pairwise sequence alignment basic algorithmsbccg131/wiki.files/2 pariwise...3 literature list •...
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Pairwise Sequence alignment Basic Algorithms
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Agenda - Previous Lesson: Minhala
- + Biological Story on Biomolecular Sequences
- + General Overview of Problems in Computational
Biology
Today:
- Reminder: Dynamic Programming
-Algorithms for Global and Local Sequence Alignment + variants
-Bioinformatic Motivation for Sequence Alignment
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Literature list
• Alberts, B et al. Essential Cell Biology: An introducton to the Molecular Biology of the Cell.
• Mount, D.W. Bioinformatics: Sequence and Genome Analysis.
• Jones N.C & Pevzner P.A. An introduction to Bioinformatics algorithms.
• R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids.
• Dan Gusfield. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology.
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• Move to Slides On Dynamic Programming…
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Sequence Comparison (cont)
• We seek the following similarities between sequences :
• Find similar proteins – Allows to predict function & structure
• Locate similar subsequences in DNA – Allows to identify (e.g) regulatory elements
• Locate DNA sequences that might overlap – Helps in sequence assembly
g1
g2
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Sequence Modifications
• Three types of changes
– Substitution (point mutation)
– Insertion
– Deletion
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TCAGT TCGAGT
TCCGT
TCGT
TCAGT
Indel (replication slippage)
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Choosing Alignments
There are many possible alignments For example, compare:
-GCGC-ATGGATTGAGCGA
TGCGCCATTGAT-GACC-A
to ------GCGCATGGATTGAGCGA
TGCGCC----ATTGATGACCA--
Which one is better?
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Another example
Given two sequences:
X: TGCATAT
Y: ATCCGAT
Question:
How can X be transformed into Y?
Or,
How did Y evolve from X?
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TGCATAT
TGCATA
TGCAT
ATGCAT
ATCCAT
ATCCGAT
delete T
delete A
insert A
G C
insert G
One possible transformation
Alignment:
-TGC-ATAT
ATCCGAT--
5 operations
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-TGCATAT
ATCCG-AT
TGCATAT
ATGCATAT
ATGCAAT
ATGCGAT
ATCCGAT
insert A
delete T
A G
Another possible transformation
Alignment:
4 operations
G C
Which one is better?
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In order to align two sequences we need a quantitive model to evaluate similarity between sequences.
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How do we quantitate sequence similarity?
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Scoring Similarity
• Assume independent mutation model
– Each site considered separately
• Score at each site
– Positive if the same
– Negative if different
• Sum to make final score
– Can be positive or negative
– Significance depends on sequence length
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GTAGTC
CTAGCG
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Pairwise Alignment - Identity
(HH) VLSPADKTNVKAAWGKVGAHAGYEG
||| | | || | |
(SWM) VLSEGEWQLVLHVWAKVEADVAGHG
• Percent Identity: 36.000 (| only)
Human Hemoglobin (HH) vs Sperm Whale Myoglobin (SWM):
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Pairwise Alignment - Similarity
(HH) VLSPADKTNVKAAWGKVGAHAGYEG
||| . | | || | |
(SWM) VLSEGEWQLVLHVWAKVEADVAGHG
• Percent Similarity: 40.000 (| and .)
• Percent Identity: 36.000 (| only)
D and E are similar:
1. structure is similar.
2. both are acidic and hydrophilic
3. one mutation can separate them
from one to the other.
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Pairwise Alignment – Gap insertion
(HH) VLSPADKTNVKAAWGKVGAH-AGYEG
.
(SWM) VLSEGEWQLVLHVWAKVEADVAGH-G
• Gaps: 2
• Percent Similarity: 54.167
• Percent Identity: 45.833 (12/26)
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Pairwise Alignment - Scoring
• The final score of the alignment is the sum of the positive scores and penalty scores:
+ Number of Identities
+ Number of Similarities
- Number of gap insertions
Alignment score
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Pairwise Alignment - Scoring
(HH) VLSPADKTNVKAAWGKVGAH-AGYEG
||| . | | || || |
(SWM) VLSEGEWQLVLHVWAKVEADVAGH-G
Final score:
(V,V) + (L,L) + (S,S) + (D,E) + … - (penalty for gap insertion)*(number of gaps) - (penalty for gap extension)*(extension length)
We are interested in both the score and the alignment trace.
