1. CSE 5290: Algorithms for Bioinformatics Fall 2011
Suprakash Datta
Office: CSEB 3043
Phone: ext 77875
Course page:
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2. Last time Dynamic programming algorithms Next: Sequence alignment
The following slides are based on slides by the authors of our text.
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3. Alignment with Affine Gap Penalties
Sequence Alignment
More realistic sequence alignment algorithms
Types:
Global Alignment
Scoring Matrices
Local Alignment
Alignment with Affine Gap Penalties
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4. From LCS to Alignment: Change the Scoring
The Longest Common Subsequence (LCS) problem—the simplest form of sequence alignment – allows only insertions and deletions (no mismatches).
In the LCS Problem, we scored 1 for matches and 0 for indels
Consider penalizing indels and mismatches with negative scores
Simplest scoring schema:
+1 : match premium
-μ : mismatch penalty
-σ : indel penalty
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5. Simple Scoring
When mismatches are penalized by –μ, indels are penalized by –σ,
and matches are rewarded with +1,
the resulting score is:
#matches – μ(#mismatches) – σ (#indels)
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6. The Global Alignment Problem
Find the best alignment between two strings under a given scoring schema
Input : Strings v and w and a scoring schema
Output : Alignment of maximum score
↑→ = -б
= 1 if match
= -µ if mismatch
si-1,j if vi = wj
si,j = max s i-1,j-1 -µ if vi ≠ wj
s i-1,j - σ
s i,j-1 - σ
m : mismatch penalty
σ : indel penalty {
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7. Scoring Matrices
To generalize scoring, consider a (4+1) x(4+1) scoring matrix δ.
In the case of an amino acid sequence alignment, the scoring matrix would be a (20+1)x(20+1) size. The addition of 1 is to include the score for comparison of a gap character “-”.
This will simplify the algorithm as follows:
si-1,j-1 + δ (vi, wj)
si,j = max s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj) {
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8. Measuring Similarity
Measuring the extent of similarity between two sequences
Based on percent sequence identity
Based on conservation
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9. Percent Sequence Identity
The extent to which two nucleotide or amino acid sequences are invariant
A C C T G A G – A G
A C G T G – G C A G
mismatch
indel
70% identical
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10. Making a Scoring Matrix
Scoring matrices are created based on biological evidence.
Alignments can be thought of as two sequences that differ due to mutations.
Some of these mutations have little effect on the protein’s function, therefore some penalties, δ(vi , wj), will be less harsh than others.
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11. Scoring Matrix: ExampleK 5 -2 -1 - 7 3 6
Notice that although R and K are different amino acids, they have a positive score.
Why? They are both positively charged amino acids will not greatly change function of protein.
AKRANR
KAAANK
-1 + (-1) + (-2) = 11
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12. Conservation
Amino acid changes that tend to preserve the physico-chemical properties of the original residue
Polar to polar
aspartate  glutamate
Nonpolar to nonpolar
alanine  valine
Similarly behaving residues
leucine to isoleucine
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13. Scoring matrices Amino acid substitution matrices PAM BLOSUM
DNA substitution matrices
DNA is less conserved than protein sequences
Less effective to compare coding regions at nucleotide level
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14. PAM some residues may have mutated several times
Point Accepted Mutation (Dayhoff et al.)
1 PAM = PAM1 = 1% average change of all amino acid positions
After 100 PAMs of evolution, not every residue will have changed
some residues may have mutated several times
some residues may have returned to their original state
some residues may not changed at all
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15. PAMX PAMx = PAM1x PAM250 = PAM1250
PAM250 is a widely used scoring matrix:
Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys ...
A R N D C Q E G H I L K ...
Ala A
Arg R
Asn N
Asp D
Cys C
Gln Q
...
Trp W
Tyr Y
Val V
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16. BLOSUM Blocks Substitution Matrix
Scores derived from observations of the frequencies of substitutions in blocks of local alignments in related proteins
Matrix name indicates evolutionary distance
BLOSUM62 was created using sequences sharing no more than 62% identity
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17. The Blosum50 Scoring Matrix
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18. Local vs. Global Alignment
The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph.
The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph.
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19. Local vs. Global Alignment
The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph.
The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph.
