Pairwise sequence comparison. Global alignment with linear gap cost

Transcription

1 Pirwise sequence comprison Globl linment with liner p cost

2 DNA sequences The humn enome 5 cells, ech contins 3 pirs of chromosomes, DNA molecules, which stores enetic informtion... GGCCTAAAGGCGCCGGTCTT TCGTACCCCAAAATCTCGGC ATTTTAAGATAAGTGAGTGT TGCGTTACACTAGCGATCTA CCGCGTCTTATACTTAAGCG TATGCCCAGATCTGACTAAT CGTGCCCCCGGATTAGACGG GCTTGATGGGAAAGAACAGC TCGTCTGTTTACGTATAAAC AGAATCGCCTGGGTTA... Totl: 3... ACGT's

3 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC

4 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC ins GTTACTC

5 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC ins GTTACTC del TTACTC

6 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC ins GTTACTC del TTACTC sub TTGCTC

7 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC ins GTTACTC del TTACTC sub TTGCTC G T T A - T C - T T G C T C Prsimony principle: The simplest explntion is ood estimte of evolution, i.e sequences tht look similr re relted... A resonble question: How similr re two sequences?

8 Usin sequence dt Evolution: DNA evolve by muttions... The most frequent muttions re: Insertion, deletion nd substitution of symbols GTTATC ins GTTACTC del G T T A - T C TTACTC - T T G C T C sub TTGCTC Note: lso pplies to proteins, since muttions in codin DNA implies chnes to the encoded Prsimony principle: The sequence simplest of explntion mino cids... is ood estimte of evolution, i.e sequences tht look similr re relted... A resonble question: How similr re two sequences?

9 Wht is pirwise linment c t t c c t c t c t - - c t A pirwise linment of cttcct nd ctctct

10 Wht is pirwise linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t p / indel columns A pirwise linment of cttcct nd ctctct

11 Wht is pirwise linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t p / indel columns An linment cn be interpreted s: Emphsizin sequence similrity: objective is to mximize the number of mtch-columns of similr/identicl symbols... Explinin sequence difference: objective is to minimize the number of indels nd subs (of different symbols)...

12 Wht is pirwise linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t p / indel columns - An linment cn be interpreted - columns re not llowed s: Emphsizin sequence similrity: objective is to mximize the number of mtch-columns of similr/identicl symbols... Explinin sequence difference: objective is to minimize the number of indels nd subs (of different symbols)...

13 How to find n optiml linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t p / indel columns To computtionlly find n optiml linment, we must: Define the cost of n linment (typiclly score mtrix nd p cost) Define n optiml linment (typiclly n linment of mx (or min) cost) Construct n efficient lorithm for computin n optiml linment

14 Cost of n linment Mtch / Mis-mtch / sub columns c t c t c t t c c t c t c t - - c t p / indel columns pcost: Cost of linment = sum of the cost of ech column Wht is the cost of the bove linment? 6

15 Cost of n linment Mtch / Mis-mtch / sub columns c t c t c t t c c t c t c t - - c t p / indel columns pcost: Cost of linment = sum of the cost of ech column Wht is the cost of the bove linment? 6

16 Cost of n linment Mtch / Mis-mtch / sub columns c t c t c t t c c t c t c t - - c t p / indel columns pcost: Cost of linment = sum of the cost of ech column Wht is the cost of the bove linment? 6

17 Cost of n linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t c t Score mtrix: c c t c t t pcost: - p / indel columns pcost: Cost of linment = sum of the cost of ech column Wht is now the cost of the bove linment? 8

18 Cost of n linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t c t Score mtrix: c c t c t t pcost: - p / indel columns pcost: Cost of linment = sum of the cost of ech column Wht is now the cost of the bove linment? 8

19 Cost of n linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t c t Score mtrix: c c t c t t Note bout p cost pcost: - p / indel columns pcost: In enerl: cost of p block = (k), where k is the p lenth Our exmples: (k) = k zero p cost (k) = - k liner p cost Mny prorms: (k) = + b k ffine p cost

20 Cost of n linment Mtch / Mis-mtch / sub columns c t t c c t c t c t - - c t c t Score mtrix: c c t c t t Note bout p cost pcost: - p / indel columns pcost: In enerl: cost of p block = (k), where k is the p lenth Our exmples: (k) = k zero p cost (k) = - k liner p cost Mny prorms: (k) = + b k ffine p cost

21 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost.

