Alignment Algorithms

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Alignment Algorithms
Instructor Yao-Ting Huang
Bioinformatics Laboratory, Department of Computer
Science Information Engineering, National Chung
Cheng University.
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Review of Big O Notation

• Definition of Big-O
• A function f(n) is O(g(n)) if there exist
  positive constants c and n0 such that f(n) ? c ?
  g(n) for all n ? n0.
• e.g., f(n) 100n2 30n 2000 is O(n2)

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Big O Notation
A function f(n) is O(g(n)) if there exist
positive constants c and n0 such that f(n) ? c ?
g(n) for all n ? n0
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How functions grow?
function
n
(Assume one million operations per second.)
For large data sets, algorithms with a complexity
greater than O(n log n) are often impractical!
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Practical Complexity
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orzs sequence evolution

• orz (kid)
• OTZ (adult)
• Orz (big head)
• Crz (motorcycle driver)
• or2 (bottom up)
• oO (back high)
• STO (the other way around)

• the origin?
• their evolutionary relationships?

Chaos illustration
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orzs sequence evolution
orz
Orz
Crz
or2
OTZ
oO
STO
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Homology

• How to measure the homology (similarity) of two
  sequences/genes/species?

AGGCGCGGGGGGTTAAGAGCTATGCCATTTATATAAAATTTAAAGCGTA
AGAGCTATGCCATTTATATAAAATTTAAAGGCGCGGGGGGTTAAGAGGCG
CGGGGGGTTAAGAGCTATGCCATTTATATAAAATTTAAAGCGTAAGAAGC
TATGCCATTTATATAAAATTTAAAGCGTAAGAGCTATGCCATTTATATAA
AATTTAAAGAGGCGCGGGGGGTTAAGAGCTATGCCATTTATATAAAATTT
AAAGCGTAAGAGCTATGCCATTTATTTATATAAAATTTAAAGATACAATC
TAAAGTTAAAGCGTAAGAGCTATGCAGGCGCGGGGGGTTAAGAGCTATGC
CATTTATATAAAATTTAAAGCGTAAGAGCTATGCCATTTATATAAAATTT
AAAGAGGCGCGGGGGGT
AAAGCGTAAGAGCTATGCCATTTATATAAAATTTAAAGGGCGCGGGGGG
TTAAGAGCTATGCCATTAGGCGCGGGGGGTTAAGAGCTATGCCATTTATA
TAAAATTTAAAGCGTAAGAGCTATGCCATTTATATAAAATTTAAATTTAT
ATAAAATTTAAAGCGTAAGAGCTATGCCATTTATATAAAATTTAAAGAAA
GCGTAAGAGCTATGCCATTTATATAAAATTTAAAGGGCGCGGGGGGTTAA
GAGCTATGCCATTTATATAAGGGGGTTAAGAGCTATGCCATTTTAAGAGC
TATGCCATTTATATAAAATTTAAAGCGTAAGAAGCTATGCCATTTATATA
AAATTTAAGAGCTATGCCATTTATATAAAATTTAAAGAGGCGCGGGGGGT
TAAGAGCTATGCCATTTATATAAAATTTAAAGCGTAAGAGCTATGCCATT
TATATAAAATTTAAGCTATGCAGGCGCGGGGAGCTGGGTTTATATAAAAT
TTA
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Genetic Variants

• Genetic variants decrease homology of sequences.
• Point mutations, insertions/deletions, and
  inversions.
• It is difficult to measure the homology with the
  existence of these genetic variants.

AAGAAGCTATGCC - G - G -
AAGAGACTATGCCGG
AA - - AGCTATGCCAGTGA
AAAGCTATGCCAGTGA
Alignment
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Genetic Variants
Variations observed in a population
Mutations over time
Disease Mutation
Common Ancestor
time
present
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Milestones of Alignment Algorithms

• 1970 Needleman-Wunsch algorithm
• 1981 Smith-Waterman algorithm developed
• 1985 FASTP/FASTN fast sequence similarity
  searching
• 1990 BLAST fast sequence similarity searching
• 2001 Mummer

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The Impact of Alignment Algorithms

• BLAST the most cited paper so far.
• Biologists may not know anything of algorithm but
  they probably know BLAST.

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Dynamic Programming

• Dynamic programming is a classical method for
  solving sequential decision problems with a
  compositional cost structure.
• Alignment algorithms are originally derived from
  dynamic programming.

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Two key ingredients

• Two key ingredients for an optimization problem
  to be suitable for a dynamic-programming solution

1. optimal substructures
2. overlapping subproblems
Subproblems are dependent. (otherwise, a
divide-and-conquer approach is the choice.)
Each substructure is optimal. (Principle of
optimality)
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Three Basic Components

• A dynamic-programming algorithm has three basic
  components
• The recurrence relation (for defining the value
  of an optimal solution)
• The tabular computation (for computing the value
  of an optimal solution)
• The traceback (for delivering an optimal
  solution).

