[Paper Review] Quartet-Based Inference Methods are Statistically Consistent Under the Unified Duplication-Loss-Coalescence Model

Alexey Markin, Oliver Eulenstein|arXiv (Cornell University)|Apr 8, 2020

Genomics and Phylogenetic Studies15 references4 citations

TL;DR

This paper proves that ASTRAL-one, a quartet-based species tree inference method, is statistically consistent under the unified duplication-loss-coalescence (DLCoal) model, which integrates gene duplication, loss, and incomplete lineage sorting. By analyzing gene tree probabilities conditioned on quartets and leveraging reconciliation scenarios, the authors show that a randomly selected quartet from a gene tree is more likely to match the species tree quartet than either of the other two possible quartets, providing theoretical justification for ASTRAL's empirical robustness in complex evolutionary scenarios.

ABSTRACT

The classic multispecies coalescent (MSC) model provides the means for theoretical justification of incomplete lineage sorting-aware species tree inference methods. A large body of work in phylogenetics is dedicated to the design of inference methods that are statistically consistent under MSC. One of such particularly popular methods is ASTRAL, a quartet-based species tree inference method. A few recent studies suggested that ASTRAL also performs well when given multi-locus gene trees in simulation studies. Further, Legried et al. recently demonstrated that ASTRAL is statistically consistent under the gene duplication and loss model (GDL). Note that GDL is prevalent in evolutionary histories and is a part of the powerful duplication-loss-coalescence evolutionary model (DLCoal) by Rasmussen and Kellis. In this work we prove that ASTRAL is statistically consistent under the general DLCoal model. Therefore, our result supports the empirical evidence from the simulation-based studies. More broadly, we prove that a randomly chosen quartet from a gene tree (with unique taxa) is more likely to agree with the respective species tree quartet than any of the two other quartets.

Motivation & Objective

• To establish theoretical statistical consistency of ASTRAL-one under the unified DLCoal model, which integrates gene duplication, loss, and incomplete lineage sorting.
• To resolve the gap in theoretical justification for ASTRAL’s strong empirical performance in multi-locus data with complex evolutionary events.
• To extend prior results on statistical consistency under MSC and GDL models to the more comprehensive DLCoal framework.
• To demonstrate that quartet agreement with the species tree is stochastically favored under DLCoal, even when gene trees are discordant due to multiple evolutionary processes.

Proposed method

• Derives gene tree probabilities under the bounded multispecies coalescent model, focusing on quartet topologies.
• Systematically partitions the probability space based on duplication-loss scenarios (e.g., (a,b,c), (ab,c), etc.) to analyze lineage coalescence patterns.
• Applies the law of total probability to compute marginal quartet probabilities across all reconciliation scenarios.
• Uses symmetry and stochastic dominance arguments to compare probabilities of different quartet topologies.
• Leverages existing results from Legried et al. on GDL consistency and extends them to the full DLCoal model.
• Employs mathematical inequalities and lemmas (e.g., Lemma 5, 6, 7) to prove that P[ab|cd ∈ G] ≥ P[ac|bd ∈ G] across all scenarios.

Experimental results

Research questions

• RQ1 Is ASTRAL-one statistically consistent under the full DLCoal model, which combines duplication, loss, and incomplete lineage sorting?
• RQ2 Does a randomly selected quartet from a gene tree have a higher probability of matching the species tree quartet than either of the other two possible quartets under DLCoal?
• RQ3 Can the statistical consistency of ASTRAL-one be formally proven when gene trees are generated under a model that includes both GDL and coalescent processes?
• RQ4 How do different duplication-loss configurations affect the likelihood of quartet agreement with the species tree?
• RQ5 Does the theoretical advantage of ASTRAL-one in quartet agreement hold across both balanced and caterpillar species tree topologies?

Key findings

• ASTRAL-one is statistically consistent under the unified DLCoal model, meaning its species tree estimate converges to the true species tree as the number of gene trees increases.
• For any species tree quartet, a randomly chosen gene tree quartet is more likely to match the species tree quartet than either of the other two possible quartets.
• In all DLCoal scenarios, P[ab|cd ∈ G] ≥ P[ac|bd ∈ G], with strict inequality in at least one case (e.g., when the locus tree displays ab|cd), ensuring a stochastic preference for the correct quartet.
• The proof holds for both balanced and caterpillar species tree topologies, confirming robustness across tree shapes.
• The result provides theoretical validation for ASTRAL’s strong empirical performance in simulation studies involving duplications, losses, and incomplete lineage sorting.
• The analysis confirms that the core principle of ASTRAL—favoring quartets that match the species tree—remains valid even under the most comprehensive gene tree evolution model.