The impracticalities of multiplicatively-closed codon models: a retreat to linear alternatives

TL;DR

This paper explores the limitations of multiplicatively-closed codon models in phylogenetics, demonstrating that their Lie closures are impractical due to excessive parameters and proposing linear alternatives as feasible solutions.

Contribution

The study introduces linear closure and linear version methods to approximate codon models, highlighting their advantages over Lie closures in phylogenetic modeling.

Findings

Lie closures of codon models require thousands of parameters.

Partial solutions violate stochasticity constraints.

Linear alternatives are more practical than Lie closures.

Abstract

A matrix Lie algebra is a linear space of matrices closed under the operation $ [A, B] = AB-BA $. The "Lie closure" of a set of matrices is the smallest matrix Lie algebra which contains the set. In the context of Markov chain theory, if a set of rate matrices form a Lie algebra, their corresponding Markov matrices are closed under matrix multiplication; this has been found to be a useful property in phylogenetics. Inspired by previous research involving Lie closures of DNA models, it was hypothesised that finding the Lie closure of a codon model could help to solve the problem of mis-estimation of the non-synonymous/synonymous rate ratio, $ \omega $. We propose two different methods of finding a linear space from a model: the first is the \emph{linear closure} which is the smallest linear space which contains the model, and the second is the \emph{linear version} which changes multiplicative constraints in the model to additive ones. For each of these linear spaces we then find the Lie closures of them. Under both methods, it was found that closed codon models would require thousands of parameters, and that any partial solution to this problem that was of a reasonable size violated stochasticity. Investigation of toy models indicated that finding the Lie closure of matrix linear spaces which deviated only slightly from a simple model resulted in a Lie closure that was close to having the maximum number of parameters possible. Given that Lie closures are not practical, we propose further consideration of the two variants of linearly closed models.