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Fibonacci Numbers

• Fibonacci numbers F(n), for n 0, 1, 2, , are
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
  233,
• Sunflowers have spirals in pair of adjacent
  Fibonacci numbers for two directions and (e.g.,
  34 and 55) Wikipedia

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Recursion

• Formal definition 0, 1, 1, 2, 3, 5, 8, 13, 21,
  34,

int fib (int n) if(n 0 n 1) return
1 else return ( fib (n-1) fib (n-2) )
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Recursion
int fib (int n) if(n 0 n 1) return
1 else return ( fib (n-1) fib (n-2) )
1 20
fib(100)
2 21
4 22
8 23
100
Exponential to n
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Tabular Computation

• The tabular computation can avoid recompuation.

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Locating Conserved Genes

• Genes conserved in two different species usually
  have the same orientation.
• e.g., Mummer (http//mummer.sourceforge.net/).

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Locating Conserved Genes

• Genes conserved in two different species usually
  have the same orientation.
• This problem is often formulated to find the
  longest increasing subsequence (LIS).

1
2
3
4
5
6
7
8
1
5
6
2
3
4
7
8
LIS 1, 2, 3, 4, 7, 8
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Longest Increasing Subsequence (LIS)

• The longest increasing subsequence is to find a
  longest increasing subsequence of a given
  sequence of distinct integers a1a2an .

• e.g. 9 2 5 3 7 11 8 10 13 6
• 3 7
• 7 10 13
• 7 11
• 3 5 11 13

are increasing subsequences.
We want to find a longest one.
are not increasing subsequences.
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A naive approach for LIS

• Let Li be the length of a longest increasing
  subsequence ending at position i.

Li 1 max j 0..i-1Lj aj lt ai(use a
dummy a0 minimum, and L00)
9 2 5 3 7 11 8 10 13 6
Li 1 1 2 2 3 4 ?
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A naive approach for LIS

• Li 1 max j 0..i-1 Lj aj lt ai

9 2 5 3 7 11 8 10 13 6
Li 1 1 2 2 3 4 4 5
6 3
The maximum length
The subsequence 2, 3, 7, 8, 10, 13 is a longest
increasing subsequence. This method runs in O(n2)
time.
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An O(n log n) method for LIS

• Define BestEndk as the smallest number of an
  increasing subsequence of length k.

9 2 5 3 7 11 8 10 13 6
9
2
2
2
BestEnd1
5
3
BestEnd2
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An O(n log n) method for LIS

• Define BestEndk as the smallest number of an
  increasing subsequence of length k.

9 2 5 3 7 11 8 10 13 6
9
2
2
2
2
2
2
BestEnd1
5
3
3
3
3
BestEnd2
7
7
7
BestEnd3
11
8
BestEnd4
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An O(n log n) method for LIS

• Define BestEndk as the smallest number of an
  increasing subsequence of length k.

9 2 5 3 7 11 8 10 13 6
9
2
2
2
2
2
2
2
2
BestEnd1
5
3
3
3
3
3
3
BestEnd2
7
7
7
7
7
BestEnd3
11
8
8
8
BestEnd4
10
10
BestEnd5
13
BestEnd6
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An O(n log n) method for LIS

• Define BestEndk to be the smallest number of an
  increasing subsequence of length k.

9 2 5 3 7 11 8 10 13 6
9
2
2
2
2
2
2
2
2
2
BestEnd1
5
3
3
3
3
3
3
3
BestEnd2
7
7
7
7
7
6
BestEnd3
11
8
8
8
8
BestEnd4
10
10
10
BestEnd5
For each position, we perform a binary search to
update BestEnd.
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13
BestEnd6
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Binary search

• Given an ordered sequence x1x2 ... xn, where
  x1ltx2lt ... ltxn, and a number y, a binary search
  finds the largest xi such that xilt y in O(log n)
  time.

n/2
...
n/4
n
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Binary search

• How many steps would a binary search reduce the
  problem size to 1?n n/2 n/4 n/8 n/16
  ... 1

How many steps? O(log n) steps.
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An O(n log n) method for LIS

• Define BestEndk to be the smallest number of an
  increasing subsequence of length k.