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Optimum Alignment
The score of an alignment is a measure of its
quality
Optimum alignment problem: Given a pair of
sequences X and Y, find an alignment (global or
local) with maximum score
The similarity between X and Y, denoted
sim(X,Y), is the maximum score of an alignment of X and Y
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Computing Optimal Score • How can we compute the optimal score ?
– If |s| = n and |t| = m, the number A(m,n) of possible “legal” alignments is large!
• we perform dynamic programming to compute the optimal score efficiently.
222( , ) ( , )
n
n
nA m n A n n
n
Stirling’s formula: Exercise 1 2! 2 x xx x e m of the
order of n
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Manhattan Tourist Problem (MTP)
Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Sink
*
*
*
*
*
* *
* *
*
*
Source
*
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Manhattan Tourist Problem (MTP)
Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Sink
*
*
*
*
*
* *
* *
*
*
Source
*
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Manhattan Tourist Problem: Formulation
Goal: Find the longest (highest scoring) path in a weighted grid.
Input: A weighted grid G with two distinct vertices, one labeled “source” and the other labeled “sink”
Output: A longest path in G from “source” to “sink”
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MTP: An Example
3 2 4
0 7 3
3 3 0
1 3 2
4
4
5
6
4
6
5
5
8
2
2
5
0 1 2 3
0
1
2
3
j coordinate i c
oo
rdin
ate
13
source
sink
4
3 2 4 0
1 0 2 4 3
3
1
1
2
2
2
4 19
9 5
15
23
0
20
3
4
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MTP: Greedy Algorithm Is Not Optimal 1 2 5
2 1 5
2 3 4
0 0 0
5
3
0
3
5
0
10
3
5
5
1
2 promising start, but leads to bad choices!
source
sink 18
22
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1
5
0 1
0
1
i
source
1
5
S1,0 = 5
S0,1 = 1
• Calculate optimal path score for each vertex in the graph
• Each vertex’s score is the maximum of the prior vertices score plus the weight of the respective edge in between
MTP: Dynamic Programming j
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MTP: Dynamic Programming
(cont’d)
1 2
5
3
0 1 2
0
1
2
source
1 3
5
8
4
S2,0 = 8
i
S1,1 = 4
S0,2 = 3 3
-5
j
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MTP: Dynamic Programming
(cont’d)
1 2
5
3
0 1 2 3
0
1
2
3
i
source
1 3
5
8
8
4
0
5
8
10 3
5
-5
9
13
1 -5
S3,0 = 8
S2,1 = 9
S1,2 = 13
S3,0 = 8
j
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MTP: Dynamic Programming (cont’d)
greedy alg. fails!
1 2 5
-5 1 -5
-5 3
0
5
3
0
3
5
0
10
-3
-5
0 1 2 3
0
1
2
3
i
source
1 3 8
5
8
8
4
9
13 8
9
12
S3,1 = 9
S2,2 = 12
S1,3 = 8
j
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MTP: Dynamic Programming
(cont’d)
1 2 5
-5 1 -5
-5 3 3
0 0
5
3
0
3
5
0
10
-3
-5
-5
2
0 1 2 3
0
1
2
3
i
source
1 3 8
5
8
8
4
9
13 8
12
9
15
9
j
S3,2 = 9
S2,3 = 15
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MTP: Dynamic Programming
(cont’d)
1 2 5
-5 1 -5
-5 3 3
0 0
5
3
0
3
5
0
10
-3
-5
-5
2
0 1 2 3
0
1
2
3
i
source
1 3 8
5
8
8
4
9
13 8
12
9
15
9
j
0
1
16 S3,3 = 16
(showing all back-traces)
Done!
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MTP: Recurrence
Computing the score for a point (i,j) by the recurrence relation:
si, j = max
si-1, j + weight of the edge between (i-1, j) and (i, j)
si, j-1 + weight of the edge between (i, j-1) and (i, j)
The running time is n x m for a n by m grid
(n = # of rows, m = # of columns)
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What about diagonals?
• The score at point B is given by:
sB = max of
sA1 + weight of the edge (A1, B)
sA2 + weight of the edge (A2, B)
sA3 + weight of the edge (A3, B)
B
A3
A1
A2
Adding Diagonal Edges to the Grid
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More generally, computing the score for point x is given by the recurrence relation:
sx = max
of
sy + weight of vertex (y, x) where
y є Predecessors(x)
• Predecessors (x) – set of vertices that have edges leading to x
Adding Diagonal Edges to the Grid
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Traveling in the Grid •The only hitch is that one must decide on the order in which visit the vertices
•By the time the vertex x is analyzed, the values sy for all its predecessors y should be computed – otherwise we are in trouble.