22 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: c t c t pcost:

23 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: Is this linment optiml? c t c t pcost: c t t c c t c t c t - - c t Cost: 6

24 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: Is this linment optiml? c t c t pcost: c t t c c t c t c t - - c t Cost: 6 c t t c c t - c t c t c t Cost: 9

25 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: Is this linment optiml? c t c t pcost: c t t c c t c t c t - - c t Cost: 6 c t t c c t - c t c t c t Cost: 9 Ide: Compute the cost of every linment of A nd B nd pick the mx...

26 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: Is this linment optiml? c t c t pcost: c t t c c t c t c t - - c t Ide: Compute the cost of every linment of A nd B nd Cost: 6 c t t c c t - pick the mx... c t c t c t How mny linments Cost: re there 9 of two strins? E. nd ct?

27 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. Exmple: Find n optiml linment of A=cttcct nd B=ctctct usin: Is this linment optiml? c t c t pcost: c t t c c t c t c t - - c t Cost: 6 Ide: Compute the cost of every linment of A nd B nd pick the mx... c t t c c t - c t c t c t How mny linments Cost: re there 9 of two strins? E. nd ct?

28 Computin n optiml linment Objective: Given two sequences A nd B, score mtrix nd p cost, find n linment of A nd B of mximum (or minimum) cost. GGCCTAAAGG CGCCGGTCTT TCGTACCCCA AAATCTCGGC ATTTTAAGAT AAGTGAGTGT TGCGTTACAC TAGCGATCTA CCGCGTCTTA TACTTAAGCG TATGCCCAGA TCTGACTAAT CGTGCCCCCG GATTAGACGG GCTTGATGGG AAAGAACAGC TCGTCTGTTT ACGTATAAAC AGAATCGCCT GGGTTCGC GGGCTAAAGG TTAGGGTCTT TCACACTAAA GAGTGGTGCG TATCGTGGCT AATGTACCGC TTCTGGTATC GTGGCTTACG GCCAGACCTA CAAGTACTAG ACCTGAGAAC TAATCTTGTC GAGCCTTCCA TTGAGGGTAA TGGGAGAGAA CATCGAGTCA GAAGTTATTC TTGTTTACGT AGAATCGCCT GGGTCCGC Needs n efficient computtionl tool. One of the first nd most studied problems in bioinformtics...

29 Computin n optiml linment V. I. Levenshtein. Binry codes cpble of correctin deletions, insertions nd reversls. Soviet Physics Dokldy, 966. T. K. Vintsyuk. Speech discrimintion by dynmic prormmin. Kibernetik, 968. S. B. Needlemn nd C. D. Wunsch. A enerl method pplicble to the serch for similrities in the mino cid sequence of two proteins. Journl of Moleculr Bioloy, 97. D. Snkoff. Mtchin sequence under deletion/insertion constrints. Proceedins of the Ntionl Acdemy of Science of the USA, 97. R. A. Wner nd M. J. Fisher. The Strin to Strin Correction Problem. Journl of the ACM, 973. P. H. Sellers. On the theory nd computtion of evolutionry distnce. SIAM Journl of Applied Mthemtics, nd mny more...

