PAM distance.

In the definition of mutation, we use the matrix M which implies certain amount of mutation. For all our work it is important to quantify mutation. We will use the terminology introduced by Dayhoff et al. [6] where amount of mutation is measured in PAM units.

A 1-PAM mutation matrix describes an amount of evolution which will change, on the average, 1% of the amino acids. In mathematical terms this is expressed as a matrix M such that

$$\sum_{i \in \Sigma} f_i (1 - M_{ii} ) = 0.01$$

The diagonal elements of M are the probabilities that a given amino acid does not change, so (1-M_ii) is the probability of mutating away from i.

If we have a probability or frequency vector p, the product Mp gives the probability vector or the expected frequency of p after an evolution equivalent to 1-PAM unit. Or, if we start with amino acid i (a probability vector which contains a 1 in position i and 0s in all others) M_*i (the i^th column of M) is the corresponding probability vector after one unit of random evolution. Similarly, after k units of evolution (what is called k-PAM evolution) a frequency vector p will be changed into the frequency vector M^k p. Notice that chronological time is not linearly dependent on PAM distance. Evolution rates may be very different for different species and different proteins.