the xercd-ftsk system unlinks replication …ichihara/knots2010/slides/...the xercd-ftsk system...

The XerCD-FtsK system unlinks replicationcatenanes in a stepwise manner

Koya Shimokawa, Kai Ishihara, Ian Grainge, David J. Sherratt, andMariel Vazquez

Department of Mathematics, Saitama University

結び目の数学 III2010年 12月 22日

Koya Shimokawa (Saitama University) Xer system unlinking 2010 年 12 月 22 日 1 / 36

§1 Introduction

Wasserman, Dungan, Cozaarelli, Science (1985)


Application of knot theory to site-specific recombination

.Idea [Ernst-Sumners, 1990].... ..

.

.

Site-specific recombination may change topology or geometry of DNA.

⇒ ⇒

Substrate recombination Product

D.W.Sumners, Notices of AMS, 42 (1995)

Characterization of the mechanism of some enzymatic actions are givenusing knot theory.


Tangle model of recombinationRecombination of two DNA strands can be considered as a tangle surgery.[Ernst-Sumners, 1990]

Substrate

Product

O = Of + Ob


Problem and assumption

.Problem..

.

. ..

.

.

Given substrate and product knots and links, characterize tangles O, Pand R.

Substrate

Product

.Biological Assumption..

.

. ..

.

.

We can assume P and R arerational tangles.(Sites are short.)


Known mathematical results

Tn3 resolvaseErnst-Sumners, Math. Proc. Camb. Phil. Soc.(1990)

Xer-psi (trivial knot → 4-cat)Vazquez-Colloms-Sumners, J. Mol. Biol. (2005)

Darcy, J. Knot Ramifications (2001),

GinVazquez-Sumners, Math. Proc. Camb. Phil. Soc. (2005)

General resultsBuck-Marcotte, Math. Proc. Camb. Phil. Soc. (2005), J. Knot

Ramifications (2007)

Buck-Flapan, J. Phys. A. : Math. Theor. (2007)


Known mathematical results

Xer-psi (6-cat → 7 crossing knot)Darcy-Ishihara-Medikonduri-S, preprint

Vazquez, dissertation (2000)

Darcy, J. Knot Ramifications (2008),

Xer/FtsK unlinkingS-Ishihara-Vazquez Bussei Kenkyu (2009)


§2 Tangle analysis of DNA catenane unlinking byXer/FtsK system


Unlinking DNA catenanes by topoisomerase

Catenanes appear during replication of closed circular DNA

=replicatoin

closed circular DNA catenane

“DNA topology / A.D. Bates and A. Maxwell”

Topoisomerases unlink replication catenanes


Unlinking by Xer-dif-Ftsk system in E. coli.

Xer-dif-FtsK recombination can unlink replication catenanes (2m-cat)with parallel diff sites formed in vivo.

Grainge et al. EMBO J. (2007)

.Problem.... ..

.

.Characterize this recombination.


Stepwise unlinking model [EMBO. J. 2007]

6-cat → 5-torus knot → 4-cat → · · ·→ trivial knot → trivial link

Grainge et al., EMBO J. 2007

Stepwise unlinking model is consistent with experimental data.


§2.1 Tangle model I

(iterative recombination model)


Iterative recombination

Xer recombination is non-processive.Here we introduce the iterative recombination model.


Unlinking by iterative recombination

.Problem 1..

.

. ..

.

.

P = (0), R = (k) (iterative recombination)



.Theorem [S-Ishihara-Vazquez, Bussei Kenkyu 2009]..

.

. ..

.

.

P = (0), R = (k) (iterative recombination)⇒ O is rational.



.Theorem [S-Ishihara-Vazquez, Bussei Kenkyu 2009]..

.

. ..

.

.


§2.2 Tangle model II

(reducing crossing number model)


Tangle equation 2

.Problem 2 (6-cat case)..

.

. ..

.

.

P = (0), R = (1b)


Rational tangle sugery and band surgery

1b-tangle surgery

band surgery

P = (0)R = (1

b)

Rational tangle surgeries has 2 classes:“Band surgery” and “Non-band surgery”


Unlinking by Xer/FtsK system

.Theorem (6-cat case) [S-Ishihara-Vazquez, 2010]..

.

. ..

.

.

P = (0), R = (1b)


Tangle equation 2

.Problem 2 (general case)..

.

. ..

.

.

P = (0), R = (1b), n > 0



.Theorem [S-Ishihara-Vazquez, 2010]..

.

. ..

.

.

P = (0), R = (1b), n > 0


Tangle equation 2’

.Problem 2’ (general case)..

.

. ..

.

.

P = (0), R = (1b), n > 0



.Theorem [S-Ishihara-Vazquez, 2010]..

.

. ..

.

.

P = (0), R = (1b), n > 0


Unique pathway

.Theorem..

.

. ..

.

.

Suppose recombinations are band surgeries. If the recombination fromparallel 2m-cat reduces the crossing number in each step,

T (2, 2m) → T (2, 2m-1) → · · · → trivial knot → trivial link

is the only pathway.

Figure: 6-cat to unlink


Proof

.Proposition..

.

. ..

.

.

|σ(L)| ≤ c(L) − 1

|σ(L)| = c(L) − 1 ⇐⇒ L = T (2, c(L))

.Theorem [Murasugi, 1965]..

.

. ..

.

.

L : substrateLb : productSuppose band surgery is coherent.⇒ |σ(L) − σ(Lb)| ≤ 1


Next problem

.Problem.... ..

.

.Characterize each recombination.


Characterization of band surgeries

.Theorem [Darcy-Ishihara-Medikonduri-S, preprint]..

.

. ..

.

.

Substrate : trefoil knotProduct : Hopf linkSuppose the recombination is a band surgery=⇒ Band surgery is unique up to isotopy.

=⇒ =⇒



.Theorem [Darcy-Ishihara-Mediconduri-S]..

.

. ..

.

.

Substrate : N(4mn−12m

) = b(2m, 2n)Product : (2, 2k)-torus link (k 6= ±2)Band surgery from b(2m, 2n) to (2, 2k)-torus link (k 6= ±2) is one ofthe followings

b(2m, 2n) (2, 2k)-torus link



.Theorem [Darcy-Ishihara-Mediconduri-S]..

.

. ..

.

.

2n

2m

2n

2m

2m

2n 2n

2m


Band surgery and minimal genus Seifert surface

.Theorem [Hirasawa-S, 2000]..

.

. ..

.

.

Suppose a coherent band surgery on L along b yields Lb.

χ(L) ≤ χ(Lb) − 1⇔∃F : minimal genus Seifert surface for L s.t. b ⊂ F



.Theorem [Thompson, 1989], [Hirasawa-S, 2000]..

.

. ..

.

.

Substrate : Hopf linkProduct : trivial knotSuppose the recombination is a band surgery=⇒ Band surgery is unique up to isotopy.

=⇒ =⇒



.Theorem [Scharlemann, 1985]..

.

. ..

.

.

Substrate : trivial knotProduct : trivial linkSuppose the recombination is a band surgery=⇒ Band surgery is unique up to isotopy.

=⇒ =⇒


Characterization of recombinations

[DIMS] [T][HS]

[Sc]


Conclusion

If we assume...1 each recombination is a band surgery...2 each recombination reduces the crosssing number

then we can conclude that

XerCD-FtsK system unlinks replication link in a stepwise manner

and several recombinations are characterized.


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