www.bioalgorithms.infoan introduction to bioinformatics algorithms sequence alignment

www.bioalgorithms.infoAn Introduction to Bioinformatics Algorithms

Sequence Alignment

An Introduction to Bioinformatics Algorithms www.bioalgorithms.info

Outline

• Global Alignment • Scoring Matrices• Local Alignment• Alignment with Affine Gap Penalties


From LCS to Alignment: Change Scoring

• The Longest Common Subsequence (LCS) problem—the simplest form of sequence alignment – allows only insertions and deletions (no mismatches).

• In the LCS Problem, we scored 1 for matches and 0 for indels

• Consider penalizing indels and mismatches with negative scores

• Simplest scoring schema: +1 : match premium -μ : mismatch penalty -σ : indel penalty


Simple Scoring

• When mismatches are penalized by –μ, indels are penalized by –σ,

and matches are rewarded with +1,

the resulting score is:

#matches – μ(#mismatches) – σ (#indels)


The Global Alignment ProblemFind the best alignment between two strings under a given scoring

schema

Input : Strings v and w and a scoring schemaOutput : Alignment of maximum score

→ = -б = 1 if match = -µ if mismatch

si-1,j-1 +1 if vi = wj

si,j = max s i-1,j-1 -µ if vi ≠ wj

s i-1,j - σ s i,j-1 - σ

: mismatch penaltyσ : indel penalty


Scoring Matrices

To generalize scoring, consider a (4+1) x(4+1) scoring matrix δ.

In the case of an amino acid sequence alignment, the scoring matrix would be a (20+1)x(20+1) size. The addition of 1 is to include the score for comparison of a gap character “-”.

This will simplify the algorithm as follows:

si-1,j-1 + δ (vi, wj)

si,j = max s i-1,j + δ (vi, -)

s i,j-1 + δ (-, wj)


Measuring Similarity

• Measuring the extent of similarity between two sequences• Based on percent sequence identity• Based on conservation


Percent Sequence Identity

• The extent to which two nucleotide or amino acid sequences are invariant

A C C T G A G – A G A C G T G – G C A G

70% identical

mismatchindel


Making a Scoring Matrix

• Scoring matrices are created based on biological evidence.

• Alignments can be thought of as two sequences that differ due to mutations.

• Some of these mutations have little effect on the protein’s function, therefore some penalties, δ(vi , wj), will be less harsh than others.


Scoring Matrix: ExampleA R N K

A 5 -2 -1 -1

R - 7 -1 3

N - - 7 0

K - - - 6

• Notice that although R and K are different amino acids, they have a positive score.

• Why? They are both positively charged amino acids will not greatly change function of protein.


Scoring matrices

• Amino acid substitution matrices• PAM• BLOSUM

• DNA substitution matrices• DNA is less conserved than protein

sequences• Less effective to compare coding regions at

nucleotide level


PAM• Point Accepted Mutation (Dayhoff et al.)• 1 PAM = PAM1 = 1% average change of all amino

acid positions• After 100 PAMs of evolution, not every residue will

have changed• some residues may have mutated several

times• some residues may have returned to their

original state• some residues may not changed at all


PAMX

• PAMx = PAM1x

• PAM250 = PAM1250

• PAM250 is a widely used scoring matrix:

Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys ... A R N D C Q E G H I L K ...Ala A 13 6 9 9 5 8 9 12 6 8 6 7 ...Arg R 3 17 4 3 2 5 3 2 6 3 2 9Asn N 4 4 6 7 2 5 6 4 6 3 2 5Asp D 5 4 8 11 1 7 10 5 6 3 2 5Cys C 2 1 1 1 52 1 1 2 2 2 1 1Gln Q 3 5 5 6 1 10 7 3 7 2 3 5...Trp W 0 2 0 0 0 0 0 0 1 0 1 0Tyr Y 1 1 2 1 3 1 1 1 3 2 2 1Val V 7 4 4 4 4 4 4 4 5 4 15 10

