Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

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.

An assembling procedure for DNA bipyramids with minimum component strands

“…To put it simply, isotopic links can be obtained from each other by applying symmetry operations. By means of mathematical or computer method, these isotopic links could be removed or filtered out [37,38,40] . After removing the isotopic links, 6 unique types of DNA triangular bipyramids remain, showed in Table 2, and 16 triangular bipyramids are classified into six groups:{{1}, {2, 15}, {3, 5, 10}, {4, 6, 8, 9, 11, 13}, {7, 12, 14}, {16}} [38] .…”

“…By means of mathematical or computer method, these isotopic links could be removed or filtered out [37,38,40] . After removing the isotopic links, 6 unique types of DNA triangular bipyramids remain, showed in Table 2, and 16 triangular bipyramids are classified into six groups:{{1}, {2, 15}, {3, 5, 10}, {4, 6, 8, 9, 11, 13}, {7, 12, 14}, {16}} [38] . 6 Intuitively, with the increase of edges with odd half-twists, the number of strands to construct triangular bipyramid decreases.…”

“…In the previous study, we have constructed and analysed DNA polyhedra from through computation program [37,38] . Here, we present an assembling procedure for DNA bipyramids with minimal component strands based on computer program.…”