title: On the group theoretical background of assigning stepwise mutations onto phylogenies.
abstract: 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.
authors.affiliation: Department of Statistics and School of Biological Sciences, University of Auckland, Private Bag 92019, Auckland, New Zealand. steffen.klaere@gmail.com.
authors.forename: Mareike
authors.initials: M
authors.lastname: Fischer
authors.forename: Steffen
authors.initials: S
authors.lastname: Klaere
authors.forename: Minh Anh
authors.initials: MA
authors.lastname: Thi Nguyen
authors.forename: Arndt
authors.initials: A
authors.lastname: von Haeseler