Computational Genomics (0382.3102)

Lecture 3

Sequence Similarity and Pairwise Alignment II:

Affine Gap Penalties, Local Alignment,

BLAST and FASTA Hueristics

Prof. Benny Chor

School of Computer Science

Tel-Aviv University

Based in part on chapter *** in Gusfield’s book, chapter 3 in Kanehisa’s book,

and on a ppt presentation by Terry Speed (UC Berkeley)c

�

Benny Chor – p.1

DistancesLet

�

be a (finite or infinite) set. A distance on

�

is afunction � � � ��� � �

satisfying the followingthree properties:

� Symmetry:

��� � �

,

� �� � � � � �

.

Non-negativity: , , andif and only if .

Triangle inequality: ,

c

�

Benny Chor – p.2

DistancesLet

�

be a (finite or infinite) set. A distance on

�

is afunction � � � ��� � �

satisfying the followingthree properties:

� Symmetry:

��� � �

,

� �� � � � � �

.

� Non-negativity:

��� � �,

� �� � �

, and� �� � � �

if and only if � .

Triangle inequality: ,

c

�

Benny Chor – p.2

DistancesLet

�

be a (finite or infinite) set. A distance on

�

is afunction � � � ��� � �

satisfying the followingthree properties:

� Symmetry:

��� � �

,

� �� � � � � �

.

� Non-negativity:

��� � �,

� �� � �

, and� �� � � �

if and only if � .

� Triangle inequality:� �� �� � �

,� �� � � �� � � �� ��

c

�

Benny Chor – p.2

Famous DistancesA distance on

�

, � � � ��� � �

is also called anorm in math jargon. Example of norms (for some ofthese it is not immediate to verify that triangleinequality holds).

� � �� � � �

if � (this norm is a bit boring).

Let ( -dim. real vectors) and .

In math jargon, this is known as the norm.

c

�

Benny Chor – p.3

Famous DistancesA distance on

�

, � � � ��� � �

is also called anorm in math jargon. Example of norms (for some ofthese it is not immediate to verify that triangleinequality holds).

� � �� � � �

if � (this norm is a bit boring).

� Let

� �

�

(

�

-dim. real vectors) and � �

.� � �� � � � � � �� � � �� � � � � � � � �

� � ��� �

� � � � � �

In math jargon, this is known as the

� norm.

c

�

Benny Chor – p.3

More Famous Distances

� � with � � �

is the “regular” Euclidean distance.

with (sum of absolute values ofdifferences).

The norm: (the limit of as).

Let be a finite, undirected, connectedgraph,with positive edges’ lengths. Letbe a pair of vertices.

Define the length of the shortest pathfrom to in . This is a norm.

c

�

Benny Chor – p.4

More Famous Distances

� � with � � �

is the “regular” Euclidean distance.

� � with � � �

(sum of absolute values ofdifferences).

The norm: (the limit of as).

Let be a finite, undirected, connectedgraph,with positive edges’ lengths. Letbe a pair of vertices.

Define the length of the shortest pathfrom to in . This is a norm.

c

�

Benny Chor – p.4

More Famous Distances

� � with � � �

is the “regular” Euclidean distance.

� � with � � �

(sum of absolute values ofdifferences).

� The

� norm: � �� ��� �

� � � � (the limit of

� as

� ).

Let be a finite, undirected, connectedgraph,with positive edges’ lengths. Letbe a pair of vertices.

Define the length of the shortest pathfrom to in . This is a norm.

c

�

Benny Chor – p.4

More Famous Distances

� � with � � �

is the “regular” Euclidean distance.

� � with � � �

(sum of absolute values ofdifferences).

� The

� norm: � �� ��� �

� � � � (the limit of

� as

� ).

� Let � ��

�

be a finite, undirected, connectedgraph,with positive edges’ lengths. Let �� �

be a pair of vertices.

Define the length of the shortest pathfrom to in . This is a norm.

c

�

Benny Chor – p.4

More Famous Distances

� � with � � �

is the “regular” Euclidean distance.

� � with � � �

(sum of absolute values ofdifferences).

� The

� norm: � �� ��� �

� � � � (the limit of

� as

� ).

