Our method for estimating tree distributions and identifying outlying trees is implemented as the R package kdetrees and is available for download from CRAN.
SummaryEvolutionary relationships are represented by phylogenetic trees, and a phylogenetic analysis of gene sequences typically produces a collection of these trees, one for each gene in the analysis. Analysis of samples of trees is difficult due to the multi-dimensionality of the space of possible trees. In Euclidean spaces, principal component analysis is a popular method of reducing high-dimensional data to a low-dimensional representation that preserves much of the sample’s structure. However, the space of all phylogenetic trees on a fixed set of species does not form a Euclidean vector space, and methods adapted to tree space are needed. Previous work introduced the notion of a principal geodesic in this space, analogous to the first principal component. Here we propose a geometric object for tree space similar to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k$\end{document}th principal component in Euclidean space: the locus of the weighted Fréchet mean of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k+1$\end{document} vertex trees when the weights vary over the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k$\end{document}-simplex. We establish some basic properties of these objects, in particular showing that they have dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$k$\end{document}, and propose algorithms for projection onto these surfaces and for finding the principal locus associated with a sample of trees. Simulation studies demonstrate that these algorithms perform well, and analyses of two datasets, containing Apicomplexa and African coelacanth genomes respectively, reveal important structure from the second principal components.
As costs of genome sequencing have dropped precipitously, development of efficient bioinformatic methods to analyze genome structure and evolution have become ever more urgent. For example, most published phylogenomic studies involve either massive concatenation of sequences, or informal comparisons of phylogenies inferred on a small subset of orthologous genes, neither of which provides a comprehensive overview of evolution or systematic identification of genes with unusual and interesting evolution (e.g., horizontal gene transfers, gene duplication, and subsequent neofunctionalization). We are interested in identifying such "outlying" gene trees from the set of gene trees and estimating the distribution of trees over the "tree space". This paper describes an improvement to the kdetrees algorithm, an adaptation of classical kernel density estimation to the metric space of phylogenetic trees (Billera-Holmes-Vogtman treespace), whereby the kernel normalizing constants, are estimated through the use of the novel holonomic gradient methods. As in the original kdetrees paper, we have applied kdetrees to a set of Apicomplexa genes. The analysis identified several unreliable sequence alignments that had escaped previous detection, as well as a gene independently reported as a possible case of horizontal gene transfer. The updated version of the kdetrees software package is available both from CRAN (the official R package system), as well as from the official development repository on Github. ( github.com/grady/kdetrees).
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