BCB 444/544
Lecture 7
Still more: Dynamic Programming
Global vs Local Alignment
Scoring Matrices & Alignment Statistics
BLAST nope
#7_Sept5
BCB 444/544 F07 ISU Dobbs #7 - Still more DP, Scoring Matrices
9/5/07
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Required Reading
(before lecture)
√Last week: - for Lectures 4-7
Pairwise Sequence Alignment, Dynamic Programming,
Global vs Local Alignment, Scoring Matrices, Statistics
• Xiong: Chp 3
• Eddy: What is Dynamic Programming? 2004 Nature Biotechnol 22:909
http://www.nature.com/nbt/journal/v22/n7/abs/nbt0704-909.html
Wed Sept 5 - for Lecture 7 & Lab 3
Database Similarity Searching: BLAST
• Chp 4 - pp 51-62
Fri Sept - for Lecture 8
BLAST variations; BLAST vs FASTA
• Chp 4 - pp 51-62
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Assignments & Announcements
√Tues Sept 4 - Lab #2 Exercise Writeup due by 5 PM
Send via email to Pete Zaback petez@iastate.edu
(For now, no late penalty - just send ASAP)
√Wed Sept 5 - Notes for Lecture 5 posted online
- HW#2 posted online & sent via email
& handed out in class
Fri Sept 14
- HW#2 Due by 5 PM
Fri Sept 21
- Exam #1
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Chp 3- Sequence Alignment
SECTION II
SEQUENCE ALIGNMENT
Xiong: Chp 3
Pairwise Sequence Alignment
•
•
•
•
•
•
√Evolutionary Basis
√Sequence Homology versus Sequence Similarity
√Sequence Similarity versus Sequence Identity
Methods - cont
Scoring Matrices
Statistical Significance of Sequence Alignment
Adapted from Brown and Caragea, 2007, with some slides from:
Altman, Fernandez-Baca, Batzoglou, Craven, Hunter, Page.
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Methods
• √Global and Local Alignment
• √Alignment Algorithms
• √Dot Matrix Method
• Dynamic Programming Method - cont
• Gap penalities
• DP for Global Alignment
• DP for Local Alignment
• Scoring Matrices
• Amino acid scoring matrices
• PAM
• BLOSUM
• Comparisons between PAM & BLOSUM
• Statistical Significance of Sequence Alignment
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Global vs Local Alignment
Global alignment
• Finds best possible alignment across entire length of 2 sequences
• Aligned sequences assumed to be generally similar over entire length
Local alignment
• Finds local regions with highest similarity between 2 sequences
• Aligns these without regard for rest of sequence
• Sequences are not assumed to be similar over entire length
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Global vs Local Alignment - example
1 = CTGTCGCTGCACG
2 = TGCCGTG
Global alignment
CTGTCGCTGCACG
-TG-C-C-G--TG
Local alignment
CTGTCGCTGCACG
-TGCCG-TG----
CTGTCGCTGCACG
-TGCCG-T----G
Which is better?
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Global vs Local Alignment
Which should be used when?
It is critical to choose correct method!
Global Alignment
vs
Local Alignment?
Shout out the answers!! Which should we use for?
1.
2.
3.
4.
5.
Searching for conserved motifs in DNA or protein sequences?
Aligning two closely related sequences with similar lengths?
Aligning highly divergent sequences?
Generating an extended alignment of closely related sequences?
Generating an extended alignment of closely related sequences
with very different lengths?
Hmmm - we'll work on that
Excellent!
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Global vs Local Alignment
Which should be used when?
It is critical to choose correct method!
Global Alignment
vs
Local Alignment?
Shout out the answers!! Which should we use for?
1. Searching for conserved motifs in DNA or protein sequences? Local
2. Aligning two closely related sequences with similar lengths?
Global
3. Aligning highly divergent sequences? Local (at least initially)
4. Generating an extended alignment of closely related sequences? Global
5. Generating an extended alignment of closely related sequences
with very different lengths? Hmmm - we'll work on that
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Alignment Algorithms
3 major methods for pairwise sequence alignment:
1. Dot matrix analysis √ - practice in HW2
2. Dynamic programming - more today & in HW2
3. Word or k-tuple methods (later, in Chp 4)
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Dynamic Programming
For Pairwise sequence alignment
Idea: Display one sequence above another with
spaces inserted in both to reveal similarity
CAT-TCA-C
| | || |
C-TCGCAGC
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Global Alignment: Scoring
CTGTCGCTGCACG
-TGC-CG-TG---Reward for matches: 
Mismatch penalty:

