More genes or more taxa? The relative contribution of gene number and taxon number to phylogenetic accuracy

The relative contribution of taxon number and gene number to accuracy in phylogenetic inference is a major issue in phylogenetics and of central importance to the choice of experimental strategies for the successful reconstruction of a broad sketch of the tree of life. Maximization of the number of taxa sampled is the strategy favored by most phylogeneticists, although its necessity remains the subject of debate. Vast increases in gene number are now possible due to advances in genomics, but large numbers of genes will be available for only modest numbers of taxa, raising the question of whether such genome-scale phylogenies will be robust to the addition of taxa. To examine the relative benefit of increasing taxon number or gene number to phylogenetic accuracy, we have developed an assay that utilizes the symmetric difference tree distance as a measure of phylogenetic accuracy. We have applied this assay to a genome-scale data matrix containing 106 genes from 14 yeast species. Our results show that increasing taxon number correlates with a slight decrease in phylogenetic accuracy. In contrast, increasing gene number has a significant positive effect on phylogenetic accuracy. Analyses of an additional taxon-rich data matrix from the same yeast clade show that taxon number does not have a significant effect on phylogenetic accuracy. The positive effect of gene number and the lack of effect of taxon number on phylogenetic accuracy are also corroborated by analyses of two data matrices from mammals and angiosperm plants, respectively. We conclude that, for typical data sets, the number of genes utilized may be a more important determinant of phylogenetic accuracy than taxon number.     

Using the tangle: a consistent construction of phylogenetic distance matrices for quartets

Distance based algorithms are a common technique in the construction of phylogenetic trees from taxonomic sequence data. The first step in the implementation of these algorithms is the calculation of a pairwise distance matrix to give a measure of the evolutionary change between any pair of the extant taxa. A standard technique is to use the log det formula to construct pairwise distances from aligned sequence data. We review a distance measure valid for the most general models, and show how the log det formula can be used as an estimator thereof. We then show that the foundation upon which the log det formula is constructed can be generalized to produce a previously unknown estimator which improves the consistency of the distance matrices constructed from the log det formula. This distance estimator provides a consistent technique for constructing quartets from phylogenetic sequence data under the assumption of the most general Markov model of sequence evolution.     

Towards building the tree of life: a simulation study for all angiosperm genera

Comprehensive phylogenetic trees are essential tools to better understand evolutionary processes. For many groups of organisms or projects aiming to build the Tree of Life, comprehensive phylogenetic analysis implies sampling hundreds to thousands of taxa. For the tree of all life this task rises to a highly conservative 13 million. Here, we assessed the performances of methods to reconstruct large trees using Monte Carlo simulations with parameters inferred from four large angiosperm DNA matrices, containing between 141 and 567 taxa. For each data set, parameters of the HKY85+G model were estimated and used to simulate 20 new matrices for sequence lengths from 100 to 10,000 base pairs. Maximum parsimony and neighbor joining were used to analyze each simulated matrix. In our simulations, accuracy was measured by counting the number of nodes in the model tree that were correctly inferred. The accuracy of the two methods increased very quickly with the addition of characters before reaching a plateau around 1000 nucleotides for any sizes of trees simulated. An increase in the number of taxa from 141 to 567 did not significantly decrease the accuracy of the methods used, despite the increase in the complexity of tree space. Moreover, the distribution of branch lengths rather than the rate of evolution was found to be the most important factor for accurately inferring these large trees. Finally, a tree containing 13,000 taxa was created to represent a hypothetical tree of all angiosperm genera and the efficiency of phylogenetic reconstructions was tested with simulated matrices containing an increasing number of nucleotides up to a maximum of 30,000. Even with such a large tree, our simulations suggested that simple heuristic searches were able to infer up to 80% of the nodes correctly.     

The systematic component of phylogenetic error as a function of taxonomic sampling under parsimony

The effect of taxonomic sampling on phylogenetic accuracy under parsimony is examined by simulating nucleotide sequence evolution. Random error is minimized by using very large numbers of simulated characters. This allows estimation of the consistency behavior of parsimony, even for trees with up to 100 taxa. Data were simulated on 8 distinct 100-taxon model trees and analyzed as stratified subsets containing either 25 or 50 taxa, in addition to the full 100-taxon data set. Overall accuracy decreased in a majority of cases when taxa were added. However, the magnitude of change in the cases in which accuracy increased was larger than the magnitude of change in the cases in which accuracy decreased, so, on average, overall accuracy increased as more taxa were included. A stratified sampling scheme was used to assess accuracy for an initial subsample of 25 taxa. The 25-taxon analyses were compared to 50- and 100-taxon analyses that were pruned to include only the original 25 taxa. On average, accuracy for the 25 taxa was improved by taxon addition, but there was considerable variation in the degree of improvement among the model trees and across different rates of substitution.     

