On the group theoretical background of assigning stepwise mutations onto phylogenies

TL;DR

This paper explores the mathematical foundation of assigning mutations to phylogenies using group theory, proving minimal substitution calculations and generalizing the approach to multistate alphabets like amino acids.

Contribution

It provides a proof for minimal substitution computation and extends the OSM matrix concept to multistate alphabets using group theory.

Findings

Proof of minimal substitution calculation method

Generalization of OSM matrix to amino acids

Discussion on biological relevance of the group structure

Abstract

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.