The Mathematics of Knotting and Linking in Polymer Physics and Molecular Biology (07w5095)

Arriving in Banff, Alberta Sunday, May 20 and departing Friday May 25, 2007


Kenneth Millett (University of California, Santa Barbara)

(University of Saint Thomas)

(University of Saskatchewan)

(University of Lausanne)

Stuart Whittington (University of Toronto)


This workshop will focus on the mathematics associated with a very specific array of cutting edge problems that show the most promise for immediate progress at the interfaces between mathematical, physical, and life sciences. We note that the intense focus on only a few cutting edge applications makes this proposed workshop independent from the Mathematics of Molecular and Cellular Biology year, September 1, 2007 – June 30, 2008, at the Institute for Mathematics and its Applications in its intense focus on only these few applications. In addition our proposed timing, Spring 2007, will serve as a launching pad for the research of some graduate students, post docs, and new professors who will be invited to participate in the workshop.

The first target stems from the observation that DNA knotting is quite common in living cells and is easily observed in systems such as bacterial plasmids or in replication bubbles of bacterial chromosomes. DNA topoisomerases, whose action enables DNA segment passage, inadvertently create knotted constituents which are relatively harmless for cells as long as they are short-lived and of low density. It is apparently evolutionarily easier to tolerate low levels of such than to develop a system to eliminate their presence. Nevertheless, the steady-state level of such DNA knotting in living cells is lower than the thermodynamic equilibrium expected for a system where inter-segmental passages within long DNA molecules occurs at random. Much effort has lead to understanding of mechanisms by which type II topoisomerases mediate passages of double-stranded DNA segments to attain a low level of DNA knotting. A remarkable recent observation revealed that human topoisomerase Iia acting on 5_2 knots selectively relaxed them directly to an unknot although it could also have proceeded via a 3_1 knot requiring two catalytic events to reach the unknot. Is this selective mechanism connected to an objective of keeping knotting below a topological equilibrium or is there a specific constraining mechanism for this relaxation of knots? The study of the characteristics of equilibrium now include geometric, spatial, and topological facets that may be implicated in these mechanisms as well as the characteristics of polymers, for example in a theta condition. While these studies require advances in computational methods to fully illuminate the equilibrium properties, sufficient information appears already to be available to inform an understanding of experimental observations.

Second, new generation mathematical, statistical, and computational tools are under development to study the knotting and linking of open and closed macromolecules in open and confined environments. For example, several strategies to quantify and characterize the entanglement of open macromolecules are currently being investigated and show promising contributions to the practical question of describing the entanglement complexity of polymers. The mathematical facet of this work brings together topologists, geometers, statisticians, and computational scientists. One thrust is the development of new computer methods for rapidly identifying knots and links that are generated in the course of Monte Carlo simulations of a wide range of phenomena in biology and polymer physics. One of the most powerful tools is the calculation of the HOMFLY polynomial. The program was written in 1985 for applications to knots and links with no more than 128 crossings. While it has stood the test of time well, we are now using this tool to study polygonal models with 1500 edges (or Kuhn statistical segments) and many thousands of crossings by using special pre-analysis programs due to Millett and to Thistlethwaite. It is the moment to revisit the strategies that have been employed in the smaller ranges and to develop new ones that will bring us all to more rapidly study knotted and linked configurations that arise in actual physical experimental conditions. Perhaps it is time to capitalize on the intrinsic parallelism inherent in the now classical Jones, HOMFLY and Kauffman polynomial invariants or to explore those at the present interface of mathematics and physics, the Khovanov invariants and their extensions. A second, distinct computational thrust concerns efforts to achieve optimal spatial configurations of knotted and linked components. Lead by Cantaralla et al., a new approach to this problem is represented by the Ridgerunner program. The initial results of this effort are quite exciting and demonstrate the opportunity that will be afforded by wider applications that, alas, require additional theoretical and computational work.

Thirdly, a principal application of the technology discussed above and new methods will be to macromolecules in confined geometries, for example polymers between two parallel planes as in models of steric stabilization of dispersions or in DNA molecules contained in a capsid. Macromolecules so confined exhibit significantly different average and individual structure in comparison with those in free environments. Effective confining arises in the case of macromolecules have specific hydrophobic and hydrophilic regions or when regions have restricted flexibility or torsion. While, in general, one might believe that great progress has occurred in the understanding of knotting of polymers, in fact rather little is known rigorously and many fundamental questions seem just beyond our grasp, both theoretically or via numerical studies. Promising steps are being taken in both areas.