In the edit graph with negatively-scored edges, Local Alignment may score higher than Global Alignment
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20. Local vs. Global Alignment (cont’d)
Local Alignment—better alignment to find conserved segment
--T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC
| || | || | | | ||| || | | | | |||| |
AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C
tccCAGTTATGTCAGgggacacgagcatgcagagac
||||||||||||
aattgccgccgtcgttttcagCAGTTATGTCAGatc
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21. Local Alignment: Example
Compute a “mini” Global Alignment to get Local
Local alignment
Global alignment
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22. Local Alignments: Why?
Two genes in different species may be similar over short conserved regions and dissimilar over remaining regions.
Example:
Homeobox genes have a short region called the homeodomain that is highly conserved between species.
A global alignment would not find the homeodomain because it would try to align the ENTIRE sequence
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23. The Local Alignment Problem
Goal: Find the best local alignment between two strings
Input : Strings v, w and scoring matrix δ
Output : Alignment of substrings of v and w whose alignment score is maximum among all possible alignment of all possible substrings
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24. The Problem with this Problem
Long run time O(n4):
- In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source.
- For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time.
This can be remedied by giving free rides
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25. Local Alignment: Example
Compute a “mini” Global Alignment to get Local
Local alignment
Global alignment
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26. Local Alignment: Example
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27. Local Alignment: Example
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28. Local Alignment: Example
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29. Local Alignment: Example
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30. Local Alignment: Example
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31. Local Alignment: Running Time
Long run time O(n4):
- In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source.
- For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time.
This can be remedied by giving free rides
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32. Local Alignment: Free Rides
Yeah, a free ride!
Vertex (0,0)
The dashed edges represent the free rides from (0,0) to every other node.
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33. The Local Alignment Recurrence
The largest value of si,j over the whole edit graph is the score of the best local alignment.
The recurrence:
Notice there is only this change from the original recurrence of a Global Alignment
si,j = max si-1,j-1 + δ (vi, wj)
s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj) {
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34. The Local Alignment Recurrence
The largest value of si,j over the whole edit graph is the score of the best local alignment.
The recurrence:
Power of ZERO: there is only this change from the original recurrence of a Global Alignment - since there is only one “free ride” edge entering into every vertex
si,j = max si-1,j-1 + δ (vi, wj)
s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj) {
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35. Scoring Indels: Naive Approach
A fixed penalty σ is given to every indel:
-σ for 1 indel,
-2σ for 2 consecutive indels
-3σ for 3 consecutive indels, etc.
Can be too severe penalty for a series of 100 consecutive indels
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36. Affine Gap Penalties ATA__GC ATATTGC ATAG_GC AT_GTGC
In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events:
ATA__GC
ATATTGC
ATAG_GC
AT_GTGC
This is more likely.
This is less likely.
Normal scoring would give the same score for both alignments
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37. Accounting for Gaps
Gaps- contiguous sequence of spaces in one of the rows
Score for a gap of length x is:
-(ρ + σx)
where ρ >0 is the penalty for introducing a gap: gap opening penalty
ρ will be large relative to σ:
gap extension penalty
because you do not want to add too much of a penalty for extending the gap.
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38. Affine Gap Penalties Gap penalties: -ρ-σ when there is 1 indel
-ρ-2σ when there are 2 indels
-ρ-3σ when there are 3 indels, etc.
-ρ- x·σ (-gap opening - x gap extensions)
Somehow reduced penalties (as compared to naïve scoring) are given to runs of horizontal and vertical edges
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39. Affine Gap Penalties and Edit Graph
To reflect affine gap penalties we have to add “long” horizontal and vertical edges to the edit graph. Each such edge of length x should have weight
- - x *
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40. Adding “Affine Penalty” Edges to the Edit Graph
There are many such edges!
Adding them to the graph increases the running time of the alignment algorithm by a factor of n (where n is the number of vertices)
So the complexity increases from O(n2) to O(n3)
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41. Manhattan in 3 Layers ρ δ δ σ δ ρ δ δ σ
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42. Affine Gap Penalties and 3 Layer Manhattan Grid
The three recurrences for the scoring algorithm creates a 3-layered graph.
The top level creates/extends gaps in the sequence w.
The bottom level creates/extends gaps in sequence v.
The middle level extends matches and mismatches.
