DNA unknotting and unlinking · 8/8/2012 · (DNA knots or links, mainly UNLINKS) August 8, 2012...

DNA unknotting and unlinking

Mariel Vazquez Mathematics Department

San Francisco State University [email protected]

August 8, 2012 1 SIAM Life Sciences 2012

DNA double-helix

"It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material” Watson and Crick (1953) A Structure for Deoxyribose Nucleic Acid

October 6, 2011 Convexity, Topology, Combinatorics and Beyond 2

DNA double-helix “Since the two chains in our model intertwine, it is essential for them to untwist if they are to separate. [...] Although it is difficult at the moment to see how these processes occur without everything getting tangled , we do not feel that this objection will be insuperable." Watson and Crick (1953) General Implications of the structure of

deoxyribonucleic acid Nature


Postow et al., PNAS (2001)

Replication of circular DNA produces links



Replication links are removed by type II topoisomerases


Type II topo


In the bacteria Escherichia coli, in the absence of topoisomerases

site-specific recombinases XerCD can unlink replication links

Grainge, Bregu, Vazquez, Sivanathan, Ip and Sherratt, EMBO J. (2007) 26(19)

Site-specific recombination

August 8, 2012 SIAM Life Sciences 2012 7

Site-specific recombination Site-specific recombinases bind two short identical DNA sites. They act by a cut-recombine-paste mechanism.

Changes in the DNA mediated by these enzymes can have important phenotypic effects.


λ- Int from bacteriophage λ integrates the viral DNA into the bacterial host genome

λ- Int


Aihara et al. (2003)

Healy and Elaydi

λ- Int from bacteriophage λ integrates the viral DNA into its bacterial host genome

λ- Int


Healy and Elaydi

integration

excision

August 8, 2012 SIAM Life Sciences 2012 11 http://www.pdbj.org/eprots/index_en.cgi?PDB%3A2IUU Massey et al., 2006

XerC / XerC recombination at dif resolved chromosome dimers

XerC / XerC recombination at dif resolved chromosome dimers


In Escherichia coli, XerC/D work at the last stages of chromosome segregation.

resolution

fusion

Reviewed in Barre et al., 2001

Site-specific recombinases can knot and link circular DNA


XerC/D

RH 4-cat Colloms et al., 1997

λ-Int

RH torus links (4-cat, 6-cat, 8-cat, etc…) Mizuuchi et al. (1980); Pollock and Nash (1983), Spengler et al. (1985)

XerC/D can also unlink DNA


Ip et al., 2003

λ-Int

RH torus links (4-cat, 6-cat, 8-cat, etc…)

Final knot/link distribution

XerCD-FtsK at dif

How do site-specific recombinases induce topological

changes on DNA?



The local action of recombination is relatively well understood

The local action of recombination is relatively well understood


The global action is not…

The product topology is a direct consequence of the geometrical

conformation adopted by the substrate prior to recombination


GOAL

Understand the topological mechanism of binding and strand-exchange

using geometry and topology


The mathematics of site-specific recombination



DNA is modeled as a curve


Circular DNA molecule

Smooth embedding of a circle in 3-space

= KNOT

DNA knot mathematical knot


Site-specific recombination is modeled as a 2-step reaction


Substrate Products (DNA knot or link) recombination

The synaptic complex appears as a ball with two emanating arcs



Enzyme = 3D ball

Ernst and Sumners (1990)

Enzyme + bound DNA = 2-string tangle

We use tangles to model the recombination reaction



We use tangles to model the recombination reaction


Substrate Products (DNA knots or links) recombination

N(O+P) = substrate N(O+R) = product O, P and R are 2-string tangles


•  Initial configuration: DNA and enzyme(s) are fixed

•  Recombination goes by tangle surgery

Tangle method assumptions

= N(E), where E = enzyme+bound DNA


•  Initial configuration: DNA and enzyme(s) are fixed

E=O+P O=outside tangle (remains fixed) P= parental tangle

•  Recombination turns P into R: O+R

R= recombinant tangle

Tangle method assumptions

= N(E), where E = enzyme + bound DNA

The enzymatic action is translated into a system of two tangle equations


N(O+P) = substrate N(O+R) = product

recombination

O, P and R are 2-string tangles

August 8, 2012 SIAM Life Sciences 2012

Rational

Most tangles in biology are rational

31


Rational

Rational tangles admit a classification

32

Classification Theorem (Conway 1970): There is a 1-1 correspondence between rational tangles and the set .

