Topological Analysis of Enzymatic Actions on DNA Polyhedral Links

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

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.

Configuration of DNA polyhedra of truncated tetrahedron, cuboctahedron, truncated octahedron

The synthesis of DNA polyhedra has attracted more interest because of its wide application prospect, but its formation mechanism from mathematical viewpoint remains poorly understood. This paper presents the assembly process and mechanism of DNA truncated tetrahedron, cuboctahedron and truncated octahedron by the means of mathematics and computer program. Firstly, based on the assumption that the total number of all the crossings within each face of a DNA polyhedron must be an even number, potential types for three DNA polyhedra above are calculated by computer programs; Secondly, the projections of the truncated tetrahedron, the cuboctahedron and the truncated octahedron are plotted based on data fetched by the program; Thirdly, the component number, the odd-crossing edge number and even-crossing edge number for the corresponding polyhedral links are computed by analysis of their projections. This study gets some assembly mechanism on the structure of DNA double helix, promoting the comprehending and design efficiency of DNA polyhedra.

A new Euler's formula for DNA polyhedra

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.