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43. Switching between 3 Layers
Levels:
The main level is for diagonal edges
The lower level is for horizontal edges
The upper level is for vertical edges
A jumping penalty is assigned to moving from the main level to either the upper level or the lower level (-r- s)
There is a gap extension penalty for each continuation on a level other than the main level (-s)
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44. The 3-leveled Manhattan Grid
Gaps in w
Matches/Mismatches
Gaps in v
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45. Affine Gap Penalty Recurrences
si,j = s i-1,j - σ
max s i-1,j –(ρ+σ)
si,j = s i,j-1 - σ
max s i,j-1 –(ρ+σ)
si,j = si-1,j-1 + δ (vi, wj)
max s i,j
s i,j
Continue Gap in w (deletion)
Start Gap in w (deletion): from middle
Continue Gap in v (insertion)
Start Gap in v (insertion):from middle
Match or Mismatch
End deletion: from top
End insertion: from bottom
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46. Next: Multiple Alignment
Dynamic Programming in 3-D
Progressive Alignment
Profile Progressive Alignment (ClustalW)
Scoring Multiple Alignments
Entropy
Sum of Pairs Alignment
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47. Multiple Alignment vs Pairwise Alignment
Up until now we have only tried to align two sequences.
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48. Multiple Alignment vs Pairwise Alignment
Up until now we have only tried to align two sequences.
What about more than two? And what for?
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49. Multiple Alignment vs Pairwise Alignment
Up until now we have only tried to align two sequences.
What about more than two? And what for?
A faint similarity between two sequences becomes significant if present in many
Multiple alignments can reveal subtle similarities that pairwise alignments do not reveal
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50. Generalizing Pairwise Alignment
Alignment of 2 sequences is represented as a
2-row matrix
In a similar way, we represent alignment of 3 sequences as a 3-row matrix
A T _ G C G _
A _ C G T _ A
A T C A C _ A
Score: more conserved columns, better alignment
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51. Alignments = Paths in… Align 3 sequences: ATGC, AATC,ATGC A -- T G C A
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52. Alignment Paths x coordinate 1 2 3 4 A -- T G C A T -- C -- A T G C1 2 3 4
x coordinate A -- T G C A T -- C -- A T G C
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53. Alignment Paths Align the following 3 sequences: ATGC, AATC,ATGC1 2 3 4
x coordinate A -- T G C
y coordinate 1 2 3 4 A T -- C -- A T G C
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54. Alignment Paths Resulting path in (x,y,z) space:1 2 3 4
x coordinate A -- T G C
y coordinate 1 2 3 4 A T -- C 1 2 3 4
z coordinate -- A T G C
Resulting path in (x,y,z) space:
(0,0,0)(1,1,0)(1,2,1) (2,3,2) (3,3,3) (4,4,4)
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55. Aligning Three Sequences
source
Same strategy as aligning two sequences
Use a 3-D “Manhattan Cube”, with each axis representing a sequence to align
For global alignments, go from source to sink
sink
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56. 2-D vs 3-D Alignment Grid V W 2-D edit graph 3-D edit graph 12/7/2018
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57. 2-D cell versus 2-D Alignment Cell
In 2-D, 3 edges in each unit square
In 3-D, 7 edges in each unit cube
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58. Architecture of 3-D Alignment Cell
(i-1,j,k-1)
(i-1,j-1,k-1)
(i-1,j-1,k)
(i-1,j,k)
(i,j,k-1)
(i,j-1,k-1)
(i,j,k)
(i,j-1,k)
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59. Multiple Alignment: Dynamic Prog.
cube diagonal: no indels
si,j,k = max
(x, y, z) is an entry in the 3-D scoring matrix
si-1,j-1,k-1 + (vi, wj, uk)
si-1,j-1,k +  (vi, wj, _ )
si-1,j,k  (vi, _, uk)
si,j-1,k  (_, wj, uk)
si-1,j,k +  (vi, _ , _)
si,j-1,k +  (_, wj, _)
si,j,k  (_, _, uk)
face diagonal: one indel
edge diagonal: two indels
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60. Multiple Alignment: Running Time
For 3 sequences of length n, the run time is 7n3; O(n3)
For k sequences, build a k-dimensional Manhattan, with run time (2k-1)(nk); O(2knk)
Conclusion: dynamic programming approach for alignment between two sequences is easily extended to k sequences but it is impractical due to exponential running time
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61. Multiple Alignment Induces Pairwise Alignments
Every multiple alignment induces pairwise alignments
x: AC-GCGG-C
y: AC-GC-GAG
z: GCCGC-GAG
Induces:
x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAG
y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG
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62. Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments
Given 3 arbitrary pairwise alignments:
x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC-GAG
y: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG
can we construct a multiple alignment that induces them?