30 Constructin n lorithm Problem: Wht is the cost, Cost(i, j), of n optiml linment of the first i symbols in A, A[..i ], nd the first j symbols in B, B[..j ]? Cost(n,m)

31 Constructin n lorithm Problem: Wht is the cost, Cost(i, j), of n optiml linment of the first i symbols in A, A[..i ], nd the first j symbols in B, B[..j ]? ATACAACGC ATCTCCACC A T ATACAACGC ATCTCCACCT A - ATACAACGCA ATCTCCACC - T Cost(n,m) Solution: Look t the lst column, there re 3 cses, pick the best

32 Constructin n lorithm Problem: Wht is the cost, Cost(i, j), of n optiml linment of the first i symbols in A, A[..i ], nd the first j symbols in B, B[..j ]? ATACAACGC ATCTCCACC A T Cost(n-,m-) + subcost(a[n], B[m]) An optiml linment of the first n- symbols in A nd the first m- symbols in B Cost(n,m) Solution: Look t the lst column, there re 3 cses, pick the best

33 Constructin n lorithm Problem: Wht is the cost, Cost(i, j), of n optiml linment of the first i symbols in A, A[..i ], nd the first j symbols in B, B[..j ]? ATACAACGC ATCTCCACC A T Cost(n-,m-) + subcost(a[n], B[m]) ATACAACGC ATCTCCACCT A - Cost(n-,m) + pcost ATACAACGCA ATCTCCACC - T Cost(n,m-) + pcost Cost(n,m) Solution: Look t the lst column, there re 3 cses, pick the best A simple recursive solution

34 Implementin the lorithm Possible cses: ATACAACGC ATCTCCACC ATACAACGC ATCTCCACCT A T A - func Cost(i,j): v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + ATACAACGCA ATCTCCACC or nothin Cost(,)= - T if (i = ) nd (j = ) then v4 = return mx(v,v,v3,v4) end print Cost(n,m) Is it ood solution? Is it correct? Is it efficient?

35 Experiment Mesure the runnin time for comprin sequences of lenth,, 3,... Slow! Wht is the bottleneck?

36 Anlysin the lorithm func Cost(i,j): v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = return mx(v,v,v3,v4) end print Cost(n,m) Computin Cost(,) involves Cost(,), Cost(,), Cost(,)... We compute the sme thin in nd in... Rule of thumb: Remember wht you hve done!!

37 Anlysin the lorithm func Cost(i,j): v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = return mx(v,v,v3,v4) end print Cost(n,m) Computin Cost(,) involves Cost(,), Cost(,), Cost(,)... We compute the sme thin in nd in... Rule of thumb: Cost(n,m) Remember wht you hve done!!

38 A better lorithm Store intermedite results Dynmic prormmin func Cost(i,j): end v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = return mx(v,v,v3,v4) print Cost(n,m) func Cost(i, j): end if T[i,j] = undef then endif v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = T[i,j] = mx(v,v,v3,v4) return T[i,j] T[..n][..m]=undef print Cost(n,m) How does this influence the runnin time?

39 Experiment Mesure the runnin time for comprin sequences of lenth,, 3,... Much fster!! The runnin time nd spce use is prop. to the size of the tble O(nm)

40 Experiment The runnin time of both lorithms implemented in C nd Python The improved lorithm mkes the solution usble in prctice

41 Globl linment recursion Cost(i, j) = mx Cost(i-, j-) + subcost(a[i ], B[j ]) Cost(i-, j) + pcost Cost(i, j-) + pcost if i= nd j= A[i] B[j] i-,j- i-,j i,j- i,j To compute the score of n optiml linment of A[..n] nd B[..m], fill out n (n+) x (m+) tble cf. bove recursion. The optiml linment score is in entry (n,m).

