ASTRAL-III: Increased Scalability and Impacts of Contracting Low Support Branches

Abstract

Discordances between species trees and gene trees can complicate phylogenetics reconstruction. ASTRAL is a leading method for inferring species trees given gene trees while accounting for incomplete lineage sorting. It finds the tree that shares the maximum number of quartets with input trees, drawing bipartitions from a predefined set of bipartitions X. In this paper, we introduce ASTRAL-III, which substantially improves on ASTRAL-II in terms of running time by handling polytomies more efficiently, exploiting similarities between gene trees, and trimming unnecessary parts of the search space. The asymptotic running time in the presence of polytomies is reduced from $O(n^3k|X|^{{1.726}})$ for n species and k genes to $O(D|X|^{1.726})$ where $D=O(nk)$ is the sum of degrees of all unique nodes in input trees. ASTRAL-III enables us to test whether contracting low support branches in gene trees improves the accuracy by reducing noise. In extensive simulations and on real data, we show that removing branches with very low support improves accuracy while overly aggressive filtering is harmful.

Appendices

A Derivations

Derivation of Equation 6 : First note that:

$$\begin{aligned} \begin{aligned} QI((A|B|C), M)=&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}}\frac{a_i+b_j+c_k-3}{2} a_i b_j c_k\\ =&\sum _{i\in [d]}\left( {\begin{array}{c}a_i\\ 2\end{array}}\right) \sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}} b_j c_k\\ +&\sum _{i\in [d]}\left( {\begin{array}{c}b_i\\ 2\end{array}}\right) \sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}} a_j c_k\\ +&\sum _{i\in [d]}\left( {\begin{array}{c}c_i\\ 2\end{array}}\right) \sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}} a_j b_k \; . \end{aligned} \end{aligned}$$

(10)

Now, we note that:

$$\begin{aligned} \begin{aligned}&\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}} b_j c_k\\ =&\sum _{j\in [d]-\{i\}}b_j\sum _{k\in [d]-\{i,j\}} c_k\\ =&\sum _{j\in [d]-\{i\}}b_j(S_c-c_i-c_j)\\ =&-b_i(S_c-c_i-c_i)+\sum _{j\in [d]}b_j(S_c-c_i-c_j)\\ =\,&2b_i c_i - S_c b_i + S_b S_c - S_b c_i - S_{b,c}\\ =\,&(S_b - b_i) (S_c - c_i) - S_{b,c} + b_i c_i \end{aligned} \end{aligned}$$

(11)

Replacing this (and similar calculations for other terms) in Eq. 10 directly gives us the Eq. 6:

$$\begin{aligned} \begin{aligned} QI((A|B|C), M)=&\sum _{i\in [d]}\left( {\begin{array}{c}a_i\\ 2\end{array}}\right) ((S_b - b_i) (S_c - c_i) - S_{b,c} + b_i c_i)\\ +&\sum _{i\in [d]}\left( {\begin{array}{c}b_i\\ 2\end{array}}\right) ((S_a - a_i) (S_c - c_i) - S_{a,c} + a_i c_i)\\ +&\sum _{i\in [d]}\left( {\begin{array}{c}c_i\\ 2\end{array}}\right) ((S_a - a_i) (S_b - b_i) - S_{a,b} + a_i b_i) \; \end{aligned} \end{aligned}$$

(12)

Derivation of the Upperbound ${{\varvec{U(Z)}}}$ : In ASTRAL, V(Z) denotes the total contribution to the support of the best rooted tree $T_Z$ on taxon set Z, where each quartet tree in the set of input gene trees contributes 0 if it conflicts with $T_Z$ or only intersects it with one leaf, and otherwise contributes 1 or 2, depending on the number of nodes in $T_Z$ it maps to. Let U(Z) be the sum of max possible support of each quartet tree in the gene trees with respect to any resolution $T_Z$ of set Z, allowing the resolution to change for each gene tree. In other words, let Q(Z) be the set of quartets that would be resolved one way or another in any resolution of Z, and note that these are quartets that include two or leaves in Z; then, U(Z) is the number of resolved gene tree quartets that would match some resolution of Z and are included in Q(Z). More formally,

$$\begin{aligned} U(Z)=\sum _{g\in G}\sum _{M\in N(g)}\sum _{T\in Q(Z)}QI(T,M)\;, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} Q_1(Z)&=\{\{\{v,w\},\{x\},\{y\}\}|\{x,y\}\subset Z, \{v,w\}\subset L-\{x,y\}\}\;,\\ Q_2(Z)&=\{\{\{v,w\},\{x\},\{y\}\}|\{v,w,x\}\subset Z, y\in L-Z\}\;, \text { and}\\ Q(Z)&=Q_1(Z)\cup Q_2(Z)\;,Q_1(Z)\cap Q_2(Z)=\emptyset \;. \end{aligned} \end{aligned}$$

