An experimental study of Quartets MaxCut and other supertree methods

Asymmetric Cluster-Based Measures for Comparative Phylogenetics

Phylogenetic inference and reconstruction methods generate hypotheses on evolutionary history. Competing inference methods are frequently used, and the evaluation of the generated hypotheses is achieved using tree comparison costs. The Robinson-Foulds (RF) distance is a widely used cost to compare the topology of two trees, but this cost is sensitive to tree error and can overestimate tree differences. To overcome this limitation, a refined version of the RF distance called the Cluster Affinity (CA) distance was introduced. However, CA distances are symmetric and cannot compare different types of trees. These asymmetric comparisons occur when gene trees are compared with species trees, when disparate datasets are integrated into a supertree, or when tree comparison measures are used to infer a phylogenetic network. In this study, we introduce a relaxation of the original Affinity distance to compare heterogeneous trees called the asymmetric CA cost. We also develop a biologically interpretable cost, the Cluster Support cost that normalizes by cluster size across gene trees. The characteristics of these costs are similar to the symmetric CA cost. We describe efficient algorithms, derive the exact diameters, and use these to standardize the cost to be applicable in practice. These costs provide objective, fine-scale, and biologically interpretable values that can assess differences and similarities between phylogenetic trees.

A Guide to Phylogenomic Inference

Phylogenomics aims at reconstructing the evolutionary histories of organisms taking into account whole genomes or large fractions of genomes. Phylogenomics has significant applications in fields such as evolutionary biology, systematics, comparative genomics, and conservation genetics, providing valuable insights into the origins and relationships of species and contributing to our understanding of biological diversity and evolution. This chapter surveys phylogenetic concepts and methods aimed at both gene tree and species tree reconstruction while also addressing common pitfalls, providing references to relevant computer programs. A practical phylogenomic analysis example including bacterial genomes is presented at the end of the chapter.

Insertions and deletions as phylogenetic signal in an alignment-free context

Most methods for phylogenetic tree reconstruction are based on sequence alignments; they infer phylogenies from substitutions that may have occurred at the aligned sequence positions. Gaps in alignments are usually not employed as phylogenetic signal. In this paper, we explore an alignment-free approach that uses insertions and deletions (indels) as an additional source of information for phylogeny inference. For a set of four or more input sequences, we generate so-called quartet blocks of four putative homologous segments each. For pairs of such quartet blocks involving the same four sequences, we compare the distances between the two blocks in these sequences, to obtain hints about indels that may have happened between the blocks since the respective four sequences have evolved from their last common ancestor. A prototype implementation that we call Gap-SpaM is presented to infer phylogenetic trees from these data, using a quartet-tree approach or, alternatively, under the maximum-parsimony paradigm. This approach should not be regarded as an alternative to established methods, but rather as a complementary source of phylogenetic information. Interestingly, however, our software is able to produce phylogenetic trees from putative indels alone that are comparable to trees obtained with existing alignment-free methods.

Phylogenetic tree inference based on DNA or protein sequence comparison is a fundamental task in computational biology. Given a multiple alignment of a set of input sequences, most approaches compare aligned sequence positions to each other, to find a suitable tree, based on a model of molecular evolution. Insertions and deletions that may have happened since the input sequences evolved from their last common ancestor are ignored by most phylogeny methods. Herein, we show that insertions and deletions can provide an additional source of information for phylogeny inference, and that such information can be obtained with a simple alignment-free approach. We provide an implementation of this idea that we call Gap-SpaM. The proposed approach is complementary to existing phylogeny methods since it is based on a completely different source of information. It is, thus, not meant to be an alternative to those existing methods but rather as a possible additional source of information for tree inference.

Using Robinson-Foulds supertrees in divide-and-conquer phylogeny estimation

One of the Grand Challenges in Science is the construction of the Tree of Life, an evolutionary tree containing several million species, spanning all life on earth. However, the construction of the Tree of Life is enormously computationally challenging, as all the current most accurate methods are either heuristics for NP-hard optimization problems or Bayesian MCMC methods that sample from tree space. One of the most promising approaches for improving scalability and accuracy for phylogeny estimation uses divide-and-conquer: a set of species is divided into overlapping subsets, trees are constructed on the subsets, and then merged together using a "supertree method". Here, we present Exact-RFS-2, the first polynomial-time algorithm to find an optimal supertree of two trees, using the Robinson-Foulds Supertree (RFS) criterion (a major approach in supertree estimation that is related to maximum likelihood supertrees), and we prove that finding the RFS of three input trees is NP-hard. Exact-RFS-2 is available in open source form on Github at https://github.com/yuxilin51/GreedyRFS .

Topological Metrizations of Trees, and New Quartet Methods of Tree Inference

Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here, the rooted triple and quartet metrizations are introduced, and applied to formulate novel methods of inferring large trees from rooted triple and quartet data. These methods lead to new statistically consistent procedures for inference of a species tree from gene trees under the multispecies coalescent model.

Statistically consistent divide-and-conquer pipelines for phylogeny estimation using NJMerge

BACKGROUND

Divide-and-conquer methods, which divide the species set into overlapping subsets, construct a tree on each subset, and then combine the subset trees using a supertree method, provide a key algorithmic framework for boosting the scalability of phylogeny estimation methods to large datasets. Yet the use of supertree methods, which typically attempt to solve NP-hard optimization problems, limits the scalability of such approaches.

RESULTS

In this paper, we introduce a divide-and-conquer approach that does not require supertree estimation: we divide the species set into pairwise disjoint subsets, construct a tree on each subset using a base method, and then combine the subset trees using a distance matrix. For this merger step, we present a new method, called NJMerge, which is a polynomial-time extension of Neighbor Joining (NJ); thus, NJMerge can be viewed either as a method for improving traditional NJ or as a method for scaling the base method to larger datasets. We prove that NJMerge can be used to create divide-and-conquer pipelines that are statistically consistent under some models of evolution. We also report the results of an extensive simulation study evaluating NJMerge on multi-locus datasets with up to 1000 species. We found that NJMerge sometimes improved the accuracy of traditional NJ and substantially reduced the running time of three popular species tree methods (ASTRAL-III, SVDquartets, and "concatenation" using RAxML) without sacrificing accuracy. Finally, although NJMerge can fail to return a tree, in our experiments, NJMerge failed on only 11 out of 2560 test cases.

CONCLUSIONS

Theoretical and empirical results suggest that NJMerge is a valuable technique for large-scale phylogeny estimation, especially when computational resources are limited. NJMerge is freely available on Github (http://github.com/ekmolloy/njmerge).

BCD Beam Search: considering suboptimal partial solutions in Bad Clade Deletion supertrees

Supertree methods enable the reconstruction of large phylogenies. The supertree problem can be formalized in different ways in order to cope with contradictory information in the input. Some supertree methods are based on encoding the input trees in a matrix; other methods try to find minimum cuts in some graph. Recently, we introduced Bad Clade Deletion (BCD) supertrees which combines the graph-based computation of minimum cuts with optimizing a global objective function on the matrix representation of the input trees. The BCD supertree method has guaranteed polynomial running time and is very swift in practice. The quality of reconstructed supertrees was superior to matrix representation with parsimony (MRP) and usually on par with SuperFine for simulated data; but particularly for biological data, quality of BCD supertrees could not keep up with SuperFine supertrees. Here, we present a beam search extension for the BCD algorithm that keeps alive a constant number of partial solutions in each top-down iteration phase. The guaranteed worst-case running time of the new algorithm is still polynomial in the size of the input. We present an exact and a randomized subroutine to generate suboptimal partial solutions. Both beam search approaches consistently improve supertree quality on all evaluated datasets when keeping 25 suboptimal solutions alive. Supertree quality of the BCD Beam Search algorithm is on par with MRP and SuperFine even for biological data. This is the best performance of a polynomial-time supertree algorithm reported so far.

Efficient Quartet Representations of Trees and Applications to Supertree and Summary Methods

Quartet trees displayed by larger phylogenetic trees have long been used as inputs for species tree and supertree reconstruction. Computational constraints prevent the use of all displayed quartets in many practical problems with large numbers of taxa. We introduce the notion of an Efficient Quartet System (EQS) to represent a phylogenetic tree with a subset of the quartets displayed by the tree. We show mathematically that the set of quartets obtained from a tree via an EQS contains all of the combinatorial information of the tree itself. Using performance tests on simulated datasets, we also demonstrate that using an EQS to reduce the number of quartets in both summary method pipelines for species tree inference as well as methods for supertree inference results in only small reductions in accuracy.

The Performance of Two Supertree Schemes Compared Using Synthetic and Real Data Quartet Input

Despite impressive advancements in technological and theoretical tools, construction of phylogenetic (evolutionary) trees is still a challenging task. The availability of enormous quantities of molecular data has made large-scale phylogenetic reconstruction involving thousands of species, a more viable goal. For this goal, separate trees over different, overlapping subsets of species, representing histories of various markers of these species, are collected. These trees, typically with conflicting signals, are subsequently combined into a single tree over the full set, an operation denoted as supertree construction. The amalgamation of such trees into a single tree lies at the heart of many tasks in phylogenetics, yet remains a daunting endeavor, especially in light of conflicting signals. In this work, we study the performance of matrix representation with parsimony (MRP), the most widely used supertree method to date, when confronted with quartet trees. Quartet trees are the most basic informational unit when amalgamation of unrooted trees is attempted, and they remain relevant in more general settings even though standard supertree methods are not necessarily confined to quartets. This study involves both real and simulated data, and the effects of several parameters on the results are evaluated, revealing a number of anomalies associated with MRP. We show that these anomalies are surmountable when using a recently introduced supertree method, weighted quartet MaxCut (wQMC).