∪ {∞}


Rational

Rational tangle – Rational number

33

= (-5,-2)

Classification Theorem (Conway 1970): There is a 1-1 correspondence between rational tangles and the set .

∪ {∞}


A and B rational N(A+B) is a 4-plat

Tangle equations

(-7)-twist knot

= =

⇒


Tangle equations

(-7)-twist knot

AIM: Given any tangle P and 4-plat knot or link K, find

tangle solutions O to tangle equations of the form N(O +P) = K

= =

A and B rational N(A+B) is a 4-plat ⇒

The tangle equations can be solved (Ernst and Sumners 1990)

O=x/y and P=u/v rational tangles

N(O+P) =K=b(p,q) is a 4-plat where:

p=|yu+xv|


Tangle equations


Goal: Given known knots or links K0 and K1, find tangle solutions O, P

and R to these tangle equations

N(O+P) = substrate = K0 N(O+R) = product = K1

Dimer resolution by Xer recombination on unknotted

plasmids


Xer recombination at psi yields a unique product topology


XerC/D

RH 4-cat Colloms et al., 1997

Xer recombination at psi yields a unique product topology


Facts: -Xer acts only on directly repeated sites

-Xer recombination leads to a unique product topology -DNA wraps around accessory proteins approximately 3 times

(2 copies of PepA, 1 copy of ArgR) Colloms et al. (1997), Alén et al. (1997)

XerC/D

RH 4-cat


Xer tangle equations

Substrate = N(O+P) = = <1> =

Product = N(O+R) = = <1,2,1> =

If O+P and O+R are assumed rational or sums of rational tangles, then the tangle equations can be solved.

Ernst and Sumners (1990, 1999), Ernst (1998)

Problem: We don’t know anything about P and R


Bio. assumptions on P and R •  P=(0)

•  R=(0,0) if P anti-parallel

•  R = (k) = , if P parallel

P contains the core regions of the recombination sites


If P=(0) parallel

O = (-3,0) = and R= (-1) =

Results N(O+P) =

N(O+R) =

Pep A psi sites Arg R


If P=(0) parallel

O = (-3,0) = and R= (-1) =

and

O = (-5,0) = and R= (+1) =

If P=(0) anti-parallel

O = (-4,0) = and R=(0,0) =

Results N(O+P) =

N(O+R) =


Unique topological mechanism for Xer recombination at psi

Pep A psi sites Arg R


Xer synapse: molecular model

XerCD/DNA complex modeled using the Cre/loxP synapse (Gopaul et al. 1998)

PepA and ArgR structures from (Sträter et al. 1999)

Vazquez, Colloms and Sumners J. Mol. Biol 346 (2005)

Configuration created with PyMol

The molecular model includes the 3 solutions



Using a theorem by Hirasawa and Shimokawa (2000) on Dehn surgeries on strongly invertible knots we can prove that:

Theorem If O, P and R are tangles that satisfy:

N(O+P)=b(1,1) N(O+R)=b(4,3)

then P and are locally unknotted and O is rational.

The O tangle involved in Xer at psi is rational


TangleSolve

Saka and Vazquez, Bioinformatics (2002)

DNA unlinking by XerCD/FtsK at dif sites



Replication of circular DNA produces links


parCts

ori dif

30o

42o

XerCD-FtsK

30 min

sc cats

sc dimer

4 cat

6 cat

8 cat

10 cat12 cat

2 cat

oclin

14 cat

7 knot

9 knot

XerCDFtsKNucleotide

AT

P

++

++

A

BTime

Fig 1 Grainge et al

30o

42o

rep

lica

tio

n c

ate

na

ne

s

Free

circle

products

1 2 3 4 5 6 7 8 9A

TP

S

Free circle

products

0 2 5 10 30 30

25o 37o

5 knot

3 knotdimer

oc


In the absence of topoIV, XerCD can unlink replication catenanes

Dimeric DNA knots and unknotted dimeric DNA (oc dimer) appear as transient reaction intermediates in the course of catenane unlinking.

