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Overview of DNA Topology. 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 . DNA Tertiary Structure, I: Circular. - PowerPoint PPT Presentation
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Overview of DNA Topology
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
DNA Tertiary Structure, I: Circular
Central axis of DNA molecule can be:
• Circular ex: bacterial chromosomal DNA chloroplast DNA viral genomic DNA
• Supercoiled: the DNA axis coils upon itself.Measured in terms of Writhe
DNA Tertiary Structure, II: Supercoiled
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 S3 is a knot if K is homeomorphic to S1 and K is the unknot if it is ambient isotopic to S1.
Images from Rob Scharein’s KnotPlot
Def: L S3 is an n-component link if L is homeomorphic to n S1 .
Ex 2: Recombination (DNA rearrangement):
Ex 1: Replication (DNA copying)
DNA Knots and Links Occur Naturally
Ex 1: DNA Knots inhibit strand separationaffects, e.g. DNA replication
gene transcriptionDNA recombination
Ex 2: Circular DNA Linked after copying
DNA Knots/Links Important Biologically
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.
Def: A tangle is a pair (B3, t) B3 = 3-ball with 4 distinguished boundary points t = pair of properly embedded unoriented arcs
Topological Paradigm: Modeling (regions of supercoiled) DNA with Tangles
TanglesThere are 3 mutually exclusive families of tangles:
Locally Knotted Rational Prime 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.
Many protein-DNA interactions act by cutting, rearranging and resealing DNA in a localised way:
Modelling DNA-Protein Interactions via Tangle Surgery
old new
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
Localised DNA Transformations Ex 1: Site-Specific Recombination
Localised DNA Transformations Ex 2: Type-II Topoisomerase Mediated Crossing Changes
from Stuchinskaya et al JMB2009
Guiding Question:
SubQuestion 1: What is the enzyme mechanism or choreography?
?
How to unveil salient features of this process?
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?
known
known ?
?
SubQuestion 2: pre- or post- reaction local DNA segment conformations:
Guiding Question:How to unveil salient features of this process?
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
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.
Then Maths + Biochem => SSR sites oriented antiparallel
Antiparallel Reaction Pathway:
unknot trefoil
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
Crossing Change
P R
Idea: Represent crossing change by a tangle replacement P for R:
Ex: Modelling Type-II Topoisomerase-Mediated Crossing Changes
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.
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
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.
Modeling Protein Action as Tangle Surgery:Rational Tangles surgery Dehn surgery in dbc
Theorem (jt w/ Ken Baker):Classifies all rational tangles adjacent to a given rational tanglevia replacement of a rational subtangle.
Δ
T’ = (1/3) T S’ S
Modeling Protein Action as Tangle Surgery:
Model for Sin Recombinase Synaptic Complex
From Mouw, Marshall Rice Mol Cell 2008
Simple Application of Subrational Tangle Replacement
?
site-specificrecombination Δ = ?
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
?
Next: How to model global DNA topology (& global topological changes).
(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
Ex where DNA becomes Knotted: Site-Specific Recombination
Big Question: Can we have a atomic-level movie of this process?
Supercoiling DNA + Recombination = Knotted DNA
Site-Specific Recombination
31 6131 # 31
DNA knots courtesy of Shailja Pathania
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!
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
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.
PO
RO
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?
Global Model 2.n of Recombination:Tangle Surgery on known knots
Recombination
N(O + P) = 31
Idea: Given the particular knots, find the tangles.
N(O + R) = 31 # 31
PO
RO
Ex: (Distributive) Recombination by Hin – see Mauro Mauricio
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:
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.
Main Idea of Proofs: