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. What is mathematics good for? “No longer just the study of number and space, mathematical science has become the science of patterns, with theory built on relations among patterns and on applications derived from the fit between pattern and observation.” L.A. Steen, SCIENCE (1988)

6. Mathematics at the Beach –Farside--Larsen

7. Drinking and Deriving

8. What is Topology?

9. Isomers

10. Using Topology in Science

11. SYNTHETIC KNOT Dietrich-Buchecker & Sauvage, Ang. Chemie 28 (1989), 189

12. KNOT IN A PROTEIN J. Am. Chem. Soc. 118(1996), 8945

13. What is Knot Theory?

14. KNOT HISTORY

15. Knots and Catenanes

16. Crossover Number

17. Peter Guthrie Tait Photo: Roger Hendrix @ICMS

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

19. Torus Knots Rob Scharien A Knot Zoo—Knot Plot

20. Twist Knots Rob Scharien A Knot Zoo—Knot Plot

21. Prime and Composite Knots

22. CHIRALITY

23. CHIRAL AND ACHIRAL KNOTS

24. CROSSING SIGN CONVENTION

25. LINKING NUMBERS

26. UNKNOTTING NUMBER

27. STRAND PASSAGE METRIC Distance between two knots {K 1,K 2 } = minimum number of strand passages to convert from K 1 to K 2 Darcy &Sumners, Math. Proc. Camb Phil. Soc. 128 (2000), 497

28. STRAND PASSAGE METRIC TABLE Isabel Darcy http://www.math.uiowa.edu/~idarcy/

29. DNA Replication

30. Adenine-Thymine Base Pair

31. Cytosine-Guanine Base Pair

32. Chemical Orientation of Backbone Strands: 3’  5’

33. Mathematical Orientation of Backbone Strands Forget the chemical orientation 3’  5’: orient the axis and the strands in parallel to the axis DNA forms a ribbon Covalently closed circular DNA forms an orientable ribbon (because of the chemistry 3’  5’)

34. DNA Structure

35. DNA is Crowded in the Cell

36. Radial Loop Chromosome

37. Replication Obstruction

38. DNA ENTANGLEMENT: A DILEMMA FOR CELLS DNA must be long enough to encode organism complexity (50kb for e Coli; 3x10 9 base pairs for humans) DNA must be thin and flexible enough to fit inside the cell (or inside the cell nucleus) THE PROBLEM: DNA collisions can knot, link and drive DNA recombination, potentially destroying cellular control of DNA geometry and topology

39. HOW DOES THE CELL CONTROL DNA TOPOLOGY? The cell produces and uses a number of types of the enzyme TOPOISOMERASE to control cellular DNA topology and geometry

40. Strand Passage Topoisomerase

41. TOPOISOMERASE Hyscience August 9, 2007

42. TOPO I MECHANISM

44. 2-Gate Model for Topoisomerase II

45. Crystal Structure of Topoisomerase

46. ROCA TOPOISOMERASE MODELS http://www.cid.csic.es/homes/roca/TOPOI I.html

47. Site-specific Recombination Recombinase

48. LINEAR, RELAXED, SUPERCOILED DNA

49. TWIST

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

51. LK = TW + WR

54. TOPOISOMERSE AND LINKING

55. TOPO I vs TOPO II

56. DNA PLASMID REPLICATION

57. Enzyme Bound to DNA

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

59. Information We Seek

60. 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

61. GEL ELECTROPHORESIS

62. Rec A Coating Enhances EM

63. RecA Coated DNA

64. DNA Trefoil Knot

65. DNA (2,13) TORUS KNOT

66. T4 TWIST KNOTS

67. GIN KNOTS Kanaar et al. CELL 62(1990), 553

68. TOPOISOMERASE CAN MAKE OR BREAK DNA KNOTS AND LINKS Stoichiometry (ratio of topoisomerase to DNA substrate) determines outcome: Many copies of topoisomerase to each copy of DNA plasmid--knots and links are created Few copies of topoisomerase to each copy of DNA plasmid (cell physiology conditions)--knots and links are destroyed (below thermodynamic equilibrium values!)

69. Topoisomerase I Experiment Dean et al. J. Biol. Chem. 260 (1985), 4795

70. Topoisomerase Knots

72. Right and Left Hand Trefoils

73. Torus and Square Knots

74. Gel Mobility of DNA Knots

75. Conclusions

76. GEL VELOCITY IDENTIFIES KNOTS

77. Toposides--Chemotherapy Replication Fork Topoisomerase

78. LANCE ARMSTRONG Testicular cancer (metastatic) Oct. 1996 Topotecan Hydrochloride (topoisomerase inhibitor) Tour de France winner 1999-2005

79. © 2008 Jennifer K. Mann

80. Effect of DNA knotting on gene function © 2008 Jennifer K. Mann

81. Hin Substrate 26 bp Recognition & Cleavage Sites Wild-type (Inversion, no knot) TTCTTGAAAACCAAGGTTTTTGATAA AAGAACTTTTGGTTCCAAAAACTATT Mutant (knot) TTCTTGAAAACCATGGTTTTTGATAA AAGAACTTTTGGTACCAAAAACTATT (2 bp overhangs) © 2008 Jennifer K. Mann

82. Processive Hin Recombination © 2008 E. Lynn Zechiedrich 0101 3131

83. Hin Recombination © 2008 E. Lynn Zechiedrich & Jennifer K. Mann 0 1 0 1 0 1 3 1

84. Processive Hin Recombination 1(360 o )  Trefoil or 3-Noded Twist 2(360 o )  5-Noded Twist 3(360 o )  7-Noded Twist In theory: n(360 o )  2+(2n-1) = (2n+1) Noded Twist Distributive Hin Recombination 1(360 o ) + 1(360 o )  6-Noded Composite 1(360 o ) + 2(360 o )  8-Noded Composite 1(360 o ) + 1(360 o ) + 1(360 o )  9-Noded Composite 3131 5252 7272 3 1 # 3 1 3 1 # 5 2 3 1 # 3 1 # 3 1 © 2008 Jennifer K. Mann

85. DNA KNOTS ARE BAD FOR THE CELL Plasmid containing ampicillin-resistance gene is knotted by in vivo Hin recombination (producing DNA trefoil knots) Knotting promotes replicon loss by blocking DNA replication Knotting blocks gene transcription Knotting causes mutation at rate 3-4 orders of magnitude higher than unrecombined substrate Knotted DNA is potentially toxic and may drive genetic evolution R.W. Deibler, J.K. Mann, D.W. Sumners, L. Zechiedrich. Hin-Mediated DNA Knotting and Recombining Promote Replicon Dysfunction and Mutation, BMC Molecular Biology 8 (2007), 44

86. DNA KNOTTING IS REPLICATION OBSTRUCTION

87. DNA KNOTTING IS LETHAL!

88. Potential Models for how knots affect DNA metabolism © 2008 Jennifer K. Mann

89. 1. Knots inhibit gene function by blocking replication and transcription Conclusions: 2. Knots induce DNA rearrangements -In an essential gene, knots can be lethal -Do knots promote evolution? -Do knots account for the genomic instability associated with chemotherapy? © 2008 Jennifer K. Mann

90. TOPOISOMERASE RESOLVES KNOTS IN HEALTHY CELLS In wild-type eColi no endogenous knotted DNA plasmids are found In eColi with a point mutation on a TOPO II gene, there is a compensatory mutation on another TOPO I gene, the cell lives, and endogenous DNA knots appear (about 3- 5%, all trefoils) Shishido et al JMB (1987)

91. Thank You National Science Foundation Burroughs Wellcome Fund