9 2 5 3 7 11 8 10 13 6
9
2
2
2
2
2
2
2
2
2
BestEnd1
5
3
3
3
3
3
3
3
BestEnd2
7
7
7
7
7
6
BestEnd3
11
8
8
8
8
BestEnd4
For each position, we perform a binary search to
update BestEnd. Therefore, the running time is
O(n log n).
10
10
10
BestEnd5
13
13
BestEnd6
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LCS

• The longest common subsequence problem is to find
  a maximum-length common subsequence between two
  sequences.
• Sequence 1 president
• Sequence 2 providence
• Its LCS is priden.

president providence
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LCS

• Another example
• Sequence 1 algorithm
• Sequence 2 alignment
• One of its LCS is algm.

a l g o r i t h m a l i g n m e n t
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How to compute LCS?

• Let Aa1a2am and Bb1b2bn .
• len(i, j) the length of an LCS between
  a1a2ai and b1b2bj
• With proper initializations, len(i, j)can be
  computed as follows.

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Pairwise Alignment

• Sequence A CTTAACT
• Sequence B CGGATCAT
• An alignment of A and B

C---TTAACTCGGATCA--T
Sequence A
Sequence B
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Pairwise Alignment

• Sequence A CTTAACT
• Sequence B CGGATCAT
• An alignment of A and B

Mismatch
Match
C---TTAACTCGGATCA--T
Deletion gap
Insertion gap
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Dot Matrix

• Sequence ACTTAACT
• Sequence BCGGATCAT

C G G A T C A T
CTTAACT
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Alignment Graph

• Sequence A CTTAACT Sequence B CGGATCAT

C G G A T C A T
CTTAACT
C---TTAACTCGGATCA--T
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A simple scoring scheme

• Match 8 (w(x, y) 8, if x y)
• Mismatch -5 (w(x, y) -5, if x ? y)
• Each gap symbol -3 (w(-,x)w(x,-)-3)

C - - - T T A A C TC G G A T C A - - T
8 -3 -3 -3 8 -5 8 -3 -3
8 12
Alignment score
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An optimal alignment-- the alignment of maximum
score

• Let Aa1a2am and Bb1b2bn .
• Si,j the score of an optimal alignment between
  a1a2ai and b1b2bj
• With proper initializations, Si,j can be computed
  by dynamic programming.

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Computing Si,j

j
w(ai,bj)
w(ai,-)
i
w(-,bj)
Sm,n
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Initializations

C G G A T C A T
CTTAACT
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S3,5 ?

• Match 8 (w(x, y) 8, if x y)
• Mismatch -5 (w(x, y) -5, if x ? y)
• Each gap symbol -3 (w(-,x)w(x,-)-3)

C G G A T C A T
CTTAACT
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S3,5 5

• Match 8 (w(x, y) 8, if x y)
• Mismatch -5 (w(x, y) -5, if x ? y)
• Each gap symbol -3 (w(-,x)w(x,-)-3)

C G G A T C A T
CTTAACT
optimal score
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C T T A A C TC G G A T C A T
8 5 5 8 -5 8 -3 8 14

C G G A T C A T
CTTAACT
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Now try this example in class

• Sequence A CAATTGA
• Sequence B GAATCTGC
• Their optimal alignment?

• Match 8 (w(x, y) 8, if x y)
• Mismatch -5 (w(x, y) -5, if x ? y)
• Each gap symbol -3 (w(-,x)w(x,-)-3)

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Initializations

G A A T C T G C
CAATTGA
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S4,2 ?

G A A T C T G C
CAATTGA
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S5,5 ?

G A A T C T G C
CAATTGA
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S5,5 14

G A A T C T G C
CAATTGA
optimal score
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C A A T - T G AG A A T C T G C

-5 8 8 8 -3 8 8 -5 27
G A A T C T G C
CAATTGA
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Expressed Sequence Tags (ESTs)

• Expressed sequence tags (ESTs) are a set of
  sequenced cDNAs reverse-transcribed from mRNA.
• ESTs are usually short (400bp) and only
  represent parts of a gene.

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Genome Annotation

• The huge amount of ESTs is often used to identify
  genes in the genome.
• e.g., 8,134,045 ESTs in human.
• http//www.ncbi.nlm.nih.gov/dbEST/dbEST_summary.ht
  ml

ESTs
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Global Alignment vs. Local Alignment

• Global alignment
• Local alignment

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An optimal local alignment

• Mutations may have added too much noise in the
  sequences of interest.
• Local alignment avoids these regions altogether
  and focuses on those with a positive score.
• Si,j the score of an optimal local alignment
  ending at ai and bj

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local alignment
Match 8 Mismatch -5 Gap symbol -3

C G G A T C A T
CTTAACT
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local alignment
Match 8 Mismatch -5 Gap symbol -3

C G G A T C A T
CTTAACT
The best score
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A C - TA T C A T 8-38-38 18

C G G A T C A T
CTTAACT
The best score