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On Computing the Breakpoint Reuse Rate in Rearrangement Scenarios

On Computing the Breakpoint Reuse Rate in Rearrangement Scenarios. Anne Bergeron Julia Mixtacki Jens Stoye. Paris, le 14 octobre 2008. 1. An inspiring figure Mammalian chromosome 17. 2. Classical view of breakpoint reuse rate Operations that make 2 cuts.

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On Computing the Breakpoint Reuse Rate in Rearrangement Scenarios

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  1. On Computing the Breakpoint Reuse Ratein Rearrangement Scenarios Anne Bergeron Julia Mixtacki Jens Stoye Paris, le 14 octobre 2008

  2. 1. An inspiring figure Mammalian chromosome 17 2. Classical view of breakpoint reuse rate Operations that make 2 cuts 3. New view of breakpoint reuse rate Operations that make 0 or 1 cut 4. Understanding reuse rate behavior The adjacency graph Looking for extreme scenarios 5. Conclusions and open problems

  3. Mammalian chromosome 17 Reconstructing the Genomic Architecture of Ancestral Mammals: Lessons From Human, Mouse, and Rat Genomes Guillaume Bourque, Pavel A. Pevzner and Glenn Tesler (2004)

  4. Mammalian chromosome 17 Reconstructing the Genomic Architecture of Ancestral Mammals: Lessons From Human, Mouse, and Rat Genomes Guillaume Bourque, Pavel A. Pevzner and Glenn Tesler (2004)

  5. Mammalian chromosome 17 1 4 3 2 Reconstructing the Genomic Architecture of Ancestral Mammals: Lessons From Human, Mouse, and Rat Genomes Guillaume Bourque, Pavel A. Pevzner and Glenn Tesler (2004)

  6. 1. An inspiring figure Mammalian chromosome 17 2. Classical view of breakpoint reuse rate Operations that make 2 cuts 3. New view of breakpoint reuse rate Operations that make 0 or 1 cut 4. Understanding reuse rate behavior The adjacency graph Looking for extreme scenarios 5. Conclusions and open problems

  7. Classical view of breakpoint reuse rate Note: the value of r lies between 1 and 2. 1 2 No reuse: each cut repairs one adjacency Max reuse: two cut repair one adjacency Let D be the minimum number of rearrangement operations that transform Genome A into Genome B, and b the number of adjacencies in Genome B that are not adjacencies in Genome A. Assuming that each operation makes 2 cuts, the breakpoint reuse rate is defined as: r = 2D / b

  8. Classical view of breakpoint reuse rate Let D be the minimum number of rearrangement operations that transform Genome A into Genome B, and b the number of adjacencies in Genome B that are not adjacencies in Genome A. Assuming that each operation makes 2 cuts, the breakpoint reuse rate is defined as: r = 2D / b What are the consequences of the 2 cuts assumption?

  9. Operations that make 2 cuts Translocation Inversion Inversion ... and do not change the number of chromosomes.

  10. Operations that make 2 cuts Fission Fusion Fission Fusion ... and change the number of chromosomes, always involve circular chromosomes.

  11. Classical view of breakpoint reuse rate Let D be the minimum number of rearrangement operations that transform Genome A into Genome B, and b the number of adjacencies in Genome B that are not adjacencies in Genome A. Assuming that each operation makes 2 cuts, the breakpoint reuse rate is defined as: r = 2D / b What are the consequences of this assumption? • If the genomes are linear, then they must have the same number of chromosomes. • The chromosomes must also be co-tailed.

  12. Classical view of breakpoint reuse rate Let D be the minimum number of rearrangement operations that transform Genome A into Genome B, and b the number of adjacencies in Genome B that are not adjacencies in Genome A. Assuming that each operation makes 2 cuts, the breakpoint reuse rate is defined as: r = 2D / b What is done in practice? • Empty chromosomes are added as necessary. • Caps are added to make the chromosomes co-tailed. (such as in GRIMM) • Chromosomes are ‘circularized’. (Alekseyev and Pevzner)

  13. Classical view of breakpoint reuse rate Cap 1 3 5 6 4 2 1 3 5 2 4 6 1 3 4 2 5 6 1 3 4 2 5 6 1 2 4 3 5 6 1 2 3 4 5 6 Genome A Genome B D = 5, Cuts = 10, b = 6 Breakpoint reuse rate = 10/6 = 1,66

  14. From 24 chromosomes To 21 chromosomes [Source: Linda Ashworth, LLNL] DOE Human Genome Program Report

  15. Classical view of breakpoint reuse rate Human and mouse whole genome comparison data (Pevzner et Tesler, 2003.): D = 246, Cuts = 492, b = 300 Breakpoint reuse rate = 492/300 = 1,64

  16. 1. An inspiring figure Mammalian chromosome 17 2. Classical view of breakpoint reuse rate Operations that make 2 cuts 3. New view of breakpoint reuse rate Operations that make 0 or 1 cut 4. Understanding reuse rate behavior The adjacency graph Looking for extreme scenarios 5. Conclusions and open problems

  17. Operations that make 0 or 1 cut Semi translocation Semi inversion Fission Fusion Linear chromosomes

  18. Operations that make 0 or 1 cut Fission Fusion Linearization Circularization Linear and circular chromosomes

  19. New view of breakpoint reuse rate Let D be the minimum number of rearrangement operations that transform Genome A into Genome B, and b the number of adjacencies and telomeres in Genome B that are not adjacencies or telomeres in Genome A. Given a rearrangement scenario that needs to cut Genome AC times, the breakpoint reuse rate is defined as: r = C / b Note: the value of r now lies between 0 and 2. 1 0 2 No reuse: each cut repairs one adjacency No cuts needed (fusions) Max reuse: two cut repair one adjacency

