PAM matrices

Log-odds matrices

We express the probabilities of transformation in what are called log-odds scores. The scores matrix S is defined as

S_{i,j}=\log \frac {p_i \cdot M_{i,j}} {p_i \cdot p_j}=\log \frac {M_{i,j}}{p_j}=\log \frac {observed\;frequency} {expected\;frequency}

where Mi,j is the probability that amino acid i transforms into amino acid j and pi is the frequency of amino acid i. The base of the logarithm is not important, and you will often see the same substitution matrix expressed in different bases.
[edit] PAM

One of the first amino acid substitution matrices, the PAM (Point Accepted Mutation) matrix was developed by Margaret Dayhoff in the 1970s. This matrix is calculated by observing the differences in closely related proteins. The PAM1 matrix estimates what rate of substitution would be expected if 1% of the amino acids had changed. The PAM1 matrix is used as the basis for calculating other matrices by assuming that repeated mutations would follow the same pattern as those in the PAM1 matrix, and multiple substitutions can occur at the same site. Using this logic, Dayhoff derived matrices as high as PAM250. Usually the PAM 30 and the PAM70 are used.

A matrix for divergent sequences can be calculated from a matrix for closely related sequences by taking the second matrix to a power. For instance, we can roughly approximate the WIKI2 matrix from the WIKI1 matrix by saying W_2 = W_1^2 where W1 is WIKI1 and W2 is WIKI2. This is how the PAM250 matrix is calculated.
[edit] BLOSUM

Dayhoff's methodology of comparing closely related species turned out not to work very well for aligning evolutionarily divergent sequences. Sequence changes over long evolutionary time scales are not well approximated by compounding small changes that occur over short time scales. The BLOSUM (BLOck SUbstitution Matrix) series of matrices rectifies this problem. Henikoff and Henikoff constructed these matrices using multiple alignments of evolutionarily divergent proteins. The probabilities used in the matrix calculation are computed by looking at "blocks" of conserved sequences found in multiple protein alignments. These conserved sequences are assumed to be of functional importance within related proteins. To reduce bias from closely related sequences, segments in a block with a sequence identity above a certain threshold were clustered giving weight 1 to each such cluster (Henikoff and Henikoff). For the BLOSUM62 matrix, this threshold was set at 62%. Pairs frequencies were then counted between clusters, hence pairs were only counted between segments less than 62% identical. One would use a higher numbered BLOSUM matrix for aligning two closely related sequences and a lower number for more divergent sequences.

It turns out that the BLOSUM62 matrix does an excellent job detecting similarities in distant sequences, and this is the matrix used by default in most recent alignment applications such as BLAST.
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