DNA unknotting and unlinking
Mariel Vazquez Mathematics Department
San Francisco State University [email protected]
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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
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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
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Postow et al., PNAS (2001)
Replication of circular DNA produces links
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Postow et al., PNAS (2001)
Replication links are removed by type II topoisomerases
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Type II topo
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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
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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.
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λ- Int from bacteriophage λ integrates the viral DNA into the bacterial host genome
λ- Int
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Aihara et al. (2003)
Healy and Elaydi
λ- Int from bacteriophage λ integrates the viral DNA into its bacterial host genome
λ- Int
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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
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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
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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
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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?
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The local action of recombination is relatively well understood
The local action of recombination is relatively well understood
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The global action is not…
The product topology is a direct consequence of the geometrical
conformation adopted by the substrate prior to recombination
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GOAL
Understand the topological mechanism of binding and strand-exchange
using geometry and topology
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The mathematics of site-specific recombination
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DNA is modeled as a curve
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Circular DNA molecule
Smooth embedding of a circle in 3-space
= KNOT
DNA knot mathematical knot
DNA is modeled as a curve
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Circular DNA molecule
Smooth embedding of a circle in 3-space
= KNOT
DNA knot mathematical knot
DNA is modeled as a curve
Site-specific recombination is modeled as a 2-step reaction
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Substrate Products (DNA knot or link) recombination
The synaptic complex appears as a ball with two emanating arcs
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Substrate Products (DNA knot or link) recombination
Enzyme = 3D ball
Ernst and Sumners (1990)
Enzyme + bound DNA = 2-string tangle
We use tangles to model the recombination reaction
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Substrate Products (DNA knot or link) recombination
We use tangles to model the recombination reaction
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Substrate Products (DNA knots or links) recombination
N(O+P) = substrate N(O+R) = product O, P and R are 2-string tangles
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• Initial configuration: DNA and enzyme(s) are fixed
• Recombination goes by tangle surgery
Tangle method assumptions
= N(E), where E = enzyme+bound DNA
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• 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
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N(O+P) = substrate N(O+R) = product
recombination
O, P and R are 2-string tangles
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Rational
Most tangles in biology are rational
31
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Rational
Rational tangles admit a classification
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Classification Theorem (Conway 1970): There is a 1-1 correspondence between rational tangles and the set .
∪ {∞}
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Rational
Rational tangle – Rational number
33
= (-5,-2)
Classification Theorem (Conway 1970): There is a 1-1 correspondence between rational tangles and the set .
∪ {∞}
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A and B rational N(A+B) is a 4-plat
Tangle equations
(-7)-twist knot
= =
⇒
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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|
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Tangle equations
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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
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Xer recombination at psi yields a unique product topology
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XerC/D
RH 4-cat Colloms et al., 1997
Xer recombination at psi yields a unique product topology
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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
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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
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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
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If P=(0) parallel
O = (-3,0) = and R= (-1) =
Results N(O+P) =
N(O+R) =
Pep A psi sites Arg R
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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) =
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Unique topological mechanism for Xer recombination at psi
Pep A psi sites Arg R
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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
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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
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TangleSolve
Saka and Vazquez, Bioinformatics (2002)
DNA unlinking by XerCD/FtsK at dif sites
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Postow et al., PNAS (2001)
Replication of circular DNA produces links
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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
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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
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Experimental Data
XerCD-FtsK
Replication links Products (DNA knots or links,
mainly UNLINKS)
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Replication links Products (DNA knots or links,
mainly UNLINKS)
recombination
XerCD / FtsK at dif
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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)
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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
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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
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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)
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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
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Acknowledgements
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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