Dresden 2003 – scientific programme
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DY: Dynamik und Statistische Physik
DY 36: Nonlinear dynamics II
DY 36.2: Talk
Wednesday, March 26, 2003, 16:45–17:00, G\"OR/226
Equilibrium shapes of knots — •Andreas Hanke1,2, Ralf Metzler3,2, Paul G. Dommersnes4, Yacov Kantor5, and Mehran Kardar2 — 1Institute of Theoretical Physics, University of Stuttgart - Pfaffenwaldring 57, D-70550 Stuttgart, Germany — 2Department of Physics, MIT - 77 Mass. Ave., Cambridge, Massachusetts 02139, USA — 3NORDITA - Blegdamsvej 17, DK-2100 Copenhagen, Denmark — 4Institut Curie - 11, rue Pierre et Marie Curie, F-75231 Paris Cedex 5, France — 5School of Physics, Tel Aviv University - IL-69978 Tel Aviv, Israel
In the biological domain, the topology of knotted DNA can be actively modified through energy-consuming enzymes (topoisomerases), the understanding of their detection methods requires knowledge on the equilibrium shapes of knotted polymers.
Given this motivation, we study the equilibrium shapes of prime and composite knots firstly confined to two dimensions, using rigorous scaling arguments originally developed for general polymer networks. For dilute polymers in a good solvent, i.e., above the Theta point, we show that, due to self-avoiding effects, the topological details of prime knots are localised on a small portion of the larger ring polymer. Within this region, the original knot configuration can assume a hierarchy of contracted shapes, the dominating one given by just one small loop. This hierarchy is investigated in detail for the flat trefoil knot.
For polymers in the dense state and at the Theta point, a similar analysis shows, in contrast, that all prime knots are delocalised over the entire ring polymer. A similar behavior is conjectured also for knots in three dimensions. Our results are corroborated by Monte Carlo simulations.