Spatial Graphs and Their Application to Complex Molecular Structures

该研究项目的总体目标是使用拓扑和几何工具来帮助分子生物学家和化学家更好地理解DNA、蛋白质和复杂合成分子的结构和行为。所研究的拓扑模型将有助于分子生物学家简化对闭合环状DNA分子的位点特异性重组机制的分析。研究者还旨在确定结，链接和非平面图形的形式，出现在蛋白质中，并模拟这些复杂的结构是如何发生的。这些信息可能为蛋白质折叠机制和降解途径提供有价值的见解。合成有机分子通常太小，无法用电子显微镜看到；当化学家合成一个复杂的结构时，他们使用来自核磁共振（NMR）光谱的数据来提供分子结构具有特定形式的证据。由于这些结构足够大，具有一定的灵活性，因此在将核磁共振数据的对称性与物理模型的对称性进行比较时，必须考虑拓扑结构和几何结构。研究人员正在与有机化学家合作，以确定复杂结构所表现出的不同类型的对称性，并设计具有有趣对称性的新结构。与结点和链路的拓扑结构完全取决于它们在三维球面上的嵌入不同，某些图的内在结构会影响图在给定三维流形中的每次嵌入的拓扑性质。例如，某些图具有这样的性质：对于图的任意嵌入G在三流形M中，不存在（M，G）对的方向反转同胚。这样的图在m中被称为本质手性。研究者将研究在三球面和其他三维流形中哪些图具有本质手性，以及确定与图的特定嵌入无关的嵌入图的其他性质。该项目借鉴了三流形的结果，包括Jaco-Shalen和Johannson特征分解、Mostow的刚性定理、Thurstons的夸张化定理、Seifert流形的分类，以及结理论和缠结理论的技术。

The broad goal of this research project is to use the tools of topology and geometry to help molecular biologists and chemists better understand the structure and behavior of DNA, proteins, and complex synthetic molecules. The topological model under study would help molecular biologists by simplifying the analysis of the site-specific recombination mechanism for closed circular DNA molecules. The investigator also aims to identify the forms of knots, links, and non-planar graphs that arise in proteins, and to model how these complex structures may have occurred. This information may offer valuable insights into protein folding mechanisms and degradation pathways. Synthetic organic molecules are normally too small to see with an electron microscope; when chemists synthesize a complex structure they use data from nuclear magnetic resonance (NMR) spectroscopy to provide evidence that the molecular structure has a particular form. Since these structures are large enough to be somewhat flexible, both topology and geometry have to be taken into account when comparing the symmetry properties of the NMR data to those of a physical model. The investigator is working with organic chemists to identify different types of symmetries exhibited by complex structures and to design new structures with interesting symmetry properties. In contrast with knots and links, whose topology depends exclusively on their embedding in the three dimensional sphere, the intrinsic structure of some graphs can affect the topological properties of every embedding of the graph in a given three dimensional manifold. For example, some graphs have the property that for any embedding G of the graph in a three-manifold M, there is no orientation reversing homeomorphism of the pair (M,G). Such a graph is said to be intrinsically chiral in M. The investigator will work on characterizing which graphs are intrinsically chiral in the three-sphere and in other three-dimensional manifolds, as well as determining other properties of embedded graphs which are independent of the particular embedding of the graph. The project draws on three-manifold results including Jaco-Shalen and Johannson characteristic decompositions, Mostow's rigidity theorem, Thurstons' hyperbolization theorem, and the classification of Seifert manifolds, as well as techniques from knot theory and the theory of tangles.