Topological structures of DNA octahedrons determined by the number of ssDNA strands

How to control the nontrivial topological structures of DNA nanocages by the number of ssDNA strands is a fundamental problem for polyhedra assembly. In this paper, oriented octahedral links have been established as topological structure models to characterize DNA octahedrons with double helix edges assembled from one or more ssDNA strands. Here a program "Octa-links" has been developed in Python language to give all octahedral links and their components (number). There are 1566 types of octahedral links generated by considering all orientations calculated and the allowable tangle types on each edge. Note that all same orientations and links have been excluded by introducing 24 symmetric operations of octahedra. Furthermore, a new algorithm is proposed to give each component of each octahedral link as a series of vertex arcs, obtaining that the component number of all links fills the integer interval [1, 8]. Hence our program designed according to this algorithm paves an effective approach to calculate the component number of polyhedral links, making up the gap in the computer program on this aspect. Also, our work provides a complete list of topological structures with different component number for new DNA octahedrons assembled by adjusting the number of ssDNA strands.