42 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell depends on three of its neihbor cells (left, upper-dionl, nd bove)... c t c t pcost:

43 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, nd bove)... 6 c t c t pcost:

44 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, nd 6+ bove)... 6 c t c t pcost:

45 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, nd 6+ bove) c t c t pcost:

46 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) c t c t pcost:

47 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) c t c t pcost:

48 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) Cn we fill out the tble in other 6+ wys (e.. non-recursively)? c t c t pcost:

49 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) Cn we fill out the tble in other 6+ wys (e.. non-recursively)? c t c t Row by row, or column by column, or dionl by dionl pcost:

50 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) How much spce is consumed? Cn it be improved? 6+ c t c t pcost:

51 Bsic ide: Fillin out tble The lorithm essentilly fills out tble (dynmic prormmin) Observtion: The vlue in cell 6depends 6 on three of its neihbor cells (left, upper-dionl, 6+ nd 6+ bove) How much spce is consumed? Cn it be improved? 6+ c t c t O(n ) cn be improved to O(n) by keepin only two rows in memory pcost:

52 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t - t

53 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t - - t

54 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t - - t

55 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

56 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

57 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

58 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

59 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

60 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

61 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

62 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

63 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

64 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

65 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - c t t

66 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

67 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

68 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

69 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

70 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

71 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

72 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

73 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

74 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

75 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t

76 Exercise Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t The optiml score, but wht bout n optiml linment?

77 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

78 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

79 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

80 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

81 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

82 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: c t c t pcost: - t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

83 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: Also optiml: t - c t t c t t c t c t c t pcost: - t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

84 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: Also optiml: t - c t t c t c t Runnin time? O(n) Spce - consumption? O(nm) t -4-3 c t c t pcost: - t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

85 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: Also optiml: t - c t t c t c t Runnin time? O(n) Spce - consumption? O(nm) t -4-3 c t c t pcost: - t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

86 Bcktrckin Compute the optiml score of n optiml linment of t nd ct usin: Also optiml: t - c t c t c t c t pcost: Runnin time? O(n) Spce - consumption? O(nm) Another lecture: Spce t consumption cn be improve to O(n) -4-3 t c t t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

87 func Cost(i, j): end if T[i,j] = undef then Bcktrckin Compute the optiml score of n optiml linment v = Cost(i-, of t j-) nd + d(a[i],b[j]) ct usin: v = v = v3 = v4 = undef if (i > ) nd (j > ) then if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + Also optiml: if (i = ) nd (j = ) then v4 = endif t - return T[i,j] c t T[i,j] = mx(v,v,v3,v4) c t c t c t pcost: Runnin time? O(n) Spce - consumption? O(nm) Another lecture: Spce t consumption cn be improve to O(n) -4-3 T[..n][..m]=undef t c t print Cost(n,m) t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

88 Bcktrckin func Cost(i, j): func RecurBckTrck(i, j): if T[i,j] = undef then if (i > ) nd (j > ) then c t v = v = v3 = v4 = undef Compute the optiml score if T[i,j] of n == optiml T[i-i, j-] + subcost(a[i], B[j]) then if (i > ) nd (j > ) then c linment v = Cost(i-, of t j-) nd + d(a[i],b[j]) output column(a[i], B[j]) ct usin: if (i > ) nd (j >= ) then RecurBckTrck(i-, j-) t v = Cost(i-, j) + endif if (i >= ) nd (j > ) then pcost: - v3 = Cost(i, j-) + else if (i > ) nd (j > =) then Also if optiml: (i = ) nd (j = ) then if T[i,j] == T[i-,j] + then v4 = c output t column(a[i], -) T[i,j] = mx(v,v,v3,v4) RecurBckTrck(i-, j) endif t - endif end else if (i>=) nd (j > ) then - if T[i,j] == T[i,j-] + then Runnin time? O(n) output column(-, B[j]) Spce - consumption? RecurBckTrck(i, O(nm) j-) endif Another lecture: Spce endif t consumption cn be improve to O(n) end -4-3 return T[i,j] c t T[..n][..m]=undef t c t print Cost(n,m) RecurBckTrck(n,m) t - c t We find n optiml linment by decidin which choice of columns hs resulted in the optiml cost (lower riht entry).