A 2 R -2 6 N 0 0 2 D 0 -1 2 4 C -2 -4 -4 -5 12 Q 0 1 1 2 -5 4 E 0 -1 1 3 -5 2 4 G 1 -3 0 1 -3 -1 0 5 H -1 2 2 1 -3 3 1 -2 6 I -1 -2 -2 -2 -2 -2 -2 -3 -2 5 L -2 -3 -3 -4 -6 -2 -3 -4 -2 -2 6 K -1 3 1 0 -5 1 0 -2 0 -2 -3 5 M -1 0 -2 -3 -5 -1 -2 -3 -2 2 4 0 6 F -3 -4 -3 -6 -4 -5 -5 -5 -2 1 2 -5 0 9 P 1 0 0 -1 -3 0 -1 0 0 -2 -3 -1 -2 -5 6 S 1 0 1 0 0 -1 0 1 -1 -1 -3 0 -2 -3 1 2 T 1 -1 0 0 -2 -1 0 0 -1 0 -2 0 -1 -3 0 1 3 W -6 2 -4 -7 -8 -5 -7 -7 -3 -5 -2 -3 -4 0 -6 -2 -5 17 Y -3 -4 -2 -4 0 -4 -4 -5 0 -1 -1 -4 -2 7 -5 -3 -3 0 10 V 0 -2 -2 -2 -2 -2 -2 -1 -2 4 2 -2 2 -1 -1 -1 0 -6 -2 4 A R N D C Q E G H I L K M F P S T W Y V

PAM250 log oddsscoring matrix

Why do we go from a mutation probabilitymatrix to a log odds matrix?

• We want a scoring matrix so that when we do a pairwise alignment (or a BLAST search) we know what score to assign to two aligned amino acid residues.

• Logarithms are easier to use for a scoring system. They allow us to sum the scores of aligned residues (rather than having to multiply them).

How do we go from a mutation probabilitymatrix to a log odds matrix?

• The cells in a log odds matrix consist of an “odds ratio”:

the probability that an alignment is authenticthe probability that the alignment was random

The score S for an alignment of residues a,b is given by:

S(a,b) = 10 log10 ( Mab / pb )

As an example, for tryptophan,

S( W, W ) = 10 log10 ( 0.55 / 0.01 ) = 17.4

What do the numbers meanin a log odds matrix?

S( W, W ) = 10 log10 ( 0.55 / 0.010 ) = 17.4

A score of +17 for tryptophan means that this alignmentis 50 times more likely than a chance alignment of twotryptophan residues.

S(W, W) = 17Probability of replacement ( Mab / pb ) = xThen

17 = 10 log10 x1.7 = log10 x101.7 = x = 50

What do the numbers meanin a log odds matrix?

A score of +2 indicates that the amino acid replacementoccurs 1.6 times as frequently as expected by chance.

A score of 0 is neutral.

A score of –10 indicates that the correspondence of two amino acids in an alignment that accurately representshomology (evolutionary descent) is one tenth as frequentas the chance alignment of these amino acids.

PAM10 log oddsscoring matrix

A 7

R -10 9

N -7 -9 9

D -6 -17 -1 8

C -10 -11 -17 -21 10

Q -7 -4 -7 -6 -20 9

E -5 -15 -5 0 -20 -1 8

G -4 -13 -6 -6 -13 -10 -7 7

H -11 -4 -2 -7 -10 -2 -9 -13 10

I -8 -8 -8 -11 -9 -11 -8 -17 -13 9

L -9 -12 -10 -19 -21 -8 -13 -14 -9 -4 7

K -10 -2 -4 -8 -20 -6 -7 -10 -10 -9 -11 7

M -8 -7 -15 -17 -20 -7 -10 -12 -17 -3 -2 -4 12

F -12 -12 -12 -21 -19 -19 -20 -12 -9 -5 -5 -20 -7 9

P -4 -7 -9 -12 -11 -6 -9 -10 -7 -12 -10 -10 -11 -13 8

S -3 -6 -2 -7 -6 -8 -7 -4 -9 -10 -12 -7 -8 -9 -4 7

T -3 -10 -5 -8 -11 -9 -9 -10 -11 -5 -10 -6 -7 -12 -7 -2 8

W -20 -5 -11 -21 -22 -19 -23 -21 -10 -20 -9 -18 -19 -7 -20 -8 -19 13

Y -11 -14 -7 -17 -7 -18 -11 -20 -6 -9 -10 -12 -17 -1 -20 -10 -9 -8 10

V -5 -11 -12 -11 -9 -10 -10 -9 -9 -1 -5 -13 -4 -12 -9 -10 -6 -22 -10 8

A R N D C Q E G H I L K M F P S T W Y V

BLOSUM Outline• Idea: use aligned ungapped regions of protein

families.These are assumed to have a common ancestor. Similar ideas but better statistics and modeling.

• Procedure:– Cluster together sequences in a family whenever more than

L% identical residues are shared, for BLOSUM-L.– Count number of substitutions across different clusters (in

the same family).– Estimate frequencies using the counts.

• Practice: BlOSUM-50 and BLOSOM62 are wildly used.

Considered the state of the art nowadays.

BLOSUM matrices are based on local alignments.

BLOSUM stands for blocks substitution matrix.

BLOSUM62 is a matrix calculated from comparisons of sequences with less than 62% identical sites.

BLOSUM Matrices

BLOSUM Matrices

100

62

30

Per

cent

am

ino

acid

iden

tity

BLOSUM62

colla

pse

BLOSUM Matrices

100

62

30

Per

cent

am

ino

acid

iden

tity

BLOSUM62

100

62

30

BLOSUM30

100

62

30

BLOSUM80

colla

pse

colla

pse

colla

pse

All BLOSUM matrices are based on observed alignments; they are not extrapolated from comparisons of closely related proteins.

The BLOCKS database contains thousands of groups ofmultiple sequence alignments.

BLOSUM62 is the default matrix in BLAST 2.0. Though it is tailored for comparisons of moderately distant proteins, it performs well in detecting closer relationships. A search for distant relatives may be more sensitive with a different matrix.

BLOSUM Matrices

A 4R -1 5N -2 0 6D -2 -2 1 6C 0 -3 -3 -3 9Q -1 1 0 0 -3 5E -1 0 0 2 -4 2 5G 0 -2 0 -1 -3 -2 -2 6H -2 0 1 -1 -3 0 0 -2 8I -1 -3 -3 -3 -1 -3 -3 -4 -3 4L -1 -2 -3 -4 -1 -2 -3 -4 -3 2 4K -1 2 0 -1 -1 1 1 -2 -1 -3 -2 5M -1 -2 -2 -3 -1 0 -2 -3 -2 1 2 -1 5F -2 -3 -3 -3 -2 -3 -3 -3 -1 0 0 -3 0 6P -1 -2 -2 -1 -3 -1 -1 -2 -2 -3 -3 -1 -2 -4 7S 1 -1 1 0 -1 0 0 0 -1 -2 -2 0 -1 -2 -1 4T 0 -1 0 -1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -2 -1 1 5W -3 -3 -4 -4 -2 -2 -3 -2 -2 -3 -2 -3 -1 1 -4 -3 -2 11Y -2 -2 -2 -3 -2 -1 -2 -3 2 -1 -1 -2 -1 3 -3 -2 -2 2 7V 0 -3 -3 -3 -1 -2 -2 -3 -3 3 1 -2 1 -1 -2 -2 0 -3 -1 4

A R N D C Q E G H I L K M F P S T W Y V

Blosum62 scoring matrix

Rat versus mouse RBP

Rat versus bacteriallipocalin


Local vs. Global Alignment

• The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph.

• The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph.


Local vs. Global Alignment• The Global Alignment Problem tries to find the

longest path between vertices (0,0) and (n,m) in the edit graph.

• The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph.

• In the edit graph with negatively-scored edges, Local Alignmet may score higher than Global Alignment


Local vs. Global Alignment (cont’d)• Global Alignment

• Local Alignment—better alignment to find conserved segment

--T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC | || | || | | | ||| || | | | | |||| | AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C

tccCAGTTATGTCAGgggacacgagcatgcagagac ||||||||||||

aattgccgccgtcgttttcagCAGTTATGTCAGatc


Local Alignment: Example

Global alignment

Local alignment

Compute a “mini” Global Alignment to get Local


Local Alignments: Why?

• Two genes in different species may be similar over short conserved regions and dissimilar over remaining regions.

• Example:• Homeobox genes have a short region

called the homeodomain that is highly conserved between species.

• A global alignment would not find the homeodomain because it would try to align the ENTIRE sequence


The Local Alignment Problem

• Goal: Find the best local alignment between two strings

• Input : Strings v, w and scoring matrix δ• Output : Alignment of substrings of v and w

whose alignment score is maximum among all possible alignment of all possible substrings


The Problem with this Problem• Long run time O(n4):

- In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source.