Maximum Parsimony and the Skewness Test: A Simulation Study of the Limits of Applicability

The maximum parsimony (MP) method for inferring phylogenies is widely used, but little is known about its limitations in non-asymptotic situations. This study employs large-scale computations with simulated phylogenetic data to estimate the probability that MP succeeds in finding the true phylogeny for up to twelve taxa and 256 characters. The set of candidate phylogenies are taken to be unrooted binary trees; for each simulated data set, the tree lengths of all (2n − 5)!! candidates are computed to evaluate quantities related to the performance of MP, such as the probability of finding the true phylogeny, the probability that the tree with the shortest length is unique, the probability that the true phylogeny has the shortest tree length, and the expected inverse of the number of trees sharing the shortest length. The tree length distributions are also used to evaluate and extend the skewness test of Hillis for distinguishing between random and phylogenetic data. The results indicate, for example, that the critical point after which MP achieves a success probability of at least 0.9 is roughly around 128 characters. The skewness test is found to perform well on simulated data and the study extends its scope to up to twelve taxa.     

Root location in random trees: A polarity property of all sampling consistent phylogenetic models except one

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we show that among models that treat all taxa equally, and are sampling consistent (i.e. the distribution on trees is not affected by taxa yet to be included), all such models, except one (the so-called PDA model), convey some information as to the location of the ancestral root in an unrooted tree.     

Effects of nucleotide composition bias on the success of the parsimony criterion in phylogenetic inference

Convergence in nucleotide composition (CNC) in unrelated lineages is a factor potentially affecting the performance of most phylogeny reconstruction methods. Such convergence has deleterious effects because unrelated lineages show similarities due to similar nucleotide compositions and not shared histories. While some methods (such as the LogDet/paralinear distance measure) avoid this pitfall, the amount of convergence in nucleotide composition necessary to deceive other phylogenetic methods has never been quantified. We examined analytically the relationship between convergence in nucleotide composition and the consistency of parsimony as a phylogenetic estimator for four taxa. Our results show that rather extreme amounts of convergence are necessary before parsimony begins to prefer the incorrect tree. Ancillary observations are that (for unweighted Fitch parsimony) transition/transversion bias contributes to the impact of CNC and, for a given amount of CNC and fixed branch lengths, data sets exhibiting substantial site-to-site rate heterogeneity present fewer difficulties than data sets in which rates are homogeneous. We conclude by reexamining a data set originally used to illustrate the problems caused by CNC. Using simulations, we show that in this case the convergence in nucleotide composition alone is insufficient to cause any commonly used methods to fail, and accounting for other evolutionary factors (such as site-to-site rate heterogeneity) can give a correct inference without accounting for CNC.     

Increased taxon sampling greatly reduces phylogenetic error

Several authors have argued recently that extensive taxon sampling has a positive and important effect on the accuracy of phylogenetic estimates. However, other authors have argued that there is little benefit of extensive taxon sampling, and so phylogenetic problems can or should be reduced to a few exemplar taxa as a means of reducing the computational complexity of the phylogenetic analysis. In this paper we examined five aspects of study design that may have led to these different perspectives. First, we considered the measurement of phylogenetic error across a wide range of taxon sample sizes, and conclude that the expected error based on randomly selecting trees (which varies by taxon sample size) must be considered in evaluating error in studies of the effects of taxon sampling. Second, we addressed the scope of the phylogenetic problems defined by different samples of taxa, and argue that phylogenetic scope needs to be considered in evaluating the importance of taxon-sampling strategies. Third, we examined the claim that fast and simple tree searches are as effective as more thorough searches at finding near-optimal trees that minimize error. We show that a more complete search of tree space reduces phylogenetic error, especially as the taxon sample size increases. Fourth, we examined the effects of simple versus complex simulation models on taxonomic sampling studies. Although benefits of taxon sampling are apparent for all models, data generated under more complex models of evolution produce higher overall levels of error and show greater positive effects of increased taxon sampling. Fifth, we asked if different phylogenetic optimality criteria show different effects of taxon sampling. Although we found strong differences in effectiveness of different optimality criteria as a function of taxon sample size, increased taxon sampling improved the results from all the common optimality criteria. Nonetheless, the method that showed the lowest overall performance (minimum evolution) also showed the least improvement from increased taxon sampling. Taking each of these results into account re-enforces the conclusion that increased sampling of taxa is one of the most important ways to increase overall phylogenetic accuracy.     

