The breakpoint distance for signed sequences

Transcription

1 The breakpoint distance for signed sequences Guillaume Blin 1, Cedric Chauve 2 Guillaume Fertin 1 and 1 LINA, FRE CNRS LACIM et Département d'informatique, Université de Nantes, Université du Québec à Montréal 2 rue de la houssinière CP 8888, Succ. Centre-ville BP Nantes Cedex 3 H3C 3P8, Montréal (QC) FRANCE CANADA {blin,fertin}@lina.univ-nantes

2 Outline Minimum Breakpoint Matching Problem result

3 Given alphabet A (gene families) a genome is a sequence of signed elements (genes) of A cardinality of f A is the number of occurrences of f in genomes f A is trivial if card(f) 2, non-trivial otherwise G t b c c d e f g e h i j k l l l m o n o p c q r s a H a -c -b c d e i i f g e -h k l l m o o n -l -p -c r q -s t A={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t} card(i)=3, card(l)=5, card(c)=4

4 Given alphabet A (gene families) non-trivial segment : substring containing only non-trivial genes balanced genomes : for any gene family f, number occurrences in both genomes is equal. G t b c c d e f g e h i j k l l l m o n o p c q r s a H a -c -b c d e i i f g e -h k l l m o o n -l -p -c r q -s t A={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t}

5 Given G=g 1,...,g n and H=h 1,...,h m a gene matching M between G and H: maximal matching gene-to-gene s.t. for every pair (g i,h j ) M, g i and h j belong to the same family G t b c c d e f g e h i j k l l l m o n o p c q r s a H a -c -b c d e i i f g e -h k l l m o o n -l -p -c r q -s t A={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t}

6 Given G, H and a gene matching M, a breakpoint between G and H is: either g i or g i+1 M (deletion breakpoint) h j or h j+1 M (insertion breakpoint) G t b c c d e f g e h i j k l l l m o n o p c q r s a H a -c -b c d e i i f g e -h k l l m o o n -l -p -c r q -s t

7 Given G, H and a gene matching M, a breakpoint between G and H is: neither M(g i )=h j, M(g i+1 )=h k and h j =g i, h k =g i+1 and k=j+1 nor M(g i )=h j, M(g i+1 )=h k and h j =-g i, h k =-g i+1 and k=j-1 G t b c c d e f g e h i j k l l l m o n o p c q r s a H a -c -b c d e i i f g e -h k l l m o o n -l -p -c r q -s t

8 Breakpoint distance between G and H: Db(G,H,M): number of breakpoints between G and H with respect to M. Db(G,H): minimum of Db(G,H,M) among all matchings M. Minimum Breakpoint Matching: Given G and H Find the gene matching M between G and H such that Db(G,H,M)=Db(G,H).

9 Given two genomes G and H, computing Db(G,H) is NP-complete, even when the number of non-trivial gene families is equal to 1 Reduction from Minimum Bin Packing problem... K...

10 Given two genomes G and H, computing Db(G,H) is NP-complete, even when the number of non-trivial gene families is equal to 1 Reduction from Minimum Bin Packing problem K...

11 MBM(L,F): An integer P and two genomes G and H on a set of gene families with F non-trivial families and s.t. non-trivial segments contains at most L genes. We prove that even if F=1 then MBM(L,F) is NP-complete

12 α xxxxx A 1 xxxx A 2 xxx A 3 xx A 4 xxxxx A 5 xxxx A 6 x A 7 B 1 B 2 B 3 B 4 β G α B 1 xxxxxxxx B 2 xxxxxxxx B 3 xxxxxxxx B 4 A 1 A 2 A 3 A 4 A 5 A 6 A 7 β H

13 At least there are those Q breakpoints α xxxxx A 1 xxxx A 2 xxx A 3 xx A 4 xxxxx A 5 xxxx A 6 x A 7 B 1 B 2 B 3 B 4 β G α B 1 xxxxxxxx B 2 xxxxxxxx B 3 xxxxxxxx B 4 A 1 A 2 A 3 A 4 A 5 A 6 A 7 β H

