Intro to Alignment Algorithms: Global and Local Algorithmic - - PowerPoint PPT Presentation

Intro to Alignment Algorithms: Global and Local

Algorithmic Functions of Computational Biology Professor Istrail

Sequence Comparison

Biomolecular sequences

□

DNA sequences (string over 4 letter alphabet {A, C, G, T})

□

RNA sequences (string over 4 letter alphabet {ACGU})

□

Protein sequences (string over 20 letter alphabet {Amino Acids}) Sequence similarity helps in the discovery of genes, and the prediction of structure and function of proteins.

Algorithmic Functions of Computational Biology Professor Istrail

The Basic Similarity Analysis Algorithm

Global Similarity

• Scoring Schemes
• Edit Graphs
• Alignment = Path in the Edit Graph
• The Principle of Optimality
• The Dynamic Programming Algorithm
• The Traceback

Algorithmic Functions of Computational Biology – Professor Istrail

The Sequence Alignment Problem

• Input. : two sequences over the same alphabet and a scoring scheme

Output: an alignment of the two sequences of maximum score Example:

□

GCGCATTTGAGCGA

□

TGCGTTAGGGTGACCA A possible alignment:

• GCGCATTTGAGCGA - -

TGCG - - TTAGGGTGACC

match mismatch indel

Algorithmic Functions of Computational Biology – Professor Istrail

CSCI2820 - Class 4

Mismatch, Deletion, Insertion

5

mismatch deletion (in template) insertion (in template)

TCAGGGGGCTATT AGTCCTCCGATAA TCAGGGGGCTATT AGTCC-CCGATAA TCAGGGGG-CTATT AGTCCCCCCGATAA

m n

y y y Y x x x X

... 2 1 ... 2 1

= =

Consider two sequences Over the alphabet

} , , , T G C A { = Σ

j i y

x,

belong to Σ

Algorithmic Functions of Computational Biology – Professor Istrail

Scoring Schemes

δ

Unit-score A C G T A C G T 1 1 1 1

• Algorithmic Functions of Computational Biology –

Professor Istrail

Alignment

ACG | | | AGG

δ

Score = (A,A) (C,G) (G,G) + + = 1 + 0 + 1 = 2 Unit-cost A | A A is aligned with A C | G C is aligned with G G is aligned with G G | G

Algorithmic Functions of Computational Biology – Professor Istrail

δ δ

Gaps

ACATGGAAT ACAGGAAAT ACAT GG - AAT ACA - GG AAAT OPTIMAL ALIGNMENTS SCORE 7 8 AAAGGG GGGAAA SCORE 3

• - - AAAGGG

GGGAAA - - -

“-” is the gap symbol

Algorithmic Functions of Computational Biology – Professor Istrail

δ δ δ(x,y) = the score for aligning x with y

(-,y) = the score for aligning - with y (x,-) = the score for aligning x with -

Algorithmic Functions of Computational Biology – Professor Istrail

A-CG - G ATCGTG Alignment Score

δ δ δ δ δ δ

(A,A) + (G,G) + (C,C) + (-,T) + (-,T ) + (G,G) THE SUM OF THE SCORES OF THE PAIRWISE ALIGNED SYMBOLS

Algorithmic Functions of Computational Biology – Professor Istrail

Margaret Dayhoff & PAM Similarity Matrices

ARTEMIS Summer 2008 Professor Istrail

• Dr. Margaret Oakley Dayhoff

The Mother & Father of Bioinformatics

ARTEMIS Summer 2008 Professor Istrail

Scoring Scheme

Dayhoff PAM scoring matrices

...

δ

PTIPLSRLFDNAMLRAHRLHQ SAIENQRLFNIAVSRVQHLHL Partial alignment for Monkey and Trout somatotropin proteins

• A R N D C Q E G H I L K M F P S T W Y V
• 8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -8
• 8 3 -3 0 0 -3 -1 0 1 -3 -1 -3 -2 -2 -4 1 1 1 -7 -4 0

A R N D 6 4

Algorithmic Functions of Computational Biology – Professor Istrail

Scoring Functions

Scoring function = a sum of a terms each for a pair of

aligned residues, and for each gap The meaning = log of the relative likelihood that the sequences are related, compared to being unrelated Identities and conservative substitutions are Positive terms Non-conservative substitutions are Negative terms

Mutations= Substitutions, Insertions, Deletions

Algorithmic Functions of Computational Biology – Professor Istrail

CSCI2820 - Class 4

Global alignment problem

• Input: Sequences X and Y of length m and

n respectively and a similarity matrix

• Output: An optimal global alignment of X

and Y

– Global alignments require all bases in both sequences are aligned

16

CSCI2820 - Class 4

Local alignment problem

• Input: Sequences X and Y of length m and

n respectively and a similarity matrix

• Output: An optimal local alignment of X

and Y

– Local alignments do not require using all bases in either sequence in the alignment

• Applicable when looking for subsequences
• f similarity

17

The Edit Graph

Suppose that we want to align AGT with AT We are going to construct a graph where alignments between the two sequences correspond to paths between the begin and and end nodes of the graph. This is the Edit Graph

Algorithmic Functions of Computational Biology – Professor Istrail

1

2 3 1 2

AGT has length 3 AT has length 2 The Edit graph has (3+1)*(2+1) nodes The sequence AGT The sequence AT

Algorithmic Functions of Computational Biology – Professor Istrail

1 2 3 1 2

A G T A T

AGT indexes the columns, and AT indexes the rows of this “table”

Begin End

Algorithmic Functions of Computational Biology – Professor Istrail

1 2 3 1 2

A G T A T

Begin End

The Graph is directed. The nodes (i,j) will hold values.