Is it better to add taxa or characters to a difficult phylogenetic problem?

The effects on phylogenetic accuracy of adding characters and/or taxa were explored using data generated by computer simulation. The conditions of this study were constrained but allowed for systematic investigation of certain parameters. The starting point for the study was a four-taxon tree in the "Felsenstein zone," representing a difficult phylogenetic problem with an extreme situation of long branch attraction. Taxa were added sequentially to this tree in a manner specifically designed to break up the long branches, and for each tree data matrices of different sizes were simulated. Phylogenetic trees were reconstructed from these data using the criteria of parsimony and maximum likelihood. Phylogenetic accuracy was measured in three ways: (1) proportion of trees that are completely correct, (2) proportion of correctly reconstructed branches in all trees, and (3) proportion of trees in which the original four-taxon statement is correctly reconstructed. Accuracy improved dramatically with the addition of taxa and much more slowly with the addition of characters. If taxa can be added to break up long branches, it is much more preferable to add taxa than characters.     

Model misspecification and bias for inverse probability weighting estimators of average causal effects

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood lead levels.     

Measures of clade confidence do not correlate with accuracy of phylogenetic trees

Metrics of phylogenetic tree reliability, such as parametric bootstrap percentages or Bayesian posterior probabilities, represent internal measures of the topological reproducibility of a phylogenetic tree, while the recently introduced aLRT (approximate likelihood ratio test) assesses the likelihood that a branch exists on a maximum-likelihood tree. Although those values are often equated with phylogenetic tree accuracy, they do not necessarily estimate how well a reconstructed phylogeny represents cladistic relationships that actually exist in nature. The authors have therefore attempted to quantify how well bootstrap percentages, posterior probabilities, and aLRT measures reflect the probability that a deduced phylogenetic clade is present in a known phylogeny. The authors simulated the evolution of bacterial genes of varying lengths under biologically realistic conditions, and reconstructed those known phylogenies using both maximum likelihood and Bayesian methods. Then, they measured how frequently clades in the reconstructed trees exhibiting particular bootstrap percentages, aLRT values, or posterior probabilities were found in the true trees. The authors have observed that none of these values correlate with the probability that a given clade is present in the known phylogeny. The major conclusion is that none of the measures provide any information about the likelihood that an individual clade actually exists. It is also found that the mean of all clade support values on a tree closely reflects the average proportion of all clades that have been assigned correctly, and is thus a good representation of the overall accuracy of a phylogenetic tree.     

Bayesian phylogenetic estimation of fossil ages

Recent advances have allowed for both morphological fossil evidence and molecular sequences to be integrated into a single combined inference of divergence dates under the rule of Bayesian probability. In particular, the fossilized birth–death tree prior and the Lewis-Mk model of discrete morphological evolution allow for the estimation of both divergence times and phylogenetic relationships between fossil and extant taxa. We exploit this statistical framework to investigate the internal consistency of these models by producing phylogenetic estimates of the age of each fossil in turn, within two rich and well-characterized datasets of fossil and extant species (penguins and canids). We find that the estimation accuracy of fossil ages is generally high with credible intervals seldom excluding the true age and median relative error in the two datasets of 5.7% and 13.2%, respectively. The median relative standard error (RSD) was 9.2% and 7.2%, respectively, suggesting good precision, although with some outliers. In fact, in the two datasets we analyse, the phylogenetic estimate of fossil age is on average less than 2 Myr from the mid-point age of the geological strata from which it was excavated. The high level of internal consistency found in our analyses suggests that the Bayesian statistical model employed is an adequate fit for both the geological and morphological data, and provides evidence from real data that the framework used can accurately model the evolution of discrete morphological traits coded from fossil and extant taxa. We anticipate that this approach will have diverse applications beyond divergence time dating, including dating fossils that are temporally unconstrained, testing of the ‘morphological clock'', and for uncovering potential model misspecification and/or data errors when controversial phylogenetic hypotheses are obtained based on combined divergence dating analyses.This article is part of the themed issue ‘Dating species divergences using rocks and clocks’.     