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63. Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments
Given 3 arbitrary pairwise alignments:
x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC-GAG
y: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG
can we construct a multiple alignment that induces them?
NOT ALWAYS
Pairwise alignments may be inconsistent
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64. Inferring Multiple Alignment from Pairwise Alignments
From an optimal multiple alignment, we can infer pairwise alignments between all pairs of sequences, but they are not necessarily optimal
It is difficult to infer a ``good” multiple alignment from optimal pairwise alignments between all sequences
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65. Combining Optimal Pairwise Alignments into Multiple Alignment
Can combine pairwise alignments into multiple alignment
Cannot combine pairwise alignments into multiple alignment
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66. Profile Representation of Multiple Alignment
- A G G C T A T C A C C T G
T A G – C T A C C A G
C A G – C T A C C A G
C A G – C T A T C A C – G G
C A G – C T A T C G C – G G A C G T
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67. Profile Representation of Multiple Alignment
- A G G C T A T C A C C T G
T A G – C T A C C A G
C A G – C T A C C A G
C A G – C T A T C A C – G G
C A G – C T A T C G C – G G A C G T
In the past we were aligning a sequence against a sequence
Can we align a sequence against a profile?
Can we align a profile against a profile?
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68. Aligning alignments Given two alignments, can we align them?
x GGGCACTGCAT
y GGTTACGTC-- Alignment 1
z GGGAACTGCAG
w GGACGTACC-- Alignment 2
v GGACCT-----
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69. Aligning alignments Given two alignments, can we align them?
Hint: use alignment of corresponding profiles
x GGGCACTGCAT
y GGTTACGTC-- Combined Alignment
z GGGAACTGCAG
w GGACGTACC--
v GGACCT-----
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70. Multiple Alignment: Greedy Approach
Choose most similar pair of strings and combine into a profile , thereby reducing alignment of k sequences to an alignment of of k-1 sequences/profiles. Repeat
This is a heuristic greedy method
u1= ACg/tTACg/tTACg/cT…
u2 = TTAATTAATTAA… …
uk = CCGGCCGGCCGG…
u1= ACGTACGTACGT…
u2 = TTAATTAATTAA…
u3 = ACTACTACTACT… …
uk = CCGGCCGGCCGG
k-1 k
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71. Greedy Approach: Example
Consider these 4 sequences
s1 GATTCA
s2 GTCTGA
s3 GATATT
s4 GTCAGC
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72. Greedy Approach: Example (cont’d)
There are = 6 possible alignments
s2 GTCTGA
s4 GTCAGC (score = 2)
s1 GAT-TCA
s2 G-TCTGA (score = 1)
s3 GATAT-T (score = 1)
s1 GATTCA--
s4 G—T-CAGC(score = 0)
s2 G-TCTGA
s3 GATAT-T (score = -1)
s3 GAT-ATT
s4 G-TCAGC (score = -1)
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73. Greedy Approach: Example (cont’d)
s2 and s4 are closest; combine:
s2 GTCTGA
s4 GTCAGC
s2,4 GTCt/aGa/cA (profile)
new set of 3 sequences:
s1 GATTCA
s3 GATATT
s2,4 GTCt/aGa/c
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74. Progressive Alignment
Progressive alignment is a variation of greedy algorithm with a somewhat more intelligent strategy for choosing the order of alignments.
Progressive alignment works well for close sequences, but deteriorates for distant sequences
Gaps in consensus string are permanent
Use profiles to compare sequences
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75. ClustalW Popular multiple alignment tool today
‘W’ stands for ‘weighted’ (different parts of alignment are weighted differently).
