# Substitution matrix

In evolutionary biology, a substitution matrix describes the rate at which one character in a sequence changes to other character states over time. Substitution matrices are usually seen in the context of amino acid or DNA sequence alignments, where the similarity between sequences depends on their divergence time and the substitution rates as represented in the matrix.

## Background

In the process of evolution, from one generation to the next the amino acid sequences of an organism's proteins are gradually altered through the action of DNA mutations. For example, the sequence

ALEIRYLRD

could mutate into the sequence

ALEINYLRD

in one generation and possibly

AQEINYQRD

over a longer period of evolutionary time. Each amino acid is more or less likely to mutate into various other amino acids. For instance, a hydrophobic residue such as valine is more likely to stay hydrophobic than not, since replacing it with a hydrophilic residue could affect the folding and/or activity of the protein.

If we have two amino acid sequences in front of us, we should be able to say something about how likely they are to be derived from a common ancestor, or homologous. If we can line up the two sequences using a sequence alignment algorithm such that the mutations required to transform a hypothetical ancestor sequence into both of the current sequences would be evolutionarily plausible, then we'd like to assign a high score to the comparison of the sequences.

To this end, we will construct a 20x20 matrix where the $(i,j)$ th entry is equal to the probability of the $i$ th amino acid being transformed into the $j$ th amino acid in a certain amount of evolutionary time. There are many different ways to construct such a matrix, called a substitution matrix. Here are the most commonly used ones:

## Identity matrix

The simplest possible substitution matrix would be one in which each amino acid is considered maximally similar to itself, but not able to transform into any other amino acid. This matrix would look like:

${\begin{bmatrix}1&0&\cdots &0&0\\0&1&&0&0\\\vdots &&\ddots &&\vdots \\0&0&&1&0\\0&0&\cdots &0&1\end{bmatrix}}$ This identity matrix will succeed in the alignment of very similar amino acid sequences but will be miserable at aligning two distantly related sequences. We need to figure out all the probabilities in a more rigorous fashion. It turns out that an empirical examination of previously aligned sequences works best.

## 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 $M_{i,j}$ is the probability that amino acid $i$ transforms into amino acid $j$ and $p_{i}$ 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.

### 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.

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 $W_{1}$ is WIKI1 and $W_{2}$ is WIKI2. This is how the PAM250 matrix is calculated.

### 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. The BLOSUM62 matrix is calculated from observed substitutions between proteins that share 62% sequence identity or more - clustering together proteins showing 62% sequence identity or more and giving weight 1 to each such cluster (Henikoff and Henikoff), (and the BLOSUM100 matrix is calculated from alignments between proteins showing 100% identity the proteins in the BLOCKS database (since all have 100% sequence identity or less)?). 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.

### Differences between PAM and BLOSUM

1. PAM matrices are based on an explicit evolutionary model (i.e. replacements are counted on the branches of a phylogenetic tree), whereas the BLOSUM matrices are based on an implicit model of evolution.
2. The PAM matrices are based on mutations observed throughout a global alignment, this includes both highly conserved and highly mutable regions. The BLOSUM matrices are based only on highly conserved regions in series of alignments forbidden to contain gaps.
3. The method used to count the replacements is different, unlike the PAM matrix, the BLOSUM procedure uses groups of sequences within which not all mutations are counted the same.
4. Higher numbers in the PAM matrix naming scheme denote larger evolutionary distance, while larger numbers in the BLOSUM matrix naming scheme denote higher sequence similarity and therefore smaller evolutionary distance. Example: PAM150 is used for more distant sequences than PAM100; BLOSUM62 is used for closer sequences than Blosum50.

## Current research

Current innovative approaches include incorporating secondary structure information into the sequences and substitution matrices. See this paper for an example of this direction of research.