1. Overview of DNA Topology

2. DNA Primary and Secondary Structure Primary: Composed of repeated units: nucleotides (nt) nt = sugar U phosphate U base Sugar-phosphate backbone. Bases pair as GC or AT. Secondary: Double helix

3. DNA Tertiary Structure, I: Circular Central axis of DNA molecule can be: Circular ex: bacterial chromosomal DNA chloroplast DNA viral genomic DNA

4. Supercoiled: the DNA axis coils upon itself. Measured in terms of Writhe DNA Tertiary Structure, II: Supercoiled

5. Central DNA axis can be Knotted or Linked: ex: viral DNA replication (DNA copying) recombination (DNA rearranging) Tertiary Structure of DNA, III: Knots and Links Defs: K  S 3 is a knot if K is homeomorphic to S 1 and K is the unknot if it is ambient isotopic to S 1. Images from Rob Scharein’s KnotPlot Def: L  S 3 is an n-component link if L is homeomorphic to  n S 1.

6. Ex 2: Recombination (DNA rearrangement): Ex 1: Replication (DNA copying) DNA Knots and Links Occur Naturally

7. Ex 1: DNA Knots inhibit strand separation affects, e.g. DNA replication gene transcription DNA recombination Ex 2: Circular DNA Linked after copying DNA Knots/Links Important Biologically

8. Modelling the DNA Axis DNA in vivo and in vitro is (negatively) plectonemically supercoiled. Occasionally branched: Visualized by 1. electron microscopy. 2. AFM in situ (at physiological conditions). Shlyakhtenko, Ultramicroscopy 2003 So DNA axis naturally forms rows of twists, broken by branch points, where additional rows of twists can emanate in another direction.

9. Def: A tangle is a pair (B 3, t) B 3 = 3-ball with 4 distinguished boundary points t = pair of properly embedded unoriented arcs Topological Paradigm: Modeling (regions of supercoiled) DNA with Tangles

10. Tangles There are 3 mutually exclusive families of tangles: Locally Knotted Rational Prime 1+1/(1 + 1/3) = 7/4 Locally knotted: There exists S 2  B 3 meeting t in 2 points, s.t. int(S 2 ) contains a knotted spanning arc. Rational: A = (p/q). {Equivalence classes}  {   Rational Numbers}, via a continued fraction expansion (Conway). Prime: Neither Locally knotted nor Rational.

11. Many protein-DNA interactions act by cutting, rearranging and resealing DNA in a localised way: Modelling DNA-Protein Interactions via Tangle Surgery old new

12. Localised DNA Transformations Ex 1: Site-Specific Recombination Def: Recombination: the rearranging of the DNA sequence. e.g. GATTACTA  ATCATTAG Site-specific recombination mediated by a protein, a site-specific recombinase

13. Localised DNA Transformations Ex 1: Site-Specific Recombination

14. Localised DNA Transformations Ex 2: Type-II Topoisomerase Mediated Crossing Changes from Stuchinskaya et al JMB2009

15. Guiding Question: SubQuestion 1: What is the enzyme mechanism or choreography? ? How to unveil salient features of this process?

16. Guiding Question: SubQuestion 1: What is the enzyme mechanism or choreography? ? How to unveil salient features of this process? Ex: Site-specific recombination proceeds through which pathway?

17. known ? ? SubQuestion 2: pre- or post- reaction local DNA segment conformations: Guiding Question: How to unveil salient features of this process?

18. known ? SubQuestion 2: pre- or post- reaction local DNA segment conformations: Guiding Question: How to unveil salient features of this process? action Ex: Xray of site-specific recombinase-DNA complex known ? action

19. IF localised action yields change in DNA knot type, THEN can answer Questions, using maths + biochemistry Ex: site-specific recombination on supercoiled circular substrates yields particular DNA knots.

20. Then Maths + Biochem => SSR sites oriented antiparallel Antiparallel Reaction Pathway: unknottrefoil

21. But so far unknown what to do if 1. DNA not knotted or linked 2. DNA knots or links not 4-plats 3. Localised axis doesn’t change knot/link type: But Previous Methods Incomplete

22. Crossing Change PR Idea: Represent crossing change by a tangle replacement P for R: Ex: Modelling Type-II Topoisomerase-Mediated Crossing Changes

23. Paradigm: Look UPSTAIRS in double branch covers. Figures from SketchesofTopology.Wordpress.com Modeling Protein Action as Tangle Surgery: P dbc of P Solid torus is double cover of the tangle 3-ball branched over its 2 properly embedded arcs.