Three-step process
1.) Construct pairwise alignments
2.) Build Guide Tree
3.) Progressive Alignment guided by the tree
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76. Step 1: Pairwise Alignment
Aligns each sequence again each other giving a similarity matrix
Similarity = exact matches / sequence length (percent identity)
v1 v2 v3 v4
v1 - v v v
(.17 means 17 % identical)
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77. Step 2: Guide Tree Create Guide Tree using the similarity matrix
ClustalW uses the neighbor-joining method
Guide tree roughly reflects evolutionary relations
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78. Step 2: Guide Tree (cont’d)v1 v3 v4 v2
v1 v2 v3 v4
v1 - v v v
Calculate: v1,3 = alignment (v1, v3) v1,3,4 = alignment((v1,3),v4) v1,2,3,4 = alignment((v1,3,4),v2)
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79. Step 3: Progressive Alignment
Start by aligning the two most similar sequences
Following the guide tree, add in the next sequences, aligning to the existing alignment
Insert gaps as necessary
FOS_RAT PEEMSVTS-LDLTGGLPEATTPESEEAFTLPLLNDPEPK-PSLEPVKNISNMELKAEPFD
FOS_MOUSE PEEMSVAS-LDLTGGLPEASTPESEEAFTLPLLNDPEPK-PSLEPVKSISNVELKAEPFD
FOS_CHICK SEELAAATALDLG----APSPAAAEEAFALPLMTEAPPAVPPKEPSG--SGLELKAEPFD
FOSB_MOUSE PGPGPLAEVRDLPG-----STSAKEDGFGWLLPPPPPPP LPFQ
FOSB_HUMAN PGPGPLAEVRDLPG-----SAPAKEDGFSWLLPPPPPPP LPFQ
. . : ** :.. *:.* * . * **:
Dots and stars show how well-conserved a column is.
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80. Multiple Alignments: Scoring
Number of matches (multiple longest common subsequence score)
Entropy score
Sum of pairs (SP-Score)
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81. Multiple LCS Score
A column is a “match” if all the letters in the column are the same
Only good for very similar sequences
AAA
AAT
ATC
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82. Entropy
Define frequencies for the occurrence of each letter in each column of multiple alignment
pA = 1, pT=pG=pC=0 (1st column)
pA = 0.75, pT = 0.25, pG=pC=0 (2nd column)
pA = 0.50, pT = 0.25, pC=0.25 pG=0 (3rd column)
Compute entropy of each column
AAA
AAT
ATC
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83. Entropy: Example
Best case
Worst case
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84. Multiple Alignment: Entropy Score
Entropy for a multiple alignment is the sum of entropies of its columns:
 over all columns  X=A,T,G,C pX logpX
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85. Entropy of an Alignment: Example
column entropy: -( pAlogpA + pClogpC + pGlogpG + pTlogpT) A C G T
Column 1 = -[1*log(1) + 0*log0 + 0*log0 +0*log0] = 0
Column 2 = -[(1/4)*log(1/4) + (3/4)*log(3/4) + 0*log0 + 0*log0] = -[ (1/4)*(-2) + (3/4)*(-.415) ] =
Column 3 = -[(1/4)*log(1/4)+(1/4)*log(1/4)+(1/4)*log(1/4) +(1/4)*log(1/4)] = 4* -[(1/4)*(-2)] = +2.0
Alignment Entropy = =
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86. Multiple Alignment Induces Pairwise Alignments
Every multiple alignment induces pairwise alignments
x: AC-GCGG-C
y: AC-GC-GAG
z: GCCGC-GAG
Induces:
x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAG
y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG
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87. Inferring Pairwise Alignments from Multiple Alignments
From a multiple alignment, we can infer pairwise alignments between all sequences, but they are not necessarily optimal
This is like projecting a 3-D multiple alignment path on to a 2-D face of the cube
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88. Multiple Alignment Projections
A 3-D alignment can be projected onto the 2-D plane to represent an alignment between a pair of sequences
All 3 Pairwise Projections of the Multiple Alignment
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89. Sum of Pairs Score(SP-Score)
Consider pairwise alignment of sequences
ai and aj
imposed by a multiple alignment of k sequences
Denote the score of this suboptimal (not necessarily optimal) pairwise alignment as
s*(ai, aj)
Sum up the pairwise scores for a multiple alignment:
s(a1,…,ak) = Σi,j s*(ai, aj)
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90. Computing SP-Score Aligning 4 sequences: 6 pairwise alignments
Given a1,a2,a3,a4:
s(a1…a4) = s*(ai,aj) = s*(a1,a2) + s*(a1,a3) s*(a1,a4) + s*(a2,a3) s*(a2,a4) + s*(a3,a4)
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91. SP-Score: Example a1 ATG-C-AAT . A-G-CATAT ak ATCCCATTT
To calculate each column: s
s*(
Pairs of Sequences A G 1
Score=3 1 -m 1
Score = 1 – 2m A A C G 1 -m
Column 1
Column 3
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92. Multiple Alignment: History
1975 Sankoff
Formulated multiple alignment problem and gave dynamic programming solution
1988 Carrillo-Lipman
Branch and Bound approach for MSA
1990 Feng-Doolittle
Progressive alignment
1994 Thompson-Higgins-Gibson-ClustalW
Most popular multiple alignment program
1998 Morgenstern et al.-DIALIGN
Segment-based multiple alignment
2000 Notredame-Higgins-Heringa-T-coffee
Using the library of pairwise alignments
2004 MUSCLE
What’s next?
