Overview of DNA Topology

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

5. Tertiary Structure of DNA, III: Knots and Links Central DNA axis can be Knotted or Linked: ex: viral DNA replication (DNA copying) recombination (DNA rearranging) Defs:KS3 is a knot if K is homeomorphic to S1 and K isthe unknot if it is ambient isotopic to S1. Def: LS3 is an n-component link if L is homeomorphic to nS1 . Images from Rob Scharein’s KnotPlot

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

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

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). So DNA axis naturally forms rows of twists, broken by branch points, where additional rows of twists can emanate in another direction. Shlyakhtenko, Ultramicroscopy 2003

9. Topological Paradigm: Modeling (regions of supercoiled) DNA with Tangles Def: A tangle is a pair (B3, t) B3 = 3-ball with 4 distinguished boundary points t = pair of properly embedded unoriented arcs

10. Tangles There are 3 mutually exclusive families of tangles: Locally KnottedRationalPrime 1+1/(1 + 1/3) = 7/4 Locally knotted: There exists S2  B3 meeting t in 2 points, s.t. int(S2) 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. Modelling DNA-Protein Interactions via Tangle Surgery Many protein-DNA interactions act by cutting, rearranging and resealing DNA in a localised way: new old

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-specificrecombinase

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

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

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

18. Guiding Question: How to unveil salient features of this process? SubQuestion 2: pre- or post- reaction local DNA segment conformations: known ? 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 unknot trefoil Antiparallel Reaction Pathway:

21. But Previous Methods Incomplete 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:

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

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

24. ModelingProtein Action as Tangle Surgery: Crossing change in dbc: R P Crossing Change Constructing dbc Slice along orangediscs 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.

25. ModelingProtein Action as Tangle Surgery: Rational Tangles surgery  Dehn surgery in dbc Def:Dehn Surgery: K in M3 K(p/q) = 3-manifold obtained from M3 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.

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

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

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 S’  S T’ = (1/2)  T

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

30. Determining DNA knots and links (1) Electron Microscopy (2) Atomic Force Microscopy = = ??? = But can’t always tell. 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 31 # 31 31 61 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. but NOT e.g. Now Generalised by Karin Valencia – see her poster! Exact Knot known  helps illuminate structural & mechanistic features

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(S3\B). 3. Glue each of the post-recombinant forms of D  B to each form of D  cl(S3\B) to classify possible product knots and links. D

35. Global Model 2 of Recombination: Tangle Surgery on 4-plats Pioneered by Ernst and Sumners – now many. O P O P Recombination: P replaced by R R 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. R = (±1) or (±2) Theorem (Sumners, Ernst, Spengler, Cozzarelli): Predicts all 4-plats from recombination on the unknot. Next: Given knot products, what does that tell you about recombination?

37. Global Model 2.n of Recombination: Tangle Surgery on known knots Idea: Given the particular knots, find the tangles. Ex: (Distributive) Recombination by Hin – see Mauro Mauricio N(O + P) = 31 O P Recombination N(O+ R) = 31 # 31 O R

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. Main Idea of Proofs: Model recombination as Tangle surgery: pulling out P and replacing with R. If tangles are rational, corresponds to Dehn surgery on core(VP). 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.