Genome Rearrangements

1. Genome Rearrangements (Lecture for CS498-CXZ Algorithms in Bioinformatics) Dec. 6, 2005 ChengXiang Zhai Department of Computer Science University of Illinois, Urbana-Champaign

2. Outline • The Problem of Genome Rearrangements • Sorting By Reversals • Greedy Algorithm for Sorting by Reversals • Position-based • Breakpoint-based

3. Genome Rearrangements Example I:Turnip vs Cabbage • Although cabbages and turnips share a recent common ancestor, they look and taste different

4. Turnip vs Cabbage: Comparing Gene Sequences Yields No Evolutionary Information

5. Turnip vs Cabbage: Almost Identical mtDNA gene sequences • In 1980s Jeffrey Palmer studied evolution of plant organelles by comparing mitochondrial genomes of the cabbage and turnip • 99% similarity between genes • These surprisingly identical gene sequences differed in gene order • This study helped pave the way to analyzing genome rearrangements in molecular evolution

6. Turnip vs Cabbage: Different mtDNA Gene Order • Gene order comparison:

7. Turnip vs Cabbage: Different mtDNA Gene Order • Gene order comparison:

8. Turnip vs Cabbage: Different mtDNA Gene Order • Gene order comparison:

9. Turnip vs Cabbage: Different mtDNA Gene Order • Gene order comparison:

10. Turnip vs Cabbage: Different mtDNA Gene Order • Gene order comparison: Before After Evolution is manifested as the divergence in gene order

11. Transforming Cabbage into Turnip

12. Genome Rearrangements Example II:Human vs. Mouse Mouse (X chrom.) • What are the similarity blocks and how to find them? • What is the architecture of the ancestral genome? • What is the evolutionary scenario for transforming one genome into the other? Unknown ancestor~ 75 million years ago Human (X chrom.)

13. History of Chromosome X Rat Consortium, Nature, 2004

14. Mouse vs Human Genome • Humans and mice have similar genomes, but their genes are ordered differently • ~245 rearrangements • Reversal, fusion, fission, translocation • Reversal: flipping a block of genes within a genomic sequence

15. Types of Rearrangements Reversal 1 2 3 4 5 6 1 2 -5 -4 -3 6 Translocation 1 2 3 45 6 1 2 6 4 5 3 Chromosome 1: Chromosome 2: Fusion 1 2 3 4 5 6 Chromosome 1: Chromosome 2: 1 2 3 4 5 6 Fission

16. 1 2 3 9 10 8 4 7 5 6 Reversals • Blocks represent conserved genes. • In the course of evolution blocks 1,…,10 could be misread as 1, 2, 3, -8, -7, -6, -5, -4, 9, 10. • Rearrangements occurred about once-twice every million years on the evolutionary path between human and mouse. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

17. Reversals 1 2 3 9 10 8 4 7 5 6 1, 2, 3, -8, -7, -6, -5, -4, 9, 10 • Blocks represent conserved genes. • In the course of evolution or in a clinical context, blocks 1,…,10 could be misread as 1, 2, 3, -8, -7, -6, -5, -4, 9, 10. • Reversals occurred one-two times every million years on the evolutionary path between human and mouse.

18. Reversals 1 2 3 9 10 8 4 7 5 6 1, 2, 3, -8, -7, -6, -5, -4, 9, 10 The reversion introduced two breakpoints(disruptions in order).

19. Reversals • Let’s first assume that genes in genome p do not have direction p = p1------ pi-1 pi pi+1 ------pj-1 pj pj+1 -----pn p` = p1------ pi-1pj pj-1 ------pi+1 pipj+1 -----pn • Reversal r( i, j ) reverses the elements from i to j inpand transforms p into p` r(i,j)

20. Reversals: Example • Example: p = 1 2 3 4 5 6 7 8 r(3,5) p’= 1 2 5 4 3 6 7 8

21. Reversal Distance Problem • Goal: Given two permutations, find the shortest series of reversals that transforms one into another • Input: Permutations pand s • Output: A series of reversals r1,…rttransforming p into s, such that t is minimum • t - reversal distance between p and s • d(p, s) - smallest possible value of t, given p and s

22. Sorting By Reversals Problem[s = (1 2 … n )] • Goal: Given a permutation, find a shortest series of reversals that transforms it into the identity permutation (1 2 … n ) • Input: Permutation p • Output: A series of reversals r1,… rt transforming p into the identity permutation such that t is minimum

23. Sorting By Reversals: Example • t =d(p ) - reversal distance between p and s • Example : input: p = 3 4 2 1 5 6 7 10 9 8 output: 4 3 2 1 5 6 7 10 9 8 4 3 2 1 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 So d(p ) = 3

24. Sorting by reversals

25. Sorting by reversalsMost parsimonious scenarios The reversal distance is the minimum number of reversals required to transform p into g.Here, the reversal distance is d=4.

