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Algorithms Mol Biol. 2012 Dec 15;7(1):36. doi: 10.1186/1748-7188-7-36.

On the group theoretical background of assigning stepwise mutations onto phylogenies.

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1
Department of Statistics and School of Biological Sciences, University of Auckland, Private Bag 92019, Auckland, New Zealand. steffen.klaere@gmail.com.

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.

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