1. DNA TOPOLOGY: EXPERIMENTS AND ANALYSIS
De Witt Sumners
Department of Mathematics
Florida State University
Tallahassee, FL

2. Isomers

3. Using Topology in Science

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

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

6. What is Knot Theory?

7. Knots and Catenanes

8. Crossover Number

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

10. Prime and Composite Knots

11. CHIRALITY

12. CROSSING SIGN CONVENTION

13. LINKING NUMBERS

14. TWIST

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

16. LK = TW + WR

17. LK = TW + WR

18. LK = TW + WR

19. DNA Replication

20. DNA is Crowded in the Cell

21. Radial Loop Chromosome

22. Replication Obstruction

23. Strand Passage
Topoisomerase

24. Strand Exchange
Recombinase

25. *
07/16/96
Enzyme Bound to DNA *

26. Topological Enzymology*
07/16/96
Topological Enzymology
Mathematics: Deduce enzyme binding and mechanism from observed products *

27. Information We Seek

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. Rec A Coating Enhances EM

31. RecA Coated DNA

32. *
07/16/96
DNA Trefoil Knot *

33. DNA (2,13) TORUS KNOT

34. TOPOISOMERSE AND LINKING

35. TOPO I vs TOPO II

36. DNA PLASMID REPLICATION

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

38. Topoisomerase Knots

39. Topoisomerase Knots

40. Right and Left Hand Trefoils

41. Torus and Square Knots

42. Gel Mobility of DNA Knots

43. Conclusions

44. Crystal Structure of Topoisomerase

45. GEL VELOCITY IDENTIFIES KNOTS

46. Toposides--Chemotherapy
Topoisomerase
Replication Fork

47. SITE-SPECIFIC RECOMBINATION

48. *
07/16/96
Enzyme Bound to DNA *

49. DIRECT vs INVERTED REPEATS

50. RESOLVASE SYNAPTIC COMPLEX

51. DNA 2-STRING TANGLES

52. 2-STRING TANGLES

53. 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:

54. RATIONAL TANGLES

55. RATIONAL TANGLE CLASSIFICATION
q/p = a2k + 1/(a2k-1 + 1(a 2k-2 +1/…)…)
Two tangles are equivalent iff q/p = q’/p’

56. TANGLE OPERATIONS

57. RATIONAL TANGLES AND 4-PLATS

58. 4-PLATS

59. 4-PLATS

60. 4-PLAT CLASSIFICATION 4-plat is b(a,b) where b/a = 1/(c1+1/(c2+1/…)…)
b(a,b) = b(a’,b’) iff
a = a’ and b+1 = b’ (mod a )

61. TANGLE EQUATIONS

62. SOLVING TANGLE EQUATIONS

63. SOLVING TANGLE EQUATIONS

64. RECOMBINATION TANGLES

65. SUBSTRATE EQUATION

66. PRODUCT EQUATION

67. TANGLE MODEL SCHEMATIC

68. ITERATED RECOMBINATION
DISTRIBUTIVE: multiple recombination events in multiple binding encounters between DNA circle and enzyme
PROCESSIVE: multiple recombination events in a single binding encounter between DNA circle and enzyme

69. DISTRIBUTIVE RECOMBINATION

70. PROCESSIVE RECOMBINATION

71. RESOLVASE PRODUCTS

72. 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

73. 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 for processive recombination

74. 1st and 2nd ROUND PRODUC TS

75. Of = 0

76. THEOREM 1

77. PROOF OF THEOREM 1
Analyze 2-fold branched cyclic cover T* of tangle T--T is rational iff T* = S1 x D2
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

78. 3rd ROUND PRODUCT

79. THEOREM 2

80. 4th ROUND PRODUCT

81. THEOREM 3

82. 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

83. REFERENCES

84. JMB COVER