Genome Rearrangement SORTING BY REVERSALS

1. Genome RearrangementSORTING BY REVERSALS Ankur Jain Hoda Mokhtar CS290I – SPRING 2003

2. Comparative Genomics The practice of analyzing and comparing the genetic material of different species for the purpose of studying evolution, the function of genes and inherited diseases. Chromosome breakage and mistakes in repair, along with a number of other processes, give rise to changes in gene order.  These have important consequences for the evolution of species. 

3. Problem Definition During biological evolution, inter- and intra-chromosomal  exchanges of chromosomal fragments disrupt the order of genes on a  chromosome. The genome rearrangements approach, is the use of  combinatorial optimization techniques, to infer a sequence of rearrangement events to account for the differences among the  genomes. 

4. Outline • Problem definition • Genome Comparison • Possible chromosomal changes • Sorting by reversals : - Previous work - Definitions - Duality Theorem • Our technique : - Bit Vector Method - Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique • Conclusions and Future work

5. Genome Comparison • In the late 1980 was discovered remarkable and novel pattern of evolutionary change in plant organelles. • Jeffrey Palmer and his collegues compared the mitochondrial genomes of cabbage and turnip, which are very closely related. Molecules which are almost identical in gene sequences, differ dramatically in gene order. {Sridhar, Pevzner 1995} • This discovery and many other studies proved that genome rearrangements represent a common mode of molecular evolution.

6. Cabbage and Turnip Gene orientation

7. Single Chromosome Operations • Reversal: A section of a chromosome is excised, reversed in orientation, and re-inserted. (abc1c2c3c4de -> ab-c4-c3-c2-c1de) • Transposition: A section of a chromosome is excised and inserted at new position in the chromosome, without changing orientation. (abcd -> cdab) • Inverted transposition: Exactly like transposition, except that the transposed segment changes orientation. (abcd -> -c-dab) • Gene duplication: A section of a chromosome is duplicated, so that multiple copies exist of every gene in that section. (abc -> abcb, abc -> abbc) • Gene loss: A section of a chromosome is excised and lost. (abc->ac )

8. Operations on 2 Chromosomes • Translocation: The end of one chromosome is broken and attached to the end of another chromosome. • Fusion: two chromosomes merge. • Fission: one chromosome splits up into two chromosomes.

9. and Genomic Sorting Problem and • Given genomes the genomic sorting problem is to find a series of reversals where and t is minimal. • We call t the genomic distance between

10. Sorting by Reversals Genome rearrangements can be modelled by a combinatorical problem of sorting by reversals. A T GC C T G T AC T A Reversal Break and Invert A T GA T G T C CC T A

11. Sorting by Reversals (Cont.) Minimum Sorting by Reversals: Given a permutation , what is the shortest sequence (12….t ) of reversals that sorts ? Complexity remains open. (NP-Hard) {Caprara ‘97} Minimum Signed Sorting by Reversals: Given a signed permutation , what is the shortest sequence (12….t ) of reversals that sorts ? Solvable in polynomial time.

12. Sorting of Signed Permutations • Transforming cabbage into turnip. {Hannenhalli, S., and Pevzner, P. ‘95} - Polynomial algorithm for sorting signed permutations by reversals • A Very Elementray Presentation of the Hannenhalli-Pevzner Theory, {A. Bergeron’95} – Polynomial algorithm for sorting signed permutations, efficiently implemented using bit vectors. • A Very Elementray Presentation of the Hannenhalli-Pevzner Theory, {A. Bergeron’95} – Polynomial algorithm for sorting signed permutations, efficiently implemented using bit vectors. • Experiments in Computing Sequences of Reversals, {A. Bergeron and F. Strasbourg’95} – Polynomial algorithm for sorting signed permutations. • Fast Sorting by Reversal, {Berman, P., Hannenhalli, S. ‘96. }- exploit a few combinatorial properties of the cycle graph of a permutation and provided a polynomial algorithm. • A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals, {Kaplan, H., Shamir, R., and Tarjan, R. ‘99.} – O(n2) using hurdles, cycles and fortress. • A Linear-Time Algorithm for Computing Inversion Distance between Signed Permutations with an Experimental Study, {Moret, and Yan’ 00} - Computes reversal distance (without actually sorting) in O(n) time. Computes the connected components using stack rather than Union-Find. {Hannenhalli-Pevzner ’96} (GRAPPA program)

13. Outline • Problem definition • Genome Comparison • Possible chromosomal changes • Sorting by reversals : - Previous work - Definitions • Our technique : - Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique • Conclusions and Future work