Space/gap penalty: 
Score = w – x - y
w = #matches
x = #mismatches
y = #spaces
Note: I changed symbols
& colors on this slide!
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Global Alignment: Scoring
Reward for matches:
Mismatch penalty:
Space/gap penalty:
10
-2
-5
C T G T C G – C T G C
- T G C – C G – T G -5 10 10 -2 -5 -2 -5 -5 10 10 -5
Note: I changed symbols
& colors on this slide!
Total = 11
We could have done better!!
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Alignment Algorithms
• Global: Needleman-Wunsch
• Local: Smith-Waterman
• Both NW and SW use dynamic programming
• Variations:
• Gap penalty functions
• Scoring matrices
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Dynamic Programming - Key Idea:
The score of the best possible alignment that ends at a
given pair of positions (i, j) is equal to:
the score of best alignment ending just previous to
those two positions (i.e., ending at i-1, j-1)
PLUS
the score for aligning xi and yj
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Global Alignment:
DP Problem Formulation & Notations
Given two sequences (strings)
• X = x1x2…xN of length N
x = AGC
N=3
• Y = y1y2…yM of length M
y = AAAC
M=4
Construct a matrix with (N+1) x (M+1) elements, where
S(i,j) = Score of best alignment of x[1..i]=x1x2…xi with y[1..j]=y1y2…yj
x1
x2
x3
Which means:
S(i,j) = Score of best alignment of
a prefix of X and a prefix of Y
y1
y2
y3
S(2,3) = score of best alignment
of AG (x1x2) to AAA (y1y2y3)
y4
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Dynamic Programming - 4 Steps:
1. Define score of optimal alignment, using recursion
2. Initialize and fill in a DP matrix for storing optimal
scores of subproblems, by solving smallest
subproblems first (bottom-up approach)
3. Calculate score of optimal alignment(s)
4. Trace back through matrix to recover optimal
alignment(s) that generated optimal score
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1- Define Score of Optimal Alignment
using Recursion
Define:
x1..i  Prefix of length i of x
y1.. j  Prefix of length j of y
S(i, j)  Score of optimal alignment of x1..i and y1..j

Initial
conditions:
S(i,0)  i   S(0, j)   j  
 = Match Reward
 = Mismatch Penalty
 = Gap penalty
Recursive definition:
For 1  i  N, 1  j  M:

S(i 1, j 1)   (xi , y j )

S(i, j)  max S(i 1, j)

S(i, j 1)


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(xi,yj) = 

or

= Gap penalty
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2- Initialize & Fill in DP Matrix for
Storing Optimal Scores ofSubproblems
• Construct sequence vs sequence matrix
• Fill in from [0,0] to [N,M] (row by row), calculating best
possible score for each alignment ending at residues at [i,j]
0
0
1
1
N
S(0,0)=0
S(i,j)
M
S(N,M)
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How do we calculate S(i,j)?
i.e., Score for alignment of x[1..i] to y[1..j]?
1 of 3 cases  optimal score for this subproblem:
xi aligns to yj
xi aligns to a gap
yj aligns to a gap
x1 x2 . . . xi-1 xi
x1 x2 . . . xi-1 xi
x1 x2 . . . x i
y1 y2 . . . yj-1 yj
y1 y2 . . . yj
y1 y2 . . . yj-1 yj
S(i-1,j-1) + (xi,yj)
S(i-1,j)
—
-
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S(i,j-1)
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—
-
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Note: I changed sequences
on this slide (to match the
rest of DP example)
Specific Example:
Case 1: Line up xi with yj
x: C
y: C
A
T
T
T
i-1
C G
C A
j-1
Case 2: Line up xi with space
x: C
y: C
A
-
T
T
T
T
C
C
Case 3: Line up yj with space
x: C
y: C
A
-
T
T
T
T
C
C
i
C
C
j
i-1
A A G
j
Scoring Consequence?
A
Mismatch Penalty
i
C
-
Space Penalty
i
A C A - G
j -1 j
Space Penalty
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Ready? Fill in DP Matrix
Keep track of dependencies of scores (in a pointer matrix)
0
1
0
S(0,0)=0
1
 = Match Reward
 = Mismatch Penalty
 = Gap penalty
M
Initialization
S(i,0)  i  
S(0, j)   j  
N
+
(xi,yj) = 
or

