A Survey on Mathematical Models for DNA Polyhedra

Mathematical models for DNA polyhedra represent important and imminent challenges for mathematical chemistry. In recent years, there has been growing interest in applying knot theory and polyhedral topology to meet this challenge. The knot approach transforms a DNA polyhedron into a polyhedral link by connecting vertices and edge building blocks. Thus, it is possible to describe DNA polyhedra by understanding the construction method of polyhedral links, and studying their knot invariants and relationships. As an alternative mathematical mode, DNA cages have also proved to be useful tools for describing DNA polyhedra, in terms of topological graph analysis and the molecular design based on octet truss. This minireview aims to summarize recent progress in these two kinds of mathematical models for DNA polyhedra, and hope to arouse broader interests in this area for both biochemists and mathematicians.