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RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS

RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS. De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 sumners@math.fsu.edu. RANDOM KNOTTING.

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RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS

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  1. RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 sumners@math.fsu.edu

  2. 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

  3. A Knot Zoo By Robert G. Scharein http://www.pims.math.ca/knotplot/zoo/ © 2005 Jennifer K. Mann

  4. TOPOLOGAL ENTANGLEMENT IN POLYMERS

  5. 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

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

  7. 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

  8. 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

  9. 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

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

  11. SIMPLE CUBIC LATTICE

  12. 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

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

  14. TIGHT KNOTS

  15. Z3

  16. TIGHT KNOT ON Z3 19 vertices, 18 edges

  17. The Smallest Trefoil on Z3

  18. 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?

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

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

  21. LONG RANDOM KNOTS

  22. MEASURING KNOT COMPLEXITY

  23. 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.

  24. GROWTH OF WRITHE FOR FIGURE 8 KNOT

  25. RANDOM KNOTS ARE CHIRAL

  26. DNA Replication

  27. Topological Enzymology Mathematics: Deduce enzyme binding and mechanism from observed products

  28. TOPOLOGICAL ENZYMOLOGY • React circular DNA plasmids in vitro (in vivo) with purified enzyme • Gel electrophoresis to separate products (DNA knots & links) • Electron microscopy of RecA coated products • Use topology and geometry to build predictive models

  29. GEL ELECTROPHORESIS

  30. RecA Coated DNA

  31. DNA Trefoil Knot Dean et al., J BIOL. CHEM. 260(1985), 4975

  32. DNA (2,13) TORUS KNOT Spengler et al. CELL 42 (1985), 325

  33. GEL VELOCITY IDENTIFIES KNOT COMPLEXITY Vologodskii et al, JMB 278 (1988), 1

  34. CHIRALITY

  35. CROSSING SIGN CONVENTION

  36. WRITHE & AVERAGE CROSSING NUMBER Writhe --average the sum of signed crossings over all projections (average number of crossings over all projections)

  37. VIRUS LIFE CYCLE

  38. VIRUS ATTACKS!

  39. VIRUS ATTACKS!

  40. T4 EM

  41. VIRUS ATTACK

  42. T4 ATTACK

  43. HOW IS THE DNA PACKED?

  44. SPOOLING MODEL

  45. FOLD MODEL

  46. RANDOM PACKING

  47. P4 DNA has cohesive ends that form closed circular molecules …. GGCGAGGCGGGAAAGCAC …... CCGCTCCGCCCTTTCGTG GGCGAGGCGGGAAAGCAC CCGCTCCGCCCTTTCGTG

  48. VIRAL KNOTS REVEAL PACKING • Compare observed DNA knot spectrum to simulation of knots in confined volumes

  49. Experimental Data: Tailless vs Mature Phage Knot Probability

  50. EFFECTS OF CONFINEMENT ON DNA KNOTTING • No confinement--3% knots, mostly trefoils • Viral knots--95% knots, very high complexity--average crossover number 27!

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