Mathematical Sciences: Applications of Topology to Biology and Chemistry

9403454 Sumners和他的同事开发了拓扑学方法来描述和计算大分子的空间构型，以及由于化学和生物制剂而引起的构型变化。这些问题出现在分子生物学和合成聚合物化学的实验数据分析中。这些应用包括酶作用于DNA的结构和机制的模型建立和理论证明，随机聚合物体系中分子的拓扑纠缠的量化，大分子结构的数值计算和模拟的算法的发展，以及这些算法的实施以产生人们可以与实验数据相比较的数学数据。在酶学的拓扑学方法中，细胞代谢中至关重要的酶与环状DNA底物反应。在由该反应产生的DNA结和链家族中观察到了拓扑酶标记。这项研究的一个长期目标是开发一套完整的实验可观察的拓扑参数，用来描述和计算酶机制和活性酶-DNA突触中间体的结构，从而能够对三维酶-DNA结构进行数学证明，并能够预测未来实验的结果。最近在三维几何和拓扑学方面的许多进展(如循环运算定理)在这些拓扑酶实验的数据分析中是有用的。这一基础研究项目的成果在生物技术、合理的药物设计和合成聚合物的设计和制造中具有潜在的实用价值。在利用细菌细胞的制造能力之前，必须首先了解这些细胞的能力，然后如何控制这些过程。为了对抗人类疾病中的病毒，并利用病毒控制细胞过程的能力，人们必须从根本上了解病毒在做什么。这一理解过程的一个组成部分是对大分子柔性空间构型的研究。例如，细胞DNA、RNA和蛋白质的三维形状对它们的生物功能至关重要。细胞DNA是高度扭曲和缠绕的，为了信息检索和细胞生存，必须引入一些几何和拓扑特征，而另一些则迅速移除。一些酶通过DNA中酶桥的瞬时断裂将一条DNA链穿过另一条链，从而维持细胞中DNA的正确几何和拓扑结构；这种酶作用在细胞新陈代谢中发挥着关键作用，包括在复制终止时分离子代染色体。其他酶通过交换DNA来分解DNA，并将末端重新组合。这些酶调节特定基因的表达，介导病毒与宿主基因组的整合和切除，介导DNA的转位和修复，并产生抗体和遗传多样性。这些酶在分子水平上执行令人难以置信的拓扑学壮举；这种酶作用的描述和量化绝对需要拓扑学的语言和计算机制。

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.