[Paper Review] Species tree estimation using ASTRAL: how many genes are enough?

Shubhanshu Shekhar, Sébastien Roch+1 more|arXiv (Cornell University)|Apr 22, 2017

Genomics and Phylogenetic Studies3 citations

TL;DR

This paper derives theoretical sample complexity bounds for ASTRAL, showing it requires $O(f^{-2} \log n)$ gene trees to reconstruct the correct species tree with high probability, where $f$ is the shortest branch length in coalescent units and $n$ is the number of species. The results are validated via simulations, revealing counterintuitive patterns such as longer basal branches increasing required gene counts due to quartet redundancy.

ABSTRACT

Species tree reconstruction from genomic data is increasingly performed using methods that account for sources of gene tree discordance such as incomplete lineage sorting. One popular method for reconstructing species trees from unrooted gene tree topologies is ASTRAL. In this paper, we derive theoretical sample complexity results for the number of genes required by ASTRAL to guarantee reconstruction of the correct species tree with high probability. We also validate those theoretical bounds in a simulation study. Our results indicate that ASTRAL requires $\mathcal{O}(f^{-2} \log n)$ gene trees to reconstruct the species tree correctly with high probability where n is the number of species and f is the length of the shortest branch in the species tree. Our simulations, which are the first to test ASTRAL explicitly under the anomaly zone, show trends consistent with the theoretical bounds and also provide some practical insights on the conditions where ASTRAL works well.

Motivation & Objective

• To establish theoretical sample complexity bounds for ASTRAL in reconstructing species trees from gene tree topologies under the multispecies coalescent model.
• To understand how species tree topology, particularly branch length and shape, affects the number of gene trees needed for accurate inference.
• To validate theoretical bounds through simulation studies under varying tree topologies and branch length regimes, including the anomaly zone.
• To provide practical insights into ASTRAL's performance and limitations, especially regarding long branch attraction-like effects.
• To compare ASTRAL-II with NJst in terms of data requirements across different tree shapes.

Proposed method

• Theoretical analysis derives the sample complexity of ASTRAL as $O(f^{-2} \log n)$, based on the probability of quartet topology agreement between gene trees and the species tree.
• The analysis leverages statistical consistency results from Allman et al., showing that the species tree topology has at least a 1/3 probability of matching any quartet in a gene tree under the multispecies coalescent.
• Simulations are conducted under the multispecies coalescent model with varying species tree topologies, including balanced trees and double-quartet trees with long basal branches.
• Gene trees are simulated under different branch lengths ($f = 0.1$ and $f = 0.005$) and tree shapes, and ASTRAL-II is used to infer species trees from these topologies.
• Quartet scores are computed to assess agreement between inferred and true species trees, and convergence to the correct topology is evaluated across increasing numbers of gene trees.
• The study compares ASTRAL-II with NJst, particularly in performance on caterpillar and balanced trees, and examines the impact of long branches on resolution accuracy.

Experimental results

Research questions

• RQ1 What is the theoretical minimum number of gene trees required for ASTRAL to reconstruct the correct species tree with high probability?
• RQ2 How does the shortest internal branch length ($f$) influence the number of required gene trees in ASTRAL?
• RQ3 How do different species tree topologies, such as balanced trees versus double-quartet trees with long basal branches, affect ASTRAL's performance and required sample size?
• RQ4 Why does ASTRAL-II require more genes for double-quartet trees despite lower overall gene tree discordance?
• RQ5 Can the sample complexity of NJst also be bounded by $\mathcal{O}(f^{-2})$, as suggested by simulation trends?

Key findings

• ASTRAL requires $O(f^{-2} \log n)$ gene trees to reconstruct the correct species tree with high probability, establishing a theoretical upper bound on sample complexity.
• Simulations confirm that ASTRAL-II performs better on balanced trees than on double-quartet trees, even though the latter have less gene tree discordance.
• The double-quartet tree with a long basal branch requires significantly more genes (e.g., ~2000) than the balanced tree (e.g., ~500) to achieve similar accuracy, despite lower discordance.
• The increased data requirement in the double-quartet tree is attributed to quartet redundancy: long branches cause highly correlated quartet frequencies, reducing information content.
• The study reveals a counterintuitive pattern resembling long branch attraction, where long basal branches impede accurate resolution of shorter terminal branches.
• The simulation results suggest that NJst may also require $\mathcal{O}(f^{-2})$ gene trees, though this remains a conjecture pending formal proof.