Protein Sequence ComparisonPatrice Koehl

http://koehllab.genomecenter.ucdavis.edu/teaching/ecs129/06/lecture-notes

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 solutionfor 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 givenpair of positions in the sequences is the score of the best alignment previous to these positions plus the score ofaligning 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 thatA(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

A T G C T G C

A 10 0 0 0 0 0 0

G 0

C 0

C 0

1) Initialize

Example 1

A T G C T G C

A 10 0 0 0 0 0 0

G 0 10

C 0

C 0

2) Propagate

Example 1

A T G C T G C

A 10 0 0 0 0 0 0

G 0 10 20

C 0

C 0

2) Propagate

Example 1

A T G C T G C

A 10 0 0 0 0 0 0

G 0 10 20 10 10 20 10

C 0 10

C 0

2) Propagate

Example 1

A T G C T G C

A 10 0 0 0 0 0 0

G 0 10 20 10 10 20 10

C 0 10 10 30 20 20 30

C 0 10 10 30 30

2) Propagate

Example 1

A T G C T G C

A 10 0 0 0 0 0 0

G 0 10 20 10 10 20 10

C 0 10 10 30 20 20 30

C 0 10 10 30 30 30 40

3) Trace Back

ATGCTGCAXGCXXC

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 thealignment 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 storingthe best score for each row and column.

True if gap penalty is linear!

Example 2

Alignments:

High Score: 30

A A T G C

A 10 10 0 0 0

G 0 10 10 20 10

G 0 10 10 20 20

C 0 10 10 10 30

AATGCAG GC

AATGCA GGC

AATGC AGGC

AATG CA GGC

AATG C A GGC

Example 2 with Gap

Alignments:

High Score: 28

A A T G C

A 10 8 -2 -2 -2

G -2 10 8 18 8

G -2 8 10 18 16

C -2 8 8 10 28

AATGCAG GC

AATGCA GGC

AATGC AGGC

Gap cost: -2

Complexity (2)

1) The traceback routine can be quite costly in computingtime if all possible optimal paths are required, sincethere may be many branches.

2) Usually, an arbitrary choice is made about whichbranch 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

ijij ff

qS log

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

C 9 -1 -1 -3 0 -3 -3 -3 -4 -3 -3 -3 -3 -1 -1 -1 -1 -2 -2 -2

S -1 4 1 -1 1 0 1 0 0 0 -1 -1 0 -1 -2 -2 -2 -2 -2 -3

T -1 1 4 1 -1 1 0 1 0 0 0 -1 0 -1 -2 -2 -2 -2 -2 -3

P -3 -1 1 7 -1 -2 -1 -1 -1 -1 -2 -2 -1 -2 -3 -3 -2 -4 -3 -4

A 0 1 -1 -1 4 0 -1 -2 -1 -1 -2 -1 -1 -1 -1 -1 -2 -2 -2 -3

G -3 0 1 -2 0 6 -2 -1 -2 -2 -2 -2 -2 -3 -4 -4 0 -3 -3 -2

N -3 1 0 -2 -2 0 6 1 0 0 -1 0 0 -2 -3 -3 -3 -3 -2 -4

D -3 0 1 -1 -2 -1 1 6 2 0 -1 -2 -1 -3 -3 -4 -3 -3 -3 -4

E -4 0 0 -1 -1 -2 0 2 5 2 0 0 1 -2 -3 -3 -3 -3 -2 -3

Q -3 0 0 -1 -1 -2 0 0 2 5 0 1 1 0 -3 -2 -2 -3 -1 -2

H -3 -1 0 -2 -2 -2 1 1 0 0 8 0 -1 -2 -3 -3 -2 -1 2 -2

R -3 -1 -1 -2 -1 -2 0 -2 0 1 0 5 2 -1 -3 -2 -3 -3 -2 -3

K -3 0 0 -1 -1 -2 0 -1 1 1 -1 2 5 -1 -3 -2 -3 -3 -2 -3

M -1 -1 -1 -2 -1 -3 -2 -3 -2 0 -2 -1 -1 5 1 2 -2 0 -1 -1

I -1 -2 -2 -3 -1 -4 -3 -3 -3 -3 -3 -3 -3 1 4 2 1 0 -1 -3

L -1 -2 -2 -3 -1 -4 -3 -4 -3 -2 -3 -2 -2 2 2 4 3 0 -1 -2

V -1 -2 -2 -2 0 -3 -3 -3 -2 -2 -3 -3 -2 1 3 1 4 -1 -1 -3

F -2 -2 -2 -4 -2 -3 -3 -3 -3 -3 -1 -3 -3 0 0 0 -1 6 3 1

Y -2 -2 -2 -3 -2 -3 -2 -3 -2 -1 2 -2 -2 -1 -1 -1 -1 3 7 2

W -2 -3 -3 -4 -3 -2 -4 -4 -3 -2 -2 -3 -3 -1 -3 -2 -3 1 2 11

Example of a Scoring matrix

Gap penalty

• Most common model:

WN = G0 + N * G1

WN : gap penalty for a gap of size NG0 : cost of opening a gapG1 : cost of extending the gap by oneN : 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 variationof 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

A C C T G S

A 1 0 0 0 0 0

C 0 2 1 0 0 0

C 0 1 3 0 0 0

N 0 0 0 1 0 0

S 0 0 0 0 0 1

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

A C C T G S

A 1 -3 -3 -3 -3 -3

C -3 2 1 -2 -2 -2

C -3 1 3 -1 -1 -1

N -3 -2 -1 1 0 0

S -3 -2 -1 0 -1 1

Global: ACCTGSACC-NS

Local: ACCACC

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

Seq1 Seq2 Seq3 Seq4 Seq5 Seq6 Seq7 … SeqN

Word1

Word2

Word3

…

WordP

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

kNk ppk

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 in the given interval, the

probability of observing exactly X success is given by:

Basic Statistics: Poisson Distribution

!

exp)(

XXp X

=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)=

Var(X)=

A normal distribution in a variable X with mean and variance 2 is a statistical

distribution with probability function:

Basic Statistics: Normal Distribution

2

2

2

)(exp

2

1)(

X

Xp

00.020.040.060.080.1

0.120.140.160.18

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 limitingcase of a binomial distribution P(n,N,p)with:

)1(2 pNp

Np

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 ofHSP with score at least S isgiven by:

HSP scores follow an extreme value distribution, characterized by two parameters, K and .

SKmnE exp

m, n : sequence lengthsE : E-value

The Bit Score of a sequence alignment

2ln

ln'

KSS

Raw scores have little meaning without knowledge of the scoring scheme used for the alignment, or equivalently of the parameters K and .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 aPoisson distribution:

(E: E-value)Then:

ESscorewithHSPrandomleastatPPval exp11

Note: when E <<1, P ≈E

Database search, where database contains NS sequencescorresponding 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 exp

SKmNE RDB 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