Martin Bader June 25, On Reversal and Transposition Medians

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1 Martin Bader June 25, 2009 On Reversal and Transposition Medians

2 Page 2 On Reversal and Transposition Medians Martin Bader June 25, 2009 Genome Rearrangements During evolution, the gene order in a chromosome can change Gene order of two land snail mitochondrial DNAs Cepaea nemoralis cox1 V rrnl L1 A nad6 P nad5 nad1 nad4 L cob D C F cox2 Y W G H Q L2 atp8 N atp6 R E rrns M nad3 S2 T cox3 S1 nad4 I nad2 K Albinaria coerulea cox1 V rrnl L1 P A nad6 nad5 nad1 nad4 L cob D C F cox2 Y W G H Q L2 atp8 N atp6 R E rrns M nad3 S2 S1 nad4 T cox3 I nad2 K

3 Page 2 On Reversal and Transposition Medians Martin Bader June 25, 2009 Genome Rearrangements During evolution, the gene order in a chromosome can change Gene order of two land snail mitochondrial DNAs Cepaea nemoralis cox1 V rrnl L1 A nad6 P nad5 nad1 nad4 L cob D C F cox2 Y W G H Q L2 atp8 N atp6 R E rrns M nad3 S2 T cox3 S1 nad4 I nad2 K Albinaria coerulea cox1 V rrnl L1 P A nad6 nad5 nad1 nad4 L cob D C F cox2 Y W G H Q L2 atp8 N atp6 R E rrns M nad3 S2 S1 nad4 T cox3 I nad2 K Reconstruct evolutionary events Use as distance measure Use for phylogenetic reconstruction

4 Page 3 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Median Problem Given gene orders π 1,π 2,π 3 Find M where 3 i=1 d(πi,m) is minimized π 1 d(π 1,M) M d(π 2,M) π 2 d(π 3,M) π 3

5 Page 3 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Median Problem Given gene orders π 1,π 2,π 3 Find M where 3 i=1 d(πi,m) is minimized π 1 d(π 1,M) M d(π 2,M) π 2 d(π 3,M) π 3 NP-hard even for the most simple distance measures

6 Page 4 On Reversal and Transposition Medians Martin Bader June 25, 2009 Our contribution Exact algorithms for the Transposition Median Problem Exact algorithm for the weighted Reversal and Transposition Median Problem (Extension of Reversal Median solver, Caprara 2003) Improved exact algorithm for pairwise distances (Improvement of Christie 1998)

7 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order

8 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

9 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

10 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

11 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

12 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

13 Page 5 On Reversal and Transposition Medians Martin Bader June 25, 2009 The Multiple Breakpoint Graph Edge-colored graph Contains neighborhood relations for each gene order Example: 1 h 5 t 1 5 π 1 = ( ) π 2 = ( ) π 3 = ( ) M = ( ) 1 t 2 h 2 2 t 5 h 4 t 4 h 4 3 h 3 3 t

14 Page 6 On Reversal and Transposition Medians Martin Bader June 25, 2009 Cycles and distances Edges of two colors form cycles 1 h 5t 1 5 π 1 = ( ) M = ( ) 1t 2 h 2 2t 5 h 4t 4 h 4 3 h 3 3t

15 Page 6 On Reversal and Transposition Medians Martin Bader June 25, 2009 Cycles and distances Edges of two colors form cycles 1 h 5t 1 5 π 2 = ( ) M = ( ) 1t 2 h 2 2t 5 h 4t 4 h 4 3 h 3 3t

16 Page 6 On Reversal and Transposition Medians Martin Bader June 25, 2009 Cycles and distances Edges of two colors form cycles 1 h 5t 1 5 π 3 = ( ) M = ( ) 1t 2 h 2 2t 5 h 4t 4 h 4 3 h 3 3t

17 Page 6 On Reversal and Transposition Medians Martin Bader June 25, 2009 Cycles and distances Edges of two colors form cycles 1 h 5t 1 5 π 3 = ( ) M = ( ) 1t 2 h 2 2t 5 h 4t 4 h 4 3t 3 Distances closely related to number of cycles 3 h d r = n c + h + f d t n c odd 2 d w w t 2 (n c odd (2 2w r w t )c even )

18 Page 7 On Reversal and Transposition Medians Martin Bader June 25, 2009 Sketch of the algorithm Solve Cycle Median Problem Verify solution

19 Page 7 On Reversal and Transposition Medians Martin Bader June 25, 2009 Sketch of the algorithm Solve Cycle Median Problem Start with empty M Subsequently add edges Estimate lower bound for partial solution Continue with partial solution with least lower bound (branch and bound) NEW: Consider cycle lengths Verify solution

20 Page 7 On Reversal and Transposition Medians Martin Bader June 25, 2009 Sketch of the algorithm Solve Cycle Median Problem Start with empty M Subsequently add edges Estimate lower bound for partial solution Continue with partial solution with least lower bound (branch and bound) NEW: Consider cycle lengths Verify solution Calculate edge weights... either by an exact algorithm for pairwise distances... or by an approximation algorithm (faster)

21 Page 8 On Reversal and Transposition Medians Martin Bader June 25, 2009 Experiments Create random input Start with id of size n (n = 37 and n = 100) Create 3 sequences of operations of length r (2 r 15) Use these sequences to obtain π 1, π 2, and π 3

22 Page 8 On Reversal and Transposition Medians Martin Bader June 25, 2009 Experiments Create random input Start with id of size n (n = 37 and n = 100) Create 3 sequences of operations of length r (2 r 15) Use these sequences to obtain π 1, π 2, and π 3 Testing Most inputs could be solved within a few seconds Verifying solutions with approximation algorithm is very accurate Much faster than previous algorithm for the Transposition Median Problem (Yue et al. 2008)

23 Page 9 On Reversal and Transposition Medians Martin Bader June 25, 2009 Conclusion We presented an algorithm that... can solve the TMP and wrtmp exactly is fast enough for practical use is FREE SOFTWARE (GPL v3.0) download it from inst.190/mitarbeiter/bader/phylo tar.gz

24 Page 9 On Reversal and Transposition Medians Martin Bader June 25, 2009 Conclusion We presented an algorithm that... can solve the TMP and wrtmp exactly is fast enough for practical use is FREE SOFTWARE (GPL v3.0) download it from inst.190/mitarbeiter/bader/phylo tar.gz Thanks for your attention!

On Reversal and Transposition Medians

On Reversal and Transposition Medians Martin Bader International Science Index, Computer and Information Engineering waset.org/publication/7246 Abstract During the last years, the genomes of more and more