•We construct compact orientable surfaces the number of whose boundary components is one (called thickened graphs), which are used to synthesize DNA and polypeptide molecules.•We specifically give an algorithm to obtain thickened graphs by using two types of junctions: the 0-crossing junction (called τ-junction) and a special d(v)-crossing junction (called π-junction-junction), where d(v) is the degree of the vertex v at which the d(v)-crossing junction is to be placed.•We use 2-connected plane graphs (allowed to have multiple edges, and not have loops) instead of platforms seen as polyhedral graphs in biosynthesis. 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.