A New Euler's Formula for DNA Polyhedra

PLOS ONE, Oct 2011

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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A New Euler's Formula for DNA Polyhedra

Citation: Hu G, Qiu W-Y, Ceulemans A ( A New Euler's Formula for DNA Polyhedra Guang Hu 0 Wen-Yuan Qiu 0 Arnout Ceulemans 0 Piero Andrea Temussi, Universita` di Napoli Federico II, Italy 0 1 Department of Chemistry, State Key Laboratory of Applied Organic Chemistry, Lanzhou University, People's Republic of China, 2 Department of Chemistry and INPAC institute for Nanoscale Physics and Chemistry, Katholieke Universiteit Leuven , Leuven , Belgium 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 m, of crossings c, and of Seifert circles s are related by a simple and elegant formula: szm~cz2. 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. - Funding: This work was supported by grants from the National Natural Science Foundation of China (Nos. 20973085, 10831001, and 21173104) and Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20090211110006). Financial support from the Leuven INPAC institute is gratefully acknowledged. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. synthesis of the DNA cube [6], a rich variety of DNA polyhedra, including tetrahedron [7], octahedron [8], dodecahedron [9], icosahedrons [10], and buckyball [11] have now been reported in the literature. In these nano-constructions, each face is made of closed, interlocked DNA rings, each edge is made of double-helix [12] or quadruplex-helix [1315] DNA strands, and each vertex represents an immobile multi-arm junction. The interest in these species is rapidly increasing not only for their potential properties but also for their intriguing architectures and topologies. The unresolved conflict has impelled a search for an even deeper understanding of nature. To address these structural puzzles, we were led to the mathematical models of so-called polyhedral links [1621], the rigorous mathematical definition of which was investigated by Jablan et al [22]. Polyhedral links are not simple, classical polyhedra, but consist of interlinked and interlocked structures, which require an extended understanding of traditional geometrical descriptors. Links, knots, helices, and holes replace the traditional structural relationships of vertices, faces and edges. The stereochemical control of these curious objects is still in its infancy, and would greatly benefit from clear theoretical models which express the relationships between the constituent descriptors, much in the same way as Eulers formula has done for the classical polyhedra. A challenge that is just now being addressed concerns how to ascertain and comprehend some of the mysterious characteristics of the DNA polyhedral folding. The needs of such a progress will spur the creation of better tools and better theories. Our treatment is based on the standard apparatus of knot theory [23,24]. A convenient way to facilitate the study of knots and links in terms of geometry makes use of the Seifert algorithm [25,26], which provides a surprisingly simple connection between knots and links and 2D surfaces. Eulers polyhedral formula has already provided a powerful tool to study the geometry of classical Polyhedral structures are basic markers of space, which have been known and celebrated for thousands of years [1]. They are encountered not only in art and architecture, but also in matter and many forms of life. The study of polyhedra has guided scientists to the discovery of spatial symmetry and geometry. A great theorem, which descends from geometry to topology, is Eulers polyhedral formula [2,3] where V, F and E are the respective total numbers of vertices, faces and edges of the polyhedron. Separate relations may also be established between pairs of these structural elements. As an example, let ni denote the degree of the i-th vertex, and let pj denote the number of sides to face j, with ni3 and pj 3. Then, we have: In particular, for the regular polyhedron where n edges radiate from every vertex, and every face is a p-gon, this becomes Eulers formula also provides a simple way for characterizing symmetry properties of polyhedral molecules [4]. In recent years entirely new types of polyhedral molecules, based on DNA, have em (...truncated)


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Guang Hu, Wen-Yuan Qiu, Arnout Ceulemans. A New Euler's Formula for DNA Polyhedra, PLOS ONE, 2011, 10, DOI: 10.1371/journal.pone.0026308