Tangle analysis of difference topology experiments: Applications to a Mu protein-DNA complex

Darcy, Isabel¹ ; Luecke, John² ; Vazquez, Mariel³

¹Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
²Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA
³Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2247-2309

Résumé

We develop topological methods for analyzing difference topology experiments involving 3–string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number.

Détail

DOI : 10.2140/agt.2009.9.2247

Keywords: $3$–string tangle, DNA topology, tangle method, difference topology, Mu transpososome, graph planarity, Dehn surgery, handle addition lemma

Bibliographie

[1] C Alén, D Sherratt, S Colloms, Direct interaction of aminopeptidase A with recombination site DNA in Xer site-specific recombination, EMBO J. 16 (1997) 5188

[2] J Arsuaga, Y Diao, M Vazquez, Mathematical methods in DNA topology : Applications to chromosome organization and site-specific recombination, from: "Mathematics of DNA Structure, Function, and Interactions" (editors C J Benham, S Harvey, W K Olson, D W Sumners), IMA Vol. Math. Appl. 150, Springer (2009) 7

[3] T Baker, K Mizuuchi, DNA-promoted assembly of the active tetramer of the Mu transposase, Genes Dev. 6 (1992) 2221

[4] A Bates, A Maxwell, DNA Topology, Oxford Univ. Press (2005) 220

[5] W Bauer, F Crick, J White, Supercoiled DNA, Sci. Am. 243 (1980) 100

[6] S A Bleiler, R A Litherland, Lens spaces and Dehn surgery, Proc. Amer. Math. Soc. 107 (1989) 1127

[7] D Buck, C Verjovsky Marcotte, Tangle solutions for a family of DNA-rearranging proteins, Math. Proc. Cambridge Philos. Soc. 139 (2005) 59

[8] H Cabrera-Ibarra, On the classification of rational 3–tangles, J. Knot Theory Ramifications 12 (2003) 921

[9] H Cabrera-Ibarra, Results on the classification of rational 3–tangles, J. Knot Theory Ramifications 13 (2004) 175

[10] C Calladine, H Drew, B Luisi, A Travers, Understanding DNA, Academic Press (2004)

[11] A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275

[12] G Chaconas, R Harshey, Transposition of phage Mu DNA, from: "In Mobile DNA II" (editors H Craig, R Craigie, M Gellert, A Lambowitz), Amer. Soc. Microbiology (2002) 384

[13] S Colloms, J Bath, D Sherrat, Topological selectivity in Xer site-specific recombination, Cell 88 (1997) 855

[14] I K Darcy, Biological distances on DNA knots and links : applications to XER recombination, from: "Knots in Hellas ’98, Vol. 2 (Delphi)", J. Knot Theory Ramifications 10 (2001) 269

[15] I K Darcy, Modeling protein-DNA complexes with tangles, Comput. Math. Appl. 55 (2008) 924

[16] I K Darcy, J Chang, N Druivenga, C Mckinney, R Medikonduri, S Mills, J Navarra-Madsen, A Ponnusamy, J Sweet, T Thompson, Coloring the Mu transpososome, BMC Bioinformatics 7 (2006) 435

[17] I K Darcy, R G Scharein, TopoICE-R : 3D visualization modelling the topology of DNA recombination, Bioinformatics 22 (2006) 1790

[18] P L Deininger, J V Moran, M A Batzer, H H Kazazian, Jr., Mobile elements and mammalian genome evolution, Curr. Opin. Genet. Dev. 13 (2003) 651

[19] J Emert, C Ernst, N–string tangles, J. Knot Theory Ramifications 9 (2000) 987

[20] C Ernst, D W Sumners, A calculus for rational tangles : applications to DNA recombination, Math. Proc. Cambridge Philos. Soc. 108 (1990) 489

[21] C Ernst, D W Sumners, Solving tangle equations arising in a DNA recombination model, Math. Proc. Cambridge Philos. Soc. 126 (1999) 23

[22] C M Gordon, On primitive sets of loops in the boundary of a handlebody, Topology Appl. 27 (1987) 285

[23] I Grainge, M Bregu, M Vasquez, V Sibanathan, S Ip, D Sherrat, Unlinking chromosome catenates in vivo by site-specific recombination, EMBO J. 26 (2007) 4228

[24] I Grainge, D Buck, M Jayaram, Geometry of site alignment during Int family recombination : antiparallel synapsis by the Flp recombinase, J. Mol. Biol. 298 (2000) 749

[25] F Guo, D Gopaul, G Van Duyne, Structure of Cre recombinase complexed with DNA in a site-specific recombination synapse, Nature 389 (1997) 40

[26] R Harshey, M Jayaram, The Mu transpososome through a topological lens, Crit. Rev. Biochem. Mol. Biol. 41 (2006) 387

[27] M Hirasawa, K Shimokawa, Dehn surgeries on strongly invertible knots which yield lens spaces, Proc. Amer. Math. Soc. 128 (2000) 3445

[28] H Hoess, M Ziese, N Sternberg, P1 site-specific recombination : Nucleotide sequence of the recombining sites (dyad symmetry/lox site/BAL-31 deletion/plasmid recombination), Proc. Natl. Acad. Sci. USA 79 (1982) 3398

[29] W Jaco, Adding a 2–handle to a 3–manifold : an application to property R, Proc. Amer. Math. Soc. 92 (1984) 288

[30] M Jayaram, R Harshey, Difference topology : Analysis of high-order DNA-protein assemblies, from: "Mathematics of DNA Structure, Function, and Interactions" (editors C J Benham, S Harvey, W K Olson, D W Sumners, D Swigon), IMA Vol. Math. Appl. 150, Springer (2009) 139

