Graph Theory: Confluences in Molecular Biology and the Physical Sciences

Solow0109738 在物理学、分子生物学、数论、Teichmueller理论、动力系统、流体动力学和3-流形不变量中有涉及图的某些汇合点，而这些汇合点又反映了它们尚未发展的理论基础之间可能更深层次的关系。 数学家们组织了一个研讨会来考虑数学结构，并将其应用于越来越多与数学相关的领域的特定问题，例如包括蛋白质和RNA在内的大分子折叠的全局几何。 最近的实验结果允许治疗的全部空间foldingproblem：如何做的物理化学性质的大分子决定其空间foldingcharacteristics？ 粗略地说，折叠可以被建模为一个加权弧族，每个弧对应于一个化学键，权重是键的玻尔兹曼强度。 由这些加权图组合而成的图给出了一个表示所有可能折叠的加权图空间。 能否在这样的空间中找到合适的能量泛函来研究总能量最小的梯度流？ 一些参与者使用这种方法来理解动态折叠。 生物学中图的相关空间是由物种形成形成的系统树的相关空间。 虽然这些空间的整体几何是理解的，局部几何的性质是未知的;一些参与者研究这些方程。 弦理论中最近的一个发现是与图空间有关的一个“操作数”的奇特结构。 研讨会上的一些研究人员研究了这些更高层次的结构，以及它们在生物学中的可能用途。 图论是研究对象图的数学的一个分支。 一个图由一组点或顶点组成，这些点或顶点由链接或边连接。 一个普通的地图，例如，是一个图，其顶点是城镇，其边缘是道路。 虽然图是抽象的对象，但它们已被用来表示各种各样的真实的对象或过程。 图形的一个类似用途是表示一组有机体之间的进化关系--家谱。 这伊萨了他们的伟大的多功能性和实用性，图形已被用于不同领域，如生物学，经济学和计算机科学。 在生物技术领域，用图形来表示由DNA产生的基本蛋白质的复杂物理结构的应用出现了爆炸性的增长。 这项工作在理解遗传疾病和设计治疗它们的药物方面有重要的应用。 因为基于图论的方法在许多领域都是独立发展的，所以在各个领域之间共享问题和结果极有可能带来进步。 本次研讨会的目的是汇集数学家和生物学家工作的问题，涉及图论描述和讨论他们的工作在一个跨学科的设置。

Solow0109738 There are certain confluences involving graphs in physics,molecular biology, number theory, Teichmueller theory, dynamicalsystems, fluid dynamics, and 3-manifold invariants, which in turnreflect possibly deeper relations among their theoreticalfoundations which have yet to be developed. The investigatorsorganize a workshop to consider mathematical structures withapplication to specific problems in fields increasingly connectedto mathematics, for example the global geometry of macromolecularfolding, including both proteins and RNA. Recent mathematicalresults allow for a treatment of the full spatial foldingproblem: how do the physico-chemical properties of macromoleculesdetermine their spatial folding characteristics? Roughly,folding can be modeled as a family of weighted arcs, each arccorresponding to a chemical bond, the weight the Boltzmannstrength of the bond. The graphs formed from combining theseweighted ars give a space of weghted graphs representing allpossible foldings. Can an appropriate energy functional be foundon such a space to study the gradient flow minimizing the totalenergy? Several participants use this approach to understanddynamic folding. Related spaces of graphs in biology are those ofphylogenetic trees formed by speciation. Though the globalgeometry of these spaces is understood, the nature of the localgeometry is not known; some of the participants study thesequestions. A recent discovery in string theory is the exoticstructure of an "operad" associated with spaces of graphs. Someof the researchers at the workshop investigate these higher orderstructures, and their possible use in biology. Graph theory is a branch of mathematics dealing with objectscalled graphs. A graph consists of a set of points or verticesthat are connected by links or edges. An ordinary map, forexample, is a graph whose vertices are towns and whose edges areroads. Although graphs are abstract objects, they have been usedto represent a wide variety of real objects or processes. Afamiliar use of graphs is to represent the evolutionaryrelationships between a group of organisms -- family trees. It isa testimony to their great versatility and usefulness that graphshave been used in fields as diverse as biology, economics, andcomputer science. In the field of biotechnology, there has beenan explosion in the use of graphs to represent the complexphysical structure of the basic proteins produced by DNA. Thiswork has important applications in understanding genetic diseasesand in designing medicines to cure them. Because methods based ongraph theory have been developed independently in many fields, itis extremely likely that sharing problems and results betweenfields will lead to advances. The purpose of this workshop is tobring together mathematicians and biologists working on problemsinvolving graph theory to describe and discuss their work in aninterdisciplinary setting.