Lie Markov models.

Area of Science:

• Computational Biology
• Mathematical Biology
• Evolutionary Biology

Background:

• Multiplicative closure is crucial for Markov models in phylogenetics.
• Lie algebras provide a sufficient condition for multiplicative closure in continuous-time Markov chains.
• Some existing models (e.g., GTR) do not form Lie algebras, necessitating new approaches.

Purpose of the Study:

• To develop a method for generating Lie Markov models using nucleotide permutation symmetries.
• To unify existing model classes like group-based and equivariant models under the Lie Markov framework.
• To present a systematic classification of Lie Markov models for different numbers of character states.

Main Methods:

• Investigating symmetries of nucleotide permutations to construct Lie Markov models.
• Analyzing the properties of rate matrices forming Lie algebras.
• Enumerating Lie Markov models with maximal symmetry for two and four character states.

Main Results:

• Lie Markov models encompass and unify group-based and equivariant models.
• New Lie Markov models, neither group-based nor equivariant, were identified for two and four character states.
• A hierarchy of models with increasing parameters is naturally generated for specific nucleotide substitution symmetries.

Conclusions:

• The proposed scheme provides a unifying framework for Markov models in phylogenetics.
• The method generates a natural hierarchy of models applicable to applied phylogenetics.
• The approach is extendable to other applications of continuous-time Markov chains beyond phylogenetics.