Point Accepted Mutations

The definition of matrix M describes mutation over a given period of evolution. In order to procede, we must quantify this change in a mathematically meaningful way. Dayhoff et. al.[9] introduced the term PAM (point accepted mutation) unit. A 1-PAM unit is the amount of evolution which will change, on average, $1\%$ of the amino acids. In mathematical terms, this is expressed as a matrix M such that

$$\sum_{i=1}^{20} f_i (1 - M_{ii} ) = 0.01$$

where f_i is the frequency of the i^th amino acid. (Recall that M_ii represents the probability amino acid i does not change.)

If we have a probability or frequency vector p, the product $M\cdot p$ gives the probability vector (or the expected frequency of p) after an evolution equivalent to 1-PAM unit.

Alternatively, if we start with amino acid i (a probability vector which contains a 1 in position i and 0s in all others), $M\cdot p = M_{*i}$ (the i^th column of M) is the corresponding probability vector after one unit of random evolution.

After k units of evolution (a k-PAM evolution), a frequency vector p will be changed into the frequency vector $M^{k}\cdot p$.

Note that PAM distance does not correlate in any immediate way to chronological time. Evolutionary rates may be very different between species and proteins.