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. http://www.pims.math.ca/knotplot/zoo/ A Knot Zoo By Robert G. Scharein © 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 Z 3, BCC, FCC, etc--curves have volume exclusion Off-lattice models: Piecewise linear arcs and circles in R 3 --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(- L)  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. Z3Z3

16. TIGHT KNOT ON Z 3 19 vertices, 18 edges

17. The Smallest Trefoil on Z 3

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 ~  n  ? 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 Z 3

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!

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!

51. MATURE vs TAILLESS PHAGE Mutants--48% of knots formed inside capsid Arsuaga et al, PNAS 99 (2002), 5373

52. P4 KNOT SPECTRUM 97% of DNA knots had crossing number > 10! Arsuaga et al, PNAS 99 (2002), 5373

53. 2D GEL

54. 2D GEL RESOLVES SMALL KNOTS Arsuaga et al, PNAS 102 (2005), 9165

55. 2D GEL RESOLVES SMALL KNOTS Arsuaga et al, PNAS 102 (2005), 9165

56. PIVOT ALGORITHM Ergodic--no volume exclusion in our simulation  as knot detector Space filling polymers in confined volumes--very difficult to simulate

57. VOLUME EFFECTS ON KNOT SIMULATION On average, 75% of crossings are extraneous Arsuaga et al, PNAS 99 (2002), 5373

58. SIMULATION vs EXPERIMENT Arsuaga et al, PNAS 102 (2005), 9165 n=90, R=4

59. EFFECT OF WRITHE-BIASED SAMPLING Arsuaga et al, PNAS 102 (2005), 9165 n=90, R=4

60. Consistent with a high writhe of the packed DNA: - Low amount of knot 4 1 and of composite 3 1 # 4 1 - Predominance of torus knots (elevated writhe) primes composites 3 1 Wr = 3.41 4 1 Wr = 0.00 5 1 Wr = 6.26 5 2 Wr = 4.54 6 1 Wr = 1.23 6 2 Wr = 2.70 6 3 Wr = 0.16 7 1 Wr = 9.15 3 1 # 3 1 Wr = 6.81 3 1 # -3 1 Wr = 0.01 3 1 # 4 1 Wr ~ 3. 41

61. CONCLUSIONS Viral DNA not randomly embedded (4 1 and 5 2 deficit, 5 1 and 7 1 excess in observed knot spectrum) Viral DNA has a chiral packing mechanism--writhe- biased simulation close to observed spectrum Torus knot excess favors toroidal or spool-like packing conformation of capsid DNA Next step--EM (AFM) of 3- and 5- crossing knots to see if they all have same chirality

62. UNKNOWN P4 KNOT

63. UNKNOWN P4 KNOTS

64. AFM Images of Simple DNA Knots (Mg 2+ ) μmμmμmμm μmμm Ercolini, Dietler, EPFL Lausanne Ercolini, Dietler EPFL Lausanne

65. NEW SIMULATION 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!

66. UNCONSTRAINED KNOTTING PROBABILITIES

67. CONSTRAINED UNKNOTTING PROBABILITY

68. CONSTRAINED UNKOTTING PROBABILITY

69. CONSTRAINED TREFOIL KNOT PROBABILITIES

70. CONSTRAINED TREFOIL PROBABILITY

71. 4 vs 5 CROSSING PHASE DIAGRAM

72. CONFINED WRITHE

73. GROWTH OF CONFINED WRITHE

74. COLLABORATORS Stu Whittington Buks van Rensburg Carla Tesi Enzo Orlandini Chris Soteros Yuanan Diao Nick Pippenger Javier Arsuaga Mariel Vazquez Joaquim Roca P. McGuirk Christian Micheletti Davide Marenduzzo

75. REFERENCES Nucleic Acids Research 29(2001), 67-71. Proc. National Academy of Sciences USA 99(2002), 5373-5377. Biophysical Chemistry 101-102 (2002), 475-484. Proc. National Academy of Sciences USA 102(2005), 9165-9169. J. Chem. Phys 124 (2006), 064903

76. Thank You National Science Foundation Burroughs Wellcome Fund