2. Overview
• The alignment problem.
• Homology, divergence, convergence.
• Amino acid substitution matrices.
• Pairwise sequence alignment algorithms: dynamic
programming, BLAST.
• Multiple sequence alignment.
• PSI-BLAST, position specific score matrices.
• Databases of multiple sequence alignments.
3. What is a sequence alignment?
• A linear, one-to-one correspondence between some of
the symbols in one sequence with some of the symbols
in another sequence.
• May be DNA or protein sequences.
• A (sequence) alignment can also be derived from a
superposition of two protein structures, it is then
sometimes called a structure alignment.
4. Exercise!
• From http://www.ncbi.nlm.nih.gov/Structure/ go to the
MMDB summary page for 1npq.
• Click on the colored bar for chain “A”.
• Scroll down and select the VAST neighbor 1KR7 A.
• Click on “View 3D Structure” at the top to view the
structural superposition using Cn3D.
• The sequence alignment will be shown in a separate
window.
5. Divergence and Convergence
• Two proteins may be similar because of divergence from
a common ancestor (i.e. they are homologous).
• … or, perhaps two proteins may be similar because they
perform similar functions and are thereby constrained,
even though they arose independently (functional
convergence hypothesis, they are then called
analogous).
6. Divergence vs. Convergence
• Convergence can occur; e.g. there exist enzymes with
different overall structures but remarkably similar
arrangements of active site residues to carry out a
similar function.
• So how can one establish homology?
7. Homology
“… whenever statistically significant sequence or structural
similarity between proteins or protein domains is
observed, this is an indication of their divergent evolution
from a common ancestor or, in other words, evidence of
homology.”
E.V. Koonin and M.Y. Galperin, Sequence – Evolution – Function,
Kluwer 2003
8. Two arguments…
• We see a continuous range of sequence similarity.
Convergence is extremely unlikely for highly similar
protein families. It then appears implausible to invoke it
for less similar families.
• The same or very similar functions may be carried out by
proteins with very different structures (folds). Therefore,
functional constraints cannot force convergence of the
sequence or structure between proteins.
10. Global and local alignment
• Alignment programs may be modified, e.g. by scoring
parameters, to produce global or local alignments.
• Local alignments tend to be more useful, as it is highly
possible that a significant similarity may encompass only
a portion of one or both sequences being compared.
11. DNA and proteins
• Much more sensitive comparison is possible between
protein sequences than DNA sequences.
• One reason is that the third codon in the genetic code is
highly redundant, and introduces noise into DNA
comparisons.
• Another reason is that the physico-chemical properties of
the amino acid residues provide information that is highly
relevant to comparison.
14. We are faced with several problems…
• How do you quantify amino acid similarity?
• How can you handle gaps in the alignment?
• How do you define the overall similarity score?
• Can you compute an optimal alignment?
• Can you compute an alignment efficiently?
• Can you calculate statistical significance?
15. Amino acid substitution matrices
• Ab initio approaches; e.g. assign scores based on
number of mutations needed to transform one codon to
another, or on other physico-chemical measures of a.a.
similarity.
• Empirical, i.e. derive statistics about a.a. substitutions
from collections of alignments. (These work the best).
16. Computation of scores, empirical approach
sij = ln(qij/(pipj))/λ
• sij is the score for substituting amino acid type i by type j.
• qij is the observed frequency of substitutions of a.a. type i
by a.a. type j (in the training set).
• pi is the background frequency for a.a. type i in the
training set.
• λ is a positive constant.
17. Substitution matrices
• There are many different matrices available.
• The most commonly used are the BLOSUM or PAM
series.
• BLAST defaults to the BLOSUM62 matrix. The
BLOSUM matrices have been shown to be more
sensitive than the PAM matrices for database searches.
18.
19. Gap penalties
• There is no suitable theory for gap penalties.
• The most common type of gap penalty is the affine gap
penalty: g = a + bx, where a is the gap opening penalty,
b is the gap extension penalty, and x is the number of
gapped-out residues.
• Typical values, e.g. a = 10 and b = 1 for BLAST.
20. Dynamic programming overview
• Dynamic programming; a computer algorithmic
technique invented in the 1940’s.
• Has applications to many types of problems.
• Key properties of problems solvable with DP: the optimal
solution typically contains optimal solutions to
subproblems, and only a “small” number of subproblems
are needed for the optimal solution. (T.H. Cormen et al.,
Introduction to Algorithms, McGraw-Hill 1990).
21. Dynamic programming algorithm for computing the
score of the optimal alignment
For a sequence S = a1, a2, …, an let Sj = a1, a2, …, aj
Align(Si,S’
j) = the score of the highest scoring alignment between S1
i,S2
j
S(ai, a’j)= similarity score between amino acids ai and a’j
given by a scoring matrix like PAM, BLOSUM
g – gap penalty
Align(Si,S’
j)= max
Align(Si-1,S’
j-1)+ S(ai, a’j)
Align(Si,S’
j-1) - g
Align(Si-1,S’
j) - g
{
22. Organizing the computation – dynamic programming table
Align(Si-1,S’
j-1)+ s(ai, a’j)
Align(Si-1,S’
j) - g
Align(Si,S’
j-1) - g
Align(Si,S’
j) = max
{
j
i
Align[i, j] =
Align
+s(ai,aj)
max
23. Example of DP computation with
g = 0; match = 1; mismatch = 0
Maximal Common Subsequence
A T T G C G C G C A T
A
T
G
C
T
T
A
A
C
C
A
+1
max
0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1
0 1 2 2 2 2 2 2 2
0 1 2 2 3
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
initialization
24. Example of DP computation with
g = 2; match = 2; mismatch = -1
A T T G C G C G C A T
A
T
G
C
T
T
A
A
C
C
A
+2
max
-2
-2
Initialization (penalty for starting with a gap)
0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22
-2 2 0 -2 -4
-4 0 4 2
-6 -2 2 3
-8 -4
-10
-12
-14
-16
-18
-20
-22
25. The iterative algorithm
m = |S|; n = |S’|
for i 0 to m do A[i,0]- i * g
for j 0 to n do A[0,j] - j * g
for i 0 to m do
for j 0 to n
A[i,j]max (
A[i-1,j] – g
A[i-1,j-1] + s(i,j)
A[i,j-1] – g
)
return(A[m,n])
26. Complexity of the DP algorithm
• Time O(nm); space O(nm) where n, m are the lengths of
the two sequences.
• Space complexity can be reduced to O(n) by not storing
the entries of dynamic programming table that are no
longer needed for the computation (keep current row and
the previous row only).
27. DP technicalities…
• One uses a separate table to trace back the computation
and determine the actual optimal alignment.
• If the gap penalty has different opening and extension
costs, then the algorithm becomes a little more
complicated (cf. Chapter 3 in Mount).
28. Exercise!
• Copy and paste the sequences for 1nqpA and 1kr7A into
a notepad.
• Go to the web site: http://pir.georgetown.edu
• Select “SSearch Alignment” from the “Search and
Retrieval” pulldown menu.
• Copy in your sequences (use FASTA format and remove
numbers) and then run SSEARCH (Smith-Waterman
algorithm, a DP alignment method).
30. BLAST (Basic Local Alignment Search Tool)
• Extremely fast, can be on the order of 50-100 times
faster than Smith-Waterman.
• Method of choice for database searches.
• Statistical theory for significance of results.
• Heuristic; does not guarantee optimal results.
• Many variants, e.g. PHI-, PSI-, RPS-BLAST.
31. Why database searches?
• Gene finding.
• Assigning likely function to a gene.
• Identifying regulatory elements.
• Understanding genome evolution.
• Assisting in sequence assembly.
• Finding relations between genes.
32. Issues in database searches
• Speed.
• Relevance of the search results (selectivity).
• Recovering all information of interest (sensitivity).
– The results depend on the search parameters, e.g. gap
penalty, scoring matrix.
– Sometimes searches with more than one matrix should be
performed.
33. Main idea of BLAST
• Homologous sequences are likely to contain a short high
scoring similarity region, a hit. Each hit gives a seed that
BLAST tries to extend on both sides.
34. Some BLAST terminology
word – substring of a sequence
word pair – pair of words of the same length
score of a word pair – score of the gapless alignment of the two
words:
V A L M R
V A K N S score = 4 + 4 – 2 – 2 - 1 = 3 (BLOSUM62)
HSP – high scoring word pair
35. Main steps of BLAST
• Parameters: w = length of a hit; T = min. score of a hit (for
proteins: w = 3, T = 13 (BLOSUM62)).
• Step 1: Given query sequence Q, compile the list of possible
words which form with words in Q high scoring word pairs.
• Step 2: Scan database for exact matching with the list of words
compiled in step 1.
• Step 3: Extend the hits from step 2.
• Step 4: Evaluate significance of extended hits from step 3.
36. Step 1: Find high scoring words
• For every word x of length w in Q make a list of words that
when aligned to x score at least T.
• Example: Let x = AIV then the score for AIA is 4+4+0
(dropped) and for AII 4+4+3 (taken).
• The number of words in the list depends on w and T, and
is usually much less than 203 (typically about 50).
37. Step 2 – Finding hits
• Scan database for exact matching with the list of
words compiled in step1 :
• Two techniques.
– Hash table.
– Keyword tree (there is a finite-automaton based method for
exact matching with a set of patterns represented as a tree).
38. Step 3: Extending hits
• Parameter: X (controlled by a user).
• Extend the hits in both ways along diagonal (ungapped
alignment) until score drops more than X relative to the
best score yet attained.
• Return the score of the highest scoring segment pair
(HSP).
extensions
39. E-values, P-values
• E-value, Expectation value; this is the expected number
of hits of at least the given score, that you would expect
by random chance for the search database.
• P-value, Probability value; this is the probability that a hit
would attain at least the given score, by random chance
for the search database.
• E-values are easier to interpret than P-values.
• If the E-value is small enough, e.g. no more than 0.10,
then it is essentially a P-value.
40. Karlin-Altschul statistics
• Expected number of HSPs with score ≥ S is:
E = KmNe –λS
• m = length of query sequence
• N = database size in residues
• K scales the search space size
• λ a scale for the scoring system
41. The bit score
• This formula “normalizes” a raw score:
S’ = (λS – ln K)/ln 2
• The E-value is then given by:
E = mN 2 –S’
• Allows direct comparison of E-values, even with differing
scoring matrices.
42. Karlin and Altschul provided a theory for computing statistical
significance
• Assumptions:
– The scoring matrix M must be such that the score for a random
alignment is negative.
– n, m (lengths of the aligned sequences) are large.
– The amino acid distribution p(x) is in the query sequence and the
data base is the same.
– Positive score is possible (i.e. M has at least one positive entry).
43. P(S<x) = exp (-e –l(x-m) ) thus:
P(S>=x) =1- exp (-e –l(x-m)) )
Score of high scoring sequence pairs follows extreme value
distribution
normal
Extreme values
l – decay constant
u – value of the peak
44. Extreme value distribution for sequence alignment
Property of extreme value distribution:
P(S<x) = exp(-e –l(x-m))
P(S>=x) =1- exp(-e –l(x-m))
m – location (zero in the fig from last slide), l scale;
For random sequence alignment:
m = ln Kmn/ l
K- constant that depends on p(x) and scoring matrix M
Since 1-exp(-x) ~ x and substituting for m and s:
P(S>=x) ~ e –l(x-m) = Kmn e –lx
45. E=value-expected number of random scores above
x
• E-value = KNme–lx
(Expected number of sequences scoring at least x observed
by chance, it is approximately same as p value for p value <
0.1 )
46. Refinement of the basic algorithm-the two hit method
• Observation: HSPs of interest are long and can
contain multiple hits a relatively short distance
apart.
• Central idea: Look for non-overlapping pairs of
hits that are of distance at most d on the same
diagonal.
• Benefits:
– Can reduce word size w from 3 two 2 without losing
sensitivity (actually sensitivity of two-hit BLAST is
higher).
– Since extending a hit requires a diagonal partner,
many fewer hits are extended, resulting in
increased speed.
47. Gapped BLAST
• Find two non-overlapping hits of score at least T and
distance at most d from one another.
• Invoke ungapped extension.
• If the HSP generated has normalized score at least Sg
bits then start gapped extension.
• Report resulting alignment if it has sufficiently
significant (very small) E-value.
48. Gapped BLAST statistics
• Theory does not work.
• Simulations indicate that for the best hits scores for local
alignments follow an extreme value distribution.
• Method approximates l and m to match experimental distribution -
l and m can be computed from the median and variation of the
experimental distribution.
• BLAST approach – simulate the distribution for set of scoring
matrices and a number of gap penalties. BLAST offers a choice of
parameters from this pre-computed set.
49. Multiple Sequence Alignment
• A multiple alignment of sequences from a protein family
makes the conserved features much more apparent.
• Much more difficult to compute than pairwise alignments.
• The most commonly used and newer programs use the
“progressive alignment strategy”.
• There are also important databases of multiple
alignments for protein families.
51. Progressive alignment
• Determine an (approximate) phylogenetic tree for the
sequences.
• Construct the multiple alignment by merging pairwise
alignments based on the phylogenetic tree, the most
closely related sequences first, etc.
• The idea is, if you are careless about the order and
merge distantly related sequences too soon in the
process, then errors in the alignment may occur and
propagate.
52. Multiple sequence alignment programs
• CLUSTALW, Thompson et al. NAR 1994.
• T-Coffee (Tree-based Consistency Objective Function for
alignment Evaluation), C. Notredame et al. JMB 2000.
• MUSCLE (Multiple sequence comparison by log
expectation), R. Edgar NAR 2004.
53.
54. PSI-BLAST
• Position Specific Iterated BLAST
• As a first step runs a (regular) BLAST.
• Hits that cross the threshold are used to construct a
position specific score matrix (PSSM).
• A new search is done using the PSSM to find more
remotely related sequences.
• The last two steps are iterated until convergence.
55. PSSM (Position Specific Score Matrix)
• One column per residue in the query sequence.
• Per-column residue frequencies are computed so that
log-odds scores may be assigned to each residue type in
each column.
• There are difficulties; e.g. pseudo-counts are needed if
there are not a lot of sequences, the sequences must be
weighted to compensate for redundancy.
56. Exercise!
• Go to the NCBI BLAST web site.
• Click on the link to PHI- and PSI- BLAST.
• Choose the search database to be “pdb”.
• Enter the sequence for 1h1bA, Human Neutrophil
Elastase.
• How many iterations does it take before 1dleB (Factor B
Serine Protease domain) shows up as a significant hit?
57. Modifying thresholds…
• On occasion, it can prove useful to modify (increase) the
inclusion threshold parameter.
• The user can also manually select hits to include in the
PSSM that do not meet the threshold, e.g. if one is
certain they are homologs to the query.
• Of course, one must always be extremely careful when
doing so!
58. HMMs
• Hidden Markov Models, a type of statistical model.
• Have numerous applications (including outside of
bioinformatics).
• HMMs were used to construct Pfam, a database of
multiple alignments for protein families (HMMer).
59. CDD Search
• Conserved Domain Database (CDD) at NCBI.
• Contains alignments from Smart, Pfam, COG, KOG, and
LOAD databases.
• Many multiple alignments are manually curated.
• PSSMs derived from multiple alignments may be
searched with RPS-BLAST (Reverse Position Specific
BLAST).
60. Thank you to Dr. Teresa Przytycka for
slides about dynamic programming
and BLAST.
