Converting quadratic entropy to diversity: Both animals and alleles are diverse, but some are more diverse than others

RESEARCH ARTICLE Converting quadratic entropy to diversity: Both animals and alleles are diverse, but some are more diverse than others Peter E. Smouse1☯, Sam C. Banks2☯*, Rod Peakall3☯ a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 1 Department of Ecology, Evolution & Natural Resources, Rutgers University, New Brunswick, New Jersey, United States of America, 2 The Fenner School of Environment and Society, The Australian National University, Acton, ACT, Australia, 3 Research School of Biology, The Australian National University, Acton, ACT, Australia ☯ These authors contributed equally to this work. * Abstract OPEN ACCESS Citation: Smouse PE, Banks SC, Peakall R (2017) Converting quadratic entropy to diversity: Both animals and alleles are diverse, but some are more diverse than others. PLoS ONE 12(10): e0185499. https://doi.org/10.1371/journal.pone.0185499 Editor: Wolfgang Arthofer, University of Innsbruck, AUSTRIA Received: May 24, 2017 Accepted: September 13, 2017 Published: October 31, 2017 Copyright: © 2017 Smouse et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data is within the paper and its Supporting Information files. The Antechinus data are archived in Excel workbook form, along with listings of the QDIVER results extracted from GenAlEx6.51 (http://biology. anu.edu.au/GenAlEx/). DC data and analyses are presented in S5 Appendix, and DR data and analyses are presented in S6 Appendix. Funding: PES was supported by the USDA National Institute of Food and Agriculture Hatch Project 1005333, and the New Jersey Agricultural Experiment Station, Hatch project NJ17160; SCB The use of diversity metrics has a long history in population ecology, while population genetic work has been dominated by variance-derived metrics instead, a technical gap that has slowed cross-communication between the fields. Interestingly, Rao’s Quadratic Entropy (RQE), comparing elements for ‘degrees of divergence’, was originally developed for population ecology, but has recently been deployed for evolutionary studies. We here translate RQE into a continuous diversity analogue, and then construct a multiply nested diversity partition for alleles, individuals, populations, and species, each component of which exhibits the behavior of proper diversity metrics, and then translate these components into [0,1]— scaled form. We also deploy non-parametric statistical tests of the among-stratum components and novel tests of the homogeneity of within-stratum diversity components at any hierarchical level. We then illustrate this new analysis with eight nSSR loci and a pair of close Australian marsupial (Antechinus) congeners, using both ‘different is different’ and ‘degree of difference’ distance metrics. The total diversity in the collection is larger than that within either species, but most of the within-species diversity is resident within single populations. The combined A. agilis collection exhibits more diversity than does the combined A. stuartii collection, possibly attributable to localized differences in either local ecological disturbance regimes or differential levels of population isolation. Beyond exhibiting different allelic compositions, the two congeners are becoming more divergent for the arrays of allele sizes they possess. Introduction The use of genetic distance matrices to estimate genetic diversity within and among populations offers a number of benefits, including the ability to accommodate different genetic distance coding schemes, and computational tractability for large datasets. Here, we elaborate Rao’s Quadratic Entropy to quantify and statistically evaluate patterns of genetic diversity, PLOS ONE | https://doi.org/10.1371/journal.pone.0185499 October 31, 2017 1 / 19 Some are more diverse than others was supported by Australian Research Council Future Fellowship FT130100043. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. both within and among strata of a multiply nested taxonomic hierarchy, which can be used for diverse types of genetic data. This approach is related to, but exhibits a variety of innovations, relative to the traditional variance-based criteria commonly applied in population genetics. Quadratic (q = 2) diversity metrics of several different types, originally developed for community ecology, have begun to infiltrate population genetic analysis, traditionally dominated by variance-based, least squares analyses [1–7]. To date, most such metrics have deployed ‘different is different’ coding of genetic markers, though sometimes measured along spanning networks [8–12], reflecting evolutionary separation. The use of ‘degree of difference’ as an evolutionary metric traces to early work [13–19], and has spawned recent efforts to elaborate diversity theory in that same vein [20–23]. In that context, Rao’s Quadratic Entropy (henceforth Q) has drawn some attention [24–29], because conversion into inverted Gini-Simpson 1/(1 − Q) form yields a well-behaved diversity metric, provided that certain conditions are met [30–35]. Our object here is to elaborate Q, incorporating the ‘degree of difference’ between pairs of individual genets into a well-behaved diversity metric. We can translate a considerable array of paired-individual Euclidean distance matrices, as deployed for Amova [8, 36–38], Permanova [39–41], or Gamova [42], into Q, and can then convert Q into diversity analogue that may prove evolutionarily and/or ecologically informative. Conversion of Q into well-behaved diversity metric is only possible if [0  Q < 1]. There are three practical issues that must be dealt with in that translation. (1) Since quadratic genetic distance increases rapidly with the ‘degree of difference’, how are we to ensure that Q is properly bounded, given the wide array of quantitative divergence measures one could imagine for pairs of genets? (2) Can we estimate a well-behaved (and multiple level) partition of that total diversity, given the limited and typically unbalanced sampling routinely available from field studies? (3) Can we use this novel treatment for useful statistical evaluation of among-stratum diversification, as well as for evaluation of homo/heterogeneity of within-stratum diversity components? To illustrate both the formalisms and the utility of diversity translation, we employ a pair of Australian marsupial (Antechinus) congeners, sampled from contiguous Australian regions in New South Wales and Victoria, presenting evolutionary / geographic / environmental contrasts. We address a trio of additional questions: (4) How has evolutionary divergence within t (...truncated)