  Various genome evolutionary models have been proposed these last decades to
predict the evolution of a DNA sequence over time, essentially described using
a mutation matrix. By essence, all of these models relate the evolution of DNA
sequences to the computation of the successive powers of the mutation matrix.
To make this computation possible, hypotheses are assumed for the matrix, such
as symmetry and time-reversibility, which are not compatible with mutation
rates that have been recently obtained experimentally on genes ura3 and can1 of
the Yeast Saccharomyces cerevisiae. In this work, authors investigate
systematically the possibility to relax either the symmetry or the
time-reversibility hypothesis of the mutation matrix, by investigating all the
possible matrices of size 2*2 and 3*3. As an application example, the
experimental study on the Yeast Saccharomyces cerevisiae has been used in order
to deduce a simple mutation matrix, and to compute the future evolution of the
rate purine/pyrimidine for $ura3$ on the one hand, and of the particular
behavior of cytosines and thymines compared to purines on the other hand.
