The Geometry And Topology Of Virus And DNA Polyhedra

The thesis interested in applications of geometry and topology to understand chemical and biological phenomena, in particular applying polyhedral theory and knot theory to viral capsids and DNA three-dimensional nanocages. The general outline of this thesis consists of three parts.I BackgroundIn a first part, we will introduce the reader some theoretical and experimental background. The theoretical knowledge includes polyhedral theory and knot theory, which provides the necessary mathematical tools for this thesis'research. The DNA polyhedra and virus polyhedra not only provides us with the relevant experimental background and research objectives, but also inspire us to develop new methods and new theory for their novel structures.II. The geometrical and topological structures of viral capsidsViruses consist two structural components, DNA or RNA genomes that carry genetic information, and protein capsids that protect these genes. In particular, my attention focuses on the understanding of the architecture of viral capsids, especially viruses with icosahedral symmetry. Combining the method coming form the geometry, graph theory and topology, a kind of novel geometrical objects with icosahedral symmetry, which are considered to explain some viral capsids, have been constructed based on Goldberg polyhedra. Then applying the "tangle-covering" method, their related polyhedral links have also been built and some structural properties including chirality and component numbers have been investigated. Moreover, fullerenes and icosahedral virus share the underlying geometry, and therefore this methodology could also be extended to the systematic theoretical study and the molecular design of novel models of fullerene and related polyhedra. III. The geometrical and topological properties of DNA nanostructuresIn structural nanotechnology, the DNA molecule has been employed to assemble a large variety of three dimensional structures which have connectivity of polyhedra. Given these recent experimental advances, it would provide a nice challenge to construct theoretical models that can describe the organization and self-assembly of DNA nanostructures. It has been proposed that polyhedral links are reasonable mathematical models for DNA polyhedra. This suggests that techniques coming from knot theory have potential applications in investigating the physico-chemical properties of these DNA nanostructures. My research is focus on the construction methods and structural rules of polyhedral links. First, various species of polyhedral links were built by developing different methods motivated by experimental results. Based on these models, it is necessary to find useful knot-theoretical descriptors to access the complexity of DNA nanostructures. These quantities include crossing numbers, unknotting numbers, and braid index if DNA strands considers as two anti-parallel ribbons, as well as genus and the number of Seifert circles if DNA polyhedral catenanes embedded in related surfaces. In addition, we have employed these quantities to describe the Euler's theorem of DNA polyhedra, and the structural transformations caused by actions of enzymes.