Phylogenetic inference under the pure drift model

When pairwise genetic distances are used for phylogenetic reconstruction, it is usually assumed that the genetic distance between two taxa contains information about the time after the two taxa diverged. As a result, upon an appropriate transformation if necessary, the distance usually can be fitted to a linear model such that it is expressed as the sum of lengths of all branches that connect the two taxa in a given phylogeny. This kind of distance is referred to as "additive distance." For a phylogenetic tree exclusively driven by random genetic drift, genetic distances related to coancestry coefficients (theta XY) between any two taxa are more suitable. However, these distances are fundamentally different from the additive distance in that coancestry does not contain any information about the time after two taxa split from a common ancestral population; instead, it reflects the time before the two taxa diverged. In other words, the magnitude of theta XY provides information about how long the two taxa share the same evolutionary pathways. The fundamental difference between the two kinds of distances has led to a different algorithm of evaluating phylogenetic trees when theta XY and related distance measures are used. Here we present the new algorithm using the ordinary- least-squares approach but fitting to a different linear model. This treatment allows genetic variation within a taxon to be included in the model. Monte Carlo simulation for a rooted phylogeny of four taxa has verified the efficacy and consistency of the new method. Application of the method to human population was demonstrated.      

A Primer to Molecular Phylogenetic Analysis in Plants

Reconstructing a tree of life by inferring evolutionary history is an important focus of evolutionary biology. Phylogenetic reconstructions also provide useful information for a range of scientific disciplines such as botany, zoology, phylogeography, archaeology and biological anthropology. Until the development of protein and DNA sequencing techniques in the 1960s and 1970s, phylogenetic reconstructions were based on fossil records and comparative morphological/physiological analyses. Since then, progress in molecular phylogenetics has compensated for some of the shortcomings of phenotype-based comparisons. Comparisons at the molecular level increase the accuracy of phylogenetic inference because there is no environmental influence on DNA/peptide sequences and evaluation of sequence similarity is not subjective. While the number of morphological/physiological characters that are sufficiently conserved for phylogenetic inference is limited, molecular data provide a large number of datapoints and enable comparisons from diverse taxa. Over the last 20 years, developments in molecular phylogenetics have greatly contributed to our understanding of plant evolutionary relationships. Regions in the plant nuclear and organellar genomes that are optimal for phylogenetic inference have been determined and recent advances in DNA sequencing techniques have enabled comparisons at the whole genome level. Sequences from the nuclear and organellar genomes of thousands of plant species are readily available in public databases, enabling researchers without access to molecular biology tools to investigate phylogenetic relationships by sequence comparisons using the appropriate nucleotide substitution models and tree building algorithms. In the present review, the statistical models and algorithms used to reconstruct phylogenetic trees are introduced and advances in the exploration and utilization of plant genomes for molecular phylogenetic analyses are discussed.     

Accurate Phylogenetic Tree Reconstruction from Quartets: A Heuristic Approach

Supertree methods construct trees on a set of taxa (species) combining many smaller trees on the overlapping subsets of the entire set of taxa. A ‘quartet’ is an unrooted tree over taxa, hence the quartet-based supertree methods combine many -taxon unrooted trees into a single and coherent tree over the complete set of taxa. Quartet-based phylogeny reconstruction methods have been receiving considerable attentions in the recent years. An accurate and efficient quartet-based method might be competitive with the current best phylogenetic tree reconstruction methods (such as maximum likelihood or Bayesian MCMC analyses), without being as computationally intensive. In this paper, we present a novel and highly accurate quartet-based phylogenetic tree reconstruction method. We performed an extensive experimental study to evaluate the accuracy and scalability of our approach on both simulated and biological datasets.     

iGLASS: an improvement to the GLASS method for estimating species trees from gene trees

Several methods have been designed to infer species trees from gene trees while taking into account gene tree/species tree discordance. Although some of these methods provide consistent species tree topology estimates under a standard model, most either do not estimate branch lengths or are computationally slow. An exception, the GLASS method of Mossel and Roch, is consistent for the species tree topology, estimates branch lengths, and is computationally fast. However, GLASS systematically overestimates divergence times, leading to biased estimates of species tree branch lengths. By assuming a multispecies coalescent model in which multiple lineages are sampled from each of two taxa at L independent loci, we derive the distribution of the waiting time until the first interspecific coalescence occurs between the two taxa, considering all loci and measuring from the divergence time. We then use the mean of this distribution to derive a correction to the GLASS estimator of pairwise divergence times. We show that our improved estimator, which we call iGLASS, consistently estimates the divergence time between a pair of taxa as the number of loci approaches infinity, and that it is an unbiased estimator of divergence times when one lineage is sampled per taxon. We also show that many commonly used clustering methods can be combined with the iGLASS estimator of pairwise divergence times to produce a consistent estimator of the species tree topology. Through simulations, we show that iGLASS can greatly reduce the bias and mean squared error in obtaining estimates of divergence times in a species tree.     