Clearly, $V(Z)\le U(Z)$ (equality can be achieved only if all gene trees are compatible with some resolution of Z). Then, letting $d=|M|$ and defining $z_i=|Z \cap M_i|$ and $l_i=|L \cap M_i|=|M_i|$, we have

$$\begin{aligned} \begin{aligned}&\sum _{\{A,B,C\}\in Q(Z)}QI((A|B|C),M)\\ =&\sum _{\{A,B,C\}\in Q_1(Z)}QI((A|B|C),M)+\sum _{\{A,B,C\}\in Q_2(Z)}QI((A|B|C),M)\\ =&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i\}-[j]}\left( {\begin{array}{c}l_i\\ 2\end{array}}\right) z_j z_k\\ +&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i\}-[j]}\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) (z_j (l_k-z_k) + (l_j-z_j) z_k) \\ =&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}}\left( {\begin{array}{c}l_i\\ 2\end{array}}\right) \frac{z_j z_k}{2}\\ +&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}}\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) \frac{z_j (l_k-z_k) + (l_j-z_j) z_k}{2}\\ =&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}}\left( {\begin{array}{c}l_i\\ 2\end{array}}\right) \frac{z_j z_k}{2}\\ +&\sum _{i\in [d]}\sum _{j\in [d]-\{i\}}\sum _{k\in [d]-\{i,j\}}\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j (l_k-z_k) \;. \end{aligned} \end{aligned}$$

(13)

Notice that based on Eq. 4,

$$\begin{aligned} \begin{aligned}&\frac{QI((Z|Z|L), M)}{2}-\frac{QI((Z|Z|Z), M)}{3} =\\&\frac{1}{2}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}}z_i z_j l_k\frac{z_i+z_j+l_k-3}{2}=\\ -&\frac{1}{3}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}}z_i z_j z_k\frac{z_i+z_j+z_k-3}{2}=\\&\frac{1}{2}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}}( \left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j l_k + z_i \left( {\begin{array}{c}z_j\\ 2\end{array}}\right) l_k + z_i z_j \left( {\begin{array}{c}l_k\\ 2\end{array}}\right) )\\ -&\frac{1}{3}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}} ( \left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j z_k + z_i \left( {\begin{array}{c}z_j\\ 2\end{array}}\right) z_k + z_i z_j \left( {\begin{array}{c}z_k\\ 2\end{array}}\right) )=\\&\frac{1}{2}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}} (\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j l_k + \left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j l_k + \left( {\begin{array}{c}l_i\\ 2\end{array}}\right) z_j z_k)\\ -&\frac{1}{3}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}} (\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j z_k + \left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j z_k + \left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j z_k )=\\&\frac{1}{2}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}}(\left( {\begin{array}{c}l_i\\ 2\end{array}}\right) z_j z_k + 2\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j l_k)\\ -&\frac{1}{3}\sum _{i\in [\,d\,]}\sum _{j\in [\,d\,]-\{i\}}\sum _{k\in [\,d\,]-\{i,j\}}3\left( {\begin{array}{c}z_i\\ 2\end{array}}\right) z_j z_k=\\&\sum _{{A,B,C}\in Q(Z)}QI((A|B|C),M)\;. \end{aligned} \end{aligned}$$

(14)

(going from the fourth term to the fifth is accomplished by changing the order of sums). Therefore,

$$\begin{aligned} \begin{aligned} U(Z)&=\sum _{g\in G}\sum _{M\in N(g)}(\frac{QI((Z|Z|L), M)}{2}-\frac{QI((Z|Z|Z), M)}{3})\\&=\frac{w(Z|Z|L)}{2}-\frac{w(Z|Z|Z)}{3}\;. \end{aligned} \end{aligned}$$

(15)

B Simulations and Commands

Simulation Setup

S100: In order to generate the gene trees and species trees using the Simphy we use this command:

Table 2. Species tree and gene tree generation parameters used for simphy [41], and sequence evolution parameters for the GTR model used for Indelible [42] for the S100 dataset.

Larege- ${\varvec{n}}$ Simulated Dataset: In order to compare running time performances of ASTRAL-II and ASTRAL-III, we created another dataset with very large numbers of species using Simphy and under the MSCM. Since we are only comparing running times, we only use true gene trees to infer the ASTRAL species trees. We have three sub-datasets with 5000, 2000, and 1000 species (plus one outgroup). Each sub-dataset has 4 replicates, and each replicate has a different species tree with 500 gene trees. Species trees are generated based on the birth-death process with birth and date rates from log uniform distributions. We sampled the number of generations and effective population size from log normal and uniform distributions respectively such that we have medium amounts of ILS. The average FN rates between the true gene trees and the species tree ranges between 4% and 23% for 1K, between 21% and 58% for 2K, and between 21% and 33% for 5K.

In order to generate the gene trees and true species trees using the Simphy we use parameters given in Table 3 and the following command.

1K:  

2K:  

Table 3. Species tree and gene tree generation parameters in Simphy [41] for 1K-taxon, 2K-taxon and 5K-taxon datasets

5K: 

Commands

Contracting Branches: In order to contract gene tree branches with bootstrap up to a certain threshold we used this command:

Drawing Bootstrap Support on ML Gene Trees: In order to draw bootstrap support on best ML gene trees we first reroot both best ML gene tree, and the bootstrap gene trees using this command:

Then we draw bootstrap supports on the branches:

Gene Tree Estimation: We used FastTree version 2.1.9 Double precision. In order to estimated best ML gene trees we used the following command:

where we have all the alignments in the PHYLIP format in the file all-genes.phylip for each replicate, and $<num>$ is the number of alignments in this file.

For bootstrapping analysis, we first generate bootstrapped sequences using RAxML version 8.2.9 with the following command:

and then we Fasttree to perform the actual ML analyses; for FastTree bootstrap runs, we use the same command and models that we used for best ML gene trees.

Running ASTRAL: ASTRAL-II in this paper refers to ASTRAL version 4.11.1 and ASTRAL-III refers to ASTRAL version 5.2.5. Both versions can be found in the link below:

Both versions of ASTRAL program were run with following command: