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On the group theoretical background of assigning stepwise mutations onto phylogenies

Mareike Fischer14, Steffen Klaere2*, Minh Anh Thi Nguyen34 and Arndt von Haeseler4

Author Affiliations

1 Department for Mathematics und Computer Science, Ernst-Moritz-Arndt-University Greifswald, Walther-Rathenau-Strasse 47, 17487 Greifswald, Germany

2 Department of Statistics and School of Biological Sciences, University of Auckland, Private Bag 92019, Auckland, New Zealand

3 Groningen Bioinformatics Centre, University of Groningen, Nijenborgh 7, 9747 AG Groningen, The Netherlands

4 , Center for Integrative Bioinformatics ViennaMax F. Perutz Laboratories, University of Vienna, Medical University of Vienna, University of Veterinary Medicine Vienna, Dr. Bohr Gasse 9, A-1030, Vienna, Austria

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Algorithms for Molecular Biology 2012, 7:36  doi:10.1186/1748-7188-7-36

Published: 15 December 2012

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.

Keywords:
Maximum likelihood; Maximum parsimony; Substitution model; Tree reconstruction; Group theory