Markov invariants for phylogenetic rate matrices derived from embedded submodels

IEEE/ACM Transactions on Computational Biology and Bioinformatics
P D Jarvis, J G Sumner

Abstract

We consider novel phylogenetic models with rate matrices that arise via the embedding of a progenitor model on a small number of character states, into a target model on a larger number of character states. Adapting representation-theoretic results from recent investigations of Markov invariants for the general rate matrix model, we give a prescription for identifying and counting Markov invariants for such “symmetric embedded” models, and we provide enumerations of these for the first few cases with a small number of character states. The simplest example is a target model on three states, constructed from a general 2 state model; the "2 --> 3" embedding. We show that for 2 taxa, there exist two invariants of quadratic degree that can be used to directly infer pairwise distances from observed sequences under this model. A simple simulation study verifies their theoretical expected values, and suggests that, given the appropriateness of the model class, they have superior statistical properties than the standard (log) Det invariant (which is of cubic degree for this case).

References

Sep 1, 1992·Molecular Phylogenetics and Evolution·H J Bandelt, A W Dress
Jan 27, 1999·Bioinformatics·D Posada, K A Crandall
Mar 23, 2004·Molecular Biology and Evolution·John P HuelsenbeckMichael E Alfaro
Jun 9, 2005·Journal of Mathematical Biology·J G Sumner, P D Jarvis
Aug 15, 2006·Mathematical Biosciences·J G Sumner, P D Jarvis
Jun 3, 2008·Journal of Theoretical Biology·J G SumnerP D Jarvis
Jun 6, 2009·Journal of Theoretical Biology·J G Sumner, P D Jarvis
Jun 9, 2009·Journal of Theoretical Biology·Ilan GronauIrad Yavneh
Oct 7, 2011·Bulletin of Mathematical Biology·J G SumnerP D Jarvis
Jan 4, 2012·Journal of Theoretical Biology·J G SumnerP D Jarvis

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Citations

Oct 7, 2011·Bulletin of Mathematical Biology·J G SumnerP D Jarvis
Jan 4, 2012·Journal of Theoretical Biology·J G SumnerP D Jarvis
Dec 15, 2015·Journal of Mathematical Biology·Peter D Jarvis, Jeremy G Sumner
Aug 4, 2018·Bulletin of Mathematical Biology·Jeremy G Sumner, Michael D Woodhams

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