1. TOPOLOGICAL METHODS IN PHYSICAL VIROLOGY FSU-UF TOPOLOGY MEETING FEB. 23, 2013 De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 sumners@math.fsu.edu

2. DNA Replication

3. TOPOLOGICAL VIROLOGY Using DNA plasmids as an assay for site-specific recombination—deduce viral enzyme binding and mechanism Using DNA knots to elucidate packing geometry and ejection of DNA in viral capsids

4. A Little Entanglement Can Go a Long Way

5. DNA KNOTTING IS LETHAL IN BACTERIA Promotes replicon loss by blocking DNA replication Blocks gene transcription Causes mutation at a rate 3 to 4 orders of magnitude higher than an unknotted plasmid Diebler et al, BMC Molecular Biology (2007) 8:44

6. Crossover Number

7. CHIRALITY

8. Knots and Catenanes

9. Prime and Composite Knots

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

11. TORUS KNOTS

12. TWIST KNOTS

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

14. Strand Passage Topoisomerase

15. Strand Exchange Recombinase

16. GEL ELECTROPHORESIS

17. RecA Coated DNA

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

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

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

21. PHAGE  GIN KNOTS Kanaar et al. CELL 62(1990), 553

22. Topoisomerase Knots Dean et al., J BIOL. CHEM. 260(1985), 4975Dean et al., J BIOL. CHEM. 260(1985), 4975Dean et al., J BIOL. CHEM. 260(1985), 4975Dean et al., J BIOL. CHEM. 260(1985), 4975

23. Dean et al., J BIOL. CHEM. 260(1985), 4975

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

25. SITE-SPECIFIC RECOMBINATION

26. Biology of Site-Specific Recombination Integration and excision of viral genome into and out of host genome DNA inversion--regulate gene expression & mediate phage host specificity Segregation of DNA progeny at cell division Plasmid copy number regulation

27. RESOLVASE SYNAPTIC COMPLEX

28. DNA 2-STRING TANGLES

29. 2-STRING TANGLES

30. 3 KINDS OF TANGLES A tangle is a configuration of a pair of strands in a 3-ball. We consider all tangles to have the SAME boundary. There are 3 kinds of tangles:

31. RATIONAL TANGLES

32. TANGLE OPERATIONS

33. RATIONAL TANGLES AND 4-PLATS

34. 4-PLATS (2-BRIDGE KNOTS AND LINKS)

35. 4-PLATS

36. TANGLE EQUATIONS

37. RECOMBINATION TANGLES

38. SUBSTRATE EQUATION

39. PRODUCT EQUATION

40. TANGLE MODEL SCHEMATIC Ernst & Sumners, Math. Proc. Camb. Phil. Soc. 108 (1990), 489

41. Tn3 RESOLVASE PRODUCTS

42. RESOLVASE MAJOR PRODUCT MAJOR PRODUCT is Hopf link [2], which does not react with Tn3 Therefore, ANY iterated recombination must begin with 2 rounds of processive recombination

43. RESOLVASE MINOR PRODUCTS Figure 8 knot [1,1,2] (2 rounds of processive recombination) Whitehead link [1,1,1,1,1] (either 1 or 3 rounds of recombination) Composite link ( [2] # [1,1,2]--not the result of processive recombination, because assumption of tangle addition for iterated recombination implies prime products (Montesinos knots and links) for processive recombination

44. 1st and 2nd ROUND PRODUC TS

45. RESOLVASE SYNAPTIC COMPLEX

46. O f = 0

47. THEOREM 1

48. PROOF OF THEOREM 1 Analyze 2-fold branched cyclic cover T* of tangle T--T is rational iff T* = S 1 x D 2 Use Cyclic Surgery Theorem to show T* is a Seifert Fiber Space Use results of Dehn surgery on SFS to show T* is a solid torus--hence T is a rational tangle Use rational tangle calculus to solve tangle equations posed by resolvase experiments

49. 3rd ROUND PRODUCT

50. THEOREM 2

51. 4th ROUND PRODUCT

52. UTILITY OF TANGLE MODEL Precise mathematical language for recombination- allows hypothesis testing Calculates ALL alternative mechanisms for processive recombination Model can be used with incomplete experimental evidence (NO EM)--crossing # of products, questionable relationship between product and round of recombination Proof shows there is NO OTHER explanation of the data

53. REFERENCES

54. JMB COVER Crisona et al, J. Mol. Biol. 289 (1999), 747

55. BACTERIOPHAGE STRUCTURE

56. T4 EM

57. HOW IS THE DNA PACKED?

58. SPOOLING MODEL

59. RANDOM PACKING

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

61. Liu et al P2 Knots (33kb)

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

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

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

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

66. 2D GEL RESOLVES SMALL KNOTS Arsuaga et al, PNAS 102 (2005), 9165

67. PIVOT ALGORITHM Ergodic—can include volume exclusion and bending rigidity Knot detector—knot polynomials (Alexander, Jones, KNOTSCAPE)

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

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

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

71. 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

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

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

74. 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

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

76. 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

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

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

79. 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

80. 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.

81. JAVIER ARSUAGA, MARIEL VAZQUEZ, CEDRIC, EITHNE

82. CHRISTIAN MICHELETTI, ENZO ORLANDINI, DAVIDE MARENDUZZO

83. ANDRZEJ STASIAK

84. COLLABORATORS Mathematics: Claus Ernst, Mariel Vazquez, Javier Arsuaga, Steve Harvey, Yuanan Diao, Christian Laing, Nick Pippenger, Stu Whittington, Chris Soteros, Enzo Orlandini, Christian Micheletti, Davide Marenduzzo Biology: Nick Cozzarelli, Nancy Crisona, Sean Colloms, Joaquim Roca, Sonja Trigeuros, Lynn Zechiedrich, Jennifer Mann, Andrzej Stasiak

85. Thank You National Science Foundation Burroughs Wellcome Fund

86. UNKNOWN P4 KNOT

87. UNKNOWN P4 KNOTS

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