Graph Theory: Confluences in Molecular Biology and the Physical Sciences

在物理学、分子生物学、数论、Teichmueller理论、动力学系统、流体力学和3-流形不变量中有某些涉及图形的汇合，这些汇合反过来又反映了它们的理论基础之间可能存在的更深层次的关系，这些关系尚未得到发展。研究人员组织了一次研讨会，以考虑数学结构，并将其应用于与数学联系越来越紧密的领域的具体问题，例如，包括蛋白质和RNA在内的大分子折叠的全球几何形状。最近的数学结果允许处理完整的空间折叠问题：大分子的物理化学性质如何决定它们的空间折叠特性？粗略地说，折叠可以被建模为一族加权的弧线，每条弧线对应于一个化学键，重量是键的玻尔兹曼强度。组合这些加权AR形成的图给出了一个表示所有可能的折叠的有向图的空间。能否在这样的空间中建立一个合适的能量泛函来研究使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。生物学中图形的相关空间是物种形成的系统发育树的相关空间。虽然这些空间的全局几何已经被理解，但局部几何的性质是未知的；一些参与者研究这些序列。弦理论中的一个最新发现是与图空间相关的“操作符”的奇异结构。研讨会上的一些研究人员研究了这些更高顺序的结构，以及它们在生物学中的可能用途。图论是处理对象比例图的数学分支。图形由一组由链接或边连接的点或顶点组成。一个普通的地图，例如，是一个图，它的顶点是城镇，它的边是弧形的。虽然图形是抽象的对象，但它们已被用来表示各种各样的真实对象或过程。通常，图的用途是表示一组有机体之间的进化关系--家谱。图形在生物学、经济学和计算机科学等领域的广泛应用证明了图形的多功能性和实用性。在生物技术领域，用图形来表示由DNA产生的碱性蛋白质的复杂物理结构已经有了爆炸性的发展。这项工作在理解遗传病和设计治愈它们的药物方面具有重要的应用。由于基于图论的方法已经在许多领域独立发展，因此领域之间共享问题和结果极有可能带来进步。这个工作坊的目的是将从事涉及图论的问题的数学家和生物学家聚集在一起，描述和讨论他们在跨学科背景下的工作。

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.