26. Sorting By Reversals: A Greedy Algorithm • If sorting permutation p = 1 2 3 6 4 5, the first three numbers are already in order so it does not make any sense to break them. These already sorted numbers of p will be defined as prefix(p) • prefix(p) = 3 • This results in an idea for a greedy algorithm: increase prefix(p) at every step

27. Doing so, p can be sorted 1 2 36 4 5 1 2 3 46 5 1 2 3 4 5 6 d(p) = 2 Number of steps to sort permutation of length n is at most (n – 1) Greedy Algorithm: An Example

28. Greedy Algorithm: Pseudocode SimpleReversalSort(p) 1 for i 1 to n – 1 2 j position of element i in p(i.e., pj = i) 3 if j != i 4 p p * r(i, j) 5 output p 6 if p is the identity permutation 7 return Progress is ensured by moving forward in the position

29. Analyzing SimpleReversalSort • Greedy algorithm; does not guarantee the smallest number of reversals • For example, let p = 6 1 2 3 4 5 • SimpleReversalSort(p) takes five steps: • Step 1: 1 6 2 3 4 5 • Step 2: 1 2 6 3 4 5 • Step 3: 1 2 3 6 4 5 • Step 4: 1 2 3 4 6 5 • Step 5: 1 2 3 4 5 6

30. But it can be done in two steps: p = 6 1 2 3 4 5 Step 1: 5 4 3 2 1 6 Step 2: 1 2 3 4 5 6 So, SimpleReversalSort(p) is not optimal Analyzing SimpleReversalSort (cont’d)

31. Breakpoints p = p1p2p3…pn-1pn • A pair of elements pi and pi + 1are adjacent if pi+1 = pi + 1 • For example: p = 1 9 3 4 7 8 2 6 5 • (3, 4) or (7, 8) and (6,5) are adjacent pairs

32. Breakpoints: An Example There is a breakpoint between any pair of non-adjacent elements: p = 1 9 3 4 7 8 2 6 5 • Pairs (1,9), (9,3), (4,7), (8,2) and (2,5) form breakpoints of permutation p

33. Extending Permutations • We put two elements p0 and pn + 1 at the ends of p p0 = 0 and pn + 1 = n + 1 • This gives us the goal to sort the elements between the end blocks to the identity permutation • Example: p = 1 9 3 4 7 8 2 6 5 Extending with 0 and 10 • p = 01 9 3 4 7 8 2 6 510 Note: A new breakpoint was created after extending

34. Reversal Distance and Breakpoints • b(p) = number of breakpoints • Each reversal eliminates at most 2 breakpoints. This implies that d(p) >= b(p) / 2 p = 2 3 1 4 6 5 0 2 3 1 4 6 5 7 b(p) = 5 01 3 24 6 5 7 b(p) = 4 0 1 2 3 4 6 57 b(p) = 2 0 1 2 3 4 5 6 7b(p) = 0

35. Sorting By Reversals: A Better Greedy Algorithm BreakPointReversalSort(p) 1while b(p) > 0 2Among all possible reversals, chooserminimizing b(p•r) 3p p•r(i, j) 4output p 5return

36. Strip: an interval between two consecutive breakpoints in p Decreasing strips: strips that are in decreasing order (e.g. 6 5 and 3 2 ). Increasing strips: strips that are in increasing order (e.g. 7 8) A single-element strip can be declared either increasing or decreasing. We will choose to declare them as decreasing with possible exception of the strips with 0 and n+1 Strips

37. Strips: An Example • For permutation 1 9 3 4 7 8 2 5 6: • There are 7 strips: 01 9 4 3 7 8 2 5 6 10

38. Things To Consider • Fact 1: • If permutation pcontains at least one decreasing strip, then there exists a reversal r which decreases the number of breakpoints (i.e. b(p •r) < b(p) )

39. Things To Consider (cont’d) • Forp = 1 4 6 5 7 8 3 2 • 01 4 6 5 7 8 3 2 9 b(p) = 5 • Choose decreasing strip with the smallest element k in p ( k = 2 in this case) • Find k – 1 in the permutation, reverse the segment between k and k-1: • 01 4 6 5 7 8 3 29b(p) = 5 • 012 3 8 7 5 6 4 9b(p) = 4

40. Things To Consider (cont’d) • Fact 2: • If there is no decreasing strip, there may be no reversal r that reduces the number of breakpoints (i.e. b(p • r) = b(p) ). • By reversing an increasing strip ( # of breakpoints stay unchanged ), we will create a decreasing strip at the next step. Then (fact 1) the number of breakpoints will be reduced in the next step.

41. Things To Consider (cont’d) • There are nodecreasing strips in p, for: p = 01 2 5 6 7 3 4 8 b(p) = 3 p•r(3,4) = 01 2 5 6 7 4 3 8b(p) = 3 • r(3,4) does not change the # of breakpoints • r(3,4) creates a decreasing strip, guaranteeing that the next step will decrease the # of breakpoints.

42. ImprovedBreakpointReversalSort ImprovedBreakpointReversalSort(p) 1 while b(p) > 0 2 ifphas a decreasing strip 3 Among all possible reversals, chooserthat minimizes b(p•r) 4 else 5 Choose a reversalr that flips an increasing strip in p 6 p p•r 7 output p 8 return

43. ImprovedBreakpointReversalSort: Performance Guarantee • ImprovedBreakPointReversalSort is an approximation algorithm with a performance guarantee of at most 4 • It eliminates at least one breakpoint in every two steps; at most 2b(p) steps • Approximation ratio: 2b(p) / d(p) • Optimal algorithm eliminates 2 breakpoints in every step: d(p) b(p) / 2 • Performance guarantee: • ( 2b(p) / d(p) )  [ 2b(p) / (b(p) / 2) ] = 4

44. 5’ 3’ Signed Permutations • Up to this point, all permutations to sort were unsigned • But genes have directions… so we should consider signed permutations p = 1-23-4

45. GRIMM Web Server • Real genome architectures are represented by signed permutations • Efficient algorithms to sort signed permutations have been developed • GRIMM web server computes the reversal distances between signed permutations: http://nbcr.sdsc.edu/GRIMM/grimm.cgi

46. GRIMM Web Server http://www-cse.ucsd.edu/groups/bioinformatics/GRIMM

47. What You Should Know • What is genome rearrangement? • How does the problem of sorting by reversal capture genome rearrangements? • How do the two greedy algorithms work?