Source-linked AI summary
Incomplete Lineage Sorting: Consistent Phylogeny Estimation From Multiple Loci
Elchanan Mossel, Sebastien Roch
TL;DR
Incomplete lineage sorting can make gene-tree topologies disagree with the species tree, undermining phylogeny reconstruction from individual or combined genes. The paper introduces GLASS, a method for estimating species trees from multiple loci, and shows it is statistically consistent with sufficiently many unlinked genes while tolerating estimation errors.
Problem
Incomplete lineage sorting can make gene trees misleading about species trees, while combining genes or using the most common gene tree can fail.
Method
The paper introduces GLASS, a technique for estimating species trees from multiple genes or loci.
Results
GLASS is statistically consistent, returning the correct species-tree topology given sufficiently many unlinked genes, and returns the correct tree under the stated estimation-error condition.
Takeaways & Limitations
GLASS provides a simple approach that avoids the inconsistency pitfalls of common gene-tree reconstruction strategies under incomplete lineage sorting.
Takeaways & Limitations
The model does not allow migration between contemporaneous populations.
Abstract
from arXiv · showhide
We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the presence of incomplete lineage sorting, that is, when the topology of the gene trees may differ from that of the species tree. We show that our technique is statistically consistent under standard stochastic assumptions, that is, it returns the correct tree given sufficiently many unlinked loci. We also show that it can tolerate moderate estimation errors.
1 Introduction
Incomplete lineage sorting can make gene-tree topologies disagree with the species tree, undermining concatenation and majority voting. GLASS addresses this problem with a statistically consistent multilocus method that also provides convergence rates and tolerates moderate estimation errors.
- Motivation: Incomplete lineage sorting can make gene trees topologically inconsistent with the species tree, especially for recently diverged populations.This discordance is less consequential for deep branchings but critical for recent population studies.
- Motivation: Concatenation and majority voting are common responses, but both can perform poorly under incomplete lineage sorting.The most likely gene tree can itself be inconsistent with the species tree on topologies with at least five species.
- Contribution: GLASS reconstructs species trees from multiple genes and is statistically consistent with sufficiently many unlinked genes.The method is designed to avoid the inconsistency highlighted for majority voting.
- Contribution: GLASS also provides explicit convergence rates and supports several alleles per population.The paper further extends Rosenberg’s topological concordance concept to multiple loci.
- Scope: The paper establishes consistency under a standard Kingman-coalescent model and studies tolerance to moderate estimation errors.It also considers removing the molecular-clock assumption and combining the method with distance-matrix procedures.
2 Basic Setup
The paper models a species tree and independent gene trees under a standard coalescent process, using interspecific coalescence times to infer the species tree. Statistical consistency means that the probability of an incorrect reconstruction vanishes as the number of unlinked loci increases.
- Species tree: The species tree contains n isolated populations, with each branch characterized by population size, generation duration, and standard coalescent-time length.Population size is assumed constant along each branch, while results allow branch-specific sizes.
- Gene trees: Each of k loci has a genealogical history represented by a gene tree whose leaves are sampled alleles from the populations.The setup permits multiple sampled alleles per population.
- Inference problem: The inference problem is to recover the species tree from gene trees and accurate estimates of pairwise coalescence times.The analysis focuses on interspecific coalescence times between alleles from different populations.
- Stochastic model: Under the stochastic model, each gene tree follows a standard coalescent process and loci are mutually independent because they are unlinked.Coalescence proceeds on merged allele sets whenever populations merge in the species tree.
- Stochastic model: An algorithm is statistically consistent when the probability of returning an incorrect reconstruction approaches 0 as the number of loci tends to infinity.This definition supplies the target property for the later analysis of GLASS.
3 Species Tree Estimation
GLASS estimates the species tree by clustering populations using the smallest interspecific coalescence times, first within one locus and then across loci. The resulting glass tree is shown to be statistically consistent.
- Single-locus inference: For one gene, GLASS repeatedly clusters the pair of populations with the smallest interspecific coalescence time.The procedure continues by defining cluster-to-cluster coalescence time as the minimum across population pairs.
