Download String-matching algorithms in BLAST and FASTA

Mathematics and computation behind
BLAST and FASTA
Xuhua Xia
[email protected]
http://dambe.bio.uottawa.ca
Why string matching?
• Early applications: Sequence similarity between an oncogene (genes in
viruses that cause a cancer-like transformation of the infected cells), v-sis,
and the platelet-derived growth factor (PDGF)
– M. D. Waterfield et al. 1983. Nature 304:35-39
– R. F. Doolittle et al. 1983. Science 221:275-277
– Implications:
• Cancer can be caused by a constitutively expressed growth factor
• Alteration of gene expression can contribute to cancer
• Growth factors and the like can be drug targets against cancer
• Fast computational methods in string matching
– FASTA
– BLAST
– Local pair-wise alignment by dynamic programming
Slide 2
FASTA
• A commonly used family of alignment and search
tools
• Generally considered to be more sensitive than
BLAST.
• Illustration with two fictitious sequences used in the
Contig Assembly lecture:
Seq1: ACCGCGATGACGAATA
Seq2: GAATACGACTGACGATGGA
Seq1: ACCGCGATGACGAATA
Seq2:
GAATACGACTGACGATGGA
Slide 3
String Match in FASTA
(a)
1
A
G
2
C
A
3
C
A
4
G
T
(b)
A
1
7
10
13
14
16
C
2
3
5
11
G
4
6
9
12
T
8
15
(c)
1
2
3
4
G A A T
-3
1
2 -4
-5 -5 -4 -11
-8 -8 -7
-11 -11 -10
-12 -11
-14 -13
Seq1
Seq2
(e)
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Left
Right
C G A T G A C G A A T A
Move N Move
A C G A C T G A C G A T G G A
-1 3
1
-2 5
2
-3 1
3
-4 3
4
-5 7
5
-6 1
6
-7 1
7
-8 4
8
-9 1
9
-10 1
10
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
-11 5
11
A C G A C T G A C G A T G G A
-12 1
12
4 4 3 7 7 2 7 11 11 10 14 8 13 14 18
-13 1
13
-2 3 1 1 6 -5 5 5 10 8 8 1 11 12 12
-14 1
14
-5 1 -2 -2 4
2 2 8 5 5
8 9 9
-15 0
15
-8 -5 -5 -5 -2
-1 -1 2 2 2
5 6 6
16
-9
-6
-2
1
5
17
-11
-8
-4
-1
3
18
(d)
Seq1:
ACCGCGATGACGAATA
Seq2: GAATACGACTGACGATGGA
Seq1:
Seq2:
ACCGCGATGACGAATA
GAATACGACTGACGATGGA
N
6
7
3
3
6
3
3
5
2
2
3
2
1
2
0
0
0
1
Word length of 2
(a)
Seq1
Seq2
1
A
G
2
C
A
3
C
A
4
G
T
5
C
A
6
G
C
7
A
G
8
T
A
(b) AA AC AG AT CA CC CG CT
13 1
7
2 3
10
14
5
11
(c)
(e)
1 2 3 4 5 6 7 8
GA AA AT TA AC CG GA AC
-5 -11 -4 -11 4 3 1 7
-8
-11
-5 1 -2 -2
-11
-5 -5
Best
Seq1:
ACCGCGATGACGAATA
Seq2: GAATACGACTGACGATGGA
9 10 11 12 13 14 15 16 17 18 19 Left
Right
G A C G A A T A
Move N Move N
C T G A C G A T G G A
-1 1
1 3
-2 2
2 5
GA GC GG GT TA TC TG TT
-3 0
3 1
6 4
15
8
-4 1
4 1
9
-5 4
5 2
12
-6 0
6 1
-7 0
7 1
-8 1
8 4
-9 0
9 1
-10 0
10 1
9 10 11 12 13 14 15 16 17 18