Algorithmic Functions of Computational Biology – Professor Istrail

1 1 2

A G A T

T

1 2 3 1 2

A G A T

Begin End

Algorithmic Functions of Computational Biology Professor Istrail

T A T

1 2 3 1 2

A G

A

• A

A A

• A
• A
• A
• A

G

• A
• A
• G
• G
• T
• T
• T
• T
• T
• T
• T

A T G T T T G A T A

Directed edges get as labels pairs of aligned letters.

Begin End

Algorithmic Functions of Computational Biology – Professor Istrail

Alignment = Path in the Edit Graph

T A T

1 2 3 1 2

A G

A

• A

A A

• A
• A
• A
• A

G

• A
• A
• G
• G
• T
• T
• T
• T
• T
• T
• T

A T G T T T G A T A

AGT A-T

Begin End

Every path from Begin to End corresponds to an alignment Every alignment corresponds to a path between Begin and End

Algorithmic Functions of Computational Biology – Professor Istrail

The Principle of Optimality

The optimal answer to a problem is expressed in terms of optimal answer for its sub-problems

Algorithmic Functions of Computational Biology – Professor Istrail

Dynamic Programming

Part 1: Compute first the optimal alignment score Part 2: Construct optimal alignment We are looking for the optimal alignment = maximal score path in the Edit Graph from the Begin vertex to the End vertex Given: Two sequences X and Y Find: An optimal alignment of X with Y

Algorithmic Functions of Computational Biology – Professor Istrail

The DP Matrix S(i,j)

1 2 3 1 2

A G T A T S(2,1) S(1,0)

Algorithmic Functions of Computational Biology – Professor Istrail

The DP Matrix

Matrix S =[S(i,j)] S(i,j) = The score of the maximal cost path from the Begin Vertex and the vertex (i,j) (i,j) (i,j-1) (i-1,j) (i-1,j-1) The optimal path to (i,j) must pass through one of the vertices (i-1,j-1) (i-1,j) (i,j-1)

O p t i m a l P a t h t

• (

i , j )

Algorithmic Functions of Computational Biology – Professor Istrail

Opt path

(i,j) (i,j-1) (i-1,j) (i-1,j-1) Optimal path to (i-1,j) + (- , yj)

• xi

yj

• S(i-1,j) + δ

δ

(- , yj)

Algorithmic Functions of Computational Biology – Professor Istrail

Optimal path

(i,j) (i-1,j) (i,j-1) (i-1,j-1) Optimal path to (i-1,j-1) + (xi,yj)

δ δ

S(i-1,j-1) + (xi , yj)

Algorithmic Functions of Computational Biology – Professor Istrail

Optimal path

(i,j) (i,j-1) (I-1,j) (i-1,j-1) Optimal path to (i,j-1) + (xi,-)

δ δ

S(i,j-1) + (xi, -)

Algorithmic Functions of Computational Biology – Professor Istrail

The Basic ALGORITHM

S(i,j) = S(i-1, j-1) + (xi, yj) S(i-1, j) + (xi, -) S(i, j-1) + (-, yj)

MAX

δ δ δ

Algorithmic Functions of Computational Biology – Professor Istrail

T A T

1 2 3 1 2

A G

A

• A

A A

• A
• A
• A
• A

G

• A
• A
• G
• G
• T
• T
• T
• T
• T
• T
• T

A T G T T T G A T A

1 1 1 1 1 2

AGT A - T Optimal Alignment

Optimal Alignment and Tracback

Algorithmic Functions of Computational Biology – Professor Istrail

S(i,j) = S(i-1, j-1) + (xi, yj), S(i-1, j) + (xi, -), S(i, j-1) + (-, yj)

MAX

δ δ δ

0,

We add this

The Basic ALGORITHM: Local Similarity

Algorithmic Functions of Computational Biology – Professor Istrail

CSCI2820 - Class 4

Protein global alignment

35

X = hlsek Y = nlsak

• X and Y represent a protein subsequence

from the BRCA2 (early onset) protein in human and chimpanzee

• Global alignments are used when the two

sequences being compared represent a similar biological sequence

CSCI2820 - Class 4

Margaret Dayhoff’s PAM 100 similarity matrix (partial)

36

A N E H L K S * A 4

• 1
• 3 -3 -3

1

• 9

N

• 1

5 1 2

• 4

1 1

• 9

E 1 5

• 1 -5 -1 -1 -9

H

• 3

2

• 1

7

• 3 -2 -2 -9

L

• 3 -4 -5 -3

6

• 4 -4 -9

K

• 3

1

• 1 -2 -4

5

• 1 -9

S 1 1

• 1 -2 -4 -1

4

• 9

*

• 9 -9 -9 -9 -9 -9 -9

1

CSCI2820 - Class 4 37

h l s e k n l s a k

X Y

• 9

2

• 7
• 16
• 25
• 34
• 9
• 18
• 27
• 36
• 45
• 27
• 16
• 1

12 3

• 6
• 36
• 25
• 10

3 12 3

• 18
• 7

8

• 1
• 10
• 19
• 45
• 34
• 19
• 6

3 17