Bioc 565, Dr. Cordes
Supplement handout to sequence evolution lecture, Sep 10, 2007
1. General facts about amino-acid substitution matrices:
The average likelihood of a particular mutation occurring and being accepted is
influenced both by codon similarity and by chemical similarity of the amino acids
(as a result of structural and functional constraints). These likelihoods are
reflected in substitution matrices. The most common methods for constructing
substitution matrices are empirical, i.e. the matrices are parametrized using
observations of what substitutions actually occur among collections of related
protein sequences. Substitution matrices are today commonly used in sequence
alignment methods and database sequence similarity searches.
A pioneer in this field was Margaret Dayhoff, who created the PAM (Per cent
Accepted point Mutations) substitution matrices. She used sets of closely related
sequences to model how evolution occurs over short distances, and then
extrapolated these findings to longer evolutionary distances. The other most
commonly used type of substitution matrix is called the BLOSUM matrix, due to
Henikoff. BLOSUM matrices use less closely related sequences for
parametrization, and only use “blocks” of the best aligned portions. Portions that
don’t align well and vary in length, such as surface loops, are not used. Thus the
PAM and BLOSUM matrices show some differences. Matrices parametrized
using soluble, globular proteins will also differ from those derived from
transmembrane proteins.
A few words on nomenclature: you will see terms like “BLOSUM 62” and
“PAM 250”. “BLOSUM 62” means that the matrix was parametrized using
sequences with 62% average sequence identity. “PAM 250” refers to the PAM
unit: 1 PAM is the amount of evolution required to change 1% of a protein’s
residues. A PAM 250 matrix is then the result of extrapolating data on closely
related sequences to an evolutionary distance of 250 PAMs.
Another thing you will note is that the numbers in the PAM matrix don’t look like
“percent accepted” values--many are negative numbers. This is because most
matrices are presented in “log odds” form. The odds are figured as the
probability of a particular change at a given evolutionary distance, normalized by
the overall frequency of occurrence of a given amino acid in the database.
These odds are then converted to logarithms.
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2. Derivation of substitution matrices from sequence conservation data:
Construction of the BLOSUM matrices
based on Henikoff & Henikoff, PNAS USA, 89, 10915 (1992)
The general procedure for deriving generalized substitution matrices from
sequence conservation data is to 1) assemble a large number of multiple
alignments from different protein families 2) tabulate the frequencies of all the
possible residue pairings 3) compare these frequencies to those expected in a
random database of the same composition, giving an odds ratio 4) convert the
odds ratio into a log format.
To make the BLOSUM matrices (which are probably the best and are certainly
the most commonly used), Henikoff and Henikoff took multiple alignments of
several hundred protein families and parsed them into a database of “blocks”,
where a block is defined as a portion of a multiple alignment which contains no
gaps. This analysis yielded a database of >2000 blocks. The block approach
contrasts with the way the PAM matrices (Dayhoff) were constructed, because in
that case regions containing gaps were included.
There are 20 different naturally occurring amino acids, and thus there are 20 + 19
+ ... 1 = 210 different possible pairings of amino acids. Henikoff and Henikoff
converted their database of blocks into a frequency table describing the number
of occurrences fij of each of the 210 possible pairings of amino acids i and j,
where 1 ≤ j ≤ i ≤ 20.
To illustrate construction of a frequency table, let’s look at a simple (and fake)
example. A given block can be described as having a width of w alignment
positions and a depth of s sequences. The very small block shown below has
w=4 (positions 1-4) and s=5 (sequences A-E). Let’s suppose that our entire
database is composed of this one block.
sequence
A
B
C
D
E
position in alignment
1
2
3
4
S
L
M
K
A
L
A
E
M
V
A
E
R
I
A
W
T
L
M
C
A block of w=4 and s=5 has a total of ws(s-1)/2 = 40 amino acid pairs. Let’s
now consider, as an illustrative exercise, the frequencies of all pairs involving
alanine. Alanine occurs at two positions in the block: positions 1 and 3. At the
first position, there are 4 alanine-containing pairs: 1 each of the pairs AM, AR, AS
and AT. At the third position, there are 9 alanine-containing pairs, including 3
different AA pairs (sequences B and C, B and D and C and D) and 6 AM pairs
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(enumerate them yourself to convince yourself). So summing the occurrences of
each pair at the two positions, the frequency of occurrence of AM pairs fAM = 1+ 6
= 7, the frequency of occurrence of AA pairs fAA = 0 + 3 = 3. The frequency of
occurrence of AR, AS and AT pairs is just 1, while pairings of alanine with all
other amino acids have frequencies of 0. If we did this same analysis with all
210 possible pairings we’d have a pair frequency table (or matrix) for our
database.
The pair frequencies are then converted into pair probabilities. The probability of
observing any particular pairing can be described as
The denominator here simply corresponds to the sum total of all pair frequencies
(which is just the total number of pairs) in the database. qAM for example, is 7/40
= 0.175, while qAA is 3/40 = 0.075.
The pair probabilities only have statistical meaning when compared to the pair
probabilities that one would expect from a random database of the same aminoacid composition. The expected probability of occurrence eij for each pair, based
on the frequency of occurrence of pairs involving each of those amino acids in
the database, is equal to pipj, where pi and pj are the individual probabilities that
a given residue pair will contain amino acid i or j, respectively.
The probability pi that a pair will contain amino acid i is generally figured as:
I find this formulation mildly confusing, and it should be noted that this number
will also come out to equal the fractional population of amino acid i in the
database. This makes sense. Intuitively, the probability of a given pair
containing alanine should be equal to the fractional population of alanine in the
database. For example, as figured from the equation above, pA = [3 + (10/2)]/40
= 0.2. The fractional population of alanine in the database is also 4/20 = 0.2.
Similarly, pM = [1 + (10/2)]/40 = 0.15. The fractional population of methionine in
the database is 3/20 = 0.15. In any case, for the example of alanine-methionine
pairs, eAM = (0.2)(0.15) = 0.03. Similarly, for alanine pairing with itself, eAA =
(0.2)(0.2) = 0.04.
The odds ratio is then computed as the observed probability qj divided by the
expected probability eij . This ratio is then converted into the log-odds ratio sij,
where
sij = log2(qij/eij).
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For the AM pairing, qAM/eAM = 0.175/0.03 = 5.82, and sAM = 2.54. For the AA
pairing, qAM/eAM = 0.075/0.04 = 1.88, and sAM = 0.9. Note that if qij is larger than
eij, the odds ratio will be greater than 1 and the log-odds score will be positive,
while if the opposite is true the odds ratio will be less than 1 and the log-odds
score will be negative. Thus, pairings which occur more frequently than
expected by chance will have positive log odds scores, while those which occur
less frequently will have negative log odds scores. Note that this particular case
is unrealistic: in any real situation, alanine would have a higher log odds score
with itself than with any other amino acid.
To convert them to final BLOSUM matrix element form, the log odds scores sij are
then all multiplied by some uniform scaling factor (2 in the case of the BLOSUM
matrices) and rounded off to integers.
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