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Markov invariants for phylogenetic rate matrices derived from embedded submodels.

Peter D Jarvis1, Jeremy G Sumner

  • 1School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart Tas 7001, Australia. Peter.Jarvis@utas.edu.au

IEEE/ACM Transactions on Computational Biology and Bioinformatics
|February 15, 2012
PubMed
Summary

This study introduces new phylogenetic models using embedded rate matrices. For a 2-state to 3-state model, two quadratic invariants were found to infer pairwise distances more effectively than the standard cubic invariant.

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Area of Science:

  • Computational Biology
  • Phylogenetics
  • Mathematical Biology

Background:

  • Phylogenetic models are crucial for inferring evolutionary relationships from sequence data.
  • Markov invariants are fundamental properties of these models, aiding in parameter estimation and model testing.
  • Existing methods for identifying invariants can be complex, especially for models with many states.

Purpose of the Study:

  • To develop novel phylogenetic models based on symmetric embeddings of rate matrices.
  • To identify and enumerate Markov invariants for these embedded models.
  • To demonstrate the utility of these invariants for inferring evolutionary distances.

Main Methods:

  • Utilizing representation-theoretic results on Markov invariants for general rate matrix models.
  • Applying these results to symmetric embedded models, specifically the 2-state to 3-state embedding.
  • Deriving and verifying quadratic invariants for pairwise distance estimation.

Main Results:

  • A prescription for identifying and counting Markov invariants in symmetric embedded models was established.
  • For the 2-state to 3-state embedding, two quadratic invariants were identified.
  • These quadratic invariants were shown to be effective for inferring pairwise distances in simulations.
  • The identified invariants demonstrated superior statistical properties compared to the standard cubic invariant.

Conclusions:

  • Symmetric embedded models offer a structured approach to developing new phylogenetic models.
  • The derived quadratic invariants provide a more efficient method for estimating pairwise distances under the 2-state to 3-state model.
  • This work contributes to the theoretical foundation of phylogenetic modeling and offers practical advantages for sequence analysis.