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

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

4. 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

5. 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

6. TOPOLOGAL ENTANGLEMENT IN POLYMERS

7. 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

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

9. SEMICRYSTALLINE POLYMERS

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

11. 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

12. MONTE CARLO RANDOM KNOT SIMULATION Thin chains--random walk in Z 3 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:  (-1) Vologodskii et al, Sov. Phys. JETP 39 (1974), 1059

13. 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  (-1). Michels & Wiegel Phys Lett. 90A (1984), 381

14. THIN CHAIN RANDOM KNOTTING

15. 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

16. 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

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

18. SIMPLE CUBIC LATTICE

19. 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

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

21. TIGHT KNOTS

22. Z3Z3

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

24. 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 B 3, and this B 3 contains a red knotted arc.

25. SMALLEST TREFOIL L = 24

26. SLIGHTLY LARGER TREFOIL L = 26

27. 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?

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

29. ALL KNOTS APPEAR Every knot type has a tight Kesten pattern representative on Z 3

30. ALL KNOTS APPEAR: PROOF Take a projection of your favorite knot on Z 2. Bump up crossovers, saturate holes to get a decorated pancake

31. LONG RANDOM KNOTS

32. MEASURING KNOT COMPLEXITY

33. 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.

34. WRITHE

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

36. WRITHE GROWS LIKE  n

37. WRITHE GROWTH: PROOF

39. GROWTH OF WRITHE FOR FIGURE 8 KNOT

40. RANDOM KNOTS ARE CHIRAL

41. ENTANGLEMENT COMPLEXITY OF GRAPHS AND ARCS

42. DETECTING KNOTTED ARCS

43. LONG RANDOM ARCS (GRAPHS) ARE VERY COMPLEX

44. 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!

45. SMOOTHING

46. UNCONSTRAINED KNOTTING PROBABILITIES

47. CONSTRAINED UNKNOTTING PROBABILITY

48. CONSTRAINED UNKOTTING PROBABILITY

49. CONSTRAINED TREFOIL KNOT PROBABILITIES

50. CONSTRAINED TREFOIL PROBABILITY

51. 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

52. RANDOM KNOTTING IN HIGHER DIMENSIONS A similar result should hold for d-spheres in Z d+2, d > 2. CONJECTURE: For d > 2, let S d be a “plaquette” d-sphere containing the origin in Z d+2. If V(S d ) denotes the d-volume of S d, and P(S d ) denotes the probability that S d is knotted, then P(S d ) goes to 1 as V(S d ) goes to infinity.

53. 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 Enzo Orlandini

54. REFERENCES J. Phys. A: Math. Gen. 21(1988), 1689 J. Phys. A: Math. Gen. 25(1992), 6557 Math. Proc. Camb. Phil. Soc. 111(1992), 75 J. Phys. A: Math. Gen. 26(1993), L981 J. Stat. Phys. 85 (1996), 103 J. Knot Theory and Its Apps. 6 (1997), 31 Proc. National Academy of Sciences USA 99(2002), 5373-5377. Proc. National Academy of Sciences USA 102(2005), 9165-9169. J. Chem. Phys. 124(2006), 064903 Biophys. J. 95 (2008), 3591-3599

55. DNA KNOTS IN VIRAL CAPSIDS De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 sumners@math.fsu.edu

56. THE PROBLEM Ds DNA of bacteriophage is packed to near- crystalline density in capsids (~800mg/ml), and the pressure inside a packed capsid is ~50 atmospheres. PROBLEM: what is the packing geometry?

57. T4 EM

58. HOW IS THE DNA PACKED?

59. THE METHOD: VIRAL KNOTS REVEAL PACKING Compare observed DNA knot spectrum to simulation of knots in confined volumes

60. Crossover Number

61. CHIRALITY

62. Knots and Catenanes

63. + TORUS KNOTS

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

65. TWIST KNOTS

66. T4 TWIST KNOTS Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567

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

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

69. EFFECTS OF CONFINEMENT ON P4 DNA KNOTTING (10.5 kb) No confinement--3% knots, mostly trefoils Viral knots--95% knots, very high complexity-- average crossover number 27!