� Let � ��

�

be a finite, undirected, connectedgraph,with positive edges’ lengths. Let �� �

be a pair of vertices.

� Define

� �� � � the length of the shortest pathfrom to � in . This is a norm.

c

�

Benny Chor – p.4

Distance vs. Similarity

� Distance and similarity are dual notions. If andare highly similar objects, than intuitively they

have small distance.

This intuition indeed holds for pairwise globalsequence alignment (see prob. 6 in assignment 1).

We can replace sequence similarity by distanceand obtain qualitatively similar results forpairwise global sequence alignment.

Dynamic programming can be used to findminimum distance alignment in time .

c

�

Benny Chor – p.5

Distance vs. Similarity

� Distance and similarity are dual notions. If andare highly similar objects, than intuitively they

have small distance.

� This intuition indeed holds for pairwise globalsequence alignment (see prob. 6 in assignment 1).

We can replace sequence similarity by distanceand obtain qualitatively similar results forpairwise global sequence alignment.

Dynamic programming can be used to findminimum distance alignment in time .

c

�

Benny Chor – p.5

Distance vs. Similarity

� Distance and similarity are dual notions. If andare highly similar objects, than intuitively they

have small distance.

� This intuition indeed holds for pairwise globalsequence alignment (see prob. 6 in assignment 1).

� We can replace sequence similarity by distanceand obtain qualitatively similar results forpairwise global sequence alignment.

Dynamic programming can be used to findminimum distance alignment in time .

c

�

Benny Chor – p.5

Distance vs. Similarity

� Distance and similarity are dual notions. If andare highly similar objects, than intuitively they

have small distance.

� This intuition indeed holds for pairwise globalsequence alignment (see prob. 6 in assignment 1).

� We can replace sequence similarity by distanceand obtain qualitatively similar results forpairwise global sequence alignment.

� Dynamic programming can be used to findminimum distance alignment in time

��� � � � .

c

�

Benny Chor – p.5

Local Sequence Alignment

� In local sequence alignment, we have two inputsequences (strings): The query

�

, and the text .

The goal is to find two substring: – substring ofand – substring of such that the (global)

alignment score between and is maximized(over all choices of pairs of substrings).

Notice that if , is an optimal alignment ofand , then , can contain indels.

c

�

Benny Chor – p.6

Local Sequence Alignment

� In local sequence alignment, we have two inputsequences (strings): The query

�

, and the text .

� The goal is to find two substring: – substring of�

and – substring of such that the (global)alignment score between and is maximized(over all choices of pairs of substrings).

Notice that if , is an optimal alignment ofand , then , can contain indels.

c

�

Benny Chor – p.6

Local Sequence Alignment

� In local sequence alignment, we have two inputsequences (strings): The query

�

, and the text .

� The goal is to find two substring: – substring of�

and – substring of such that the (global)alignment score between and is maximized(over all choices of pairs of substrings).

� Notice that if

�

,�

is an optimal alignment ofand , then

�

,�

can contain indels.

c

�

Benny Chor – p.6

Local Alignment DP Algorithm

� The global and local alignment problems seemsvery different. But a rather small change in the”global” DP algorithm yields an efficient��� � � � ”local” DP algorithm.

Goal: Fill the matrix by values , thevalue of the best (global) alignment between allsuffixes of -prefix of and suffixes of -prefix of

.

c

�

Benny Chor – p.7

Local Alignment DP Algorithm

� The global and local alignment problems seemsvery different. But a rather small change in the”global” DP algorithm yields an efficient��� � � � ”local” DP algorithm.

� Goal: Fill the � � � matrix by values

� ��

� �

, thevalue of the best (global) alignment between allsuffixes of

�

-prefix of�

and suffixes of

�

-prefix of.

c

�

Benny Chor – p.7

Local Alignment DP Algorithm

� Initialize all boundary values (upper row, leftcolumn) to

�

.

Update rule:

Keep pointers like before (no pointer if).

Pick the highest entry in the matrix. Trace backto recover optimal local alignment(s).

c

�

Benny Chor – p.8

Local Alignment DP Algorithm

� Initialize all boundary values (upper row, leftcolumn) to

�

.

