Abstract
DNA cages can be joined together to make larger 3D nanostructures on which molecular electronic circuits and tiny containers are built for drug delivery. The mathematical models for these promising nanomaterials play important roles in clarifying their assembly mechanism and understanding their structures. In this study, we propose a mathematical and computer method to construct permissible topological structures with double-helical edges for a triangular bipyramid and pentangular bipyramid. Furthermore, we remove the same topological links, without eliminating the nonrepeated ones for a triangular bipyramid and pentangular bipyramid. By analyzing characteristics of these unique links, some self-assembly and statistic rules are discussed. This study may obtain some new insights into the DNA assembly from the viewpoint of mathematics, promoting the comprehending and design efficiency of DNA polyhedra with required topological structures.
Highlights
The polyhedra made up of several triangles possess stable structures
The Biosphere for the 1967 World Fair, Expo 67 surprised the world, for it reached its maximum volume by using minimum materials
Geodesic domes are based on deltahedra which refer to polyhedra constructed from triangular faces
Summary
The polyhedra made up of several triangles possess stable structures. Of five Platonic solids constructed from one type of polygon, three are made of triangles, accounting for 60%. Benson et al proposed a general method to construct scaffolded DNA nanostructures from flat sheet meshes with the computer-aided technique [13] Qiu and his colleagues took a mathematical approach to answer the complex question of what models are suitable to describe the DNA polyhedron structure [14,15,16,17,18,19]. In consideration of significance of deltahedra-like nanostructures with lower symmetry, the triangular bipyramid and pentangular bipyramid assembled by a double helix as well as their nonrepeated isotopic links are studied. By introducing and analyzing their space diagrams, the nonrepeated triangular bipyramid and pentangular bipyramid are obtained In this process, some statistical law and assembly mechanism for deltahedra are discussed. This study may promote the comprehending and design efficiency of DNA polyhedra with required topological structures
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