•We need to traverse the vertices in some order
•Try to find such order for a directed acyclic grid graph
???
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Traversing the Manhattan Grid
• 3 different strategies:
• a) Column by column
• b) Row by row
• c) Along diagonals
a) b)
c)
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Comparison methods
• Global alignment – Finds the best alignment across the whole two sequences.
• Local alignment – Finds regions of similarity in parts of the sequences. Global Local
_____ _______ __ ____
__ ____ ____ __ ____
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Global Alignment
• Algorithm of Needleman and Wunsch (1970)
• Finds the alignment of two complete sequences: ADLGAVFALCDRYFQ
|||| |||| |
ADLGRTQN-CDRYYQ
• Some global alignment programs “trim ends”
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Local Alignment
• Algorithm of Smith and Waterman (1981).
• Makes an optimal alignment of the best segment of
similarity between two sequences.
ADLG CDRYFQ
|||| |||| |
ADLG CDRYYQ
• Can return a number of highly aligned segments.
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Global Alignment: Algorithm
1..j1..i T and S of alignment optimum of Cost),( jiC
T of jlength of Prefix
S of i length of Prefix
..1
..1
j
i
T
S
ba
babaw
if
if),(
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)1j,i(C
)j,1i(C
)T,S(w)1j,1i(C
max)j,i(Cji
j)j,0(Ci)0,i(C
Initial conditions:
Recurrence relation: For 1 i n, 1 j m:
Theorem. C(i,j) satisfies the following relationships:
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Example
Case 1: Line up Si with Tj
S: C A T T C A C
T: C - T T C A G
i - 1 i
j j -1
S: C A T T C A - C
T: C - T T C A G -
Case 2: Line up Si with space i - 1 i
j
S: C A T T C A C -
T: C - T T C A - G
Case 3: Line up Tj with space i
j j -1
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Justification: Optimal Substructure Property Followed
S1 S2 . . . Si-1 Si
T1 T2 . . . Tj-1 Tj
C(i-1,j-1) + w(Si,Tj)
S1 S2 . . . Si-1 Si
T1 T2 . . . Tj —
C(i-1,j)
S1 S2 . . . Si —
T1 T2 . . . Tj-1 Tj
C(i,j-1)
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Computation Procedure
C(n,m)
C(0,0)
C(i,j)
)1j,i(C,)j,1i(C),T,S(w)1j,1i(Cmax)j,i(C ji
C(i-1,j) C(i-1,j-1)
C(i,j-1)
![Page 44: Pairwise Sequence alignment Basic Algorithmsbccg131/wiki.files/2 pariwise...3 Literature list • Alberts, B et al. Essential Cell Biology: An introducton to the Molecular Biology](https://reader036.vdocument.in/reader036/viewer/2022070802/5f02ea917e708231d406a573/html5/thumbnails/44.jpg)
44
λ C T C G C A G C
A
C
T
T
C
A
C
+10 for match, -2 for mismatch, -5 for space
0 -5 -10 -15 -20 -25 -30 -35 -40
-5
-10
-15
-20
-25
-30
-35
10 5
λ
![Page 45: Pairwise Sequence alignment Basic Algorithmsbccg131/wiki.files/2 pariwise...3 Literature list • Alberts, B et al. Essential Cell Biology: An introducton to the Molecular Biology](https://reader036.vdocument.in/reader036/viewer/2022070802/5f02ea917e708231d406a573/html5/thumbnails/45.jpg)
45
0 -5 -10 -15 -20 -25 -30 -35 -40
-5 10 5 0 -5 -10 -15 -20 -25
-10 5 8 3 -2 -7 0 -5 -10
-15 0 15 10 5 0 -5 -2 -7
-20 -5 10 13 8 3 -2 -7 -4
-25 -10 5 20 15 18 13 8 3
-30 -15 0 15 18 13 28 23 18
-35 -20 -5 10 13 28 23 26 33
λ C T C G C A G C
A
C
T
T
C
A
C
λ
Traceback can yield both optimum alignments
*
*