- Single-locus inference: The single-gene procedure produces the collapsed gene tree, which summarizes the inferred topology on populations rather than individual alleles.This addresses the mismatch between gene-tree and species-tree leaf sets when populations have multiple samples.
- Multilocus inference: Across loci, GLASS takes the minimum coalescence time rather than averaging it.One sufficiently early cross-population coalescence can provide evidence for the corresponding species branch.
- Multilocus inference: GLASS builds the minimal tree consistent with the evidence supplied by the gene trees and is detailed in Figure 1.The algorithm takes gene trees and coalescence times as input and outputs an estimated topology.
- Concordance: Multilocus concordance is defined by agreement between the glass tree and the species tree.Showing that this agreement becomes probable as loci increase is sufficient to establish statistical consistency.
4 Sufficient Conditions
The paper gives a combinatorial sufficient condition under which GLASS merges only sets corresponding to species-tree clades. Under that condition, GLASS returns the correct species tree.
- Sufficient condition: The section extends a single-gene condition into a combinatorial sufficient condition for GLASS to recover the species tree.The condition is used to guarantee multilocus concordance.
- Sufficient condition: For any two clades, the condition requires at least one locus and allele pair whose lineages coalesce before the end of the branch above their MRCA.This supplies early cross-clade evidence for the corresponding species-tree branch.
- Consequence: Proposition 1 concludes that satisfying the condition makes the gene trees multilocus concordant with the species tree.The condition is sufficient rather than necessary, although GLASS always returns a tree even when it fails.
- Proof strategy: Under the condition, every newly created GLASS cluster is the leaf set of a clade in the species tree.The proof establishes this invariant by induction over the algorithm’s execution.
5 Statistical Consistency
GLASS is statistically consistent under the paper’s coalescent and multilocus framework, including species trees in the anomaly zone. Its convergence analysis also shows a saturation effect: with fixed loci, adding alleles cannot make correctness approach one.
- Consistency: The consistency theorem applies to any species tree, including the anomaly zone of Degnan and Rosenberg.
- Consistency: GLASS is statistically consistent, meaning its probability of returning an incorrect reconstruction goes to 0 as the number of loci increases.The proof establishes this under the paper’s standard coalescent assumptions and independent loci.
- Consistency: GLASS consistency is proved by showing that a sufficient coalescence condition yields multilocus concordance and then that this condition’s success probability approaches 1.For each branching, chosen allele pairs must coalesce before the end of the branch above across the loci.
- Rates: The convergence-rate analysis is implicit in the proof of Proposition 2 and is stated as Proposition 3.
- Multiple Alleles: Saturation Effect: For fixed loci, increasing the number of sampled alleles cannot make the probability of correct reconstruction approach 1.The probability of multilocus discordance remains at least (q∗)^k > 0, where 0 < q∗ < 1 depends only on the species tree.
6 Tolerance to Estimation Error
The paper relaxes the assumption that GLASS receives exact coalescence times by using estimated intercluster coalescence times subject to a uniform error bound. Under a sufficient noisy-case condition, GLASS remains statistically consistent.
- Noisy input: The noisy-case analysis replaces exact coalescence times with estimated coalescence times and corresponding estimated intercluster coalescence times.
- Noisy input: The shortest branch length is measured in generations and denoted by m.
- Consistency with noise: Under the noisy-case sufficient condition, GLASS returns the correct species tree.This extends the sufficient-condition argument used for exact coalescence times.
- Consistency with noise: Under the assumptions of Proposition 6, GLASS remains statistically consistent in the noisy case.
7 Generalization
The approach generalizes beyond molecular-clock distances by estimating molecular distances from minimum divergence times and applying distance-based reconstruction. Under bounded additive error, standard four-point methods recover the species tree, and sufficiently close population pairs can suffice.
- Minimum divergence times among individuals and genes provide estimates of distances between populations.
- The distance-based approach can be combined with any distance-based reconstruction algorithm under general assumptions.
- Molecular distances need not equal gene-specific divergence times, allowing branch-specific mutation rates while requiring equal rates across genes and individuals within each branch.
- Distances estimated up to an additive error of m′/4 are sufficient for standard four-point methods to reconstruct the species tree correctly.
- Estimating distances only between sufficiently close population pairs can also support correct species-tree reconstruction under suitable consistency conditions.