� Update rule:

� � �

�

� � � � � � �

� � ��

� � � � � � � � ��

� � � � ��

� � �

�

� � � � ��

� � � � ��

� ��

� � � � � � � � � �� � ��

� ��

Keep pointers like before (no pointer if).

Pick the highest entry in the matrix. Trace backto recover optimal local alignment(s).

c

�

Benny Chor – p.8

Local Alignment DP Algorithm

� Initialize all boundary values (upper row, leftcolumn) to

�

.

� Update rule:

� � �

�

� � � � � � �

� � ��

� � � � � � � � ��

� � � � ��

� � �

�

� � � � ��

� � � � ��

� ��

� � � � � � � � � �� � ��

� ��

� Keep pointers like before (no pointer if

� �� � �

).

Pick the highest entry in the matrix. Trace backto recover optimal local alignment(s).

c

�

Benny Chor – p.8

Local Alignment DP Algorithm

� Initialize all boundary values (upper row, leftcolumn) to

�

.

� Update rule:

� � �

�

� � � � � � �

� � ��

� � � � � � � � ��

� � � � ��

� � �

�

� � � � ��

� � � � ��

� ��

� � � � � � � � � �� � ��

� ��

� Keep pointers like before (no pointer if

� �� � �

).

� Pick the highest entry in the matrix. Trace backto recover optimal local alignment(s).

c

�

Benny Chor – p.8

Local Sequence Alignment IIFor example, suppose our sequences are� � �� � �� �� � �� �� �

and

� �� � � � � �� � �� � �� � �

.

Our scoring gives +2 for a match, -1 for amismatch,and -2 for indel.

A reasonable (best?) choice would be the substringsand , with the (global) alignment

between them being

_

In the context of the original sequences, we haveand

c

�

Benny Chor – p.9

Local Sequence Alignment IIFor example, suppose our sequences are� � �� � �� �� � �� �� �

and

� �� � � � � �� � �� � �� � �

.

Our scoring gives +2 for a match, -1 for amismatch,and -2 for indel.

A reasonable (best?) choice would be the substringsand , with the (global) alignment

between them being

_

In the context of the original sequences, we haveand

c

�

Benny Chor – p.9

Local Sequence Alignment IIFor example, suppose our sequences are� � �� � �� �� � �� �� �

and

� �� � � � � �� � �� � �� � �

.

Our scoring gives +2 for a match, -1 for amismatch,and -2 for indel.

A reasonable (best?) choice would be the substrings�� �� � ��

and

�� � �� � ��

, with the (global) alignmentbetween them being

_

In the context of the original sequences, we haveand

c

�

Benny Chor – p.9

Local Sequence Alignment IIFor example, suppose our sequences are� � �� � �� �� � �� �� �

and

� �� � � � � �� � �� � �� � �

.

Our scoring gives +2 for a match, -1 for amismatch,and -2 for indel.

A reasonable (best?) choice would be the substrings�� �� � ��

and

�� � �� � ��

, with the (global) alignmentbetween them being

�

_

� �� � ��

�� � �� � ��

In the context of the original sequences, we haveand

c

�

Benny Chor – p.9

Local Sequence Alignment IIFor example, suppose our sequences are� � �� � �� �� � �� �� �

and

� �� � � � � �� � �� � �� � �

.

Our scoring gives +2 for a match, -1 for amismatch,and -2 for indel.

A reasonable (best?) choice would be the substrings�� �� � ��

and

�� � �� � ��

, with the (global) alignmentbetween them being

�

_

� �� � ��

�� � �� � ��In the context of the original sequences, we have�� � �� �� � �� �� �

and�� � � � � �� � �� � �� � �

c

�

Benny Chor – p.9

Other Versions of Alignment

� Global in

�

, local in .

Affine gap penalties.

Linear space algorithm.

c

�

Benny Chor – p.10

Other Versions of Alignment

� Global in

�

, local in .

� Affine gap penalties.

Linear space algorithm.

c

�

Benny Chor – p.10

Other Versions of Alignment

� Global in

�

, local in .

� Affine gap penalties.

� Linear

��� � � space algorithm.

c

�

Benny Chor – p.10