Phylogenetic invariants for the general Markov model of sequence mutation

Allman, Elizabeth S.; Rhodes, John A.

doi:10.1016/j.mbs.2003.08.004

Cited by 83 publications

(150 citation statements)

References 19 publications

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“…Under mild additional restrictions on model parameters, changing the root location corresponds to a simple invertible change of variables in the parameterization. (See [13], [14], or [15] for details.) This justifies our slight abuse of language in referring to the GM or GM+I model on T , rather than on T r , and we omit future references to root location.…”

Section: The Gm+i Modelmentioning

confidence: 99%

“…Then multiple parameter choices give rise to the same joint distribution since the labeling of the states at the internal node can be permuted in κ! ways, as long as the Markov matrix parameters are adjusted accordingly [14]. The occurrence of this sort of 'label-swapping' non-identifiability in statistical models with hidden (unobserved) variables is well-known, but is not of great concern.…”

Section: Model Identifiabilitymentioning

confidence: 99%

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Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

Allman

Rhodes

2008

Mathematical Biosciences

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The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of 'label swapping.' The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites.

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Section: The Gm+i Modelmentioning

confidence: 99%

Section: Model Identifiabilitymentioning

confidence: 99%

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

Allman

Rhodes

2008

Mathematical Biosciences

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“…where each subscript "Σ" can be thought as either a sum over states (as with Allman & Rhodes (2003)) or the "0" component of the basis specified in Sumner et al (2008), and all remaining indices are summed from 1 to k. One can readily check that if…”

Section: Explicit Formsmentioning

confidence: 99%

“…Note that this construction requires that the Σ's are spread evenly across the legs of the tensors (one for each part of the tensor product). It is worth observing that this presentation can be related to that given by Allman & Rhodes (2003) by observing that the cofactor matrix can be expressed as…”

Section: Explicit Formsmentioning

confidence: 99%

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Markov invariants and the isotropy subgroup of a quartet tree

Sumner

Jarvis

2009

Journal of Theoretical Biology

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The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results directly to Markov invariants, thereby extending previous theoretical results by systematically identifying linear combinations that vanish for a given quartet. We also note that the theory is fully generalizable to arbitrary trees and is equally applicable to the related case of phylogenetic invariants. All results follow from elementary consideration of the representation theory of finite groups. † Alexander von Humboldt Fellow

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Quartet Sampling distinguishes lack of support from conflicting support in the green plant tree of life

Pease

Brown

Walker

et al. 2018

American J of Botany

203

244

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PREMISE OF THE STUDY : Phylogenetic support has been difficult to evaluate within the green plant tree of life partly due to a lack of specificity between conflicted versus poorly informed branches. As data sets continue to expand in both breadth and depth, new support measures are needed that are more efficient and informative. METHODS:We describe the Quartet Sampling (QS) method, a quartet-based evaluation system that synthesizes several phylogenetic and genomic analytical approaches. QS characterizes discordance in large-sparse and genome-wide data sets, overcoming issues of alignment sparsity and distinguishing strong conflict from weak support. We tested QS with simulations and recent plant phylogenies inferred from variously sized data sets.KEY RESULTS: QS scores demonstrated convergence with increasing replicates and were not strongly affected by branch depth. Patterns of QS support from different phylogenies led to a coherent understanding of ancestral branches defining key disagreements, including the relationships of Ginkgo to cycads, magnoliids to monocots and eudicots, and mosses to liverworts. The relationships of ANA-grade angiosperms (Amborella, Nymphaeales, Austrobaileyales), major monocot groups, bryophytes, and fern families are likely highly discordant in their evolutionary histories, rather than poorly informed. QS can also detect discordance due to introgression in phylogenomic data. CONCLUSIONS:Quartet Sampling is an efficient synthesis of phylogenetic tests that offers more comprehensive and specific information on branch support than conventional measures. The QS method corroborates growing evidence that phylogenomic investigations that incorporate discordance testing are warranted when reconstructing complex evolutionary histories, in particular those surrounding ANA-grade, monocots, and nonvascular plants.

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Phylogenetic invariants for the general Markov model of sequence mutation

Cited by 83 publications

References 19 publications

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

Markov invariants and the isotropy subgroup of a quartet tree

Quartet Sampling distinguishes lack of support from conflicting support in the green plant tree of life

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