isabel k. darcy - mathematical sciences home pages...

Isabel K Darcy

Mathematics Department

Applied Mathematical and Computational Sciences (AMCS)

University of Iowa

httpwwwmathuiowaedu~idarcy

copy2008 IK Darcy All rights reserved

This work was partially supported by the Joint DMSNIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285)

httpknotplotcomzoo

httpknotplotcomzoo Minimal diagrams of knots (knot with fewest number of crossings)

Unknot

Trefoil

Figure 8

9 crossing knot 7th in the list of 9 crossing knots

Pentafoil

Duplex DNA knots produced by Escherichia coli topoisomerase I Dean FB Stasiak A Koller T Cozzarelli NR J Biol Chem 1985 Apr 25260(8)4975-83

Links (math) = catenanes (biology)

Unlink Hopf link or 2-cat

4-cat 2 component

link

3 component link

6-cat 8-cat

A mathematicianrsquos introduction to a simplified view of the biology

behind DNA topology

DNA ndash The Double Helix

DNA ndash The twisted annulus

httpwwwpersonalpsuedurch8workmgStruc_Nucleic_Acids_Chpt2htm


httpwwwchcamacukmagnusmoleculesnucleicdna1jpg

httpwwwaccessexcellenceorgRCVLGGimagesdna_replicatinggif

James Watson Francis Crick and Rosalind Franklin

Rosalind Franklinrsquos notebook DNA is a double helix with antiparallel strands httpphilosophyofscienceportalblogspotcom200804rosalind-franklin-double-helixhtml

Anti-parallel strands means DNA does not (normally) form a mobius band

httpplusmathsorgissue18puzzlemobiusIIjpg

httpwwwsusquedubrakkeknotsknot32mhgif

Circular DNA normally forms a twisted annulus

httpmathforumorgmathimagesimgUploadthumbFull_twistjpg200px-Full_twistjpg

(J Mann) httpwwwsbsutexaseduherrinbio344

Postow L etal PNAS2001988219-8226

Cellular roles of DNA topoisomerases a molecular perspective James C Wang Nature Reviews Molecular Cell Biology 3 430-440 (June 2002)

Topoisomerase II performing a crossing change on DNA

Topoisomerases are proteins which cut one segment of DNA allowing a second DNA segment to pass through before resealing the break

Today 3-330 -Koya Shimokawa Saitama University Japan Tangle analysis of unlinking by XerCD-diff-FtsK system Recombinase

Recombination is mathematically equivalent to smoothing a crossing



DNA substrate = starting conformation of DNA before protein action

Usually unkotted and supercoiled




Meiotic double-strand breaks in yeast artificial chromosomes containing human DNA Grzegorz Ira Ekaterina Svetlova Jan Filipski Nucl Acids Res (1998) 26 (10)2415-2419




But can be knotted Eg Twist knots Wed 10-1030 -Karin Valencia Topological characterization of knots and links arising from site-specific recombination on twist knot substrates (or torus knotslinks)


Supercoiled DNA-directed knotting by T4 topoisomerase Wasserman SA Cozzarelli NR J Biol Chem 1991

Recombination

Recombination (2 p) torus link

(2 p) torus knot

Today 230-3 -Dorothy Buck Imperial College London UK The classification of rational subtangle replacements with applications to complex Nucleoprotein Assemblies Recombinase and Topoisomerase

Today 3-330 -Koya Shimokawa Saitama University Japan Tangle analysis of unlinking by XerCD-diff-FtsK system Recombinase Recombinase and Topoisomerase (maybe)

Tomorrow 10-1030 -Karin Valencia Imperial College London UK Topological characterization of knots and links arising from site-specific recombination on twist knot substrates Tomorrow 1-130 ndash Ken Baker University of Miami Recombination on rational knotlink substrates producing the unknotunlink (and vice versa)

Today 4-430 -Robert Scharein Hypnagogic Software Canada Computational knot theory with KnotPlot

Today 5-530 -Egor Dolzhenko University of South Florida STAGR software to annotate genome rearrangement Tomorrow Session molecular rearrangements 9-930 -Laura Landweber Princeton University USA Radical genome architectures in Oxytricha 930-10 -Gueacutenola Drillon Universiteacute Pierre et Marie Curie France Combinatorics of Chromosomal Rearrangements