14. What is a Permutation? Permutation () : an ordered arrangement of the set { 1,2,…,n} Signed Permutation (): a permutation where the elements are oriented a reversal switches element orientation {+3 -4+7 -6 +1 -5+2 } (7,-5) = {+3 -4+5 -1 +6 -7+2}

15. BreakPoint Let i ~ j if | i – j | = 1. Extend permutation by adding = 0 and = n + 1. We call pair of elements , 0 ≤ i ≤ n, of an adjacency if~ and a breakpoint if is not ( ~ ) =0 =n+1 ~ ~

16. ~ What is breakpoint graph? Thebreakpoint graphof a permutation is a edge-colored graph with 2n+2 vertices by ablackedge We join vertices and and by agrayedge if We join vertices

17. Breakpoint graph – signed case Straight edges – every other pair of consecutive elements Curved edges - every other pair of consecutive integers Every connected component of the graph is a cycle

18. Correlation between the breakpoints and reversal distance • Correlations exists between the reversal distance and the number of breakpoints • Sorting by reversals corresponds to eliminating breakpoints • Every resersal can eliminate at most 2 breakpoints {Shamir, 95}

19. Outline • Problem definition • Genome Comparison • Possible chromosomal changes • Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) • Our technique : - Vector-Method Experimental results : - Synthetic datasets - Real datasets -Breakpoints Technique • Conclusions and Future work

20. Hurdle Hurdle - an unoriented component whose elements are consecutive Simple hurdle - a hurdle whose deletion decreases the number of hurdles Super hurdles - hurdles that are not simple

21. Duality Theorem for Sorting Signed Permutations Hannenhalli and Pevzner, 1995. For every signed permutation if is a fortress = otherwise

22. For an arbitary reversal Reversal issafeif Safe reversal C=3, h=1 C = 5, h= 2

23. Outline • Problem definition • Genome Comparison • Possible chromosomal changes • Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) • Our technique : - Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique • Conclusions and Future work

24. Our Approach • Finding hurdles and fortresses in a graph are difficult and expensive {Kaplan, H., Shamir, R., and Tarjan, R. ‘99.} • Use oriented sort to remove the oriented components in a graph and then apply the breakpoint approach to perform the remaining reversals • We used the bit-vector approach to perform the oriented sort

25. Oriented Sort Choose among the several candidates, a safe reversal, that is a reversal that decreases the reversal distance. Theorem : The reversal that maximizes the number of oriented vertices is safe {A. Bergeron’95}

26. Basic Sorting – oriented pair • Anoriented pair is a pair of consecutive integers, that is with opposite signs Example: (0 3 1 6 5 -2-4 7) Oriented pairs are: (1,-2) , (3, -4)

27. Reversalscore The number of oriented pairs in the resulting permutation as a result of a reversal Example: ( 0 3 1 6 5 -2 4 7 ) ( 0 3 1 6 5 -2 4 7 ) ( 0 3 1 6 5 -2 4 7 ) (1, -2) (3, -2) ( 0-5-6-1-3-2 47 ) ( 0 3 1 2 -5-647 ) Score 4 Score 2

28. Algorithm • As long as has an oriented pair choose the oriented reversal that has maximal score (0 3 1 6 5 –2 4 7) ( 0 -5 -6 -1 -3 -2 4 7 ) (-3, 4) ( 0 -5 -6 -1 2 3 4 7 )(-1,2) ( 0 -5 -6 1 2 3 4 7 )(-6,7) ( 0 -5 -4 -3 -2 -1 6 7 )(-5,6) ( 0 1 2 3 4 5 6 7 )

29. Orientededge Let be a gray edge incident to black edges . and Then i – k = j - l isorientedif and only if Edge 20-21 is oriented (contains 3 [odd] number of vertices). I= 20, j=21, k=22, l=23 I-k = -2 = j-l = -2 Bergeron Pevzner

30. Oriented reversals • Reversals induces by an oriented pair will be: , and , if , if Reversals that create consecutive integers are always induced by oriented pairs. Such reversals are called oriented reversal. Example: The pair (1, -2) induces the reversal: (0 3 1 6 5 –2 4 7) (0 3 1 2 –5 –6 4 7)

31. Interleaving Graph C Every 2 components are adjacent if there is an overlap between them but neither of them contains the other.

