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Abstract Over the past 15 years, the D-statistic, a four-taxon test for organismal admixture (hybridization, or introgression) which incorporates single nucleotide polymorphism data with allelic patterns ABBA and BABA, has seen considerable use. This statistic seeks to discern significant deviation from either a given species tree assumption, or from the balanced incomplete lineage sorting that could otherwise defy this species tree. However, while the D-statistic can successfully discriminate admixture from incomplete lineage sorting, it is not a simple matter to determine the directionality of admixture using only four-leaf tree models. As such, methods have been developed that use 5 leaves to evaluate admixture. Among these, the DFOIL method, which tests allelic patterns on the “symmetric” tree S = (((1,2),(3,4)),5), succeeds in finding admixture direction for many five-taxon examples. However, DFOIL does not make full use of all symmetry, nor can DFOIL function properly when ancient samples are included because of the reliance on singleton patterns (such as BAAAA and ABAAA). Here, we take inspiration from DFOIL to develop a new and completely general family of five-leaf admixture tests, dubbed Δ-statistics, that can either incorporate or exclude the singleton allelic patterns depending on individual taxon and age sampling choices. We describe two new shapes that are also fully testable, namely the “asymmetric” tree A = ((((1,2),3),4),5) and the “quasisymmetric” tree Q = (((1,2),3),(4,5)), which can considerably supplement the “symmetric“ S = (((1,2),(3,4)),5) model used by DFOIL. We demonstrate the consistency of Δ-statistics under various simulated scenarios, and provide empirical examples using data from black, brown and polar bears, the latter also including two ancient polar bear samples from previous studies. Recently DFOIL and one of these ancient samples was used to argue for a dominant polar bear → brown bear introgression direction. However, we find, using both this ancient polar bear and our own, that by far the strongest signal using both DFOIL and Δ-statistics on tree S is actually bidirectional gene flow of indistinguishable direction. Further experiments on trees A and Q instead highlight what were likely two phases of admixture: one with stronger brown bear → polar bear introgression in ancient times, and a more recent phase with predominant polar bear → brown bear directionality. Code and documentation available at https://github.com/KalleLeppala/Delta-statistics. 

Abstract With populations of threatened and endangered species declining worldwide, efforts are being made to generate high quality genomic records of these species before they are lost forever. Here, we demonstrate that data from single Oxford Nanopore Technologies (ONT) MinION flow cells can, even in the absence of highly accurate short DNA-read polishing, produce high quality de novo plant genome assemblies adequate for downstream analyses, such as synteny and ploidy evaluations, paleodemographic analyses, and phylogenomics. This study focuses on three North American ash tree species in the genusFraxinus(Oleaceae) that were recently added to the International Union for Conservation of Nature (IUCN) Red List as critically endangered. Our results support a hexaploidy event at the base of the Oleaceae as well as a subsequent whole genome duplication shared bySyringa,Osmanthus,Olea, and Fraxinus. Finally, we demonstrate the use of ONT long-read sequencing data to reveal patterns in demographic history.