[31] E Kilbride, M Boocock, W Stark, Topological selectivity of a hybrid site-specific recombination system with elements from Tn3 res/resolvase and bacteriophase PL lox P/Cre, J. Mol. Biol. 289 (1999) 1219

[32] S Kim, K Darcy Isabel, Topological analysis of DNA-protein complexes, from: "Mathematics of DNA Structure, Function, and Interactions" (editors C J Benham, S Harvey, W K Olson, D W Sumners, D Swigon), IMA Vol. Math. Appl. 150, Springer (2009) 177

[33] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007) 457

[34] W Li, S Kamtekar, Y Xiong, G Sarkis, N Grindley, T Steitz, Structure of a synaptic γδ resolvase tetramer covalently linked to two cleaved DNAs, Science 209 (2005) 1210

[35] W B R Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981) 321

[36] M Miziuuchi, T Baker, K Mizuuchi, Assembly of the active form of the transposase-Mu DNA complex : a critical control point in Mu transposition, Cell 70 (1992) 303

[37] K Mouw, S Rowland, M Gajjar, M Boocock, W Stark, P Rice, Architecture of a serine recombinase-DNA regulatory complex, Molecular Cell 30 (2008) 145

[38] S Pathania, M Jayaram, R Harshey, Path of DNA within the Mu Transpososome : Transposase interaction bridging two Mu ends and the enhancer trap five five DNA supercoils, Cell 109 (2002) 425

[39] S Pathania, M Jayaram, R Harshey, A unique right end-enhancer complex precedes synapsis of Mu ends : the enhancer is sequestered within the transpososome throughout transposition, EMBO J. 22 (2003) 3725

[40] N Robertson, P D Seymour, R Thomas, Linkless embeddings of graphs in 3–space, Bull. Amer. Math. Soc. (N.S.) 28 (1993) 84

[41] S Rowland, W Stark, M Boocock, Sin recombinase from Staphylococcus aureus : synaptic complex architecture and transposon targeting, Mol. Microbiol. 44 (2002) 607

[42] Y Saka, M Vazquez, Topological analysis of site-specific recombination, Bioinformatics 18 (2002) 1011

[43] D Sankoff, Rearrangements and chromosomal evolution, Curr. Opin. Genet. Dev. 13 (2003) 583

[44] M Scharlemann, Outermost forks and a theorem of Jaco, from: "Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982)" (editors J R Harper, R Richard Mandelbaum), Contemp. Math. 44, Amer. Math. Soc. (1985) 189

[45] M Scharlemann, A Thompson, Detecting unknotted graphs in 3–space, J. Differential Geom. 34 (1991) 539

[46] W Stark, D Sherratt, M Boocock, Site-specific recombination by Tn3 resolvase : topological changes in the forward and reverse reactions, Cell 58 (1989) 779

[47] D Sumners, Lifting the curtain: using topology to probe the hidden action of enzymes, Notices Amer. Math. Soc. 42 (1995) 528

[48] D Sumners, C Ernst, N Cozzarelli, S Spengler, Mathematical analysis of the mechanisms of DNA recombination using tangles, Quarterly Reviews of Biophysics 28 (1995) 253

[49] A Thompson, A polynomial invariant of graphs in 3–manifolds, Topology 31 (1992) 657

[50] M Vazquez, S Colloms, D Sumners, Tangle analysis of Xer recombination reveals only three solutions, all consistent with a single three-dimensional topological pathway, J. Mol. Biol. 346 (2005) 493

[51] M Vazquez, D Sumners, Tangle analysis of Gin site-specific recombination, Math. Proc. Cambridge Philos. Soc. 136 (2004) 565

[52] A Vetcher, A Lushnikov, J Navarra-Madsen, R Scharein, Y Lyubchenko, I K Darcy, S Levene, DNA topology and geometry in Flp and Cre recombination, J. Mol. Biol. 357 (2006) 1089

[53] C Wang, Helical repeat of DNA in Solution, Proc. Natl. Acad. Sci. USA 76 (1979) 200

[54] S Wasserman, J Dungan, N Cozzarelli, Discovery of a predicted DNA knot substantiates a model for site-specific recombination, Science 229 (1985) 171

[55] J Watson, F Crick, The structure of DNA, Cold Spring Harb. Symp. Quant. Biol. 18 (1953) 123

[56] Y Q Wu, A generalization of the handle addition theorem, Proc. Amer. Math. Soc. 114 (1992) 237

[57] Y Q Wu, On planarity of graphs in 3–manifolds, Comment. Math. Helv. 67 (1992) 635

[58] Y Q Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996) 457

[59] Z Yin, M Jayaram, S Pathania, R Harshey, The Mu transposase interwraps distant DNA sites within a functional transposome in the absence of DNA supercoiling, J. Biol. Chem. 280 (2005) 6149

[60] Z Yin, A Suzuki, Z Lou, M Jayaram, Interactions of phage Mu enhancer and termini that specify the assembly of a topologically unique interwrapped transpososome, J. Mol. Biol. 372 (2007) 382

[61] W Zheng, C Galloy, B Hallet, M Vasquez, The tangle model for site-specific recombination : a computer interface and the TnpI-IRS recombination system, from: "Knot Theory for Scientific Objects (Proc. of Intl. Workshop, Osaka, 2006)" (editor K Kawauchi), Ocami Studies 1, Osaka Municipal Univ. Press (2007) 251