Lili Yu

A maximum pseudo-likelihood approach for estimating species trees under the coalescent model

BackgroundSeveral phylogenetic approaches have been developed to estimate species trees from collections of gene trees. However, maximum likelihood approaches for estimating species trees under the coalescent model are limited. Although the likelihood of a species tree under the multispecies coalescent model has already been derived by Rannala and Yang, it can be shown that the maximum likelihood estimate (MLE) of the species tree (topology, branch lengths, and population sizes) from gene trees under this formula does not exist. In this paper, we develop a pseudo-likelihood function of the species tree to obtain maximum pseudo-likelihood estimates (MPE) of species trees, with branch lengths of the species tree in coalescent units.ResultsWe show that the MPE of the species tree is statistically consistent as the number M of genes goes to infinity. In addition, the probability that the MPE of the species tree matches the true species tree converges to 1 at rate O(M-1). The simulation results confirm that the maximum pseudo-likelihood approach is statistically consistent even when the species tree is in the anomaly zone. We applied our method, Maximum Pseudo-likelihood for Estimating Species Trees (MP-EST) to a mammal dataset. The four major clades found in the MP-EST tree are consistent with those in the Bayesian concatenation tree. The bootstrap supports for the species tree estimated by the MP-EST method are more reasonable than the posterior probability supports given by the Bayesian concatenation method in reflecting the level of uncertainty in gene trees and controversies over the relationship of four major groups of placental mammals.ConclusionsMP-EST can consistently estimate the topology and branch lengths (in coalescent units) of the species tree. Although the pseudo-likelihood is derived from coalescent theory, and assumes no gene flow or horizontal gene transfer (HGT), the MP-EST method is robust to a small amount of HGT in the dataset. In addition, increasing the number of genes does not increase the computational time substantially. The MP-EST method is fast for analyzing datasets that involve a large number of genes but a moderate number of species.

Estimating Species Phylogenies Using Coalescence Times among Sequences

The estimation of species trees (phylogenies) is one of the most important problems in evolutionary biology, and recently, there has been greater appreciation of the need to estimate species trees directly rather than using gene trees as a surrogate. A Bayesian method constructed under the multispecies coalescent model can consistently estimate species trees but involves intensive computation, which can hinder its application to the phylogenetic analysis of large-scale genomic data. Many summary statistics-based approaches, such as shallowest coalescences (SC) and Global LAteSt Split (GLASS), have been developed to infer species phylogenies for multilocus data sets. In this paper, we propose 2 methods, species tree estimation using average ranks of coalescences (STAR) and species tree estimation using average coalescence times (STEAC), based on the summary statistics of coalescence times. It can be shown that the 2 methods are statistically consistent under the multispecies coalescent model. STAR uses the ranks of coalescences and is thus resistant to variable substitution rates along the branches in gene trees. A simulation study suggests that STAR consistently outperforms STEAC, SC, and GLASS when the substitution rates among lineages are highly variable. Two real genomic data sets were analyzed by the 2 methods and produced species trees that are consistent with previous results.

Coalescent methods for estimating phylogenetic trees

We review recent models to estimate phylogenetic trees under the multispecies coalescent. Although the distinction between gene trees and species trees has come to the fore of phylogenetics, only recently have methods been developed that explicitly estimate species trees. Of the several factors that can cause gene tree heterogeneity and discordance with the species tree, deep coalescence due to random genetic drift in branches of the species tree has been modeled most thoroughly. Bayesian approaches to estimating species trees utilizes two likelihood functions, one of which has been widely used in traditional phylogenetics and involves the model of nucleotide substitution, and the second of which is less familiar to phylogeneticists and involves the probability distribution of gene trees given a species tree. Other recent parametric and nonparametric methods for estimating species trees involve parsimony criteria, summary statistics, supertree and consensus methods. Species tree approaches are an appropriate goal for systematics, appear to work well in some cases where concatenation can be misleading, and suggest that sampling many independent loci will be paramount. Such methods can also be challenging to implement because of the complexity of the models and computational time. In addition, further elaboration of the simplest of coalescent models will be required to incorporate commonly known issues such as deviation from the molecular clock, gene flow and other genetic forces.

Estimating Species Trees from Unrooted Gene Trees

In this study, we develop a distance method for inferring unrooted species trees from a collection of unrooted gene trees. The species tree is estimated by the neighbor joining (NJ) tree built from a distance matrix in which the distance between two species is defined as the average number of internodes between two species across gene trees, that is, average gene-tree internode distance. The distance method is named NJ(st) to distinguish it from the original NJ method. Under the coalescent model, we show that if gene trees are known or estimated correctly, the NJ(st) method is statistically consistent in estimating unrooted species trees. The simulation results suggest that NJ(st) and STAR (another coalescence-based method for inferring species trees) perform almost equally well in estimating topologies of species trees, whereas the Bayesian coalescence-based method, BEST, outperforms both NJ(st) and STAR. Unlike BEST and STAR, the NJ(st) method can take unrooted gene trees to infer species trees without using an outgroup. In addition, the NJ(st) method can handle missing data and is thus useful in phylogenomic studies in which data sets often contain missing loci for some individuals.

