Mathematical Sciences: Applications of Topology to Biology and Chemistry

该项目继续早期的应用研究， 生物学和化学中的拓扑学问题。有三 主要努力领域。第一是应用 纽结理论中的分类和量化技术， 酶-DNA三维结构与酶分析 机制数学模型，如缠结模型，将是 开发和完善，目的是产生一个数学 实验分子生物学家使用的工具。的 这里使用的数学是最近的工作理论， 三维流形，生物输入是实验结果 从拓扑学方法获得的酶学。第二 区域是研究随机的拓扑纠缠 三维空间中的链。 随机链纠缠统计是 与聚合物体系的构象熵有关， 稀溶液 渐近纠缠（当长度达到 无穷大）是合理的理解;基本问题 科学兴趣的是描述和计算 有限长度的纠缠 严格的数学和 将采用模拟技术。 第三个领域是 可激发介质中螺旋波的拓扑描述 媒体 微分拓扑法和阻塞法 理论将被用来提供必要的和充分的 的存在条件和时间演化 有界可定向三维流形 分子生物学的一个中心问题是处理 长链DNA分子它们折叠、复制和 重组与其功能密切相关。的 长链聚合物在溶液中的行为是一个更普遍的 同样的问题形式。重要的应用出现在 基因工程，药物设计，以及 用于工业和技术用途的长链聚合物。 结 理论是拓扑学的一部分，它研究的正是这类 问题：结构是如何折叠、重组和复制的。

This project continues earlier studies of applications of topology to problems in biology and chemistry. There are three main areas of effort. The first is the application of classification and quantification techniques in knot theory to the analysis of enzyme-DNA three-dimensional structure and enzyme mechanism. Mathematical models, such as the tangle model, will be developed and perfected, the aim being to produce a mathematical tool of use to the experimental molecular biologist. The mathematics used here is recent work in the theory of 3-manifolds, and the biological input is experimental results obtained from the topological approach to enzymology. The second area is the study of the topological entanglement of random chains in 3-space. Random chain entanglement statistics are related to the conformational entropy of a polymer system in dilute solution. Asymptotic entanglement (as the length goes to infinity) is reasonably well understood; the fundamental problem of scientific interest is the description and computation of entanglement at finite lengths. Both rigorous mathematics and simulation techniques will be employed. The third area is the topological description of spiral wave patterns in excitable media. The methods of differential topology and obstruction theory will be used to provide necessary and sufficient conditions for the existence and time evolution of patterns on bounded orientable 3-manifolds. A central problem in molecular biology is to deal with long-chain DNA molecules. The way they fold, replicate, and recombine is intimately bound up with their functions. The behavior of long-chain polymers in solution is a more general form of the same questions. Important applications arise in genetic engineering, the design of drugs, and the development of long-chain polymers for industrial and technological uses. Knot theory is a part of topology that studies exactly these kinds of questions: how structures fold and recombine and replicate.