24. Modeling Protein Action as Tangle Surgery: Crossing change in dbc: Crossing Change Slice along orange discs Blue becomes 2 curves, each intersecting red curve twice So replacing P with R  replacing solid torus with m by a solid torus with blue meridians m’ bounding discs. P R Constructing dbc

25. Def: Dehn Surgery: K in M 3 K(p/q) = 3-manifold obtained from M 3 by p/q surgery on K, by removing a toroidal nbhd(K) and replacing it with another torus whose meridian is sent to a p/q-curve on original boundary. We measure the distance, Δ, as the number of times the p/q-curve intersects the meridian μ. Modeling Protein Action as Tangle Surgery: Rational Tangles surgery  Dehn surgery in dbc

26. Theorem (jt w/ Ken Baker): Classifies all rational tangles adjacent to a given rational tangle via replacement of a rational subtangle. Δ T’ = (1/3)  T S’  S Modeling Protein Action as Tangle Surgery :

27. Model for Sin Recombinase Synaptic Complex From Mouw, Marshall Rice Mol Cell 2008 Simple Application of Subrational Tangle Replacement ? site-specific recombination Δ = ?

28. Examples of Applications: Can elucidate all local structures 2. Type 2 Topoisomerase Reactions: Can classify all local possible structures arising from crossing change. Δ = 2 T’ = (1/2)  T S’  S ?

29. Next: How to model global DNA topology (& global topological changes).

30. (2) Atomic Force Microscopy But can’t always tell. Determining DNA knots and links (1) Electron Microscopy == ??? = To restrict knot type, need to understand how DNA knots form

31. Ex where DNA becomes Knotted: Site-Specific Recombination Big Question: Can we have a atomic-level movie of this process?

32. Supercoiling DNA + Recombination = Knotted DNA Site-Specific Recombination 3131 6161 3 1 # 3 1 DNA knots courtesy of Shailja Pathania

33. Global Topologial Model 1 of Recombination Theorem: (joint w/ Erica Flapan) Given an unknot, unlink, or torus knot/link DNA molecule, recombination can yield only very particular knots. Exact Knot known  helps illuminate structural & mechanistic features but NOT e.g. Now Generalised by Karin Valencia – see her poster!

34. Idea behind Proofs: 1. Let ball B = convex hull of the four recombinase molecules D= spanning surface D for DNA axis. and determine D  B pre- and post-recombination. 2. Characterize D  cl(S 3 \B). 3. Glue each of the post-recombinant forms of D  B to each form of D  cl(S 3 \B) to classify possible product knots and links. D

35. Recombination: P replaced by R P R O O Global Model 2 of Recombination: Tangle Surgery on 4-plats Pioneered by Ernst and Sumners – now many. P O R O

36. Global Model 2 of Recombination: Tangle Surgery on 4-plats Given tangle model, & biologically reasonable assumptions: 1. P = (0) 2. O is rational, or the sum of 2 rationals 3. Products are 4-plats: (braids on 4 strings, closed as below) ≤ 9 crossings. Theorem (Sumners, Ernst, Spengler, Cozzarelli): Predicts all 4-plats from recombination on the unknot. R = (±1) or (±2) Next: Given knot products, what does that tell you about recombination?

37. Global Model 2.n of Recombination: Tangle Surgery on known knots Recombination N(O + P) = 3 1 Idea: Given the particular knots, find the tangles. N(O + R) = 3 1 # 3 1 P O R O Ex: (Distributive) Recombination by Hin – see Mauro Mauricio

38. Global Model 2.n of Recombination: Tangle Surgery on known knots Processive Hin recombination => P = (0) R = (2). Then only 4 solns for O are:

39. Model recombination as Tangle surgery: pulling out P and replacing with R. If tangles are rational, corresponds to Dehn surgery on core(V P ). Then: 3-manifold techniques => limits type of Dehn surgeries (ex: showing dbc(O) is simple & placing distance bounds on exceptional surgeries) Uniqueness of dbc => limits type of tangle surgeries => limits type of tangles. Main Idea of Proofs: