1. Bioinformatics PhD. Course Summary (approximate) 1. Biological introduction 2. Comparison of short sequences (<10.000 bps) 4 Sequence assembly 3 Comparison of large sequences (up to 250 000 000) 5 Efficient data search structures and algorithms 6 Proteins...

2. 3. Comparison of large sequences Summary (more or less) 3.1 Overview 3.2 Suffix trees 3.3 MUMs

3. Suffix trees Algorithms on strings, trees and sequences, Dan Gusfield Cambridge University Press http://sequence.rutgers.edu/st/

4. Suffix trees Given string ababaas: 1: ababaas 2: babaas 3: abaas 4: baas 5: aas 6: as 7: s as,3 s,6 as,5 s,7 as,4 ba baas,2 a ba baas,1 a ba baas,1 ba baas,2 as,3as,4 s,6 as,5 s,7 Suffixes: What kind of queries can we do?

5. Applications of Suffix trees a ba baas,1 as,3 ba baas,2 as,4 s,6 as,5 s,7 1. Exact string matching ………………………… Does the sequence ababaas contain any ocurrence of the patterns abab, aab, and ab?

6. Applications of Suffix trees 2. Finding the repeats within a sequence. a ba baas,1 as,3 ba baas,2 as,4 s,6 as,5 s,7 …………………………

7. Queries on Suffix trees a ba baas,1 as,3 ba baas,2 as,4 s,6 as,5 s,7 Does the sequence ababaas contain any ocurrence of patterns abab, aab, and ab? Find repeats within the sequence ababaas. …………………………

8. Quadratic Insertion algorithm Given the string ababaabbs ababaabbs,1

9. Quadratic Insertion algorithm Given the string ababaabbs babaabbs,2 ababaabbs,1

10. Quadratic Insertion algorithm Given the string ababaabbs babaabbs,2 ababaabbs,1 aba baabbs,1

11. Quadratic Insertion algorithm Given the string ababaabbs babaabbs,2 aba baabbs,1 abbs,3

12. Quadratic Insertion algorithm Given the string ababaabbs babaabbs,2 aba baabbs,1 abbs,3 ba baabbs,2

13. Quadratic Insertion algorithm Given the string ababaabbs aba baabbs,1 abbs,3 ba baabbs,2 abbs,4

14. Quadratic Insertion algorithm Given the string ababaabbs aba baabbs,1 abbs,3 abbs,4 ba baabbs,2 abbs,4 abbs,3 ba a baabbs,1

15. Quadratic Insertion algorithm Given the string ababaabbs abbs,4 ba baabbs,2 abbs,4 abbs,3 ba a baabbs,1 abbs,5

16. Quadratic Insertion algorithm Given the string ababaabbs abbs,4 ba baabbs,2 abbs,4 abbs,3 ba a baabbs,1 abbs,5

17. Quadratic Insertion algorithm Given the string ababaabbs abbs,4 ba baabbs,2 abbs,4 a abbs,5 b a abbs,3 baabbs,1

18. Quadratic Insertion algorithm Given the string ababaabbs abbs,4 ba baabbs,2 abbs,4 a abbs,5 b a abbs,3 baabbs,1 bs,6

19. Quadratic Insertion algorithm Given the string ababaabbs abbs,4 ba baabbs,2 abbs,4 a abbs,5 b a abbs,3 baabbs,1 bs,6

20. Quadratic Insertion algorithm Given the string ababaabbs a abbs,5 b a abbs,3 baabbs,1 bs,6 a baabbs,2 b abbs,4 bs,7

21. Quadratic Insertion algorithm Given the string ababaabbs a abbs,5 b a abbs,3 baabbs,1 bs,6 a baabbs,2 b abbs,4 bs,7 s,7

22. Quadratic Insertion algorithm Given the string ababaabbs a abbs,5 b a abbs,3 baabbs,1 bs,6 a baabbs,2 b abbs,4 bs,7 s,7

23. Generalizad suffix tree A suffix tree of many strings … and is the suffix tree of the concatenation of strings. the generalized suffix tree of ababaabb and aabaat … is the suffix tree of ababaabαaabaatβ, : is called a generalized suffix tree … For instance,

24. Generalizad suffix tree a abbα,5 b a abbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 Given the suffix tree of ababaabα : Construction of the suffix tree of ababaabbαaabaaβ :

25. Generalizad suffix tree a abbα,5 b a abbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 Construction of the suffix tree of ababaabbαaabaaβ :

26. Generalizad suffix tree Construction of the suffix tree of ababaabbαaabaaβ : a bα,5 b a abbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 ab aaβ,1

27. Generalizad suffix tree Construction of the suffix tree of ababaabbαaabaaβ : a bα,5 b a abbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 ab aaβ,1

28. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 ab aaβ,1 a β,2 Construction of the suffix tree of ababaabbαaabaaβ :

29. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b abbα,4 bα,7 α,7 ab aaβ,1 a β,2 Construction of the suffix tree of ababaabbαaabaaβ :

30. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 ab aaβ,1 a β,2 a β,3

31. Construction of the suffix tree of ababaabbαaabaaβ : Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 ab aaβ,1 a β,2 a β,3

32. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Construction of the suffix tree of ababaabbαaabaaβ :

33. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Construction of the suffix tree of ababaabbαaabaaβ :

34. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Construction of the suffix tree of ababaabbαaabaaβ :

35. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Construction of the suffix tree of ababaabbαaabaaβ :

36. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Construction of the suffix tree of ababaabbαaabaaβ :

37. Generalizad suffix tree a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4 Generalized suffix tree of ababaabbαaabaaβ : What kind of queries can we do?

38. Applications of Suffix trees 1. The substring problem for a database of patterns DB Does the DB contain any ocurrence of patterns abab, aab, and ab? a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4

39. Applications of Suffix trees 2. The longest common substring of two strings a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4

40. Applications of Suffix trees 3. Finding MUMs. a bα,5 b a bbα,3 baabbα,1 bα,6 a baabbα,2 b bbα,4 bα,7 α,7 b aaβ,1 a β,2 a β,3 a β,4

41. Linear Insertion algorithm: Given the string …………………………...... P2: the string  is the longest string that can be spelt through the tree. P1: the leaves of suffixes from  have been inserted and the suffix-tree  …...  

42. Insertion algorithm: example Given the string ababaababb... ba baababb...,2 a ababb...,5 ba ababb...,3 baababb...,1 ababb...,4   

43. Linear Insertion algorithm: Given the string …………………………...... P2: the string  is the longest string that … P1: the leaves of suffixes from  have been inserted  …...   P3: there is a pointer,called “suffix pointer” between any node and its longest no proper suffix node. ´´ ´´

44. Insertion algorithm: example ba baababb...,2 a ababb...,5 ba ababb...,3 baababb...,1 ababb...,4  6 7... Given the string ababaababb... 

45. Insertion algorithm: example a ababb...,5 ba ababb...,3 baababb...,1 ba baababb...,2 ababb...,4 Given the string ababaababb...   6 7 8...

46. Insertion algorithm: example a ababb...,5 ba ababb...,3 baababb...,1 ba baababb...,2 ababb...,4 Given the string ababaababb...   6 7 8... baababb...,1 b b...,6 ababb...,1

47. Insertion algorithm: example a ababb...,5 ba ababb...,3 ba baababb...,2 ababb...,4 Given the string ababaababb...   7 8... b b...,6 ababb...,1

48. Insertion algorithm: example a ababb...,5 ba ababb...,3 ba baababb...,2 ababb...,4 Given the string ababaababb...   7 8… b b...,6 ababb...,1 baababb...,2 b b...,7 aababb...,2

49. Insertion algorithm: example a ababb...,5 ba ababb...,3 ba ababb...,4 Given the string ababaababb...   8… b b...,6 ababb...,1 b b...,7 aababb...,2

50. Insertion algorithm: improving time Resume: Given the string ababaababb...   6 7 8... a ababb...,5 ababb...,3 ba baababb...,1 ba baababb...,2 ababb...,4 we have pointed to the following nodes

51. a ababb...,5 ababb...,3 ba baababb...,1 ba baababb...,2 ababb...,4 Insertion algorithm: improving time Resume: Given the string ababaababb...   6 7 8... we have pointed to the following nodes ba baababb...,1 ba baababb...,2

52. Suffix tree implementation:suffix-links Given sequence ababaas a ba baas,1 as,3 ba baas,2 as,4 s,6 as,5 s,7 aa 

53. Suffix links a ba baas,1 as,3 ba baas,2 as,4 s,6 as,5 s,7 Given Suffix tree of ababaas

54. Insertion algorithm Given the string ababaabbs ababaabbs,1

55. Insertion algorithm Given the string ababaabbs babaabbs,2 ababaabbs,1

56. Insertion algorithm Given the string ababaabbs babaabbs,2 ababaabbs,1

57. Insertion algorithm Given the string ababaabbs babaabbs,2 ababaabbs,1 aba baabbs,1

58. Insertion algorithm Given the string ababaabbs babaabbs,2 aba baabbs,1 abbs,3

59. Insertion algorithm Given the string ababaabbs babaabbs,2 aba baabbs,1 abbs,3

60. Insertion algorithm Given the string ababaabbs babaabbs,2 aba baabbs,1 abbs,3 ba baabbs,2

61. Insertion algorithm Given the string ababaabbs aba baabbs,1 abbs,3 ba baabbs,2 abbs,4

62. Insertion algorithm Given the string ababaabbs baabbs,1 abbs,5 abbs,3 ba baabbs,2 abbs,4 a

63. Insertion algorithm Given the string ababaabbs ba baabbs,2 a abbs,5 ba abbs,3 baabbs,1 abbs,4

64. Insertion algorithm Given the string ababaabbs bs,6 ba baabbs,2 a b a abbs,3 baabbs,1 abbs,4 abbs,5

65. Insertion algorithm Given the string ababaabbs bs,6 ba baabbs,2 a b a abbs,3 baabbs,1 abbs,4 abbs,5 a baabbs,2 b abbs,4

66. Insertion algorithm Given the string ababaabbs bs,7 a b bs,6 a abbs,3 baabbs,1 a baabbs,2 b abbs,4 abbs,5

67. Insertion algorithm Given the string ababaabbs bs,7 a b bs,6 a abbs,3 baabbs,1 a baabbs,2 b abbs,4 abbs,5 s,8

68. Insertion algorithm Given the string ababaabbs bs,7 a b bs,6 a abbs,3 baabbs,1 a baabbs,2 b abbs,4 abbs,5 s,8 s,9