Effects of discordance between species and gene trees on phylogenetic diversity conservation

Abstract

Phylogenetic diversity indices such as the Fair Proportion (FP) index are frequently discussed as prioritization criteria in biodiversity conservation. They rank species according to their contribution to overall diversity by taking into account the unique and shared evolutionary history of each species as indicated by its placement in an underlying phylogenetic tree. Traditionally, phylogenetic trees were inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the “true” evolutionary history of the species under consideration. However, nowadays it is common to sequence whole genomes of hundreds or thousands of genes, and it is often the case that conflicting genealogical histories exist in different genes throughout the genome, resulting in discordance between individual gene trees and the species tree. Here, we analyze the effects of gene and species tree discordance on prioritization decisions based on the FP index. In particular, we consider the ranking order of taxa induced by (i) The FP index on a species tree, and (ii) The expected FP index across all gene tree histories associated with the species tree. On the one hand, we show that for particular tree shapes, the two rankings always coincide. On the other hand, we show that for all leaf numbers greater than or equal to five, there exist species trees for which the two rankings differ. Finally, we illustrate the variability in the rankings obtained from the FP index across different gene tree and species tree estimates for an empirical multilocus mammal data set.

Appendices

Appendix A FP indices for the species trees depicted in Figs. 1 and 10

In the following, we give expression for the FP indices on the species tree and the expected FP indices for the species trees ${{\mathcal {S}}}$, ${{\mathcal {S}}}'$, and ${{\mathcal {S}}}''$ on 5 leaves considered in the main part of this manuscript (Figs. 1 and 10). The expected FP indices were obtained using Mathematica (Wolfram Research I 2020) to enumerate all gene tree histories in ${{\mathcal {H}}}({{\mathcal {S}}})$, respectively ${{\mathcal {H}}}({{\mathcal {S}}}')$ and ${{\mathcal {H}}}({{\mathcal {S}}}'')$.

Species tree ${{\mathcal {S}}}$ (Fig.1).

• FP indices on the species tree:

  $$\begin{aligned} FP_{{{\mathcal {S}}}}(x_1)&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{2} + \frac{\tau _3}{2} = FP_{{{\mathcal {S}}}}(x_2) \\ FP_{{{\mathcal {S}}}}(x_3)&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{3} \\ FP_{{{\mathcal {S}}}}(x_4)&= \tau _0 + \tau _1 + \frac{\tau _2}{2} + \frac{\tau _3}{3} = FP_{{{\mathcal {S}}}}(x_5). \end{aligned}$$

• Expected FP indices:

  $$\begin{aligned} {{\mathbb {E}}}\left[ FP(x_1) \vert {{\mathcal {S}}}\right]&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{2} + \frac{\tau _3}{2} + 1 - \frac{ e^{-\tau _1 - 2 \tau _2 - 2 \tau _3}}{24} - \frac{e^{- \tau _1 - \tau _2 - 2 \tau _3}}{12} \\&\quad + \frac{e^{-\tau _2 - \tau _3}}{72} - \frac{e^{-\tau _1 - \tau _2 - \tau _3}}{12} + \frac{e^{-\tau _3}}{36} = {{\mathbb {E}}}\left[ FP(x_2) \vert {{\mathcal {S}}}\right] \\ {{\mathbb {E}}}\left[ FP(x_3) \vert {{\mathcal {S}}}\right]&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _2}}{9} - \frac{e^{-\tau _1 - \tau _2 - 2 \tau _3}}{12} \\&\quad + \frac{e^{-\tau _2 - \tau _3}}{12} + \frac{e^{-\tau _1 - \tau _2 - \tau _3}}{18} - \frac{e^{-\tau _3}}{9} \\ {{\mathbb {E}}}\left[ FP(x_4) \vert {{\mathcal {S}}}\right]&= \tau _0 + \tau _1 + \frac{\tau _2}{2} + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _2}}{9} - \frac{e^{-\tau _1 - 2 \tau _2 - 2 \tau _3}}{24} \\&\quad - \frac{e^{-\tau _1 - \tau _2 - 2 \tau _3}}{24} - \frac{e^{-\tau _2-\tau _3}}{18} + \frac{e^{-\tau _1-\tau _2-\tau _3}}{18} + \frac{e^{-\tau _3}}{36} = {{\mathbb {E}}}\left[ FP(x_5) \vert {{\mathcal {S}}}\right] . \end{aligned}$$

Species tree ${{\mathcal {S}}}'$(Fig. 10, top)

• FP indices on the species tree:

  $$\begin{aligned} FP_{{{\mathcal {S}}}'}(x_1)&= \tau _0 + \tau _1 + \frac{\tau _2}{2} + \frac{\tau _3}{2} = FP_{{{\mathcal {S}}}'}(x_2) \\ FP_{{{\mathcal {S}}}'}(x_3)&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{3} \\ FP_{{{\mathcal {S}}}'}(x_4)&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{2} + \frac{\tau _3}{3} = FP_{{{\mathcal {S}}}'}(x_5). \end{aligned}$$

