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

§ 1 Introduction Wasserman, Dungan, Cozaarelli, Science (1985) 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 2 / 36

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

�� �� � �� �� � Tangle model of recombination Recombination of two DNA strands can be considered as a tangle surgery. [Ernst-Sumners, 1990] Substrate Product O = O f + O b 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 4 / 36

� � � � Tangle model of recombination Recombination of two DNA strands can be considered as a tangle surgery. [Ernst-Sumners, 1990] Substrate Product O = O f + O b 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 4 / 36

� � � � Problem and assumption . Problem . . . Given substrate and product knots and links, characterize tangles O , P and R . . . . . . Substrate . Biological Assumption . . . We can assume P and R are rational tangles. (Sites are short.) . . . . . Product 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 5 / 36

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

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

§ 2 Tangle analysis of DNA catenane unlinking by Xer/FtsK system 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 7 / 36

= replicatoin Unlinking DNA catenanes by topoisomerase Catenanes appear during replication of closed circular DNA closed circular DNA catenane “DNA topology / A.D. Bates and A. Maxwell” Topoisomerases unlink replication catenanes 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 8 / 36

Unlinking by Xer- dif -Ftsk system in E. coli. Xer- dif -FtsK recombination can unlink replication catenanes ( 2 m -cat) with parallel diff sites formed in vivo . Grainge et al. EMBO J. (2007) . Problem . . . Characterize this recombination. . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 9 / 36

� Unlinking by Xer- dif -Ftsk system in E. coli. 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 10 / 36

� � Unlinking by Xer- dif -Ftsk system in E. coli. 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 11 / 36

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

§ 2 . 1 Tangle model I (iterative recombination model) 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 13 / 36

Iterative recombination Xer recombination is non-processive. Here we introduce the iterative recombination model . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 14 / 36

� � � � Unlinking by iterative recombination . Problem 1 . . . P = (0) , R = ( k ) (iterative recombination) . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 15 / 36

� � � � Unlinking by iterative recombination . Theorem [S-Ishihara-Vazquez, Bussei Kenkyu 2009] . . . P = (0) , R = ( k ) (iterative recombination) ⇒ O is rational. . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 16 / 36

� � Unlinking by iterative recombination . Theorem [S-Ishihara-Vazquez, Bussei Kenkyu 2009] . . . . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 17 / 36

§ 2 . 2 Tangle model II (reducing crossing number model) 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 18 / 36

� � ������ � � ����������������������� ����� � � Tangle equation 2 . Problem 2 ( 6 -cat case) . . . P = (0) , R = ( 1 b ) . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 19 / 36

Rational tangle sugery and band surgery 1 b -tangle surgery P = (0) R = ( 1 b ) band surgery Rational tangle surgeries has 2 classes: “Band surgery” and “Non-band surgery” 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 20 / 36

� � ������ ������ � � ���������� ���������� ����� � � Unlinking by Xer/FtsK system . Theorem ( 6 -cat case) [S-Ishihara-Vazquez, 2010] . . . P = (0) , R = ( 1 b ) . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 21 / 36

� � � � ������� �������������������������� ������ � � Tangle equation 2 . Problem 2 (general case) . . . P = (0) , R = ( 1 b ) , n > 0 . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 22 / 36

� � ������� ��������� � � ���������� ������������� ������ � � Unlinking by iterative recombination . Theorem [S-Ishihara-Vazquez, 2010] . . . P = (0) , R = ( 1 b ) , n > 0 . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 23 / 36

� � � � ������������������������ ��������� � � Tangle equation 2’ . Problem 2’ (general case) . . . P = (0) , R = ( 1 b ) , n > 0 . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 24 / 36

� � ������� ��������� � � ������ � � Unlinking by iterative recombination . Theorem [S-Ishihara-Vazquez, 2010] . . . P = (0) , R = ( 1 b ) , n > 0 . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 25 / 36

Unique pathway . Theorem . . . Suppose recombinations are band surgeries. If the recombination from parallel 2 m -cat reduces the crossing number in each step, T (2 , 2 m ) → T (2 , 2 m - 1) → · · · → trivial knot → trivial link is the only pathway. . . . . . Figure: 6 -cat to unlink 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 26 / 36

Proof . Proposition . . . | σ ( L ) | ≤ c ( L ) − 1 | σ ( L ) | = c ( L ) − 1 ⇐ ⇒ L = T (2 , c ( L )) . . . . . . Theorem [Murasugi, 1965] . . . L : substrate L b : product Suppose band surgery is coherent. ⇒ | σ ( L ) − σ ( L b ) | ≤ 1 . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 27 / 36

Next problem . Problem . . . . Characterize each recombination. . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 28 / 36

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. = ⇒ = ⇒ . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 29 / 36

Characterization of band surgeries . Theorem [Darcy-Ishihara-Mediconduri-S] . . . Substrate : N ( 4 mn − 1 ) = b (2 m, 2 n ) 2 m Product : (2 , 2 k ) -torus link ( k � = ± 2) Band surgery from b (2 m, 2 n ) to (2 , 2 k ) -torus link ( k � = ± 2) is one of the followings b (2 m, 2 n ) (2 , 2 k ) -torus link . . . . . 2010 年 12 月 22 日 Koya Shimokawa (Saitama University) Xer system unlinking 30 / 36