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DNA TOPOLOGY

DNA TOPOLOGY. De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL sumners@math.fsu.edu. Pedagogical School: Knots & Links: From Theory to Application. Pedagogical School: Knots & Links: From Theory to Application. De Witt Sumners: Florida State University

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DNA TOPOLOGY

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  1. DNA TOPOLOGY De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL sumners@math.fsu.edu

  2. Pedagogical School: Knots & Links: From Theory to Application

  3. Pedagogical School: Knots & Links: From Theory to Application De Witt Sumners: Florida State University Lectures on DNA Topology: Schedule • Introduction to DNA Topology Monday 09/05/11 10:40-12:40 • The Tangle Model for DNA Site-Specific Recombination Thursday 12/05/11 10:40-12:40 • Random Knotting and Macromolecular Structure Friday 13/05/11 8:30-10:30

  4. RANDOM KNOTTING • Proof of the Frisch-Wasserman-Delbruck conjecture--the longer a random circle, the more likely it is to be knotted • Knotting of random arcs • Writhe of random curves and arcs • Random knots in confined volumes

  5. TOPOLOGAL ENTANGLEMENT IN POLYMERS

  6. 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

  7. BIOCHEMICAL MOTIVATION Predict the yield from a random cyclization experiment in a dilute solution of linear polymers

  8. SEMICRYSTALLINE POLYMERS

  9. CHEMICAL SYNTHESIS OF CIRCULAR MOLECULES Frisch and Wasserman JACS 83(1961), 3789

  10. 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, persistence lengths)--for polyethylene, Kuhn length is about 3.5 monomers. For duplex DNA, persistence length is about 300-500 base pairs

  11. MONTE CARLO RANDOM KNOT SIMULATION • Thin chains--random walk in Z3 beginning at 0, no backtracking, perturb self-intersections away, keep walking. Force closure--the further you are along the chain, the more you want to go toward the origin. • Detect knot types: D(-1) Vologodskii et al, Sov. Phys. JETP 39 (1974), 1059

  12. IMPROVED MONTE CARLO • Start with phantom closed circular polymer (equilateral polygon), point mass vertices, semi-rigid edges (springs), random thermal forces (random 3D vector field), molecular mechanics, edges pass through each other, take snapshots of equilibrium distribution, detect knots with D(-1). Michels & Wiegel Phys Lett. 90A (1984), 381

  13. THIN CHAIN RANDOM KNOTTING

  14. FRISCH-WASSERMAN-DELBRUCK CONJECTURE • L = # edges in random polygon • P(L) = knot probability lim P(L) = 1 L   Frisch & Wasserman, JACS 83(1961), 3789 Delbruck, Proc. Symp. Appl. Math. 14 (1962), 55

  15. RANDOM KNOT MODELS • Lattice models: self-avoiding walks (SAW) and self-avoiding polygons (SAP) on Z3, BCC, FCC, etc--curves have volume exclusion • Off-lattice models: Piecewise linear arcs and circles in R3--can include thickness

  16. RANDOM KNOT METHODS • Small L: Monte Carlo simulation • Large L: rigorous asymptotic proofs

  17. SIMPLE CUBIC LATTICE

  18. PROOF OF FWD CONJECTURE THEOREM: P(L) ~ 1 - exp(-lL) l > 0 Sumners & Whittington, J. Phys. A: Math. Gen. 23 (1988), 1689 Pippenger, Disc Appl. Math. 25 (1989), 273

  19. KESTEN PATTERNS Kesten, J. Math. Phys. 4(1963), 960

  20. TIGHT KNOTS

  21. Z3

  22. TIGHT KNOT ON Z3 19 vertices, 18 edges

  23. TREFOIL PATTERN FORCES KNOTTING OF SAP • Any SAP which contains the trefoil Kesten pattern is knotted--each occupied vertex is the barycenter of a dual 3-cube (the Wigner-Seitz cell)--the union of the dual 3-cubes is homeomorphic to B3, and this B3 contains a red knotted arc.

  24. SMALLEST TREFOIL L = 24

  25. SLIGHTLY LARGER TREFOIL L = 26

  26. RANDOM KNOT QUESTIONS • For fixed length n, what is the distribution of knot types? • How does this distribution change with n? • What is the asymptotic behavior of knot complexity--growth laws ~bna ? • How to quantize entanglement of random arcs?

  27. KNOTS IN BROWNIAN FLIGHT • All knots at all scales Kendall, J. Lon. Math. Soc. 19 (1979), 378

  28. ALL KNOTS APPEAR Every knot type has a tight Kesten pattern representative on Z3

  29. ALL KNOTS APPEAR: PROOF Take a projection of your favorite knot on Z2. Bump up crossovers, saturate holes to get a decorated pancake

  30. LONG RANDOM KNOTS

  31. MEASURING KNOT COMPLEXITY

  32. LONG RANDOM KNOTS ARE VERY COMPLEX THEOREM: All good measures of knot complexity diverge to +  at least linearly with the length--the longer the random polygon, the more entangled it is. Examples of good measures of knot complexity: crossover number, unknotting number, genus, bridge number, braid number, span of your favorite knot polynomial, total curvature, etc.

  33. WRITHE

  34. KESTEN WRITHE Van Rensburg et al., J. Phys. A: Math. Gen 26 (1993), 981

  35. WRITHE GROWS LIKE n

  36. WRITHE GROWTH: PROOF

  37. WRITHE GROWTH: PROOF

  38. GROWTH OF WRITHE FOR FIGURE 8 KNOT

  39. RANDOM KNOTS ARE CHIRAL

  40. ENTANGLEMENT COMPLEXITY OF GRAPHS AND ARCS

  41. DETECTING KNOTTED ARCS

  42. LONG RANDOM ARCS (GRAPHS) ARE VERY COMPLEX

  43. NEW SIMULATION: KNOTS IN CONFINED VOLUMES • Parallel tempering scheme • Smooth configuration to remove extraneous crossings • Use KnotFind to identify the knot--ID’s prime and composite knots of up to 16 crossings • Problem--some knots cannot be ID’d--might be complicated unknots!

  44. SMOOTHING

  45. UNCONSTRAINED KNOTTING PROBABILITIES

  46. CONSTRAINED UNKNOTTING PROBABILITY

  47. CONSTRAINED UNKOTTING PROBABILITY

  48. CONSTRAINED TREFOIL KNOT PROBABILITIES

  49. CONSTRAINED TREFOIL PROBABILITY

  50. ANALYTICAL PROOFS FOR KNOTS IN CONFINED VOLUMES • CONJECTURE: THE KNOT PROBABILITY GOES TO ONE AS LENGTH GOES TO INFINITY FOR RADOM CONFINED KNOTS--AND THAT THE KNOTTING PROBABILITY GROWS MUCH FASTER THAN RANDOM KNOTTING IN FREE 3-SPACE

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