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DNA TOPOLOGY

DNA TOPOLOGY. De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL sumners@math.fsu.edu. Pedagogical School: Knots & Links: From Theory to Application. Pedagogical School: Knots & Links: From Theory to Application. De Witt Sumners: Florida State University

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DNA TOPOLOGY

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

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

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

  4. DNA Site-Specific Recombination • Topological Enzymology • Rational tangles and 4-plats • The Tangle Model • Analysis of Tn3 Resolvase Experiments • Open tangle problem

  5. Site-Specific Recombination Recombinase

  6. Biology of Recombination • Integration and excision of viral genome into and out of host genome • DNA inversion--regulate gene expression • Segregation of DNA progeny at cell division • Plasmid copy number regulation

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

  8. GEL ELECTROPHORESIS

  9. Rec A Coating Enhances EM

  10. RecA Coated DNA

  11. DNA Trefoil Knot Dean et al. J. Biol. Chem. 260(1985), 4795

  12. DNA (2,13) TORUS KNOT Spengler et al. Cell 42(1985), 325

  13. T4 TWIST KNOTS Wasserman & Cozzarelli, J. Biol. Chem. 30(1991), 20567

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

  15. SITE-SPECIFIC RECOMBINATION

  16. Enzyme Bound to DNA

  17. DIRECT vs INVERTED REPEATS

  18. RESOLVASE SYNAPTIC COMPLEX

  19. DNA 2-STRING TANGLES

  20. 2-STRING TANGLES

  21. 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:

  22. RATIONAL TANGLES

  23. RATIONAL TANGLE CLASSIFICATION q/p = a2k + 1/(a2k-1 + 1(a 2k-2 +1/…)…) Two tangles are equivalent iff q/p = q’/p’ J. Conway, Proc. Conf. Oxford 1967, Pergamon (1970), 329

  24. TANGLE OPERATIONS

  25. RATIONAL TANGLES AND 4-PLATS

  26. 4-PLATS (2-BRIDGE KNOTS AND LINKS)

  27. 4-PLATS

  28. 4-PLAT CLASSIFICATION 4-plat is b(a,b) where b/a = 1/(c1+1/(c2+1/…)…) b(a,b) = b(a’,b’) as unoriented knots and links) iff • = a’and b+1b’ (mod a ) Schubert Math. Z. (1956)

  29. TANGLE EQUATIONS

  30. SOLVING TANGLE EQUATIONS

  31. SOLVING TANGLE EQUATIONS

  32. RECOMBINATION TANGLES

  33. SUBSTRATE EQUATION

  34. PRODUCT EQUATION

  35. TANGLE MODEL SCHEMATIC

  36. 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

  37. DISTRIBUTIVE RECOMBINATION

  38. PROCESSIVE RECOMBINATION

  39. RESOLVASE PRODUCTS

  40. 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

  41. 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

  42. 1st and 2nd ROUND PRODUC TS

  43. RESOLVASE SYNAPTIC COMPLEX

  44. Of = 0

  45. THEOREM 1

  46. 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 (SFS) • 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

  47. Proof that Tangles are Rational 2 biological arguments • DNA tangles are small, and have few crossings—so are rational by default • DNA is on the outside of protein 3-ball, and any tangle on the surface of a 3-ball is rational

  48. Proof that Tangles are Rational THE MATHEMATICAL ARGUMENT •The substrate (unknot) and the 1st round product (Hopf link) contain no local knots, so Ob, P and R are either prime or rational. • If tangle A is prime, then ∂A* (a torus) is incompressible in A. If both A and B are prime tangles, then (AUB)* contains an incompressible torus, and cannot be a lens space.

  49. Proof that Tangles are Rational THE MATHEMATICAL ARGUMENT N(Ob+P) = [1] so N(Ob+P)* = [1]* = S3 If Ob is prime, the P is rational, and Ob* is a knot complement in S3. One can similarly argue that R and (R+R) are rational; then looking at the 2-fold branched cyclic covers of the 1st 2 product equations, we have:

  50. Proof that Tangles are Rational N(Ob+R) = [2] so N(Ob+R)* = [2]* = L(2,1) N(Ob+R+R) = [2,1,1] so N(Ob+R+R)* = [2,1,1]* = L(5,3) Cyclic surgery theorem says that since Dehn surgery on a knot complement produces two lens spaces whose fundamental group orders differ by more than one, then Ob* is a Seifert Fiber Space. Dehn surgery on a SFS cannot produce L(2,1) unless Ob* is a solid torus, hence Ob is a rational tangle. N(Ob+R+R) = [] so N(Ob+R)* = [2]* = L(5,3)

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