The XerCD-FtsK system unlinks replication catenanes in a stepwise - - PowerPoint PPT Presentation

The XerCD-FtsK system unlinks replication catenanes in a stepwise manner

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

Department of Mathematics, Saitama University

結び目の数学 III 2010 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)

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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 given using knot theory.

Koya Shimokawa (Saitama University) Xer system unlinking 2010 年 12 月 22 日 3 / 36

Tangle model of recombination

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

• Substrate

Product O = Of + Ob

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Tangle model of recombination

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

• Substrate

Product O = Of + Ob

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Problem and assumption

.

Problem

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

• Substrate

Product

.

Biological Assumption

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

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Known mathematical results

Tn3 resolvase Ernst-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), Gin Vazquez-Sumners, Math. Proc. Camb. Phil. Soc. (2005) General results Buck-Marcotte, Math. Proc. Camb. Phil. Soc. (2005), J. Knot

Ramifications (2007)

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

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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 unlinking S-Ishihara-Vazquez Bussei Kenkyu (2009)

Koya Shimokawa (Saitama University) Xer system unlinking 2010 年 12 月 22 日 6 / 36

§2 Tangle analysis of DNA catenane unlinking by Xer/FtsK system

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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

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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.

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

• Koya Shimokawa (Saitama University)

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

• Koya Shimokawa (Saitama University)

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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.

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§2.1 Tangle model I (iterative recombination model)

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Iterative recombination

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

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Unlinking by iterative recombination

.

Problem 1

. . . . . . . .

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

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Unlinking by iterative recombination

.

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

. . . . . . . .

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

⇒ O is rational.

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Unlinking by iterative recombination

.

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

. . . . . . . .

• Koya Shimokawa (Saitama University)

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§2.2 Tangle model II (reducing crossing number model)

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Tangle equation 2

.

Problem 2 (6-cat case)

. . . . . . . .

• P = (0), R = ( 1

b)

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Rational tangle sugery and band surgery

1 b-tangle surgery

band surgery P = (0) R = ( 1

b)

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

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Unlinking by Xer/FtsK system

.

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

. . . . . . . .

• P = (0), R = ( 1

b)

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Tangle equation 2

.

Problem 2 (general case)

. . . . . . . .

• P = (0), R = ( 1

b), n > 0

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Unlinking by iterative recombination

.

Theorem [S-Ishihara-Vazquez, 2010]

. . . . . . . .

• P = (0), R = ( 1

b), n > 0

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Tangle equation 2’

.

Problem 2’ (general case)

. . . . . . . .

• P = (0), R = ( 1

b), n > 0

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Unlinking by iterative recombination

.

Theorem [S-Ishihara-Vazquez, 2010]

. . . . . . . .

• P = (0), R = ( 1

b), n > 0

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Unique pathway

.

Theorem

. . . . . . . . Suppose recombinations are band surgeries. If the recombination from parallel 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

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Proof

.

Proposition

. . . . . . . . |σ(L)| ≤ c(L) − 1 |σ(L)| = c(L) − 1 ⇐ ⇒ L = T (2, c(L)) .

Theorem [Murasugi, 1965]

. . . . . . . . L : substrate Lb : product Suppose band surgery is coherent. ⇒ |σ(L) − σ(Lb)| ≤ 1

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Next problem

.

Problem

. . . . . . . . Characterize each recombination.

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Characterization of band surgeries

.

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

. . . . . . . . Substrate : trefoil knot Product : Hopf link Suppose the recombination is a band surgery = ⇒ Band surgery is unique up to isotopy. = ⇒ = ⇒

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Characterization of band surgeries

.

Theorem [Darcy-Ishihara-Mediconduri-S]

. . . . . . . . Substrate : N(4mn−1

2m

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

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Characterization of band surgeries

.

Theorem [Darcy-Ishihara-Mediconduri-S]

. . . . . . . .

2n 2m 2n 2m 2m 2n 2n 2m

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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

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Characterization of band surgeries

.

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

. . . . . . . . Substrate : Hopf link Product : trivial knot Suppose the recombination is a band surgery = ⇒ Band surgery is unique up to isotopy. = ⇒ = ⇒

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Characterization of band surgeries

.

Theorem [Scharlemann, 1985]

. . . . . . . . Substrate : trivial knot Product : trivial link Suppose the recombination is a band surgery = ⇒ Band surgery is unique up to isotopy. = ⇒ = ⇒

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Characterization of recombinations

• [DIMS]

[T] [HS] [Sc]

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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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