: Recently one step mutation matrices were introduced to model the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site. This concept works nicely for the four-state nucleotide alphabet and provides an efficient procedure conjectured to compute the minimal number of substitutions needed to transform one alignment site into another. The present paper delivers a proof of the validity of this algorithm. Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multi-state 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 this approach by looking at Abelian groups over twenty states and critically discuss their biological usefulness when investigating amino acids. © 2012 Fischer et al.; licensee BioMed Central Ltd.
Fischer, M., Klaere, S., Thi Nguyen, M. A., & Haeseler, A. von. (2012). On the group theoretical background of assigning stepwise mutations onto phylogenies. Algorithms for Molecular Biology, 7(1). https://doi.org/10.1186/1748-7188-7-36