S(i-1,j-1)
S(i-1,j)
S(i,j-1)
S(i,j)
-
-
S(N,M)
Recursion
S(i 1, j 1)   (xi , y j )

S(i, j)  max S(i 1, j)

S(i, j 1)


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Fill in the DP matrix !!
λ
λ
0
-5
C
A -10
T
-15
T
-20
C
A
-25
-30
C
-35
C
T
C
G
C
A
G
C
-5 -10 -15 -20 -25 -30 -35 -40
10
5
+10 for match, -2 for mismatch, -5 for space
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3- Calculate Score S(N,M) of Optimal
Alignment - for Global Alignment
λ
C
T
C
G
C
A
G
C
λ
0
-5
-10
-15
-20
-25
-30
-35
-40
C
A
-5
10
5
0
-5
-10
-15
-20
-25
-10
5
8
3
-2
-7
0
-5
-10
T
-15
0
15
10
5
0
-5
-2
-7
T
-20
-5
10
13
8
3
-2
-7
-4
C
A
-25
-10
5
20
15
18
13
8
3
-30
-15
0
15
18
13
28
23
18
C
-35
-20
-5
10
13
28
23
26
33
+10 for match, -2 for mismatch, -5 for space
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4- Trace back through matrix to recover
optimal alignment(s) that generated
the optimal score
How? "Repeat" alignment calculations in reverse order,
starting at from position with highest score and
following path, position by position, back through
matrix
Result? Optimal alignment(s) of sequences
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Traceback - for Global Alignment
Start in lower right corner & trace back to upper left
Each arrow introduces one character at end of alignment:
• A horizontal move puts a gap in left sequence
• A vertical move puts a gap in top sequence
• A diagonal move uses one character from each sequence
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Traceback to Recover Alignment
λ
C
λ
0
d -5
C
A
-5
10v
-10
5
d 8
T
-15
0
15
T
-20
-5
C
A
-25
C
T
C
G
C
A
G
C
-10
-15
-20
-25
-30
-35
-40
5
0
-5
-10
-15
-20
-25
-2
-7
0
-5
-10
10
d 5
0
-5
-2
-7
10d
13
8
3
-2
-7
-4
-10
5
20
13
8
3
-30
-15
0
23 d
18
-35
-20
-5
26
33
h
1
3
h
d
d
15
18
15
18
13
28
10
13
28
23
2
h
Can have >1 optimal alignment; this example has 2
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What are the 2 Global Alignments
with Optimal Score = 33?
1:
2:
C
T
C
G
C
A
G
C
A
T
T
C
A
C
C
T
C
G
C
A
G
C
C
T
C
G
C
A
G
C
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Local Alignment: Motivation
• To "ignore" stretches of non-coding DNA:
• Non-coding regions (if "non-functional") are more likely to
contain mutations than coding regions
• Local alignment between two protein-encoding sequences is
likely to be between two exons
• To locate protein domains or motifs:
• Proteins with similar structures and/or similar functions but
from different species (for example), often exhibit local
sequence similarities
• Local sequence similarities may indicate ”functional modules”
Non-coding - "not encoding protein"
Exons - "protein-encoding" parts of genes
vs Introns = "intervening sequences" - segments of eukaryotic
genes that "interrupt" exons
Introns are transcribed into RNA, but are later removed by
RNA processing & are not translated into protein
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Local Alignment: Example
G G T C T G A G
A A A C G A
Match: +2
Mismatch or space: -1
Best local alignment:
G G T C T G A G
A A A C – G A -
Score = 5
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Local Alignment: Algorithm
•S [i, j] = Score for optimally aligning a suffix of X with
a suffix of Y
• Initialize top row & leftmost column of matrix with "0"
Recall: for Global Alignment,
• S [i, j] = Score for optimally aligning a prefix of X with a prefix of Y
• Initialize top row & leftmost column of with gap penalty
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Traceback - for Local Alignment
λ
C
T
C
G
C
A
G
C
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
0
0
0
0
0
0
2
0
0
T
0
0
1
0
0
0
0
1
0
T
0
0
1
0
0
0
0
0
0
0
1
0
2
0
1
0
0
1
0
0
0
0
1
0
2
0
0
0
1
0
1
0
2
0
1
1
λ
C
A
C
A
C
+1 for a match, -1 for a mismatch, -5 for a space
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What are the 4 Local Alignments with
Optimal Score = 2?
C
C
T
A
C
T
G
T
C
C
A
A
G
C
C
1:
C
T
C
G
C
A
G
C
2:
C
T
C
G
C
A
G
C
3:
C
T
C
G
C
A
G
C
4:
C
T
C
G
C
A
G
C
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Some Results re: Alignment Algorithms
(for ComS, CprE & Math types!)
• Most pairwise sequence alignment problems can be
solved in O(mn) time
• Space requirement can be reduced to O(m+n), while
keeping run-time fixed [Myers88]
• Highly similar sequences can be aligned in O (dn)
time, where d measures the distance between the
sequences [Landau86]
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Affine Gap Penalty Functions
Affine Gap Penalties = Differential Gap Penalties
used to reflect cost differences between opening a
gap and extending an existing gap
Total Gap Penalty is linear function of gap length:
W =
where