-11 4
11 1
CT TG GA AC CG GA AT TG GG GA
-12 0
12 1
2 5 11 10 8 8 8
12
-13 0
13 0
2 2 8 5 1
9
-14 0
14 0
-1
2 2
6
15 0
16 0
17 0
(d)
One of the three 2nd best
Seq1: ACCGCGATGACGAATA
Seq2:
GAATACGACTGACGATGGA
Human COX1
RWLFSTNHKDIGTLYLLFGAWAGVLGTALSLLIRAELGQPGNLLGNDHIYNVIVTAHAFVMI
FFMVMPIMIGGFGNWLVPLMIGAPDMAFPRMNNMSFWLLPPSLLLLLASAMVEAGAGTGWTV
YPPLAGNYSHPGASVDLTIFSLHLAGVSSILGAINFITTIINMKPPAMTQYQTPLFVWSVLI
TAVLLLLSLPVLAAGITMLLTDRNLNTTFFDPAGGGDPILYQHLFWFFGHPEVYILILPGFG
MISHIVTYYSGKKEPFGYMGMVWAMMSIGFLGFIVWAHHMFTVGMDVDTRAYFTSATMIIAI
PTGVKVFSWLATLHGSNMKWSAAVLWALGFIFLFTVGGLTGIVLANSSLDIVLHDTYYVVAH
FHYVLSMGAVFAIMGGFIHWFPLFSGYTLDQTYAKIHFTIMFIGVNLTFFPQHFLGLSGMPR
RYSDYPDAYTTWNILSSVGSFISLTAVMLMIFMIWEAFASKRKVLMVEEPSMNLEWLYGCPP
PYHTFEEP
Exact match:
All sequences in the database are pre-indexed.
Cys is the rarest in this protein in the database.
If a query sequence contain a C, then go directly to C at site
494 to check; if the query has no C, then report 'No match'
BLAST
• Adapted from Crane & Raymer 2003
• Motivation: matching short sequences are faster than matching
longer ones
• Input sequence: AILVPTVIGCTVPT
• Algorithm:
– Break the query sequence into words:
AILV, ILVP, LVPT, VPTV, PTVI, TVIG, VIGC, IGCT,
GCTV, CTVP, TVPT
– Discard common words (i.e., words made entirely of common amino
acids)
– Search for matches against database sequences, assess significance and
decide whether to discard to continue with extension using dynamic
programming:
AILVPTVIGCTVPT
MVQGWALYDFLKCRAILVPTVIACTCVAMLALYDFLKC
• Critical decision: Discard or continue?
• The E-value as an answer.
Slide 7
Basic stats in string matching
• Given PA, PC, PG, PT in a target (database) sequence, the
probability of a query sequence, say, ATTGCC, having a
perfect match of the target sequence is:
prob = PAPTPT PGPCPC = PA (PC)2 PG (PT)2
• Let M be the target sequence length and N be the query
sequence length, the “matching operation” can be performed
(M – N +1) times, e.g.,
Query: ATG
Target CGATTGCCCG
• The probability distribution of the number of matches follows
(approximately) a binomial distribution with p = prob and n =
(M – N +1)
Slide 8
Basic stats in string matching
• Probability of having a sequence match: p
• Probability of having no match: q = 1-p
• Binomial distribution:
( p  q) n  p n 
n!
n!
p n 1q  ... 
p n  x q x  ...  q n
(n  1)!1!
(n  x)! x !
• When np > 50, the binomial distribution can be approximated
by the normal distribution with the mean = np and variance =
npq
( x )

2
P( x) 
1
e
2
2 2
• When np < 1 and n is very large, binomial distribution can be
approximated by the Poisson distribution with mean and
variance equal to np (i.e.,  = 2 = np).
e   x
P( x) 
x!
Slide 9
From Binomial to Poisson
( p  q) n  p n 
n!
n!
n!
p n 1q  ... 
p n  x q x ... 
p x q n  x  ...  q n
(n  1)!1!
(n  x)! x !
(n  x)! x !
P ( n)  p n
P(n  x) 
n!
p x q n x
(n  x)! x !
n!

p xq xqn
(n  x)! x !
P(0)  q n
n(n  1)(n  2)...(n  x  1)  p 
n

(1

p
)
 
x!
q
x
n x x  np n x p x  np (np ) x  np
 

pe 
e 
e e
x!
x!
x!
x!
P(n  1)  np n 1q
n!
p n x q x
(n  x)! x !
n!
P( x) 
p x q n x
(n  x)! x !
P( x) 
x
Slide 10
Matching two sequences without gap
• Assuming equal nucleotide frequencies, the probability of a
nucleotide site in the query sequence matching a site in the
target sequence is p = 0.25.
• The probability of finding an exact match of L letters is a = pL
= 0.25L = 2-2L = 2-S, where S is called the bit score in BLAST.
• M: query length; N: target length, e.g., M = 8, N = 5, L = 3
AACGGTTC
CGGTT
• A sequence of length L can move at (M – L +1) distinct sites
along the query and (N – L +1) distinct sites along the target.
• m = (M-L+1) and n = (N-L+1) are called effective lengths of
the two sequences.
• The expected number of matches with length L is mn2-S,
which is called E-value in ungapped BLAST.
• S is calculated differently in the gapped BLAST
Slide 11
Blast Output (Nuc. Seq.)
BLASTN 2.2.4 [Aug-26-2002]
...
Query= Seq1 38
Database: MgCDS
480 sequences; 526,317 total letters
Sequences producing significant alignments:
MG001 1095 bases
Score = 34.2 bits (17), Expect = 7e-004
Identities = 35/40 (87%), Gaps = 2/40 (5%)
Query: 1
Sbjct: 1
Constant gap penalty vs
affine function penalty
Score
E
(bits) Value
34
7e-004
atgaataacg--attatttccaacgacaaaacaaaaccac 38
|||||||||| ||||||||||| |||||| ||||||||
atgaataacgttattatttccaataacaaaataaaaccac 40
Lambda
K
H
1.37
0.711
1.31
Matrix: blastn matrix:1 -3
Gap Penalties: Existence: 5, Extension: 2
…
effective length of query: 26
effective length of database: 520,557
e E ( E ) x
p ( x) 
x!
Typically one would
count only 1 GE here.
Matches: 35*1 = 35
Mismatches: 3*(-3) = -9
Gap Open: 1*5 = 5
Gap extension: 2*2 =4
R = 35 - 9 - 5 - 4 = 17
S = [λR – ln(K)]/ln(2) =[1.37*17-ln(0.711)]/ln(2) = 34
E = mn2-S = 26 * 520557 * 2-34 = 7.878E-04
x
p(x)
0
0.999265217
1
0.000734513
…
Alternatively, E = KmnExp(-lambda*R)
E-Value in BLAST
• The e-value is the expected number of random
matches that is equally good or better than the
reported match. It can be a number near zero or much
larger than 1.
• It is NOT the probability of finding the reported
match.
• Only when the e-value is extremely small can it be
interpreted as the probability of finding 1 match that
is as good as the reported one (see next slide).
Slide 13
E-value and P(1)
e E ( E ) x
p ( x) 
x!
p(1)  e E E  E (when E  0)
1
P(1)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.00
0.20
0.40
0.60
0.80
1.00
E-value
Slide 14
BLAST Programs
Program
Database
Query
Typical Uses
BLASTN/ME
GABLAST
Nucleotide
Nucleotide
MEGABLAST has longer word size than
BLASTN
BLASTP
Protein
Protein
Query a protein/peptide against a protein
database.
BLASTX
Protein
Nucleotide
Translate a nuc sequence into a “protein”
in six frames and search against a protein
database
TBLASTN
Nucleotide
Protein
Unannotated nuc sequences (e.g., ESTs)
are translated in six frames against which
the query protein is searched
TBLASTX
Nucleotide
Nucleotide
6-frame translation of both query and
database
PHI-BLAST
Protein
Protein
Pattern-hit iterated BLAST
PSI-BLAST
Protein
Protein
Position-specific iterated BLAST
RPS-BLAST
Protein
Protein
Reverse PSI-BLAST
Slide 15
Comparison: BLAST and FASTA
• BLAST starts with exact string matching, while
FASTA starts with inexact string matching (or exact
string matching with a shorter words). BLAST is
faster than FASTA.
• For the examples given, both BLAST and FASTA
will find the same best match, i.e., shifting the query
sequence by 2 sites to the right.
• Both perform dynamic programming for extending
the match after the initial match.
Slide 16
Optional: BLAST Parameters
• Lambda  and Karlin-Altschul (K) parameters are important
because they directly affect the computation of E value.
• Both  and K depend on
– nucleotide (or aminon acid) frequencies
– match-mismatch matrix
• All BLAST implementations generally assume that nucleotide
(or amino acid) sequences have roughly equal frequencies.
• For nucleotide (or amino acid) sequences with strongly biased
frequencies, BLAST E value obtained with the assumption
can be quite misleading, i.e., one should use appropriate  and
K.
Lambda () and K
BLAST output includes lambda () and K. Mathematically,  is defined as follows:
4 4
 pi p j e
sij 
1
i 1 j 1
where pi, pj are nucleotide frequencies (i,j = A, C, G, or T), and sij is the match (when i = j) or
mismatch (when i  j) score. In nucleotide BLAST by default, we have sii = 1 and sij = -3. In the
simplest case with equal nucleotide frequencies, i.e., when pi = 0.25, the equation above is reduced to
4
4
 pi p j e
sij 
 4  0.252 e  12  0.252 e3  0.25e  0.75e3  1
i 1 j 1
Now insert different  values to the equation above to find which  balances the equation
(not the trivial solution of  = 0)
20 20
 pi p j e
sij 
1
(for amino acid sequences)
i 1 j 1
See the updated Chapter 1 and BLASTParameter.xlsm on how to compute K.
 implies nucleotide frequencies
(a)
A
G
C
T
A
0.25
0.0625
0.0625
0.0625
0.0625
0.25
0.25
0.25
0.25
Match-mismatch matrix
A
(b)
G
C
T
Lambda
(c)
1
-3
-3
-3
G
0.25
0.0625
0.0625
0.0625
0.0625
-3
1
-3
-3
C
0.25
0.0625
0.0625
0.0625
0.0625
-3
-3
1
-3
T
0.25
0.0625
0.0625
0.0625
0.0625
-3
-3
-3
1
1.374063
(a’)
A
G
C
T
0.49
0.01
0.01
0.49
A
0.49
0.2401
0.0049
0.0049
0.2401
Match-Mismatch matrix
(b’) A
G
C
T
1
-3
-3
-3
G
0.01
0.0049
0.0001
0.0001
0.0049
C
0.01
0.0049
0.0001
0.0001
0.0049
T
0.49
0.2401
0.0049
0.0049
0.2401
-3
1
-3
-3
-3
-3
1
-3
-3
-3
-3
1
Lambda 0.658295
0.246961
0.001013
0.001013
0.001013
0.001013
0.246961
0.001013
0.001013
0.001013
0.001013
0.246961
0.001013
0.001013
0.001013
0.001013
0.246961
1
(c’)
0.463752 0.00068 0.00068 0.03332
0.00068 0.000193 1.39E-05 0.00068
0.00068 1.39E-05 0.000193 0.00068
0.03332 0.00068 0.00068 0.463752
1
BLAST parameters , K and H are computed for each BLAST
database created.
Finding  III: Different , s/v
A
G
C
T
0.1
0.4
0.4
0.1
Match-Mismatch
A
G
C
T
Lambda
A
0.1
0.01
0.04
0.04
0.01
G
0.4
0.04
0.16
0.16
0.04
C
0.4
0.04
0.16
0.16
0.04
T
0.1
0.01
0.04
0.04
0.01
1
-1
-3
-3
-1
1
-3
-3
-3
-3
1
-1
-3
-3
-1
1
0.02691
0.014865
0.002053
0.000513
0.014865
0.430554
0.008211
0.002053
1
0.9899
0.002053 0.000513
0.008211 0.002053
0.430554 0.014865
0.014865 0.02691 1.000046
Finding K: equal , (1, -3)
A
G
C
T
0.25
0.25
0.25
0.25
A
0.25
0.0625
0.0625
0.0625
0.0625
G
0.25
0.0625
0.0625
0.0625
0.0625
C
0.25
0.0625
0.0625
0.0625
0.0625
T
0.25
0.0625
0.0625
0.0625
0.0625
1
-3
-3
-3
G
-3
1
-3
-3
C
-3
-3
1
-3
T
-3
-3
-3
1
Match-Mismatch
A
Lambda
1
0.169893 0.003112 0.003112 0.003112
0.003112 0.169893 0.003112 0.003112
0.003112 0.003112 0.169893 0.003112
0.003112 0.003112 0.003112 0.169893 0.716911
Double-click it, copy to EXCEL and find  by using solver.
Slide 21