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sumners-random

# sumners-random - RANDOM KNOTTING AND VIRAL DNA PACKING...

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RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 [email protected]

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RANDOM KNOTTING Proof of the Frisch-Wasserman-Delbruck conjecture--the longer a random circle, the more likely it is to be knotted DNA knotting in viral capsids
http://www.pims.math.ca/knotplot/zoo/ A Knot Zoo By Robert G. Scharein © 2005 Jennifer K. Mann

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TOPOLOGAL ENTANGLEMENT IN POLYMERS
WHY STUDY RANDOM ENTANGLEMENT? Polymer chemistry and physics: microscopic entanglement related to macroscopic chemical and physical characteristics--flow of polymer fluid, stress-strain curve, phase changes (gel formation) Biopolymers: entanglement encodes information about biological processes--random entanglement is experimental noise and needs to be subtracted out to get a signal

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BIOCHEMICAL MOTIVATION Predict the yield from a random cyclization experiment in a dilute solution of linear polymers
MATHEMATICAL PROBLEM If L is the length of linear polymers in dilute solution, what is the yield (the spectrum of topological products) from a random cyclization reaction? L is the # of repeating units in the chain--# of monomers, or # of Kuhn lengths (equivalent statistical lengths)--for polyethylene, Kuhn length is about 3.5 monomers. For duplex DNA, Kuhn length is about 300-500 base pairs

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FRISCH-WASSERMAN- DELBRUCK CONJECTURE L = # edges in random polygon P(L) = knot probability lim P(L) = 1 L Delbruck, Proc. Symp. Appl. Math. 14 (1962), 55
RANDOM KNOT MODELS Lattice models: self-avoiding walks (SAW) and self-avoiding polygons (SAP) on Z 3 , BCC, FCC, etc--curves have volume exclusion Off-lattice models: Piecewise linear arcs and circles in R 3 --can include thickness

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RANDOM KNOT METHODS Small L : Monte Carlo simulation Large L : rigorous asymptotic proofs
SIMPLE CUBIC LATTICE

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sumners-random - RANDOM KNOTTING AND VIRAL DNA PACKING...

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