Phylogenetic mixtures and linear invariants for equal input models

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely...

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Bibliographic Details
Published in:Journal of mathematical biology Vol. 74; no. 5; pp. 1107 - 1138
Main Authors: Casanellas, Marta, Steel, Mike
Format: Journal Article Publication
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017
Springer Nature B.V
Subjects:
05 Combinatorics
05C Graph theory
60 Probability theory and stochastic processes
60J Markov processes
92 Biology and other natural sciences
92D Genetics and population dynamics
Applications of Mathematics
Biomatemàtica
Biomathematics
Classificació AMS
linear invariants
Markov Chains
Markov processes
Markov, Processos de
Matemàtiques i estadística
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Models, Biological
Molecular Sequence Data
Phylogenetic tree
Phylogeny
Àrees temàtiques de la UPC
Markov processes
60J28
Linear invariants
92D15
Phylogenetic tree
05C05
ISSN:0303-6812, 1432-1416, 1432-1416
Online Access:Get full text
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Summary:The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the ‘equal input model’. This model generalizes the ‘Felsenstein 1981’ model (and thereby the Jukes–Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ‘random cluster’ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees—the so called ‘model invariants’), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n = 4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167–191, 1987 ).
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ISSN:0303-6812
1432-1416
1432-1416
DOI:10.1007/s00285-016-1055-8