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.
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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)