confidence: 99%

“…Benson et al proposed a general method to construct scaffolded DNA nanostructures from flat sheet meshes with the computer-aided technique [13]. Qiu and his colleagues took a mathematical approach to answer the complex question of what models are suitable to describe the DNA polyhedron structure [14][15][16][17][18][19]. Guo et al provided a topological approach to assemble DNA tetrahedra and DNA triangular prism [20,21].…”

Section: Introductionmentioning

confidence: 99%

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Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

Deng

2020

Computational and Mathematical Methods in Medicine

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DNA cages can be joined together to make larger 3D nanostructures on which molecular electronic circuits and tiny containers are built for drug delivery. The mathematical models for these promising nanomaterials play important roles in clarifying their assembly mechanism and understanding their structures. In this study, we propose a mathematical and computer method to construct permissible topological structures with double-helical edges for a triangular bipyramid and pentangular bipyramid. Furthermore, we remove the same topological links, without eliminating the nonrepeated ones for a triangular bipyramid and pentangular bipyramid. By analyzing characteristics of these unique links, some self-assembly and statistic rules are discussed. This study may obtain some new insights into the DNA assembly from the viewpoint of mathematics, promoting the comprehending and design efficiency of DNA polyhedra with required topological structures.

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Section: Dna Polyhedral Linksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

Deng

2020

Computational and Mathematical Methods in Medicine

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“…Note that along each edge, there are two anti-parallel DNA duplexes [15], thus we view double crossover polyhedral links as oriented links. See also [19,51]. Their braid indices have been determined in [52,53], which, however, are useless for detecting chirality.…”

Section: Double Crossover Dna Polyhedral Linksmentioning

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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“…oxDNA, a coarse grain model considering DNA at the nucleotide level, has been proposed to study the self-assembly of DNA nanostructures [ 18 , 19 , 20 ], mainly to simulate large systems, such as DNA origami based assembly [ 21 ]. Mathematical models based on knot and topological graph theory have been built to unravel the basic structural rules driving the assembly of these nanostructures and design the DNA nanocages topologies [ 22 , 23 , 24 , 25 , 26 ]. A computational program called Polygen has also been developed to easily engineer DNA nanocages [ 27 ].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Dynamics Analysis of a DNA Octahedron by Elastic Network Model

Iacovelli

et al. 2017

Molecules

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DNA is a fundamental component of living systems where it plays a crucial role at both functional and structural level. The programmable properties of DNA make it an interesting building block for the construction of nanostructures. However, molecular mechanisms for the arrangement of these well-defined DNA assemblies are not fully understood. In this paper, the intrinsic dynamics of a DNA octahedron has been investigated by using two types of Elastic Network Models (ENMs). The application of ENMs to DNA nanocages include the analysis of the intrinsic flexibilities of DNA double-helices and hinge sites through the calculation of the square fluctuations, as well as the intrinsic collective dynamics in terms of cross-collective map calculation coupled with global motions analysis. The dynamics profiles derived from ENMs have then been evaluated and compared with previous classical molecular dynamics simulation trajectories. The results presented here revealed that ENMs can provide useful insights into the intrinsic dynamics of large DNA nanocages and represent a useful tool in the field of structural DNA nanotechnology.

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A New Euler's Formula for DNA Polyhedra

Cited by 18 publications

References 25 publications

Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

Topological chirality of a type of DNA and protein polyhedral links

Intrinsic Dynamics Analysis of a DNA Octahedron by Elastic Network Model

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