Embeddability and rate identifiability of Kimura 2-parameter matrices

Area of Science:

• Computational Biology
• Evolutionary Biology
• Mathematical Biology

Background:

• The embeddability of substitution matrices, determining if a Markov process has a continuous-time realization, remains an open problem for 4x4 matrices.
• Understanding embeddability and rate identifiability is crucial for accurate phylogenetic inference and evolutionary modeling.

Purpose of the Study:

• To fully characterize the set of embeddable K80 Markov matrices.
• To identify the subset of embeddable K80 matrices for which evolutionary rates are identifiable.
• To investigate the implications of non-identifiable rates in parameter estimation for phylogenetics.

Main Methods:

• Mathematical analysis of K80 model matrices to define conditions for embeddability.
• Characterization of the parameter space for embeddable and identifiable K80 matrices.
• Computation of relative volumes of these sets to quantify their prevalence.

Main Results:

• Complete characterization of embeddable K80 matrices and those with identifiable rates.
• Identification of an open subset of embeddable matrices with non-identifiable rates, including those with positive eigenvalues and diagonal largest column properties.
• Calculation of the relative volumes of embeddable K80 matrices and embeddable matrices with identifiable rates.

Conclusions:

• The study resolves the embedding problem for the K80 model and its submodels.
• The findings provide critical insights into rate identifiability issues in phylogenetic parameter estimation.
• This work concludes the embedding problem for the K81 model and related submodels.