• Expected FP indices:

  $$\begin{aligned} {{\mathbb {E}}}\left[ FP(x_1) \vert {{\mathcal {S}}}'\right]&= \tau _0 + \tau _1 + \frac{\tau _2}{2} + \frac{\tau _3}{2} + 1 - \frac{e^{-\tau _1-2\tau _2-2 \tau _3}}{24} - \frac{e^{-\tau _2 - 2 \tau _3}}{12} \\&\quad - \frac{e^{-\tau _2-\tau _3}}{12} + \frac{e^{-\tau _1 - \tau _2 - \tau _3}}{72} + \frac{e^{-\tau _3}}{36} = {{\mathbb {E}}}\left[ FP(x_2) \vert {{\mathcal {S}}}'\right] \\ {{\mathbb {E}}}\left[ FP(x_3) \vert {{\mathcal {S}}}'\right]&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _1- \tau _2}}{9} - \frac{e^{-\tau _2-2 \tau _3}}{12} \\&\quad + \frac{e^{-\tau _2-\tau _3}}{18} + \frac{e^{-\tau _1-\tau _2-\tau _3}}{12} - \frac{e^{-\tau _3}}{9} \\ {{\mathbb {E}}}\left[ FP(x_4) \vert {{\mathcal {S}}}'\right]&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{2} + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _1-\tau _2}}{9} - \frac{e^{-\tau _1-2\tau _2-2\tau _3}}{24} \\&\quad - \frac{e^{-\tau _2-2 \tau _3}}{24} + \frac{e^{-\tau _2 - \tau _3}}{18} - \frac{e^{-\tau _1 - \tau _2 - \tau _3}}{18} + \frac{e^{-\tau _3}}{36} = {{\mathbb {E}}}\left[ FP(x_5) \vert {{\mathcal {S}}}'\right] . \end{aligned}$$

Species tree ${{\mathcal {S}}}''$ Fig. 10, bottom)

• FP indices on the species tree:

  $$\begin{aligned} FP_{{{\mathcal {S}}}''}(x_1)&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{2} = FP_{{{\mathcal {S}}}''}(x_2) \\ FP_{{{\mathcal {S}}}''}(x_3)&= \tau _0 + \tau _1 + \frac{\tau _2}{3} + \frac{\tau _3}{3} \\ FP_{{{\mathcal {S}}}''}(x_4)&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{3} + \frac{\tau _3}{3} = FP_{{{\mathcal {S}}}''}(x_5). \end{aligned}$$

• Expected FP indices:

  $$\begin{aligned} {{\mathbb {E}}}\left[ FP(x_1) \vert {{\mathcal {S}}}''\right]&= \tau _0 + \tau _1 + \tau _2 + \frac{\tau _3}{2} + 1 - \frac{e^{-\tau _2-2\tau _3}}{12} - \frac{e^{-\tau _1-\tau _2-2\tau _3}}{24} \\&\quad + \frac{e^{-\tau _2-\tau _3}}{36} + \frac{e^{-\tau _1-\tau _2-\tau _3}}{72} - \frac{e^{-\tau _3}}{12} = {{\mathbb {E}}}\left[ FP(x_2) \vert {{\mathcal {S}}}''\right] \\ {{\mathbb {E}}}\left[ FP(x_3) \vert {{\mathcal {S}}}''\right]&= \tau _0 + \tau _1 + \frac{\tau _2}{3} + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _1}}{9} - \frac{e^{-\tau _2-2 \tau _3}}{12} \\&\quad - \frac{e^{-\tau _2-\tau _3}}{9} + \frac{e^{-\tau _1-\tau _2-\tau _3}}{12} + \frac{e^{- \tau _3}}{18} \\ {{\mathbb {E}}}\left[ FP(x_4) \vert {{\mathcal {S}}}''\right]&= \tau _0 + \frac{\tau _1}{2} + \frac{\tau _2}{3} + \frac{\tau _3}{3} + 1 - \frac{e^{-\tau _1}}{9} - \frac{e^{-\tau _2-2 \tau _3}}{24} \\&\quad - \frac{e^{-\tau _1-\tau _2-2 \tau _3}}{24} + \frac{e^{-\tau _2-\tau _3}}{36} - \frac{e^{-\tau _1-\tau _2-\tau _3}}{18} + \frac{e^{-\tau _3}}{18} \\&= {{\mathbb {E}}}\left[ FP(x_5) \vert {{\mathcal {S}}}''\right] . \end{aligned}$$

Appendix B Species name abbreviations used in Fig. 11

Table 2 Species name abbreviations used in Fig. 11

Table 3 (continued)

Appendix C Details on species tree estimation for Sect. 4

In order to obtain species tree estimates for the multilocus mammal data set of Song et al. (2012), we downloaded the full sequences of the 447 genes from https://datadryad.org/stash/dataset/doi:10.5061%2Fdryad.3629v (file gene447_final.nexus).

After converting to PHYLIP format, a maximum liklihood tree based on the concatenated data was obtained using RAxML version 8.2.10 (Stamatakis 2014) via the following command:

raxMLHPC-AVX -s gene447_final.phy -p 12345 -m GTRGAMMA -o Gal -n raxml_GTR.tre

Afterwards, maximum likelihood branch lengths under the GTR+$\Gamma$ model and enforcing a molecular clock were computed in PAUP* (Swofford 2021) via the following commands:

exe gene447_final.nexus

gettrees file=raxml_GTR.tre

set criterion=likelihood

lset nst=6 basefreq=empirical rates=gamma rmatrix =estimate shape=estimate clock=yes

lscores

savetrees RAxML_GTR_MLbranchlengths.tre brlens

Moreover, a species tree using the species tree estimation method SVDQuartets (Chifman and Kubatko 2014) and maximum likelihood branch lengths for this species tree were obtained in PAUP* via the following commands:

exe gene447_final.nex

svdq

outgroup 35 (where species 35 corresponds to Gal)

roottrees rootmethod=outgroup

set criterion=likelihood

lset nst=6 basefreq=empirical rates=gamma rmatrix =estimate shape=estimate clock=yes

lscores

savetrees svdq_GTR_MLbranchlengths.tre brlens