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.
introduced the term PAM (point accepted mutation) unit. A
1-PAM unit is the amount of evolution which will change, on
of the amino acids. In mathematical terms, this is
expressed as a matrix M such that
If we have a probability or frequency vector p, the product 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), (the ith 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 .
Note that PAM distance does not correlate in any immediate way to chronological time. Evolutionary rates may be very different between species and proteins.