# Embeddability and rate identifiability of Kimura 2-parameter matrices

## TL;DR

This paper fully characterizes embeddable and rate-identifiable Markov matrices within the K80 nucleotide substitution model, revealing sets with non-identifiable rates and implications for phylogenetic parameter estimation.

## Contribution

It provides a complete characterization of embeddability and rate identifiability for 4x4 K80 matrices, including open subsets with non-identifiable rates.

## Key findings

- Identified the set of embeddable K80 matrices
- Described open subset with non-identifiable rates
- Computed relative volumes of embeddable matrices

## Abstract

Deciding whether a Markov matrix is embeddable (i.e. can be written as the exponential of a rate matrix) is an open problem even for $4\times 4$ matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these $4\times 4$ matrices, we fully characterize the set of embeddable K80 Markov matrices and the set of embeddable matrices for which rates are identifiable. In particular, we describe an open subset of embeddable matrices with non-identifiable rates. This set contains matrices with positive eigenvalues and also diagonal largest in column matrices, which might lead to consequences in parameter estimation in phylogenetics. Finally, we compute the relative volumes of embeddable K80 matrices and of embeddable matrices with identifiable rates. This study concludes the embedding problem for the more general model K81 and its submodels, which had been initiated by the last two authors in a separate work.