An introduction to the tangle model for protein-bound DNA

Mathematical Model

Protein =

DNA =

=

= =

Protein-DNA complex Heichman and Johnson

C Ernst D W Sumners A calculus for rational tangles applications to DNA recombination Math Proc Camb Phil Soc 108 (1990) 489-515 protein = three dimensional ball protein-bound DNA = strings

Slide (modified) from Soojeong Kim

= ne

Tangle = 3-dimensional ball containing strings where the endpoints of the strings are fixed on the boundary of the ball

Protein = 3-dimensional ball DNA = strings For geometry see 12-1230 -Mary Therese Padberg Exploring the conformations of protein-bound DNA adding geometry to known topology Wednesday March 14 12-1230pm and poster

Rational Tangles

Rational tangles alternate between vertical crossings amp horizontal crossings k horizontal crossings are right-handed if k gt 0 k horizontal crossings are left-handed if k lt 0 k vertical crossings are left-handed if k gt 0 k vertical crossings are right-handed if k lt 0 Note that if k gt 0 then the slope of the overcrossing strand is negative while if k lt 0 then the slope of the overcrossing strand is positive By convention the rational tangle notation always ends with the number of horizontal crossings

Tangles

3112012

Which tangles are rational

This one is not rational The others are all rational

Rational tangles can be classified with fractions

Why are we interested in rational tangles

1) Rational tangles are simple simplest non-rational tangles


1) Rational tangles are simple simplest non-rational tangles 2) Rational tangles are formed by adding twists Think supercoils EM courtesy of Andrzej Stasiak



3) A tangle is rational if and only if one can push the strings to lie on the boundary of the 3-ball so that the strings do not cross themselves on 3-ball

A knotlink is rational if it can be formed from a rational tangle via numerator closure

N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)


A rational knotlink is also called a 2-bridge knotlink or 4-plat

a = c

and

b ndash d is a multiple of a or

bd ndash 1 is a multiple of a

A + B = C

2 + 0 = 2

2 + -2 = 0

Most tangles donrsquot have inverses

The tangle equations corresponding to an electron micrograph

Different recombinases have different topological mechanisms

Xer recombinase on psi Unique product Uses topological filter to only perform deletions not inversions

Ex Cre recombinase can act on both directly and inversely repeated sites




There are an infinite number of solutions to

Can solve by using

TopoICE in Rob Schareinrsquos KnotPlotcom

A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A

There exists an algebraic formula for converting solutions

Protein-bound DNA vs Local Action




Today 230-3 -Dorothy Buck Imperial College London UK The classification of rational subtangle replacements with applications to complex Nucleoprotein Assemblies


More general model


Processive Recombination (multiple reactions while protein remains bound to DNA)

Processive Recombination (multiple reactions per one encounter)

Distributive Recombination (multiple encounters and one reaction per encounter)

httpknotplotcomzoo


Unknot

Trefoil

Figure 8


Pentafoil




4-cat 2 component

link

3 component link

6-cat 8-cat


behind DNA topology



































Recombination


(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model






Unknot

Trefoil

Figure 8


Pentafoil




4-cat 2 component

link

3 component link

6-cat 8-cat


behind DNA topology



































Recombination


(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model








4-cat 2 component

link

3 component link

6-cat 8-cat


behind DNA topology



































Recombination


(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model






behind DNA topology



































Recombination


(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model







































Recombination


(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model






(2 p) torus knot







Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model











Mathematical Model

Protein =

DNA =

=

= =




= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model








= ne



Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





Rational Tangles


Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





Tangles

3112012












N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model
















N(27) = N(21)

Note 7 ndash 1 = 6 = 2(3)



a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model







a = c

and



A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





A + B = C

2 + 0 = 2

2 + -2 = 0










Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model













Can solve by using


A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





A)

A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





A) B)

A) B)

C)

A

A)

D)

B)

C)

A








More general model





A) B)

C)

A

A)

D)

B)

C)

A








More general model





A)

D)

B)

C)

A








More general model












More general model





isabel k. darcy - mathematical sciences home pages...

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