Mathematical Sciences: Applications of Topology to Biology and Chemistry

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9403454 Sumners The investigator and colleagues develop topological methods to describe and compute the spatial configuration of macromolecules, and changes in configuration due to chemical and biological agents. These issues arise in the analysis of experimental data in molecular biology and synthetic polymer chemistry. The applications include model building and theoretical proofs of the structure and mechanism of enzymes that act on DNA, the quantization of topological entanglement of molecules in random polymer systems, the development of algorithms for the numerical computation and simulation of macromolecular structure, and the implementation of these algorithms to create mathematical data that one can compare with experimental data. In the topological approach to enzymology, enzymes vital in cellular metabolism are reacted with circular DNA substrate. A topological enzyme signature is observed in the family of DNA knots and links produced by the reaction. One long-range goal of this research is the development of a complete set of experimentally observable topological parameters with which to describe and compute enzyme mechanism and the structure of the active enzyme-DNA synaptic intermediate, allowing mathematical proof of 3-dimensional enzyme-DNA structure, and the ability to predict results of future experiments. Many recent advances in 3-dimensional geometry and topology (such as the Cyclic Surgery Theorem) are useful in the analysis of data from these topological enzymology experiments. The results of this basic research project have potential utility in biotechnology, rational drug design and the design and manufacture of synthetic polymers. Before one can harness the manufacturing power of bacterial cells, one must first understand what these cells are capable of, and then how to control these processes. In order to combat viruses in human disease, and to exploit the ability of viruses to control cellular processes, one must understand at a fundamental level what viruses are doing. An integral part of this understanding process is the study of the spatial configuration of large flexible molecules. For example, the 3-dimensional shape of cellular DNA, RNA and proteins is crucially important to their biological function. Cellular DNA is highly twisted and convoluted, and for information retrieval and cell viability, some geometric and topological features must be introduced, and others quickly removed. Some enzymes maintain the proper geometry and topology of DNA in the cell by passing one strand of DNA through another via an enzyme-bridged transient break in the DNA; this enzyme action plays a crucial role in cell metabolism, including segregation of daughter chromosomes at the termination of replication. Other enzymes break the DNA apart and recombine the ends by exchanging them. These enzymes regulate the expression of specific genes, mediate viral integration into and excision from the host genome, mediate transposition and repair of DNA, and generate antibody and genetic diversity. These enzymes perform incredible feats of topology at the molecular level; the description and quantization of such enzyme action absolutely requires the language and computational machinery of topology.