Protein Sequence Comparison

Why do we want to align protein

sequences?

• If two sequences align well, the corresponding

proteins are homologous; they probably share the

same structure and/or function

• Sequence Alignment is a Tool for organizing the

protein sequence space

detection of homologous proteins

build evolutionary history

Alignment Methods

• Rigorous algorithms

- Needleman-Wunsch (global)

- Smith-Waterman (local)

• Rapid heuristics

- FASTA

- BLAST

What is sequence alignment?

• Given two sequences of letters and a scoring

scheme for evaluating letter matching, find the

optimal pairing of letters from one sequence to

the other.

• Different alignments:

Favors identity Favors similarity

ACCTAGGC ACCTAGGC

AC-T-GG ACT-GG

Gaps

Aligning Biological Sequences

• Aligning DNA: 4 letter alphabet

ACGTTGGC

AC-T-GG

• Aligning protein: 20 letter alphabet

MCYTSWGC

MC-T-WG

Computing Cost

The computational complexity of aligning two

sequences when gaps are allowed anywhere is

exponential in the length of the sequences being

aligned.

Computer science offers a solution

for reducing the running time:

Dynamic Programming

Dynamic Programming (DP) Concept

A problem with overlapping sub-problems and optimal sub-structures can be solved using the following algorithm:

(1) break the problem into smaller sub-problems

(2) solve these problems optimally using this 3-step procedure recursively

(3) use these optimal solutions to construct an optimal solution to the original problem

DP and Sequence Alignment

Key idea:

The score of the optimal alignment that ends at a given

pair of positions in the sequences is the score of the best

alignment previous to these positions plus the score of

aligning these two positions.

Test all alignments that can lead to i aligned with j

i

j

?

DP and Sequence Alignment

j

i

?

Find alignment with best [previous score + score(i,j)]i

j

?

DP and Sequence Alignment

j

i

Best alignment that ends at (i,j)

Implementing the DP algorithm for sequences

1) Build a NxM alignment matrix A such that

A(i,j) is the optimal score for alignments

up to the pair (i,j)

2) Find the best score in A

3) Track back through the matrix to get

the optimal alignment of S1 and S2.

Aligning 2 sequence S1 and S2 of lengths N and M:

Example 1

Sequence 1: ATGCTGC

Sequence 2: AGCC

Score(i,j) = 10 if i=j, 0 otherwise

no gap penalty

Example 1

0C

0C

0G

00000010A

CGTCGTA

1) Initialize

Example 1

0C

0C

100G

00000010A

CGTCGTA

2) Propagate

Example 1

0C

0C

20100G

00000010A

CGTCGTA

2) Propagate

Example 1

0C

100C

1020101020100G

00000010A

CGTCGTA

2) Propagate

Example 1

303010100C

3020203010100C

1020101020100G

00000010A

CGTCGTA

2) Propagate

Example 1

4030303010100C

3020203010100C

1020101020100G

00000010A

CGTCGTA

3) Trace Back

ATGCTGC

AXGCXXC

Alignment: Score: 40

Mathematical Formulation

+−−−

+−−−

−−

+=

−≤≤

−≤≤

k2ik1

k2jk1

W1)jk,1A(i

,Wk)1j1,A(i

1),j1,A(i

j)Score(i,j)A(i,

max

maxmax

Wk: penalty for a gap of size k

Global alignment (Needleman-Wunsch):

Complexity

1) The computing time required to fill in the

alignment matrix is O(NM(N+M)), where

N and M are the lengths of the 2 sequences

2) This can be reduced to O(NM) by storing

the best score for each row and column.

True if gap penalty is linear!

Example 2

Alignments:

High Score: 30301010100C

202010100G

102010100G

0001010A

CGTAA

AATGC

AG GC

AATGC

A GGC

AATGC

AGGC

AATG C

A GGC

AATG C

A GGC

Example 2 with Gap

Alignments:

High Score: 28281088-2C

1618108-2G

818810-2G

-2-2-2810A

CGTAA

AATGC

AG GC

AATGC

A GGC

AATGC

AGGC

Gap cost: -2

Complexity (2)

1) The traceback routine can be quite costly in computing

time if all possible optimal paths are required, since

there may be many branches.

2) Usually, an arbitrary choice is made about which

branch to follow. Then computing time is O(max(N,M))

By simply following pointers.

The Scoring Scheme

• Scores are usually stored in a “weight” matrix or “substitution” matrix or “matching” matrix.

• Defining the “proper” matrix is still an active area of research

• Usually, start from known, reliable alignment. Compute fi, the frequency of occurrence of residue type i, and qij, the probability that residue types i and j are aligned; score is computed as:

=

ji

ij

ijff

qS log

1121-3-2-3-1-3-3-2-2-3-4-4-2-3-4-3-3-2W

273-1-1-1-1-2-22-1-2-3-2-3-2-3-2-2-2Y

136-1000-3-3-1-3-3-3-3-3-2-4-2-2-2F

-3-1-14131-2-3-3-2-2-3-3-30-2-2-2-1V

-2-103422-2-2-3-2-3-4-3-4-1-3-2-2-1L

-3-101241-3-3-3-3-3-3-3-4-1-3-2-2-1I

-1-10-2215-1-1-20-2-3-2-3-1-2-1-1-1M

-3-2-3-3-2-3-152-111-10-2-1-100-3K

-3-2-3-3-2-3-125010-20-2-1-2-1-1-3R

-22-1-2-3-3-2-1080011-2-2-20-1-3H

-2-1-3-2-2-301105200-2-1-100-3Q

-3-2-3-3-3-3-21002520-2-1-100-4E

-4-3-3-3-4-3-3-1-2-10261-1-2-110-3D

-4-2-3-3-3-3-200-100160-2-201-3N

-2-3-30-4-4-3-2-2-2-2-2-1-260-210-3G

-3-2-2-2-1-1-1-1-1-2-1-1-2-104-1-110A

-4-3-4-2-3-3-2-1-2-2-1-1-1-1-2-171-1-3P

-3-2-2-2-2-2-10-1000101-1141-1T

-3-2-2-2-2-2-10-1-1000101-114-1S

-2-2-2-1-1-1-1-3-3-3-3-4-3-3-30-3-1-19C

WYFVLIMKRHQEDNGAPTSC

Example of a Scoring matrix

Gap penalty

• Most common model:

WN = G0 + N * G1

WN : gap penalty for a gap of size N

G0 : cost of opening a gap

G1 : cost of extending the gap by one

N : size of the gap

Global versus Local Alignment

• Global alignment finds best arrangement that

maximizes total score

• Local alignment identifies highest scoring

subsequences, sometimes at the expense of the

overall score

Local alignment algorithm is just a variation

of the global alignment algorithm!

Modifications for local alignment

1) The scoring matrix has negative values for mismatches

2) The minimum score for any (i,j) in the alignment matrix is 0.

3) The best score is found anywhere in the filled alignment matrix

These 3 modifications cause the algorithm to search for matching sub-sequences which are not penalized by other regions (modif. 2), with minimal poor matches (modif 1), which can occur anywhere (modif 3).

Mathematical Formulation

Wk: penalty for a gap of size k

Local alignment (Smith Waterman):

+−−−

+−−−

−−

+=

−≤≤

−≤≤

0

,max

maxmax

k2ik1

k2jk1

W1)jk,1A(i

,Wk)1j1,A(i

1),j1,A(i

j)Score(i,j)A(i,

Global versus Local Alignment

100000S

001000N

000310C

000120C

000001A

SGTCCA

Match: +1; Mismatch: -2; Gap: -1

1-10-1-2-3S

001-1-2-3N

-1-1-131-3C

-2-2-212-3C

-3-3-3-3-31A

SGTCCA

Global: ACCTGS

ACC-NS

Local: ACC

ACC

Heuristic methods

• O(NM) is too slow for database search

• Heuristic methods based on frequency of shared

subsequences

• Usually look for ungapped small sequences

FASTA, BLAST

FASTA

• Create hash table of short words of the query sequence (from 2 to 6 characters)

• Scan database and look for matches in the query hash table

• Extend good matches empirically

WordP

…

Word3

Word2

Word1

SeqN…Seq7Seq6Seq5Seq4Seq3Seq2Seq1

http://www.ncbi.nlm.nih.gov/Education/BLASTinfo/information3.htmlTutorial:

BLAST

1) Break query sequence and database sequences into words

2) Search for matches (even not perfect) that scores at least T

3) Extend matches, and look for alignment that scores at least S

Summary

• Dynamic programming finds the optimal alignment

between two sequences in a computing time

proportional to NxM, where N and M are the

sequence lengths

• Critical user choices are the scoring matrix, the gap

penalties, and the algorithm (local or global)

Statistics of Sequence Alignment

Significance

• We have found that the score of the alignment between two sequences is S.

Question: What is the “significance” of this score?

• Otherwise stated, what is the probability P that the alignment of two random sequences has a score at least equal to S ?

• P is the P-value, and is considered a measure of statistical significance.

If P is small, the initial alignment is significant.

A given experiment may yield the event A or the event not(A) with probabilities p,

and q=1-p, respectively. If the experiment is repeated N times and X is the number

of time A is observed, then the probability that X takes the value k is given by:

Basic Statistics: Binomial Distribution

kNkpp

k

NkXp

−−

== )1()(

)!(!

!

kNk

N

k

N

−=

With the binomial coefficient:

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Properties:

E(X)=Np

Var(X)=Np(1-p)

N=40; p=0.2

The Poisson distribution is a discrete distribution usually defined over a volume

or time interval.

Given a process with expected number of success λ in the given interval, the

probability of observing exactly X success is given by:

Basic Statistics: Poisson Distribution

( )!

exp)(

XXp

X λλ

−=

λλλλ=8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Properties:

E(X)=λ

Var(X)=λ

A normal distribution in a variable X with mean µ and variance σ2 is a statistical

distribution with probability function:

Basic Statistics: Normal Distribution

−−=

2

2

2

)(exp

2

1)(

σ

µ

πσ

XXp

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

N=40; p=0.2

The normal distribution is the limiting

case of a binomial distribution P(n,N,p)

with:

)1(2 pNp

Np

−=

=

σ

µ

A extreme value distribution in a variable X is a statistical distribution with

probability function:

Basic Statistics: Extreme Value

Distribution

−−

−=

b

xa

b

xa

bXp expexp

1)(

a=0; b=1

-10 -8 -6 -4 -2 0 2 4 6 8 10

Note the presence of a long tail

BLAST Input

BLAST results

BLAST results (2)

Statistics of Protein Sequence Alignment

• Statistics of global alignment:

Unfortunately, not much is known! Statistics based on Monte

Carlo simulations (shuffle one sequence and recompute

alignment to get a distribution of scores)

• Statistics of local alignment

Well understood for ungapped alignment. Same theory

probably apply to gapped-alignment

Statistics of Protein Sequence Alignment

What is a local alignment ?

“Pair of equal length segments, one from each sequence, whose scores

can not be improved by extension or trimming. These are called

high-scoring pairs, or HSP”

http://www.people.virginia.edu/~wrp/cshl98/Altschul/Altschul-1.html

The E-value for a sequence alignment

-10 -8 -6 -4 -2 0 2 4 6 8 10

S

The expected number of

HSP with score at least S is

given by:

HSP scores follow an extreme value distribution, characterized

by two parameters, K and λ.

( )SKmnE λ−= exp

m, n : sequence lengths

E : E-value

The Bit Score of a sequence alignment

( )( )2ln

ln'

KSS

−=

λ

Raw scores have little meaning without knowledge of the scoring

scheme used for the alignment, or equivalently of the parameters

K and λ.

Scores can be normalized according to:

S’ is the bit score of the alignment.

The E-value can be expressed as:

'2 SmnE

−=

The P-value of a sequence alignment

( ) ( )ESscorewithHSPrandomP −=≥ exp0

( ) ( )!

expX

EESscorewithHSPrandomXP

X

−=≥

The number of random HSP with score greater of equal to S follows a

Poisson distribution:

(E: E-value)

Then:

( ) ( )ESscorewithHSPrandomleastatPPval −−=≥= exp11

Note: when E <<1, P ≈E

Database search, where database contains NS sequences

corresponding to NR residues:

1) All sequences are a priori equally likely to be related to the query:

2) Longer sequences are more likely to be related to the query:

BLAST reports EDB2

The database E-value for a sequence

alignment

( )SKmnNE SDB λ−= exp

( )SKmNE RDB λ−= exp2

Summary

• Statistics on local sequence alignment are defined by:

– Raw score

– Bit score (normalized score)

– E-value

– P-value

• Statistics on database search:

- database E-value