a new eulers formula for dna polyhedradna多面体的欧拉公式.pdfVIP

A New Euler’s Formula for DNA Polyhedra 1,2 1 2 Guang Hu , Wen-Yuan Qiu *, Arnout Ceulemans 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 Abstract 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: s zm ~c z2. 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. Citation: Hu G, Qiu W-Y, Ceulemans A (2011) A New Euler’s Formula for DNA Polyhedra. PLoS ONE 6(10): e26308. doi:10.1371/journal.pone.0026308 ` Editor: Piero Andrea Temussi, Universita di Napoli Federico II, Italy Received August 5, 2011; Accepted September 23, 2011; Published October 17, 2011 Copyright: 2011 Hu et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction