sequence alignment - unibo.it · sequence alignment. sequence alignment ... (optimal global...
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Sequence Alignment
Sequence Alignment
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- || ||||||| |||| | || ||| |||||TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC
AGGCTATCACCTGACCTCCAGGCCGATGCCCTAGCTATCACGACCGCGGTCGATTTGCCCGAC
Distance from sequencesDistance from sequences
Mutation events
Mutationwe need a score for the substitution of the symbol i with j (amino acidic residues, nucleotides, etc.)
substitution matrices s(i,j)
A: ALASVLIRLITRLYP B: ASAVHLNRLITRLYP
The substitution matrix should account for the underlying biological process (conserved structures or functions)
Substitution matricesSubstitution matrices
The basic idea is to measure the correlation between 2 sequences
Given a pair of “correlated” sequences we measure the substitution frequency of a-> b ( assuming symmetry) Pab
The null hypothesis (random correlation or independent events) is the product PaPb
We then define s (a,b) = log(Pab/PaPb)
(log is additive so that the probability => sum over pairs)
Minimal distance = Maximum score (s)
Following this idea several matrices have been derivedTheir main difference is relative to the alignment types used for computing the frequencies
PAMx: (Point Accepted Mutation). Number x % of mutation events.The matrix is built as:
A1ik = P(k|i) for sequences with 1% mutations.
Anik=(A1
ik)n n % mutation events (number of mutations NOT mutated residues.
E.g.: n=2 P(i|k) = aP(i|a) P(a|k)
PAMn = Log(Anij /Pi)
Substitution matricesSubstitution matrices
PAM10PAM10
Very stringent matrix, no out of diagonal element is > 0
PAM160PAM160
Some positive values outside the diagonal appear
PAM250PAM250
This is one of the most widely used matrix
PAM500PAM500
PAM matrices are computed starting from very similar sequences and more distance scoring matrices are derived by matrix products
BLOSUMx: This family is computed directly from blocks of alignments with a defined sequence identity threshold (> x%).
=> For very related sequences we must use PAM with low numbers and BLOSUM with large numbers. The opposite holds for distant sequences
Substitution matricesSubstitution matrices
BLOSUM62BLOSUM62
Probably the most widely used
BLOSUM90BLOSUM90
BLOSUM30BLOSUM30
Exercise:Exercise:
Write your own substitution Matrix:Given e set of aligned sequences compute thePab=frequency of mutations between a->b (assume symmetry, a->b counts also as b->a).Pa=as the marginal probability of Pab
Finally, derive the substitution matrix: s (a,b) = log(Pab/PaPb)Use the files provided at:http://www.biocomp.unibo.it/piero/P4B/Malignments/
Hints: 1.start with toy.aln to check your algorithm2. Initialize your P[a][b] matrix with pseudocounts (such as 1 instead of 0)
DotPlots:DotPlots:
DotPlot Exercise:DotPlot Exercise:
Write a program dotplot.pyThat takes as input a fasta file with two sequencesA scoring matrix and sliding window and the threshold cut-off, such as:
dotplot.py fasta.in score.mat 11 threshold
and prints on the standard oputput the dotplot.
PS: you can use matplotlib for a nicer presentation
Distance between sequencesDistance between sequences
What kind of operations we consider?
Mutation Deletion and Insertion(rarely rearrangements )
A: ALASVLIRLIT--YP B: ASAVHL---ITRLYP
The negative gap score value depends only on the number of holes(n) = -nd linear(n) = -d - (n-1)e affine (d: open e: extension)
!! Please note that all scores are position-independent along the sequence
Pairwise alignmentPairwise alignment
Given 2 sequences what is an alignment with a maximum score?
Naïf solution: try every possible alignments and select one with the best score
Using the score function :
How may alignments there are?How may alignments there are?
Case without internal gaps
AAA AAA AAA AAA AAA AAABB BB BB BB BB BB
Given 2 sequences of length m e n, the number of shifts is m +n +1
Case with internal gaps
AAA AAA AAA AAA AAABB BB BB BB BBBBAAA BABAA BAABA BAAAB ABBAA
AAA AAA AAA AAA AAABB BB BB BB BB ...ABABA ABAAB AABBA AABAB AAABB
The number of the alignments is equal at the number of all possible way of mixing 2 sequences keeping track of the original sequence order. Given 2 sequences of lengths n and m, they are ∑k=0,min(m,n)(m+n-k)!/k!(n-k)!(m-k)!
For n=m=80we get > 1043 possible alignments !!!!!!!
How may alignments there are?How may alignments there are?
We can keep a table of the precomputed substring alignments (dynamic programming)
ALSKLASPALSAKDLDSPALSALSKIADSLAPIKDLSPASLT
ALSKLASPALSAKDLDSPAL-SALSKIADSLAPIKDLSPASLT-
ALSKLASPALSAKDLDSPALS-ALSKIADSLAPIKDLSPASL-T
Basic ideaBasic idea
Building step by stepGiven the 2 sequences
ALSKLASPALSAKDLDSPALS, ALSKIADSLAPIKDLSPASLT
The best alignment between the two substrings
ALSKLASPA ALSKIAD
can be computed taking only into consideration
ALSKLASP A ALSKLASP A ALSKLASPA - ALSKIA D ALSKIAD - ALSKIA D
The best among these 3 possibilities
Basic ideaBasic idea
+ + +
The Needleman-Wunsch Matrixx1 ……………………………… xM
yN …
……
……
……
……
……
…
y1
Every nondecreasing path
from (0,0) to (M, N)
corresponds to an alignment of the two sequences
The Needleman-Wunsch Algorithm
x = AGTA m = 1
y = ATA s = -1
d = -1
20-1-1-3A
0100-2T
-2-101-1A
-4-3-2-10
ATGA
F(i,j) i = 0 1 2 3 4
j = 0
1
2
3
Optimal Alignment:
F(4,3) = 2
AGTAA - TA
The Needleman-Wunsch Algorithm
• Initialization.• F(0, 0) = 0• F(0, j) = - j d• F(i, 0) = - i d
• Main Iteration. Filling-in partial alignments• For each i = 1……M
For each j = 1……N F(i-1,j) – d [case 1]
F(i, j) = max F(i, j-1) – d [case 2] F(i-1, j-1) + s(xi, yj) [case 3]
UP, if [case 1]Ptr(i,j) = LEFT if [case 2]
DIAG if [case 3]• Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace
back optimal alignment
Exercise:
Suppose you want only to know the score of a global alignment.
=> Write a program that given two input sequence (in a single file in fasta format), a gap cost and a similarity matrix computes the score of the global alignment in O(N*M) time and in O(M) space, where M and N are the lengths of the input sequences and M<=N
The Overlap Detection variant
Changes:
• Initialization
For all i, j,
F(i, 0) = 0
F(0, j) = 0
• Termination
maxi F(i, N)
FOPT = max maxj F(M, j)
x1 ……………………………… xN
yM …
……
……
……
……
……
…
y1
The local alignment problem
Given two strings x = x1……xM,
y = y1……yN
Find substrings x’, y’ whose similarity (optimal global alignment value)is maximum
e.g. x = aaaacccccggggy = cccgggaaccaacc
Why local alignment• Genes are shuffled between genomes
• Portions of proteins (domains) are often conserved
The Smith-Waterman algorithm
Idea: Ignore badly aligning regions
Modifications to Needleman-Wunsch:
Initialization: F(0, j) = F(i, 0) = 0
0
Iteration: F(i, j) = max F(i – 1, j) – d
F(i, j – 1) – d
F(i – 1, j – 1) + s(xi, yj)
The Smith-Waterman algorithm
Termination:
• If we want the best local alignment…
FOPT = maxi,j F(i, j)
• If we want all local alignments scoring > t
For all i, j find F(i, j) > t, and trace back
Scoring the gaps more accurately
Current model:
Gap of length nincurs penalty nd
However, gaps usually occur in bunches
Concave gap penalty function:
G(n):for all n, G(n + 1) - G(n) G(n) - G(n – 1)
G(n)
G(n)
General gap dynamic programming
Initialization: same
Iteration:
F(i-1, j-1) + s(xi, yj)
F(i, j) = max maxk=0…i-1F(k,j) – (i-k)
maxk=0…j-1F(i,k) – (j-k)
Termination: same
Running Time: O(N2M) (assume N>M)
Space: O(NM)
Exercise:
Write a program that given two input sequence (in a single file in fasta format), and a choice of a general gap function and scoring matrix computes the alignments of the two sequences and returns one of the possible best alignments.
Remember that when you store that the best score is obtained using
maxk=0…i-1F(k,j) – g(i-k)
maxk=0…j-1F(i,k) – g(j-k)
You have to store this information in the corresponding pointer (back-trace) matrix.
Compromise: affine gaps
g(n) = d + (n – 1)e | |gap gapopen extend
To compute optimal alignment,
At position i,j, need to “remember” best score if gap is open best score if gap is not open
F(i, j): score of alignment x1…xi to y1…yj
ifif xi aligns to yj
G(i, j): score ifif xi, or yj, aligns to a gap
de
g(n)
Needleman-Wunsch with affine gaps
Initialization: F(0,0)=0, F(i, 0) = d + (i – 1)e, F(0, j) = d + (j – 1)eR(0,j)= -∞ , C(i,0)= -∞
Iteration: F(i – 1, j – 1) + s(xi, yj)
F(i, j) = max R(i , j) C(i , j)
F(i – 1, j) – d R(i, j) = max
R(i – 1, j) – e
F(i , j -1) – d C(i, j) = max
C(i , j -1 ) – e
Termination: same
Bounded Dynamic Programming
Assume we know that x and y are very similar
Assumption: # gaps(x, y) < k(N) ( say N>M )
xi Then, | implies | i – j | < k(N)
yj
We can align x and y more efficiently:
Time, Space: O(N k(N)) << O(N2)
Bounded Dynamic ProgrammingInitialization:
F(i,0), F(0,j) undefined for i, j > k
Iteration:
For i = 1…M
For j = max(1, i – k)…min(N, i+k)
F(i – 1, j – 1)+ s(xi, yj)
F(i, j) = max F(i, j – 1) – d, if j > i – k(N)
F(i – 1, j) – d, if j < i + k(N)
Termination: same
x1 ………………………… xM
y1 …
……
……
……
……
…
yN
k(N)Easy to extend to the affine gap case
Significant alignments
Significance of an alignmentSignificance of an alignment
Given an alignment with score S, is it significant?
How are the score of random alignments distributed?
100,000 alignments of unrelated and shuffled sequences:
Score
Occ
orre
nza
Z=(S-<S>)/s
S= Alignment score<S>= average of the scores on a random set of alignments s= Standard deviation of the scores on a random set of alignments
Significance of the alignment
Z<3 not significant3<Z<10 probably significantZ>10 significant
Z-scoreZ-score
Write a program that takes in input a fasta with two sequences, and a number N.Compute the score of the global alignment of the two sequence and the Z-score with respect N shuffled sequences (generated from the first of the fasta) against the original second sequence of the fasta.Z=(S-<S>)/s
S= Alignment score<S>= average of the scores on a random set of alignments s Standard deviation of the scores on a random set of alignments
Execise: Z-scoreExecise: Z-score
The Z-score of this alignment is 7.5 over 54 residuesSequence identity is as high as 25.9%. The sequences have a completely different structure
Citrate synthase (2cts) vs transthyritin (2paba)
Is the Z-score reliable?Is the Z-score reliable?
E-valueE-value
Expected number of random alignments obtaining a score greater or equal to a given score (s)
Relies on the Extreme Value Distribution
E=Kmn e-s
m, n: lengths of the sequencesK, : “scaling” constants
The number of high scoring random alignment increases when the sequence lengths increase and decreases in an exponential way when the score increases.
Alignment significance The significance of the E-value depends on the length of the considered database. Considering Swiss Prot,
E> 10-1 non significant10-1 > E > 10-3 probably not significant10 -3 > E > 10-8 probably significantE < 10-8 significant
E-valueE-value
P-valueP-value
Probability for random alignments to obtain a score greater or equal to a given score (s)
Given the E-value (expected number of alignments), which statistics do describe the probability of having a number a of random alignments with score ≥ S?
Poisson: P(a) =
Which is the probability of finding at least one random alignment with score ≥ S?
P(a ≥ 1) = 1 – P(0) = 1 – exp (-E)
Similarity search in Data BasesSimilarity search in Data Bases
Given a sequence, search for similar sequences in huge data sets
In principle, the alignments between the query sequence and ALL the sequences in the data sets could be tried
Too many sequences!
Heuristic algorithms can be used. They do not assure to find the optimal alignment
FASTABLAST
FASTAFASTA
The query sequence is chopped in words of k-tup characters. Usually k-tup = 2 for proteins, 6 for DNA
ADKLPTLPLRLDPTNMVFGHLRI
Words (indexed by position):AD, DK, KL, LP, PT, TL, LP, PR, RL, …,…,1 2 3 4 5 6 7 8 9 ….
The list of indexed words is compiled for each sequence in the data set (subject)
The search of the correspondence between the words is very fast.The difference between the indexes of the matches in the query and the subject sequences determines the distance from the main diagonal
FASTAFASTA
Query
Sub
ject
Many matches along the same diagonal correspond to longer identical segments along the sequences
FASTAFASTA
Query
Sub
ject
The alignment of the longest matched diagonals are evaluated with a score matrix (PAM or BLOSUM)
FASTAFASTA
Query
Sub
ject
Most similar regions on close diagonals are isolated
Query
Sub
ject
FASTAFASTA
An exact Smith-Waterman alignment is computed on a narrow band around the diagonal endowed with the highest similarity (a 32-residue band is usually adopted)
Sequence similarity with FASTA
BLASTBLAST
The sequence data set is indexed as follow:for each possible residue triplet the occurrence and the position along the sequences are stored. (FORMATDB)
AAAAACAADACA.........
BLASTBLAST
The query sequence is chopped in words, W-residue long (usually W=3 for proteins)
LSHLPTLPLRLDPTNMVFGHLRI
LSH, SHL, HLP, LPT, PTL, TLR, …,…,
For each word, all the similar proteins are generated, using the BLOSUM62 matrix and setting a similarity threshold (usually T = 11--13)
LSH 16 ISH 14MSH 14VSH 13LAH 13LTH 13LNH 13
BLASTBLAST
Each word included in the list of the similar words is retrieved in the sequence of the data set by means of the indexes
The match is extended, until the score is higher than a threshold S
Sequence similarity with BLAST (Basic Local Alignment Search Tool)
Alignment of all the retrieved sequencesAlignment of all the retrieved sequences
ATTENTION: It is not a multiple sequence alignment
1 Y K D Y H S - D K K K G E L - - 2 Y R D Y Q T - D Q K K G D L - - 3 Y R D Y Q S - D H K K G E L - - 4 Y R D Y V S - D H K K G E L - - 5 Y R D Y Q F - D Q K K G S L - - 6 Y K D Y N T - H Q K K N E S - - 7 Y R D Y Q T - D H K K A D L - - 8 G Y G F G - - L I K N T E T T K 9 T K G Y G F G L I K N T E T T K 10 T K G Y G F G L I K N T E T T K
A 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0 0 70 0 0 0 0 60 0 0 0 0 20 0 0 0 E 0 0 0 0 0 0 0 0 0 0 0 0 70 0 0 0 F 0 0 0 10 0 33 0 0 0 0 0 0 0 0 0 0 G 10 0 30 0 30 0 100 0 0 0 0 50 0 0 0 0 H 0 0 0 0 10 0 0 10 30 0 0 0 0 0 0 0 K 0 40 0 0 0 0 0 0 10 100 70 0 0 0 0 100 I 0 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 L 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 N 0 0 0 0 10 0 0 0 0 0 30 10 0 0 0 0 P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 0 0 0 0 40 0 0 0 30 0 0 0 0 0 0 0 R 0 50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 0 0 0 0 0 33 0 0 0 0 0 0 10 10 0 0 T 20 0 0 0 0 33 0 0 0 0 0 30 0 30 100 0 V 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 W 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y 70 0 0 90 0 0 0 0 0 0 0 0 0 0 0 0
Position
Sequence profileSequence profile
Usefulness of the sequence profilesUsefulness of the sequence profiles
Sequence profiles describes the basic features of all the sequence used in the alignment
Most conserved regions and most frequent mutations for each position are highlighted
Sequence-to-profile alignment
The alignment score are weighted position by position using the profile. The same mutations in different positions are scored with different values
Sequence-to-profile alignmentSequence-to-profile alignment
Given the position i along a sequence profile, it is represented by a 20-valued vector Pi = Pi(A) Pi(C) …… Pi (Y)
Given the residue in position j along the sequence to align: Sj
The score for aligning Sj to the vector Pi is:
where M is a matrix score (BLOSUM or PAM)
•How to score a profile-profile alignment?
PSI-BLASTPSI-BLAST
http://www.ncbi.nlm.nih.gov/BLAST/
Sequence
Data Base
BLAST
Sequence profile
PSI-BLAST
Until converges
• PSI-BLAST takes as an input a single protein sequence and compares it to a protein database, using the gapped BLAST program
• The program constructs a multiple alignment, and then a profile, from any significant local alignments found. The original query sequence serves as a template for the multiple alignment and profile, whose lengths are identical to that of the query. Different numbers of sequences can be aligned in different template positions
• The profile is compared to the protein database, again seeking local alignments. After a few minor modifications, the BLAST algorithm can be used for this directly.
• PSI-BLAST estimates the statistical significance of the local alignments found. Because profile substitution scores are constructed to a fixed scale, and gap scores remain independent of position, the statistical theory and parameters for gapped BLAST alignments remain applicable to profile alignments.
• Finally, PSI-BLAST iterates, by returning to step (2), an arbitrary number of times or until convergence.
The design of PSI-BLAST