Graph Theory: Confluences in Molecular Biology and the Physical Sciences

Woods Hole Oceanographic Institution, Woods Hole MA

Investigators

Abstract

Solow 0109738 There are certain confluences involving graphs in physics, molecular biology, number theory, Teichmueller theory, dynamical systems, fluid dynamics, and 3-manifold invariants, which in turn reflect possibly deeper relations among their theoretical foundations which have yet to be developed. The investigators organize a workshop to consider mathematical structures with application to specific problems in fields increasingly connected to mathematics, for example the global geometry of macromolecular folding, including both proteins and RNA. Recent mathematical results allow for a treatment of the full spatial folding problem: how do the physico-chemical properties of macromolecules determine their spatial folding characteristics? Roughly, folding can be modeled as a family of weighted arcs, each arc corresponding to a chemical bond, the weight the Boltzmann strength of the bond. The graphs formed from combining these weighted ars give a space of weghted graphs representing all possible foldings. Can an appropriate energy functional be found on such a space to study the gradient flow minimizing the total energy? Several participants use this approach to understand dynamic folding. Related spaces of graphs in biology are those of phylogenetic trees formed by speciation. Though the global geometry of these spaces is understood, the nature of the local geometry is not known; some of the participants study these questions. A recent discovery in string theory is the exotic structure of an "operad" associated with spaces of graphs. Some of the researchers at the workshop investigate these higher order structures, and their possible use in biology. Graph theory is a branch of mathematics dealing with objects called graphs. A graph consists of a set of points or vertices that are connected by links or edges. An ordinary map, for example, is a graph whose vertices are towns and whose edges are roads. Although graphs are abstract objects, they have been used to represent a wide variety of real objects or processes. A familiar use of graphs is to represent the evolutionary relationships between a group of organisms -- family trees. It is a testimony to their great versatility and usefulness that graphs have been used in fields as diverse as biology, economics, and computer science. In the field of biotechnology, there has been an explosion in the use of graphs to represent the complex physical structure of the basic proteins produced by DNA. This work has important applications in understanding genetic diseases and in designing medicines to cure them. Because methods based on graph theory have been developed independently in many fields, it is extremely likely that sharing problems and results between fields will lead to advances. The purpose of this workshop is to bring together mathematicians and biologists working on problems involving graph theory to describe and discuss their work in an interdisciplinary setting.