+

X
(k - 1)
 = gap opening penalty
 = gap extension penalty
Can also be solved in
O(nm) time using DP
k = length of gap
Sometimes, a Constant Gap Penalty is used, but it is usually
least realistic than the Affine Gap Penalty
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Methods
•
•
•
•
√Global and Local Alignment
√Alignment Algorithms
√Dot Matrix Method
√Dynamic Programming Method - cont
• Gap penalities
• DP for Global Alignment
• DP for Local Alignment
• Scoring Matrices
• Amino acid scoring matrices
• PAM
• BLOSUM
• Comparisons between PAM & BLOSUM
• Statistical Significance of Sequence Alignment
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"Scoring" or "Substitution" Matrices
2 Major types for Amino Acids: PAM & BLOSUM
PAM = Point Accepted Mutation
relies on "evolutionary model" based on observed
differences in alignments of closely related proteins
BLOSUM = BLOck SUbstitution Matrix
based on % aa substitutions observed in blocks of
conserved sequences within evolutionarily divergent
proteins
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PAM Matrix
PAM = Point Accepted Mutation
relies on "evolutionary model" based on observed
differences in closely related proteins
• Model includes defined rate for each type of
sequence change
• Suffix number (n) reflects amount of "time"
passed: rate of expected mutation if n% of amino
acids had changed
• PAM1 - for less divergent sequences (shorter time)
• PAM250 - for more divergent sequences (longer time)
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BLOSUM Matrix
BLOSUM = BLOck SUbstitution Matrix
based on % aa substitutions observed in blocks of
conserved sequences within evolutionarily divergent
proteins
• Doesn't rely on a specific evolutionary model
• Suffix number (n) reflects expected similarity:
average % aa identity in the MSA from which the
matrix was generated
• BLOSUM45 - for more divergent sequences
• BLOSUM62 - for less divergent sequences
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BLOSUM62 Substitution Matrix
s(a,b) corresponds to score of
aligning character a with character b
Match scores are often calculated
based on frequency of mutations in
very similar sequences
(more details later)
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