  In a recent paper, Klaere et al. modeled the impact of substitutions on
arbitrary branches of a phylogenetic tree on an alignment site by the so-called
One Step Mutation (OSM) matrix. By utilizing the concept of the OSM matrix for
the four-state nucleotide alphabet, Nguyen et al. presented an efficient
procedure to compute the minimal number of substitutions needed to translate
one alignment site into another.The present paper delivers a proof for this
computation.Moreover, we provide several mathematical insights into the
generalization of the OSM matrix to multistate alphabets.The construction of
the OSM matrix is only possible if the matrices representing the substitution
types acting on the character states and the identity matrix form a commutative
group with respect to matrix multiplication. We illustrate a means to establish
such a group for the twenty-state amino acid alphabet and critically discuss
its biological usefulness.