32. Bit Matrix Scores Parity Constructing the Bit Matrix Consider the sequence P = 3 1 6 5 –2 4 7 Represent Pi by 2i-1, 2i if Pi is +ve and 2i, 2i-1 otherwise Pi is -ve 3 1 6 5 -2 4 7 0 5 6 1 2 11 12 9 10 4 3 7 8 13 14 15

33. The Algorithm Step 1. Select the vertex vi with the maximum score and perform the these operations until we reach a situation when parity of all the vertices is zero Step 2. If the sequence is not sorted completely apply the breakpoint technique to complete the sorting

34. Outline • Problem definition • Genome Comparison • Possible chromosomal changes • Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) • Our technique : - Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique • Conclusions and Future work

35. Experimental Settings • 1- Synthetic Datasets: • generated random signed permutation of different lengths and evolution rate using GRIMM permutation generation module 2- Real Datasets: • Used GRAPPA test sets for different species of “Campanulaceae” (flower plant) • MGR (multiple genome rearrangement) human-mouse gene order data • Genome.org Herpes Virus that affects human

36. Experiment 1 - Synthetic 1- Generated files of random permutations of different lengths (50, 100, 200, 400, 800, 1600) each file with 50 permutations. 2- We computed the number of correctly sorted permutations. 3- Evolution rate varies : 20,30,40

37. Experiment 2 - Synthetic 1- Generated files of random permutations of different lengths (50, 100, 200, 400, 800, 1600) each file with 50 permutations. 2- We computed the time needed to obtain the correctly sorted permutations. 3- Evolution rate varies : 20,30,40

38. Experiment 3 - Synthetic 1- Generated files of random permutations of length 1000 2- We computed the time needed to obtain the correctly sorted permutations. 3- Evolution rate varies in increments of 100. Observation: Saturation state is reached as evolution rate approaches 1000

39. Experiment 1 - Real Considered Herpes simplex virus (HSV), Epstein-Barr virus (EBV), and Cytomegalovirus (CMV) gene orders (Hannenhalli et al. 1995) as well as the identity gene order (A) Observations:Our reversal results matched those obtained in optimal evolutionary scenario recovered by MGR-MEDIAN.

40. Experiment 2 - Real 1- Considered Campanulaceae species 2- Obtained reversals for Cyanathus (11 reversals), Triodanus (13 reversals), and Symphanra (12 reversals) versus Tobacco but failed to sort Platyncodon, Legousia and Codonopsis Observation: The ones we sorted were sorted with same number of reversals as GRIMM

41. 40 reversals 3 reversals Experiment 3 - Real 1- Considered Human-Mouse gene order from MGR 12 13 14 15 -9 -8 -7 -6 47 48 -46 -45 -44 -11 -10 -58 -57 -56 92 93 -95 -94 -21 -20 -5 -4 -3 -2 -1 34 35 41 42 43 36 37 38 -64 -63 61 62 65 66 67 68 90 91 -55 -54 51 52 53 39 40 -60 -59 -77 -76 -19 -18 16 17 -97 -96 -75 -74 -73 24 25 78 79 -83 -82 -81 -80 84 85 86 87 -28 -27 -26 22 23 98 99 69 70 -72 -71 -33 -32 -31 -30 -29 88 89 -50 -49 -105 -104 106 107 108 114 115 -117 -116 -103 -102 109 110 111 112 113 -101 -100 118 119 120 121 122 123 (mouse genome and human is identity) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 Identity GRIMM sorts the permutation in 41 reversals

42. Conclusions • We implemented a technique that integrates the bit-matrix oriented sorting technique together with the greedy breakpoint reversal technique. • The technique proposed was tested on both real and synthetic data and was able to sort signed permutations in a fair number of the test data • We think that such integration can yield good results beside being a simple and relatively fast technique • However, the oriented sort algorithm fails to sort permutations that have hurdles, in those cases we have to apply the breakpoint approach

43. Future Work • We really think that the technique we implemented can provide good results, we think that further experiments can strengthen our claim • We started implementing the algorithm proposed in Kaplan, H., Shamir, R., and Tarjan R. ’99} but didn’t succeed to complete the implementation. We think that having this technique implemented under that same conditions as ours can provide a good source of comparative results, and can give a better confidence about what we propose. • Applying the technique in different datasets including exon order rather than gene order • Considering different species and trying to compute reversal distance and use it to confirm phylogenetic trees

44. Oriented Pairs = (0 … ) • An oriented pair ( , ) is a pair of consecutive integers, that is with opposite signs Example: (0 3 1 6 5 –2 4 7) Oriented pairs are (1,-2) (3, 2)

45. Reversal Distance Estimation This reversal distance is very in-accurate. Bafna and Pevzner, 1996 showed that another hidden parameter ”hurdles” estimated reversal distance with much greater accuracy.

46. Proper reversal For every permutation and reversal Given an arbitary reversal denote (increase in the size of cycle decomposition) Then for every permutation and reversal We callreversalproperif = 1

47. Orientedpairs • Oriented pairs are useful because they indicate reversals that create consecutive elements of the permutation. Example: The pair (1, -2) induces the reversal: (0 3 1 6 5 –2 4 7) (0 3 1 2 –5 –6 4 7)