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93. Problems with Multiple Alignment
Multidomain proteins evolve not only through point mutations but also through domain duplications and domain recombinations
Although MSA is a 30 year old problem, there were no MSA approaches for aligning rearranged sequences (i.e., multi-domain proteins with shuffled domains) prior to 2002
Often impossible to align all protein sequences throughout their entire length
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94. Next: Gene prediction
Similarity-Based Approaches
de Novo approaches
Note: HW2 is about a de novo (or ab initio) approach.
Some of the following slides are based on slides by the authors of our text.
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95. Using Known Genes to Predict New Genes
Some genomes may be very well-studied, with many genes having been experimentally verified.
Closely-related organisms may have similar genes
Unknown genes in one species may be compared to genes in some closely-related species
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96. Similarity-Based Approach to Gene Prediction
Genes in different organisms are similar
The similarity-based approach uses known genes in one genome to predict (unknown) genes in another genome
Problem: Given a known gene and an unannotated genome sequence, find a set of substrings of the genomic sequence whose concatenation best fits the gene
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97. Comparing Genes in Two Genomes
Small islands of similarity corresponding to similarities between exons
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98. Reverse Translation
Given a known protein, find a gene in the genome which codes for it
One might infer the coding DNA of the given protein by reversing the translation process
Inexact: amino acids map to > 1 codon
This problem is essentially reduced to an alignment problem
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99. Reverse Translation (cont’d)
This reverse translation problem can be modeled as traveling in Manhattan grid with free horizontal jumps
Complexity of Manhattan is n3
Every horizontal jump models an insertion of an intron
Problem with this approach: would match nucleotides pointwise and use horizontal jumps at every opportunity
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100. Comparing Genomic DNA Against mRNA
Portion of genome
(codon sequence)
mRNA
exon3
exon1
exon2 {
intron1
intron2
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101. Using Similarities to Find the Exon Structure
The known frog gene is aligned to different locations in the human genome
Find the “best” path to reveal the exon structure of human gene
Frog Gene (known)
Human Genome
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102. Finding Local Alignments
Use local alignments to find all islands of similarity
Human Genome
Frog Genes (known)
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103. Chaining Local Alignments
Find substrings that match a given gene sequence (candidate exons)
Define a candidate exons as
(l, r, w)
(left, right, weight defined as score of local alignment)
Look for a maximum chain of substrings
Chain: a set of non-overlapping nonadjacent intervals.
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104. Exon Chaining Problem 3 4 11 9 15 5 2 6 13 16 20 25 27 28 30 32
Locate the beginning and end of each interval (2n points)
Find the “best” path
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105. Exon Chaining Problem: Formulation
Exon Chaining Problem: Given a set of putative exons, find a maximum set of non-overlapping putative exons
Input: a set of weighted intervals (putative exons)
Output: A maximum chain of intervals from this set
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106. Exon Chaining Problem: Formulation
Exon Chaining Problem: Given a set of putative exons, find a maximum set of non-overlapping putative exons
Input: a set of weighted intervals (putative exons)
Output: A maximum chain of intervals from this set
Would a greedy algorithm solve this problem?
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107. Exon Chaining Problem: Graph Representation
This problem can be solved with dynamic programming in O(n) time.
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108. Exon Chaining Algorithm
ExonChaining (G, n) //Graph, number of intervals
for i ← to 2n
si ← 0
for i ← 1 to 2n
if vertex vi in G corresponds to right end of the interval I
j ← index of vertex for left end of the interval I
w ← weight of the interval I
sj ← max {sj + w, si-1}
else
si ← si-1
return s2n
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109. Exon Chaining: Deficiencies
Poor definition of the putative exon endpoints
Optimal chain of intervals may not correspond to any valid alignment
First interval may correspond to a suffix, whereas second interval may correspond to a prefix
Combination of such intervals is not a valid alignment
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