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Phylogenetic placement of a query sequence on a backbone tree is increasingly used across biomedical sciences to identify the content of a sample from its DNA content. The accuracy of such analyses depends on the density of the backbone tree, making it crucial that placement methods scale to very large trees. Moreover, a new paradigm has been recently proposed to place sequences on the species tree using single-gene data. The goal is to better characterize the samples and to enable combined analyses of marker-gene (e.g., 16S rRNA gene amplicon) and genome-wide data. The recent method DEPP enables performing such analyses using metric learning. However, metric learning is hampered by a need to compute and save a quadratically growing matrix of pairwise distances during training. Thus, DEPP (or any distance-based method) does not scale to more than roughly ten thousand species, a problem that we faced when trying to use our recently released Greengenes2 (GG2) reference tree containing 331,270 species. Scalability problems can be addressed in phylogenetics using divide- and-conquer. However, applying divide- and-conquer to data-hungry machine learning methods needs nuance. This paper explores divide- and-conquer for training ensembles of DEPP models, culminating in a method called C-DEPP that uses carefully crafted techniques to enable quasi-linear scaling while maintaining accuracy. C-DEPP enables placing twenty million 16S fragments on the GG2 reference tree in 41 hours of computation.

16S rRNA and shotgun metagenomics studies typically yield different results, usually attributed to biases in PCR amplification of 16S rRNA genes. Here, we introduce Greengenes2 and show that differences in reference phylogeny are more important. By inserting sequences into a whole-genome phylogeny, we show that 16S rRNA and shotgun metagenomic data generated from the same samples agree in principal coordinates space, taxonomy, and in phenotype effect size when analyzed with the same tree.

Phylogenetic placement of query samples on an existing phylogeny is increasingly used in molecular ecology, including sample identification and microbiome environmental sampling. As the size of available reference trees used in these analyses continues to grow, there is a growing need for methods that place sequences on ultra-large trees with high accuracy. Distance-based placement methods have recently emerged as a path to provide such scalability while allowing flexibility to analyze both assembled and unassembled environmental samples. In this paper, we introduce a distance-based phylogenetic placement method, APPLES-2, that is more accurate and scalable than existing distance-based methods and even some of the leading maximum likelihood methods. This scalability is owed to a divide-and-conquer technique that limits distance calculation and phylogenetic placement to parts of the tree most relevant to each query. The increased scalability and accuracy enables us to study the effectiveness of APPLES-2 for placing microbial genomes on a data set of 10,575 microbial species using subsets of 381 marker genes. APPLES-2 has very high accuracy in this setting, placing 97% of query genomes within three branches of the optimal position in the species tree using 50 marker genes. Our proof of concept results show that APPLES-2 can quickly place metagenomic scaffolds on ultra-large backbone trees with high accuracy as long as a scaffold includes tens of marker genes. These results pave the path for a more scalable and widespread use of distance-based placement in various areas of molecular ecology.

In this paper, we investigate stable perturbations and their characterizations for various types of outer inverses, such as generalized, { 2 , 3 } -, { 2 , 4 } -, { 2 , 5 } -, { 1 , 2 , 3 } -, { 1 , 2 , 4 } -, Moore–Penrose, group, Drazin and generalized Drazin inverses. Some known results are improved and extended.