Constructions of DNA and polypeptide cages based on plane graphs and odd crossing [formula omitted]-junctions

Abstract

The constructions of three-dimensional synthetic DNA and polypeptide structures with a single closed DNA strand and polypeptide chain are mathematically based on strong traces of polyhedral graphs. However, a method developed for constructing such a DNA and polypeptide structure may impose additional restrictions on the types of strong traces and polyhedral graphs. In this paper, we show that strong traces for certain 2-connected plane graphs (allowed to have multiple edges) can be obtained using thickened graphs (sometimes called ribbon graphs) constructed with only two types of junctions : 0-crossing junction and special d(v)-crossing junction (called π-junction), where d(v) is the degree of the vertex v at which the d(v)-crossing junction is to be placed. The π-junctions are only applicable to vertices with odd degrees (≥3). We characterize the 2-connected plane graphs to which our approach can be applied and provide a brief guideline for the implementation of our method. This approach provides the theory, as well as a set of candidates, for designing and constructing stable DNA and polypeptide molecules needing only a method capable of creating the 0-crossing and π-junctions in a 2-connected plane graph.