  20. New view of breakpoint reuse rate No cap 1 3 5 6 4 2 1 3 5 2 4 6 1 3 4 2 5 6 1 3 4 2 5 6 1 2 4 3 5 6 1 2 3 4 5 6 Genome A Genome B D = 5, Cuts = 9, b = 6 Breakpoint reuse rate = 9/6 = 1,50

  21. New view of breakpoint reuse rate No cap 1 3 5 6 4 2 1 2 4 6 5 3 1 2 3 5 6 4 1 2 3 5 4 6 1 2 3 4 5 6 1 2 3 4 5 6 Genome A Genome B D = 5, Cuts = 7, b = 6 Breakpoint reuse rate = 7/6 = 1,17

  22. New view of breakpoint reuse rate Maximum reuse With caps No caps Scenario 1 No caps Scenario 2 No reuse Is going from high breakpoint reuse to low breakpoint reuse an artifact of the particular example we constructed ?

  23. New view of breakpoint reuse rate Scenario 1 D = 246, Cuts = 453, b = 300 Breakpoint reuse rate = 453/300 = 1,51 Scenario 2 D = 246, Cuts = 267, b = 300 Breakpoint reuse rate = 267/300 = 0,89 Human and mouse whole genome comparison data (Pevzner et Tesler, 2003.):

  24. New view of breakpoint reuse rate Maximum reuse With caps No caps Scenario 1 No caps Scenario 2 No reuse The drop is even more pronounced with realistic data !

  25. 1. An inspiring figure Mammalian chromosome 17 2. Classical view of breakpoint reuse rate Operations that make 2 cuts 3. New view of breakpoint reuse rate Operations that make 0 or 1 cut 4. Understanding reuse rate behavior The adjacency graph Looking for extreme scenarios 5. Conclusions and open problems

  26. The adjacency graph 4 1 6 3 5 2 Genome A Genome B 1 2 3 4 5 6 • A BB-path.

  27. The adjacency graph 4 1 6 3 5 2 Genome A Genome B 1 2 3 4 5 6 • A BB-path. • A BB-path. • A cycle.

  28. The adjacency graph 4 1 6 3 5 2 Genome A Genome B 1 2 3 4 5 6 • A BB-path. • A cycle. • A BB-path. • A cycle. • An AB-path.

  29. The adjacency graph 4 1 6 3 5 2 Genome A Genome B 1 2 3 4 5 6 • A BB-path. • A cycle. • An AB-path. • A BB-path. • A cycle. • An AB-path. • An AA-path.

  30. The adjacency graph 4 1 6 3 5 2 Genome A Genome B 1 2 3 4 5 6 • A BB-path. • A cycle. • An AB-path. Another AB-path. • An AA-path. • A BB-path. • A cycle. • An AB-path. • An AA-path.

  31. Looking for extreme scenarios: cycles Genome A Genome B Cycles of length 2L always need L-1 DCJ operations to be sorted, thus have a constant contribution to the reuse rate. Long cycles yield high breakpoint reuse.

  32. Looking for extreme scenarios: AB-paths Genome A Genome B The A extremity of an AB-path always corresponds to a DCJ operation that makes 1 cut. The length of the AB-path is shortened by two.

  33. Looking for extreme scenarios: AB-paths Genome A Genome B Thus one cut repairs one adjacency, and the path can be sorted without breakpoint reuse.

  34. Looking for extreme scenarios: AA-paths Genome A Genome B Both extremities of an AA-path correspond to DCJ operations that make 1 cut. The length of the AA-path is shortened by two.

  35. Looking for extreme scenarios: AA-paths Genome A Genome B Thus one cut repairs one adjacency, and the path can be sorted without breakpoint reuse.

  36. Looking for extreme scenarios: BB-paths Genome A Genome B A BB-path can always be transformed into two AB-paths with a fission.

  37. Looking for extreme scenarios: BB-paths Genome A Genome B Thus the path can be sorted without breakpoint reuse.

  38. Statistics for the human mouse comparison 1 cycle of length 6 1 cycle of length 8 1 cycle of length 10 These cycles implies that 6 breakpoints are reused, to repair 12 adjacencies 24 cycles of length 4 12 AB-paths: lengths 3 to 51 12 AA-paths: lengths 2 to 46 15 BB-paths: lengths 2 to 22 The remaining 288 adjacencies and telomeres can be repaired without any breakpoint reuse.

  39. 1. An inspiring figure Mammalian chromosome 17 2. Classical view of breakpoint reuse rate Operations that make 2 cuts 3. New view of breakpoint reuse rate Operations that make 0 or 1 cut 4. Understanding reuse rate behavior The adjacency graph Looking for extreme scenarios 5. Conclusions and open problems

  40. Conclusions and open problems • The reuse rate depends in a crucial way on how genomes and telomeres are modeled. • Finding a scenario exhibiting a particular value of reuse rate seems easy, even with complex data. • However, we do not have an algorithm to do this if we insist that the intermediate genomes remain linear. • The transformation between mouse and man with minimal breakpoint reuse can be done with linear intermediate genomes...

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