Phybase: an R package for species tree analysis

MOTIVATION Phybase is an R package for phylogenetic analysis using species trees. It provides functions to read, write, manipulate, simulate, estimate, summarize and plot species trees, which contain not only the topology and branch lengths but also population sizes. AVAILABILITY The Phybase package is available at the R repository. The manual and supporting materials including source code, sample R code and sample data files for the species tree analysis are available at http://stat.osu.edu/~liuliang/research/phybase.html.

Maximum tree: a consistent estimator of the species tree

We propose a model based approach to use multiple gene trees to estimate the species tree. The coalescent process requires that gene divergences occur earlier than species divergences when there is any polymorphism in the ancestral species. Under this scenario, speciation times are restricted to be smaller than the corresponding gene split times. The maximum tree (MT) is the tree with the largest possible speciation times in the space of species trees restricted by available gene trees. If all populations have the same population size, the MT is the maximum likelihood estimate of the species tree. It can be shown the MT is a consistent estimator of the species tree even when the MT is built upon the estimates of the true gene trees if the gene tree estimates are statistically consistent. The MT converges in probability to the true species tree at an exponential rate.

Coalescent Methods for Estimating Species Trees from Phylogenomic Data

Genome‐scale sequence data have become increasingly available in the phylogenetic studies for understanding the evolutionary histories of species. However, it is challenging to develop probabilistic models to account for heterogeneity of phylogenomic data. The multispecies coalescent model describes gene trees as independent random variables generated from a coalescence process occurring along the lineages of the species tree. Since the multispecies coalescent model allows gene trees to vary across genes, coalescent‐based methods have been popularly used to account for heterogeneous gene trees in phylogenomic data analysis. In this paper, we summarize and evaluate the performance of coalescent‐based methods for estimating species trees from genome‐scale sequence data. We investigate the effects of deep coalescence and mutation on the performance of species tree estimation methods. We found that the coalescent‐based methods perform well in estimating species trees for a large number of genes, regardless of the degree of deep coalescence and mutation. The performance of the coalescent methods is negatively correlated with the lengths of internal branches of the species tree.

A Bayesian model for gene family evolution

BackgroundA birth and death process is frequently used for modeling the size of a gene family that may vary along the branches of a phylogenetic tree. Under the birth and death model, maximum likelihood methods have been developed to estimate the birth and death rate and the sizes of ancient gene families (numbers of gene copies at the internodes of the phylogenetic tree). This paper aims to provide a Bayesian approach for estimating parameters in the birth and death model.ResultsWe develop a Bayesian approach for estimating the birth and death rate and other parameters in the birth and death model. In addition, a Bayesian hypothesis test is developed to identify the gene families that are unlikely under the birth and death process. Simulation results suggest that the Bayesian estimate is more accurate than the maximum likelihood estimate of the birth and death rate. The Bayesian approach was applied to a real dataset of 3517 gene families across genomes of five yeast species. The results indicate that the Bayesian model assuming a constant birth and death rate among branches of the phylogenetic tree cannot adequately explain the observed pattern of the sizes of gene families across species. The yeast dataset was thus analyzed with a Bayesian heterogeneous rate model that allows the birth and death rate to vary among the branches of the tree. The unlikely gene families identified by the Bayesian heterogeneous rate model are different from those given by the maximum likelihood method.ConclusionsCompared to the maximum likelihood method, the Bayesian approach can produce more accurate estimates of the parameters in the birth and death model. In addition, the Bayesian hypothesis test is able to identify unlikely gene families based on Bayesian posterior p-values. As a powerful statistical technique, the Bayesian approach can effectively extract information from gene family data and thereby provide useful information regarding the evolutionary process of gene families across genomes.

Quasi-Likelihood for Right-Censored Data in the Generalized Linear Model

This article proposes a method for estimation in a generalized linear model with right-censored data. Consider the model , μ i (β) = g(β T X i ), where v and g are known functions, e i , i = 1,…,n are identically and independently distributed with mean 0 and finite variance φ2, but the distribution is unspecified. We use Kaplan–Meier estimates to replace the censored observations and employ quasi-likelihood to estimate the parameters. Consistency and asymptotic normality of the parameter estimates are derived. The performance of the proposed method for small sample sizes is investigated by simulation study. The new method is illustrated with the Stanford heart transplant data.

A phylogenetic model for understanding the effect of gene duplication on cancer progression

As biotechnology advances rapidly, a tremendous amount of cancer genetic data has become available, providing an unprecedented opportunity for understanding the genetic mechanisms of cancer. To understand the effects of duplications and deletions on cancer progression, two genomes (normal and tumor) were sequenced from each of five stomach cancer patients in different stages (I, II, III and IV). We developed a phylogenetic model for analyzing stomach cancer data. The model assumes that duplication and deletion occur in accordance with a continuous time Markov Chain along the branches of a phylogenetic tree attached with five extended branches leading to the tumor genomes. Moreover, coalescence times of the phylogenetic tree follow a coalescence process. The simulation study suggests that the maximum likelihood approach can accurately estimate parameters in the phylogenetic model. The phylogenetic model was applied to the stomach cancer data. We found that the expected number of changes (duplication and deletion) per gene for the tumor genomes is significantly higher than that for the normal genomes. The goodness-of-fit test suggests that the phylogenetic model with constant duplication and deletion rates can adequately fit the duplication data for the normal genomes. The analysis found nine duplicated genes that are significantly associated with stomach cancer.