Grainge, Bregu, Vazquez, Sivanathan, Ip and Sherratt, EMBO J. (2007) 26(19)

XerCD-dif FtsK50C unlinking of replication catenanes

formed in vivo


Experimental Data

XerCD-FtsK

Replication links Products (DNA knots or links,

mainly UNLINKS)


Replication links Products (DNA knots or links,

mainly UNLINKS)

recombination

XerCD / FtsK at dif


Tangle equations for XerCD-FtsK unlinking

Substrate = N(O+P) = = parallel 6-cat

Product = N(O+R) = = some knot or link

Experimental observation: Over time most final products are unlinked circles

If O+P and O+R are assumed rational or sums of rational tangles, then the tangle equations can be solved.

Ernst and Sumners (1990, 1999), Ernst (1998)


Theorem [Ishihara, Shimokawa, V.] Suppose the substrate is a 6-crossing torus link and the product

is the unlink, and that recombination is iterative. If P=(0) and R=(k), then O is rational.

If Xer recombination is iterative then the mechanism is unique

Shimokawa et al. (2009)

O=(6), P=(0) R=(-1)

=

Parallel 6-cat 51 knot


Let’s not assume that the process is iterative… Theorem [Ishihara, Shimokawa, V.]

If the substrate is a 6-crossing torus link and the product is a knot or link with 5 or less crossings then the only possible product is the 51 torus link.

Likewise for 5 to 4, 4 to 3, 3 to 2 etc…

Assuming that 6-cat goes to a knot with 5 or less crossings results in a

unique product topology

Parallel 6-cat 51 knot


There is a unique unlinking pathway by XerCD/FtsK

The XerCD/FtsK complex unlinks DNA by a stepwise mechanism, taking the 6 torus link into a 5 torus knot, the 5 into a 4, 4 into 3 etc… until the substrate is unlinked.

Grainge et al. (2007); Shimokawa and Vazquez (2010); Ishihara, Shimokawa and Vazquez, preprint

Grainge et al. (2007) August 8, 2012 SIAM Life Sciences 2012 60

If we assume rationality, there are three mechanisms of unlinking the

6-cat into a 5-knot

Assume: P=(0)

O=(-5,-1,-1,0) R=(+1)

O=(-5,-1) R=(0,0)

O=(6), P=(0) R=(-1)


Conclusions XerCD at psi convert unknots into 4cats in a unique manner

Vazquez et al. (2005) XerCD-FtsK at dif unlink replication catenanes in a stepwise manner, taking (2n)-torus link to (2n-1)-torus knot, to (2n-2) link etc…

This pathway is consistent with a unique topological mechanism of action Grainge et al. (2007) Shimokawa and Vazquez (2010)

On-going work •  Analyze other possible topological pathways from

T(2,2n) to unlink

•  Study the frequency of the pathways numerically: Recombo


Acknowledgements


De Witt Sumners (FSU)

David J. Sherratt (Oxford U., UK) Sean Colloms (Glasgow U., UK) Ian Grainge (U. of Newcastle, Australia)

K. Shimokawa (Saitama U., Japan) Kai Ishihara (Imperial College, UK)

Rob Scharein (SFSU)

Students: Yuki Saka (UC Berkeley) Wenjing Zheng (UC Berkeley) Jennifer Lopez (SFSU) Masaaki Yoshida (Saitama U.)

NSF Math Biology

NSF CAREER Award

2012 PECASE Award

NIH RIMI

DNA unknotting and unlinking · 8/8/2012 · (DNA knots or links, mainly UNLINKS) August 8, 2012...

Documents

Transcript of DNA unknotting and unlinking · 8/8/2012 · (DNA knots or links, mainly UNLINKS) August 8, 2012...