Mathematical Sciences: Applications of Topology to Biology and Chemistry

这个项目继续了拓扑学在生物学和化学问题中的应用的早期研究。有三个主要的努力领域。首先是将结理论中的分类和定量技术应用于酶- dna的三维结构和酶的机理分析。数学模型，如缠结模型，将得到发展和完善，目的是为实验分子生物学家提供一种数学工具。这里使用的数学是最近在3流形理论中的工作，生物输入是从酶学的拓扑方法中获得的实验结果。第二个领域是三维空间中随机链的拓扑纠缠研究。随机链纠缠统计量与稀溶液中聚合物体系的构象熵有关。渐近纠缠（当长度趋于无穷大时）是相当容易理解的；科学兴趣的基本问题是有限长度纠缠的描述和计算。将采用严格的数学和模拟技术。第三个领域是可激介质中螺旋波的拓扑描述。利用微分拓扑学和阻塞理论的方法，给出了有界可定向3流形上图案存在和时间演化的充分必要条件。分子生物学的一个中心问题是如何处理长链DNA分子。它们折叠、复制和重组的方式与它们的功能密切相关。长链聚合物在溶液中的行为是相同问题的更一般形式。重要的应用出现在基因工程，药物设计，以及工业和技术用途的长链聚合物的发展。结理论是拓扑学的一部分，它研究的正是这类问题：结构如何折叠、重组和复制。

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.