Heuristic alignment algorithms
and cost matrices
Linda Muselaars and Miranda Stobbe
Overview chapter 2
1.
2.
3.
4.
What sorts of alignment should be
considered?
The scoring system used to rank
alignments.
The algorithm used to find optimal (or
good) scoring alignments.
The statistical methods used to evaluate
the significance of an alignment score.
Linda Muselaars and Miranda Stobbe
2
Overview chapter 2
1.
2.
3.
4.
What sorts of alignment should be
considered?
The scoring system used to rank
alignments.
The algorithm used to find optimal (or
good) scoring alignments.
The statistical methods used to evaluate
the significance of an alignment score.
Linda Muselaars and Miranda Stobbe
3
Contents

Heuristic alignment algorithms
– BLAST
– FASTA

Linear space methods
 Significance of scores
– Bayesian approach
– Classical approach

Deriving score parameters
– PAM matrices
– BLOSUM
Linda Muselaars and Miranda Stobbe
4
Contents

Heuristic alignment algorithms
– BLAST
– FASTA

Linear space methods
 Significance of scores
– Bayesian approach
– Classical approach

Deriving score parameters
– PAM matrices
– BLOSUM
Linda Muselaars and Miranda Stobbe
5
The term heuristic

A heuristic algorithm is based on empirical
information that has no explicit
rationalization.
 It does not necessarily return the exact
answer to the problem under study, but is
faster than the algorithm that does and is
still very usable.
Linda Muselaars and Miranda Stobbe
6
BLAST

Basic Linear Alignment Search Tool.
 Simplification of the Smith-Waterman algorithm.
 Uses subsequences of the query sequence to make
‘neighbourhood words’ using a threshold.
 When a neighbourhood word matches a
subsequence in the database a ‘hit extension’
process is started.
Linda Muselaars and Miranda Stobbe
7
Example
Query sequence: q l n f
All subsequences: q l, l n, n f
Creating neighbourhood words:
q l  q l, q m, h l, z l
l n  l n, l b
n f  n f, a f, n y, d f, q f, e f, g f, h f, k f, s f, t f,
b f, z f
Linda Muselaars and Miranda Stobbe
8
FASTA

FAST Alignment.
 Fast approximation of the Smith-Waterman
algorithm.
 Step 1:
– Exact short word matches with length ktup

Step 2:
– extend to ungapped alignments

Step 3:
– identify gapped alignments

Step 4:
– dynamic programming restricted to a subregion
Linda Muselaars and Miranda Stobbe
9
Linda Muselaars and Miranda Stobbe
10
BLAST versus FASTA

They both use the same extension method.
 They both can be used for both DNA and proteins.





BLAST is faster than FASTA.
BLAST is more sensitive than FASTA on proteins.
BLAST is less sensitive than FASTA for nucleic acid
sequences.
BLAST uses neighbourhood words, FASTA does not.
BLAST is mainly for ungapped alignment, FASTA for
gapped alignments.
Linda Muselaars and Miranda Stobbe
11
BLAST vs. FASTA, example

Consider the sequences: n f l and n y l
 ktup = 2 (remember: only for FASTA)
 Even though FASTA only needs a matching
word of size 2 it does not find a match.
 BLAST does find a match (of word size 3
even) on account of neighbourhood words.
Linda Muselaars and Miranda Stobbe
12
Demo at www.ebi.ac.uk
Linda Muselaars and Miranda Stobbe
13
Linda Muselaars and Miranda Stobbe
14
Contents

Heuristic alignment algorithms
– BLAST
– FASTA

Linear space methods
 Significance of scores
– Bayesian approach
– Classical approach

Deriving score parameters
– PAM matrices
– BLOSUM
Linda Muselaars and Miranda Stobbe
15
Reducing memory usage
Score matrices so far are of size nm (with
n and m the sequence lengths).
 We can reduce memory usage to n+m.
 Cost: time is doubled.
 This is done by linear space methods.

Linda Muselaars and Miranda Stobbe
16
Divide and conquer

We find a cell (u,v) in
the middle column that
is on the optimal path.
 This cell divides the
matrix in four parts of
which two are
important for the path.
 This is done
recursively to these
two parts.
Linda Muselaars and Miranda Stobbe
17
Contents

Heuristic alignment algorithms
– BLAST
– FASTA

Linear space methods
 Significance of scores
– Bayesian approach
– Classical approach

Deriving score parameters
– PAM matrices
– BLOSUM
Linda Muselaars and Miranda Stobbe
18
Short review

Letter a occurs independently with
frequency qa in the random model.
 Aligned pairs of residues occur with a joint
probability pab in the match model.
 Random model: P(x,y|R) = ΠkqxkΠlqyl
 Match model: P(x,y|M) = Πkpxkyk
Linda Muselaars and Miranda Stobbe
19
Bayesian approach: model comparison
Linda Muselaars and Miranda Stobbe
20
Comparison

For global matches compare with 0 to
determine whether the alignment is
significant.
 When setting the prior odds ratio in inverse
proportion to the size of the database N,
compare with log N.
 For local matches compare with 0.1 • log(nm)
Linda Muselaars and Miranda Stobbe
21
Extreme value distribution

Scores of a sequence
aligned to a set of
random sequences
obey EVD.
 We compute the
probability that the
best match of
unrelated sequences
has score greater than
our maximal score.
0.4
0.0
-4
-3
-2
-1
1
0
2
3
4
x
Linda Muselaars and Miranda Stobbe
22
Other alignments

For local ungapped alignments we have a
different EVD than for fixed ungapped
alignment (because we have more possible
starting points).

For gapped alignments empirically
established distributions are used.
Linda Muselaars and Miranda Stobbe
23
Correcting for length

When database
sequences are longer,
we have higher scores.
 Solutions:
– Subtract log (mi) for
length mi of the
database sequence.
– Bin all the database
entries by length and
fit a linear function.
Linda Muselaars and Miranda Stobbe
24
Notes on test statistic

Search statistic is the
same as the test
statistic.
 Advantage: both have
highly discriminative
power.
 Disadvantage:
introduction of bias in
test phase.
Linda Muselaars and Miranda Stobbe
25
Contents

Heuristic alignment algorithms
– BLAST
– FASTA

Linear space methods
 Significance of scores
– Bayesian approach
– Classical approach

Deriving score parameters
– PAM matrices
– BLOSUM
Linda Muselaars and Miranda Stobbe
26
Substitution and gap scores

Letter a occurs
independently with
frequency qa in the
random model.
 Aligned pairs of
residues occur with a
joint probability pab in
the match model.
 f(g) is a function of the
length of the gap
Linda Muselaars and Miranda Stobbe
27
Estimating probabilities

Simple approach: set the probabilities to
normalised frequencies (assessed by
counting frequences in confirmed
alignments).
 But:
– It is difficult to obtain a good random sample.
– Does not take into account different ‘distances’
to the common ancestor.
Linda Muselaars and Miranda Stobbe
28
PAM matrices

Percentage of Acceptable point Mutations
per 108 years matrices.
 Amino acid substitution matrices.
 Obtain substitution data from alignments
and estimate probabilities for longer
evolutionary distances.
 A PAMn: n accepted mutations event per
100 amino acids.
Linda Muselaars and Miranda Stobbe
29
PAM matrices (2)






Construct phylogenetic trees relating the
sequences in 71 families (at least 85% similar).
Count the number of amino acid changes with
respect to immediate ancestor.
20 x 20 amino acid substitution matrix computed.
Expected number of substitutions is 1% in PAM1.
PAMn = (PAM1)n.
PAM-matrix converted to a log-odds matrix.
Linda Muselaars and Miranda Stobbe
30
Drawbacks

Using the matrix for short time intervals to
compute the ones for longer time intervals
does not capture the true difference.
 Takes into account only single base changes
instead of all types of codon changes.
 Databases containing alignments of more
distantly related proteins are used to derive
matrix scores more directly and accurately.
Linda Muselaars and Miranda Stobbe
31
BLOCKS database

Used to derive BLOSUM matrices.
 Sequences are clustered according to
percentage of identical residues.
 Aab then is the frequency of observing a in
one cluster aligned to b in another cluster.
 Size of the clusters needs to be corrected
for.
Linda Muselaars and Miranda Stobbe
32
BLOSUM



BLOcks SUbstitution Matrix
BLOSUMn is the matrix where two sequences
are put into one cluster when more then n% of
their residues are identical (lower n
corresponds to longer evolutionary time).
From Aab qa and pab are estimated, which are
used to compute the scores for the matrix.
Linda Muselaars and Miranda Stobbe
33
PAM versus BLOSUM

Based on global
alignments.

Based on local alignments.

PAM1 is the matrix
calculated from
comparisons of
substitutions in unit time.

BLOSUMn is a matrix
calculated from sequences
with no less than n%
divergence.

Other PAM matrices are
extrapolated from PAM1.

All matrices are based on
observed alignments.
Linda Muselaars and Miranda Stobbe
34
Gap penalties

Time-dependent:
– Number of gaps increases (gap-open score d linear in
log t).
– Length distribution constant (gap-extend score e
remains constant).

In practice people choose gap costs empirically
(only two parameters).
 As gaps become more likely we could reduce the
pairwise scores.
Linda Muselaars and Miranda Stobbe
35
Notes



Objective was to
determine whether two
sequences are related.
Scoring schemes and
statistics to determine the
significance of a match.
Even so, it is not always
possible to distinguish
between two related
sequences or two
sequences that seem to be
related, but are not.
Linda Muselaars and Miranda Stobbe
36
Summary

BLAST and FASTA packages are used to
reduce the time used for finding alignments.
 Linear space alignments can be used to
reduce memory usage.
 We need the significance of scores for the
importance of a match.
 We can use the score parameters stated in
PAM and BLOSUM matrices.
Linda Muselaars and Miranda Stobbe
37