89 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... t c t c t c t pcost: -

90 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... t c t c t c t pcost: - Weiht: -

91 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... Weiht: subcost(,c)= c t t c t c t pcost: - Weiht: -

92 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... Weiht: subcost(,c)= c t t c t c t pcost: - Weiht: subcost(,)= -4 Weiht: - -

93 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... Weiht: subcost(,c)= c t t c t c t pcost: - Weiht: subcost(,)= -4 Weiht: - - The cost of n optiml linment is the lenth of lonest (shortest) pth from (,) to (n,m), nd the pth yields the linment...

94 Alinment rph An oriented weiht (rid) rph where nodes re the cells in the dynmic prormmin mtrix nd edes re the recursive dependencies. The weiht of n ede is the cost of the correspondin linment column... Weiht: subcost(,c)= c t t c t c t pcost: - Weiht: subcost(,)= -4 Weiht: - - The cost of n optiml linment is the lenth of lonest (shortest) pth from (,) to (n,m), nd the pth yields the linment...

95 Summry You hve been introduced to the nottion of pirwise linment n lorithms for computin n optiml (lobl) pirwise linment with liner p cost. You should be ble to explin how to compute the cost of n optiml linment in time O(n ) nd how to find n optiml linment by bcktrckin in time O(n).

96 Summry func Cost(i, j): end if T[i,j] = undef then endif v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = T[i,j] = mx(v,v,v3,v4) return T[i,j] T[..n][..m]=undef print Cost(n,m) Input: Sequences A nd B, nd score function... The column-score is iven s substitution mtrix d nd p cost... A C G T A 5 C 5 G 5 T 5 pcost: -5

97 Summry func Cost(i, j): end if T[i,j] = undef then endif v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = T[i,j] = mx(v,v,v3,v4) return T[i,j] T[..n][..m]=undef print Cost(n,m) Input: Sequences A nd B, nd score function... The column-score is iven s substitution mtrix d nd p cost... A C G T A 5 C 5 G 5 T 5 pcost: -5 Trnsitions (muttions between two purines (A,G) or two pyrimidines (T,C)) re more likely thn trnsversions (muttions between purine nd pyrimidine)...

98 Summry func Cost(i, j): end if T[i,j] = undef then endif v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = T[i,j] = mx(v,v,v3,v4) return T[i,j] T[..n][..m]=undef print Cost(n,m) Input: Sequences A nd B, nd score function... The column-score is iven s substitution mtrix d nd p cost... A C G T A 5 C 5 G 5 T 5 pcost: -5 Note: Cn be implemented s mximizin similrity, s bove, or minimizin cost (or distnce), with min insted of mx...

99 Summry func Cost(i, j): end if T[i,j] = undef then endif v = v = v3 = v4 = undef if (i > ) nd (j > ) then v = Cost(i-, j-) + d(a[i],b[j]) if (i > ) nd (j >= ) then v = Cost(i-, j) + if (i >= ) nd (j > ) then v3 = Cost(i, j-) + if (i = ) nd (j = ) then v4 = T[i,j] = mx(v,v,v3,v4) return T[i,j] T[..n][..m]=undef print Cost(n,m) Input: Sequences A nd B, nd score function... The column-score is iven s substitution mtrix d nd p cost... A C G T A 5 C 5 G 5 T 5 pcost: -5 Cn e.. be used to compute: () Unit cost edit-distnce, or () lenth of lonest common subsequence. How?

100 Extension Modelin pcost Bioloicl observtion: loner insertions nd deletions (indels) re more common thn shorter indels, i.e. ood linment tends to few lon indels rther thn mny short indels... Cn the simple lorithm for pirwise linment be dpted to reflect this dditionl bioloicl insith, i.e. better model of bioloy? Yes, we introduce the concept of pcost-function (k) which ives the cost/penlty for block of k consecutive insertions or deletions... Exmple A T A C A C G C A A T C T C C A C C T s(a,a) + s(t,t) + () + s(c,c) + s(a,t) + (3) + s(c,c) + () + s(c,c) + s(a,t)

101 Next time Topics Discussion of exercises Hndlin enerl nd ffine p cost Locl linment, findin the most similr pir of substrins