Molecular distance and divergence time in carnivores and primates

Numerous studies have used indices of genetic distance between species to reconstruct evolutionary relationships and to estimate divergence time. However, the empirical relationship between molecular-based indices of genetic divergence and divergence time based on the fossil record is poorly known. To date, the results of empirical studies conflict and are difficult to compare because they differ widely in their choice of taxa, genetic techniques, or methods for calibrating rates of molecular evolution. We use a single methodology to analyze the relationship of molecular distance and divergence time in 86 taxa (72 carnivores and 14 primates). These taxa have divergence times of 0.01-55 Myr and provide a graded series of phylogenetic divergences such that the shape of the curve relating genetic distance and divergence time is often well defined. The techniques used to obtain genetic distance estimates include one- and two-dimensional protein electrophoresis, DNA hybridization, and microcomplement fixation. Our results suggest that estimates of molecular distance and divergence time are highly correlated. However, rates of molecular evolution are not constant; rather, in general they decline with increasing divergence time in a linear fashion. The rate of decline may differ according to technique and taxa. Moreover, in some cases the variability in evolutionary rates changes with increasing divergence time such that the accuracy of nodes in a phylogenetic tree varies predictably with time.     

Time squared: repeated measures on phylogenies

Studies of gene expression profiles in response to external perturbation generate repeated measures data that generally follow nonlinear curves. To explore the evolution of such profiles across a gene family, we introduce phylogenetic repeated measures (PR) models. These models draw strength from 2 forms of correlation in the data. Through gene duplication, the family's evolutionary relatedness induces the first form. The second is the correlation across time points within taxonic units, individual genes in this example. We borrow a Brownian diffusion process along a given phylogenetic tree to account for the relatedness and co-opt a repeated measures framework to model the latter. Through simulation studies, we demonstrate that repeated measures models outperform the previously available approaches that consider the longitudinal observations or their differences as independent and identically distributed by using deviance information criteria as Bayesian model selection tools; PR models that borrow phylogenetic information also perform better than nonphylogenetic repeated measures models when appropriate. We then analyze the evolution of gene expression in the yeast kinase family using splines to estimate nonlinear behavior across 3 perturbation experiments. Again, the PR models outperform previous approaches and afford the prediction of ancestral expression profiles. To demonstrate PR model applicability more generally, we conclude with a short examination of variation in brain development across 4 primate species.     

Impact of missing data,gene choice,and taxon sampling on phylogenetic reconstruction: the Caryophyllales (angiosperms)

Density of taxon sampling and number/kind of characters are central to achieving the ultimate goals in phylogenetic reconstruction: tree robustness and improved accuracy. In molecular phylogenetics, DNA sequence repositories such as GenBank are potential sources for expanding datasets in two dimensions, taxa and characters, to the level of “supermatrices.” However, the issue of missing characters/genomic regions is generally considered a major impediment to this endeavor. We used here the angiosperm order Caryophyllales to systematically address the impact of missing data when expanding taxon sampling and number of characters in phylogenetic reconstruction. Our analyses show that expansion of taxon sampling by ~13-fold resulted in improved phylogenetic assessment of the Caryophyllales despite up to 38% missing data. Expanding number of characters in the dataset by allowing for up to 100-fold increase in amount of missing data and inclusion of entries with about 40% missing genomic regions did not negatively impact tree structure or robustness, but to the contrary improved both. These results are timely regarding the ongoing efforts to achieve detailed assessment of the tree of life.     

Consistency, Characters, and the Likelihood of Correct Phylogenetic Inference

Computer simulations of character-state evolution in 8, 16, 32, and 64 ingroup taxa with a known set of relationships demonstrate that the maximum probability of correct phylogenetic inference increases with the number of variable (or informative) characters and their consistency index and decreases with the number of taxa, when the consistency index has been standardized to eliminate its dependence on the number of taxa. Equations for the probability of correct phylogenetic inference and for the standardized consistency indices (including or excluding autapomorphies) are derived. Given that actual studies based on DNA restriction sites and sequences generate more characters with a higher level of consistency than comparable studies based on morphology, calculations suggest that such molecular studies may often provide a more precise guide to phylogenetic relationships.