14 α xxxxx A 1 xxxx A 2 xxx A 3 xx A 4 xxxxx A 5 xxxx A 6 x A 7 B 1 B 2 B 3 B 4 β G α B 1 xxxxxxxx B 2 xxxxxxxx B 3 xxxxxxxx B 4 A 1 A 2 A 3 A 4 A 5 A 6 A 7 β H if one set P=Q then problems are equivalent

15 Given two balanced genomes G and H, computing Db(G,H) is (L+1)- approximable, when the number of non-trivial gene families is equal to 1 and every non-trivial segment is of length at most L xxx...xx L

16 Duplicated non-trivial segements: asb in G and asb or bsa in H S is a non-trivial segment a and b are trivial genes There is an optimal matching that matches any duplicated segment of G to its copy in H. a xxxxx x xx b e x? a xxxxx x xx b e x? a xxxxxx xx b c x? a xxxxxx xx b c x?

17 Let S be a non-trivial segment of G that is not part of a duplicated segment There is at least one breakpoint involving a gene of S a xx... xxb a xx... xxb a xx... xxb a xx... xxb a xx... xxb a x... x b a xx... xxc c xx... xxb

18 Given Mopt, a matching M where each duplicated segment is paired with its copy induces at most L+1 time more breakpoints Same number of breakpoints Trivial genes Duplicated segment Non-trivial genes (not part of duplicated segment)

19 for general case Key idea: long segments that match in G and H are likely to belong to a minimum gene matching Branch and cut strategy Suffix tree

20 for general case R 1 R 2 R 3 R 4 R 5 G H ABCxxyDyExFyyGxxHIJ AxxxBxyDyyFyGxHIJCE xx is a suffix of S1 starting at position 2 -y-y is a suffix of S3 starting at position 1 S 1 S 2 S 3 S 4 S 5 x -x y -y T(H) x $ (0,1,1) x (0,1,2) $ (0,1,3)(0,5,1) (1,1,3)(1,5,1) (1,2,2) (0,3,2)(0,4,1) (0,2,2) (1,3,2)(1,4,1) y -x y -y -x $ $ $ $ $ $ $ $ (0,2,1) (1,1,2) (0,3,1) (1,3,1) (1,2,1) -x $ $ (1,1,1)

21 for general case Consider a partial matching M' of R1.. Rk[1..i-1] with Rk = l Try to extend M' Leave Rk[i] unmatched Match a prefix of Rk[i..l]

22 for general case R 1 R 2 R 3 R 4 R 5 R 1 R 2 R 3 R 4 R 5 G ABCxxyDyExFyyGxxHIJ G B m =12 ABCxxyDyExFyyGxxHIJ H AxxxBxyDyyFyGxHIJCE H AxxxBxyDyyFyGxHIJCE S 1 S 2 S 3 S 4 S 5 S 1 S 2 S 3 S 4 S 5 We look for an extension that does not increase the breakpoint distance of the best matching generated so far (we try any extension which respect this condition) We are provided with a complete matching (the best one for now), we apply the improvement to decrease the global number of breakpoints

23 for general case The theoritical complexity = brute force complexity Upper bound for b.d. Branch-and-cut strategy Better practical computation time Improvements Case where all unmatched are of size 1 -> no need to explore the solution space (minimum weighted bipartite matching)

24 Conclusion Consider breakpoint distance for signed permutations NP-Complete even in the case of simple instances L+1-approximable for the general case It is of interest to find a constant ratio for this problem which in theory does not increase brute-force complexity; but does it in practical.

25 Questions about The breakpoint distance for signed sequences Guillaume Blin 1, Cedric Chauve 2 Guillaume Fertin 1 and 1 LINA, FRE CNRS LACIM et Département d'informatique, Université de Nantes, Université du Québec à Montréal 2 rue de la houssinière CP 8888, Succ. Centre-ville BP Nantes Cedex 3 H3C 3P8, Montréal (QC) FRANCE CANADA {blin,fertin}@lina.univ-nantes