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

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 − cox 1 − − − → → V − rrnL − − → → L 1 − → A − nad 6 − − − → → P − nad 5 − − − → nad 1 − − − → nad 4 − − − → → L − cob − − → → D − → C − → F − cox 2 − − − → → Y − → W − → G − → H ← Q ← − L 2 ← − atp 8 ← − − N ← − atp 6 ← − − R ← − E ← − rrnS ← − − M ← − nad 3 ← − − − S 2 ← − − T ← cox 3 − − − − S 1 − → nad 4 − − − → → I − nad 2 − − − → → K Albinaria coerulea − cox 1 − − − → → V − rrnL − − → → L 1 − → P − → A − nad 6 − − − → nad 5 − − − → nad 1 − − − → nad 4 − − − → → L − cob − − → → D − → C − → F − cox 2 − − − → Y − → → W − → G − → H ← Q ← − L 2 ← − atp 8 ← − − N ← − atp 6 ← − − R ← − E ← − rrnS ← − − M ← − nad 3 ← − − − S 2 − − S 1 − → nad 4 ← − − → T ← − cox 3 − − − − → I − nad 2 − − − → → K

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 − cox 1 − − − → → V − rrnL − − → → L 1 − → A − nad 6 − − − → → P − nad 5 − − − → nad 1 − − − → nad 4 − − − → → L − cob − − → → D − → C − → F − cox 2 − − − → → Y − W − → → G − → H ← Q ← − L 2 ← − atp 8 ← − − N ← − atp 6 ← − − R ← − E ← − rrnS ← − − M ← − nad 3 ← − − − S 2 ← − − T ← cox 3 − − − − S 1 − → nad 4 − − − → → I − nad 2 − − − → → K Albinaria coerulea − cox 1 − − − → → V − rrnL − − → L 1 − → → P − → A − nad 6 − − − → nad 5 − − − → nad 1 − − − → nad 4 − − − → → L − cob − − → → D − → C − → F − cox 2 − − − → → Y − → W − G − → → H ← Q ← − L 2 ← − atp 8 ← − − N ← − atp 6 ← − − R ← − E ← − rrnS ← − − M ← − nad 3 ← − − − S 2 − − S 1 − → nad 4 ← − − → − T ← cox 3 − − − − → I − nad 2 − − − → → K ◮ Reconstruct evolutionary events ◮ Use as distance measure ◮ Use for phylogenetic reconstruction

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 ( π i , M ) is minimized π 1 d ( π 1 , M ) M d ( π 2 , M ) d ( π 3 , M ) π 3 π 2

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 ( π i , M ) is minimized π 1 d ( π 1 , M ) M d ( π 2 , M ) d ( π 3 , M ) π 3 π 2 ◮ NP-hard even for the most simple distance measures

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)

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

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 = ( − → 1 ← 3 − − → 5 ← − 4 − → 1 t 5 h 2 ) π 2 = ( − → 1 ← 3 ← − − 4 − → 2 − → 5 ) 2 h 4 t π 3 = ( − → 1 ← 4 ← − − 3 ← − 2 − → 5 ) 2 4 M = ( − → 1 ← 3 ← − 2 − − → 4 − → 5 ) 4 h 2 t 3 h 3 t 3

On Reversal and Transposition Medians | Martin Bader | June 25, 2009 Cycles and distances ◮ Edges of two colors form cycles 1 h 5 t 1 5 1 t 5 h π 1 = ( − → 1 ← − 5 ← − 4 − → 3 − → 2 ) 2 h 4 t M = ( − → 1 ← 3 ← − − 2 − → 4 − → 5 ) 2 4 4 h 2 t 3 h 3 t 3

On Reversal and Transposition Medians | Martin Bader | June 25, 2009 Cycles and distances ◮ Edges of two colors form cycles 1 h 5 t 1 5 1 t 5 h π 2 = ( − → 1 ← 3 ← − 4 − − → 2 − → 5 ) 2 h 4 t M = ( − → 1 ← 3 ← − − 2 − → 4 − → 5 ) 2 4 4 h 2 t 3 h 3 t 3

On Reversal and Transposition Medians | Martin Bader | June 25, 2009 Cycles and distances ◮ Edges of two colors form cycles 1 h 5 t 1 5 1 t 5 h π 3 = ( − → 1 ← 4 ← − 3 ← − − 2 − → 5 ) 2 h 4 t M = ( − → 1 ← 3 ← − − 2 − → 4 − → 5 ) 2 4 4 h 2 t 3 h 3 t 3

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

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

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

On Reversal and Transposition Medians Martin Bader | June 25, 2009 - PowerPoint PPT Presentation

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

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