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 and 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.