Topological Propertues Of Crossed DNA Polyhedral Links

DNA polyhedron is a type of novel polyhedral structure assembled with DNA catenanes. Along with the numerous syntheses of these structures, it has been an enormous challenge to build theoretical models that can describe the architectures and assembly of DNA nanostructures. Meanwhile, the development of polyhedral links provides theoretical basis for research on the DNA polyhedra. In this thesis, we focus on the crossed DNA polyhedral links and study their topological properties with the benefit of knot theory and polyhedral theory. The thesis includes the following three parts.The part of the research background consists of experimental background and theoretical background. Since1990s, the obtainments of DNA polyhedra provide new way for the construction of complex nanostructures. The discovery of some unique structures in nature also provides new basic models for theorists. We will also introduce some knowledge about the knot theory, topology, Euler’s formula and the new Euler’s formula. These theories and the polyhedral links provide tools for the description of DNA polyhedra.The Euler’s formula for polyhedra displays the geometric properties of polyhedral structures, while the new Euler’s formula for DNA polyhedra reveals the intrinsic properties of these novel structures, and offers an important tool for the description and study of the DNA polyhedra. We focus on the crossed DNA polyhedral links, study their topological properties including the number of Seifert circles and the new Euler’s characteristic, and get the expressions of their new Euler’s characteristics. The study extends the new Euler’s formula to more kinds of DNA polyhedra and provides theoretic foundations to the design and synthesis of more complex DNA polyhedra.Some regulatory enzymes like recombinases and topoisomerases can change the structures of knots or links assembled with DNA catenanes, and the inverse processes will apply new thoughts to the synthesis of complex DNA structures. We study on the topological transformation of the crossed DNA polyhedral links induced by enzymes with the mathematical tools like the numbers of Seifert circles, genus and the new Euler’s characteristics, hoping to provide theoretical guidance for the assembly of DNA polyhedra.