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 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

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

Page 3 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

The Median Problem

◮ Given gene orders π1,π2,π3 ◮ Find M where ∑3

i=1 d(πi,M) is minimized

π3

d(π3,M) d(π2,M) d(π1,M)

π2 M π1

Page 3 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

The Median Problem

◮ Given gene orders π1,π2,π3 ◮ Find M where ∑3

i=1 d(πi,M) is minimized

π3

d(π3,M) d(π2,M) d(π1,M)

π2 M π1

◮ NP-hard even for the most simple distance measures

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)

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

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

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:

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

Page 6 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

Cycles and distances

◮ Edges of two colors form cycles

π1 = (− → 1 ← − 3 − → 5 ← − 4 − → 2 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

Page 6 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

Cycles and distances

◮ Edges of two colors form cycles

π2 = (− → 1 ← − 3 ← − 4 − → 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

Page 6 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

Cycles and distances

◮ Edges of two colors form cycles

π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

Page 6 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

Cycles and distances

◮ Edges of two colors form cycles

π3 = (− → 1 ← − 4 ← − 3 ← − 2 − → 5 ) M = (− → 1 ← − 3 ← − 2 − → 4 − → 5 )

1 5 4 3 2

1h 4h 5t 3t 4t 5h 1t 2h 2t 3h

◮ Distances closely related to number of cycles

dr = n −c +h +f dt ≥ n−codd

2

dw ≥ wt

2 (n −codd −(2− 2wr wt )ceven)

Page 7 On Reversal and Transposition Medians | Martin Bader | June 25, 2009

Sketch of the algorithm

◮ Solve Cycle Median Problem ◮ Verify solution

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

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)

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 π1, π2, and π3

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 π1, π2, and π3

◮ 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)

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 http://www.uni-ulm.de/fileadmin/website_uni_ulm/iui. inst.190/Mitarbeiter/bader/phylo-1.0.1.tar.gz

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 http://www.uni-ulm.de/fileadmin/website_uni_ulm/iui. inst.190/Mitarbeiter/bader/phylo-1.0.1.tar.gz

Thanks for your attention!