The model we consider here is a Markovian model of evolution
[1], which assumes that amino acids mutate
independently of each other, with probabilities which depend only
on the amino acids and on the amount of evolution. In mathematical
terms we can describe the model with mutation matrices:
a mutation matrix, denoted by M, describes the
probabilities of amino acid mutations for a given period of
evolution.

Amino acids appear in nature with different frequencies. These
frequencies are denoted by f_{i} and correspond to the steady
state of the Markov process defined by the matrix M, that is,
the vector f is any of the columns of
(M
multiplied by itself an infinite number of times) or the
eigenvector of M whose corresponding eigenvalue is 1 (M f =
f). When we find a mutation in aligned sequences, we cannot
distinguish which one mutated into which, or whether a third amino
acid mutated into them. This implies a simple relation for the
entries of M:

M describes mutations over a given period of evolution.
We must quantify this amount of change in a
mathematically meaningful way. Dayhoff et. al.[3]
introduced the term PAM (point accepted mutation) unit.

Definition 1.1
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 f_{i} is the frequency of the i^{th}
amino acid.

If we have a probability vector p, the product Mp gives the probability vector after an evolution equivalent to a
1-PAM unit. After k units of evolution (a k-PAM
evolution), a frequency vector p will be changed into the
frequency vector M^{k}p.
There are
possible events where
an amino acid X mutates in one branch of evolution to amino acid A
and to B in another branch.
The sum of all these probabilities must be 1.

(1)

where
d = d_{XA} + d_{XB}. The above left hand side can be
summed in X and we can express the probabilities of each event
in terms of the children A and B (present day) and the total
amount of evolution between A and B.

Figure 1:
Amino acids A and B are d PAM units
apart and originate from a common ancestor X.