- For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time.

• This can be remedied by giving free rides



Global alignment

Local alignment

Compute a “mini” Global Alignment to get Local


Local Alignment: Running Time • Long run time O(n4):

- In the grid of size n x n there are ~n2 vertices (i,j) that may serve as a source.

- For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n2) time.

• This can be remedied by giving free rides


Local Alignment: Free Rides

Vertex (0,0)

The dashed edges represent the free rides from (0,0) to every other node.

Yeah, a free ride!


The Local Alignment Recurrence• The largest value of si,j over the whole edit

graph is the score of the best local alignment.

• The recurrence:

0 si,j = max si-1,j-1 + δ (vi, wj)

s i-1,j + δ (vi, -)

s i,j-1 + δ (-, wj)

Notice there is only this change from the original recurrence of a Global Alignment


The Local Alignment Recurrence• The largest value of si,j over the whole edit

graph is the score of the best local alignment.

• The recurrence:

0 si,j = max si-1,j-1 + δ (vi, wj)

s i-1,j + δ (vi, -)

s i,j-1 + δ (-, wj)

Power of ZERO: there is only this change from the original recurrence of a Global Alignment - since there is only one “free ride” edge entering into every vertex


Scoring Indels: Naive Approach

• A fixed penalty σ is given to every indel:• -σ for 1 indel, • -2σ for 2 consecutive indels• -3σ for 3 consecutive indels, etc.

Can be too severe penalty for a series of 100 consecutive indels


Affine Gap Penalties

• In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events:

Normal scoring would give the same score for both alignments

This is more likely.

This is less likely.


Accounting for Gaps• Gaps- contiguous sequence of spaces in one of the

rows

• Score for a gap of length x is: -(ρ + σx) where ρ >0 is the penalty for introducing a gap: gap opening penalty ρ will be large relative to σ: gap extension penalty because you do not want to add too much of a

penalty for extending the gap.


Affine Gap Penalties

• Gap penalties:• -ρ-σ when there is 1 indel• -ρ-2σ when there are 2 indels• -ρ-3σ when there are 3 indels, etc. • -ρ- x·σ (-gap opening - x gap extensions)

• Somehow reduced penalties (as compared to naïve scoring) are given to runs of horizontal and vertical edges


Affine Gap Penalties and Edit Graph

To reflect affine gap penalties we have to add “long” horizontal and vertical edges to the edit graph. Each such edge of length x should have weight

- - x *


Adding “Affine Penalty” Edges to the Edit Graph

There are many such edges!

Adding them to the graph increases the running time of the alignment algorithm by a factor of n (where n is the number of vertices)

So the complexity increases from O(n2) to O(n3)


Manhattan in 3 Layers

ρ

ρ

σ

σδ

δ

δ

δδ


Affine Gap Penalties and 3 Layer Manhattan Grid

• The three recurrences for the scoring algorithm creates a 3-layered graph.

• The top level creates/extends gaps in the sequence w.

• The bottom level creates/extends gaps in sequence v.

• The middle level extends matches and mismatches.


Switching between 3 Layers

• Levels:• The main level is for diagonal edges • The lower level is for horizontal edges• The upper level is for vertical edges

• A jumping penalty is assigned to moving from the main level to either the upper level or the lower level (-)

• There is a gap extension penalty for each continuation on a level other than the main level (-)


The 3 Grids

-σ

+δ(vi,wj)

-σ

+0+0

-(ρ + σ)

-(ρ + σ)

Away from mid-level

Toward mid-level


The 3-leveled Manhattan Grid


Affine Gap Penalty Recurrencessi,j = s i-1,j - σ

max s i-1,j –(ρ+σ)

si,j = s i,j-1 - σ

max s i,j-1 –(ρ+σ)

si,j = si-1,j-1 + δ (vi, wj)

max s i,j

s i,j

Continue Gap in w (deletion)

Start Gap in w (deletion): from middle

Continue Gap in v (insertion)

Start Gap in v (insertion):from middle

Match or Mismatch

End deletion: from top

End insertion: from bottom

www.bioalgorithms.infoan introduction to bioinformatics algorithms sequence alignment

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j s i

w j s i

w j slide

mismatch s i

max s i

simple scoring

penalty slide

indels slide