Deciding whether a substitution matrix is embeddable (i.e. the corresponding Markov process has a continuous-time realization) is an open problem even for [Formula: see text] matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these [Formula: see text] 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.

中文翻译：

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Kimura 2参数矩阵的可嵌入性和速率可识别性。

即使对于[公式：参见文本]矩阵，确定替换矩阵是否可嵌入（即，相应的马尔可夫过程具有连续时间实现）也是一个未解决的问题。我们研究了核苷酸替换的K80模型的嵌入问题和速率可识别性。对于这些[公式：请参见文本]矩阵，我们充分描述了可嵌入的K80马尔可夫矩阵集和可确定速率的可嵌入矩阵集。特别是，我们描述了具有不可确定比率的可嵌入矩阵的开放子集。该集合包含特征值正的矩阵，并且列矩阵的对角线最大，这可能导致系统发育参数估计的后果。最后，我们计算可嵌入K80矩阵和具有可确定速率的可嵌入矩阵的相对体积。