Ear decomposition of 3-regular polyhedral links with applications

The calculation of topological structures of strands-based DNA trigonal bipyramids

With the rapid development of DNA nanotechnology, strands-based DNA polyhedra have been reported and widely applied in chemical biology, drug delivery and materials science. A fundamental problem is to determine which topological structures such a DNA polyhedron will allow. Our goal in this paper is to determine all permissible nontrivial topological structures for DNA trigonal bipyramids with double-helical edges, which have been partially realized by one-step self-assembly of multiple DNA single-strands designed rationally. Here oriented trigonal bipyramid links (OTB links) are constructed as the mathematical models for DNA trigonal bipyramids and are further classified into 451 link types. In this process, an OTBL program is designed in the Python programming language to generate all OTB link diagrams and also to remove the same topological structures produced by the rotational symmetry of trigonal bipyramid. Our result gives a list of candidates for further synthesized DNA trigonal bipyramid prisms 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.