70. PIVOT ALGORITHM Ergodic--no volume exclusion in our simulation  as knot detector Reject swollen conformations—a problem!! Space filling polymers in confined volumes--very difficult to simulate

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

72. 2D GEL RESOLVES SMALL KNOTS Arsuaga et al, PNAS 102 (2005), 9165

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

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

75. 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 torus knots to see if they all have same chirality

76. NEW SIMULATION SCHEME Multiple chain Monte Carlo (parallel tempering) Several Monte Carlo evolutions run in parallel, each evolution labeled by temperature (pressure) Periodically, conformations are swapped (Metropolis energy probability) between adjacent runs Scheme greatly enhances exploration of phase space

77. SMOOTH CONFORMATIONS Need to reduce complexity—70% of crossings are extraneous Smoothing of conformations to reduce crossing number

78. IDENTIFYING KNOTS Use KNOTFIND to identify knots of 16 crossings or less Problem--some knots cannot be ID’d--might be complicated unknots!

79. CONCLUSIONS FROM MCMC Get precise determination of knot type (if < 16 crossings after smoothing!), volume exclusion, model tuned to P4 DNA bending rigidity Get confining volume down to 2.5 times P4 capsid volume Get excess of chiral knots Do not get excess torus over twist knots

80. NEW PACKING DATA—4.7 KB COSMID Trigeuros & Roca, BMC Biotechnology 7 (2007) 94

81. CRYO EM VIRUS STRUCTURE Jiang et al NATURE 439 (2006) 612Jiang et al NATURE 439 (2006) 612

82. DNA-DNA INTERACTIONS GENERATE KNOTTING AND SURFACE ORDER Contacting DNA strands (apolar cholosteric interaction) assume preferred twist angle Marenduzzo et al PNAS 106 (2009) 22269

83. SIMULATED PACKING GEOMETRY Marenduzzo et al PNAS 106 (2009) 22269

84. THE BEAD MODEL Semiflexible chain of 640 beads--hard core diameter 2.5 nm Spherical capsid 45 nm Kink-jump stochastic dynamic scheme for simulating packing

85. KNOTS DELOCALIZED Marenduzzo et al PNAS 106 (2009) 22269 Black—unknot; 9 1 —red; complex knot--green

86. SIMULATED KNOT SPECTRUM Marenduzzo et al PNAS 106 (2009) 22269

87. DNA-DNA INTERACTION CONCLUSIONS Reproduce cryo-em observed surface order Reproduce observed knot spectrum—excess of torus knots over twist knots Handedness of torus knots—no excess of right over left at small twist angles—some excess at larger twist angles and polar interaction

88. 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 Biophys. J. 95 (2008), 3591-3599 Proc. National Academy of Sciences USA 106(2009), 2269-2274.

89. JAVIER ARSUAGA, MARIEL VAZQUEZ, CEDRIC, EITHNE

90. CHRISTIAN MICHELETTI, ENZO ORLANDINI, DAVIDE MARENDUZZO

91. ANDRZEJ STASIAK

92. Thank You National Science Foundation Burroughs Wellcome Fund

93. UNCONSTRAINED KNOTTING PROBABILITIES

94. CONSTRAINED UNKNOTTING PROBABILITY

95. CONSTRAINED UNKOTTING PROBABILITY

96. CONSTRAINED TREFOIL KNOT PROBABILITIES

97. CONSTRAINED TREFOIL PROBABILITY

98. 4 vs 5 CROSSING PHASE DIAGRAM

99. CONFINED WRITHE

100. GROWTH OF CONFINED WRITHE

101. Thank You National Science Foundation Burroughs Wellcome Fund