Constructions of DNA and polypeptide cages based on plane graphs and odd crossing \(\pi \)-junctions. (English) Zbl 1511.92048
Summary: 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 \(\pi \)-junction), where \(d ( v )\) is the degree of the vertex \(v\) at which the \(d ( v )\)-crossing junction is to be placed. The \(\pi \)-junctions are only applicable to vertices with odd degrees \(( \geq 3)\). 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 \(\pi \)-junctions in a 2-connected plane graph.
MSC:
| 92D20 | Protein sequences, DNA sequences |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
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