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,
of the amino acids. In mathematical terms, this is
expressed as a matrix *M* such that

where

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 *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
.

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