The model-specific Markov embedding problem for symmetric group-based models

Abstract

We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an application to hachimoji models, which are eight-state models for synthetic DNA. Moreover, our main result on model embeddability enables us to compute the volume of the set of model embeddable Markov matrices relative to the volume of other relevant sets of Markov matrices within the model.

Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

“…For instance, in the case exactly of the matrices are discarded (Ardiyansyah et al. 2021 , Table 5), while in the case of matrices up to of the matrices are discarded (see Table 7 ) and in the case of matrices the amount of discarded matrices is about as indicated in Table 9 . However, when restricting to subsets of Markov matrices which are mathematically more meaningful in biological terms, such as DD or DLC matrices, the proportion of embeddable matrices is much higher so that we are discarding less matrices (e.g.…”

“…Moreover, The inequalities in Theorem 2 can be spelled out as follows: These inequalities are equivalent to the K3P-embeddability criteria presented in (Roca-Lacostena and Fernández-Sánchez 2018b , Theorem 3.1) and (Ardiyansyah et al. 2021 , Theorem 1). Moreover, they are also equivalent to the restriction to centrosymmetric-matrices of the embeddability criteria for Markov matrices with different eigenvalues given in (Casanellas et al.…”

“…Moreover, the embedding problem has been solved for special matrices with a biological interest such as equal-input and circulant matrices (Baake and Sumner 2020 ), group-based models (Ardiyansyah et al. 2021 ) and time-reversible models (Jia 2016 ). Despite the fact that there is no theoretical explicit solution for the embeddability of general Markov matrices, there are results (Casanellas et al.…”

Maximum Likelihood Estimation for Unrooted 3-Leaf Trees: An Analytic Solution for the CFN Model

“…For example, in [25] they appear as embeddability conditions for the Kimura 3-parameter model; in that setting, the inequalities are equivalent to the nonnegativity of the offdiagonal entries of the mutation rate matrix, and therefore-just as in our settingimplicitly specify that the branch lengths be nonnegative. For a generalization of these inequalities as embeddability conditions see the main result of [4].…”

Embeddability of centrosymmetric matrices

“…For the general case of n ˆn Markov matrices, there are several results; some presenting necessary conditions [17,29,36], while others sufficient conditions [25,18,20,16] for embeddability of Markov matrices. Moreover, the embedding problem has been solved for special n ˆn matrices with a biological interest such as equal-input and circulant matrices [3], group-based models [2] and time-reversible models [26]. Despite the fact that there is no theoretical explicit solution for the embeddability of general n ˆn Markov matrix, there are results [8] that enable us to decide whether a Markov matrix with different eigenvalues is embeddable or not.…”