Phylogenomics

Phylogenomics aims at reconstructing the evolutionary histories of organisms taking into account whole genomes or large fractions of genomes. The abundance of genomic data for an enormous variety of organisms has enabled phylogenomic inference of many groups, and this has motivated the development of many computer programs implementing the associated methods. This chapter surveys phylogenetic concepts and methods aimed at both gene tree and species tree reconstruction while also addressing common pitfalls, providing references to relevant computer programs. A practical phylogenomic analysis example including bacterial genomes is presented at the end of the chapter.

Bad Clade Deletion Supertrees: A Fast and Accurate Supertree Algorithm

Supertree methods merge a set of overlapping phylogenetic trees into a supertree containing all taxa of the input trees. The challenge in supertree reconstruction is the way of dealing with conflicting information in the input trees. Many different algorithms for different objective functions have been suggested to resolve these conflicts. In particular, there exist methods based on encoding the source trees in a matrix, where the supertree is constructed applying a local search heuristic to optimize the respective objective function. We present a novel heuristic supertree algorithm called Bad Clade Deletion (BCD) supertrees. It uses minimum cuts to delete a locally minimal number of columns from such a matrix representation so that it is compatible. This is the complement problem to Matrix Representation with Compatibility (Maximum Split Fit). Our algorithm has guaranteed polynomial worst-case running time and performs swiftly in practice. Different from local search heuristics, it guarantees to return the directed perfect phylogeny for the input matrix, corresponding to the parent tree of the input trees, if one exists. Comparing supertrees to model trees for simulated data, BCD shows a better accuracy (F1 score) than the state-of-the-art algorithms SuperFine (up to 3%) and Matrix Representation with Parsimony (up to 7%); at the same time, BCD is up to 7 times faster than SuperFine, and up to 600 times faster than Matrix Representation with Parsimony. Finally, using the BCD supertree as a starting tree for a combined Maximum Likelihood analysis using RAxML, we reach significantly improved accuracy (1% higher F1 score) and running time (1.7-fold speedup).

COSPEDTree: COuplet Supertree by Equivalence Partitioning of Taxa Set and DAG Formation

From a set of phylogenetic trees with overlapping taxa set, a supertree exhibits evolutionary relationships among all input taxa. The key is to resolve the contradictory relationships with respect to input trees, between individual taxa subsets. Formulation of this NP hard problem employs either local search heuristics to reduce tree search space, or resolves the conflicts with respect to fixed or varying size subtree level decompositions. Different approximation techniques produce supertrees with considerable performance variations. Moreover, the majority of the algorithms involve high computational complexity, thus not suitable for use on large biological data sets. Current study presents COSPEDTree, a novel method for supertree construction. The technique resolves source tree conflicts by analyzing couplet (taxa pair) relationships for each source trees. Subsequently, individual taxa pairs are resolved with a single relation. To prioritize the consensus relations among individual taxa pairs for resolving them, greedy scoring is employed to assign higher score values for the consensus relations among a taxa pair. Selected set of relations resolving individual taxa pairs is subsequently used to construct a directed acyclic graph (DAG). Vertices of DAG represents a taxa subset inferred from the same speciation event. Thus, COSPEDTree can generate non-binary supertrees as well. Depth first traversal on this DAG yields final supertree. According to the performance metrics on branch dissimilarities (such as FP, FN and RF), COSPEDTree produces mostly conservative, well resolved supertrees. Specifically, RF metrics are mostly lower compared to the reference approaches, and FP values are lower apart from only strictly conservative (or veto) approaches. COSPEDTree has worst case time and space complexities of cubic and quadratic order, respectively, better or comparable to the reference approaches. Such high performance and low computational costs enable COSPEDTree to be applied on large scale biological data sets.

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.

Supertrees Based on the Subtree Prune-and-Regraft Distance

Supertree methods reconcile a set of phylogenetic trees into a single structure that is often interpreted as a branching history of species. A key challenge is combining conflicting evolutionary histories that are due to artifacts of phylogenetic reconstruction and phenomena such as lateral gene transfer (LGT). Many supertree approaches use optimality criteria that do not reflect underlying processes, have known biases, and may be unduly influenced by LGT. We present the first method to construct supertrees by using the subtree prune-and-regraft (SPR) distance as an optimality criterion. Although calculating the rooted SPR distance between a pair of trees is NP-hard, our new maximum agreement forest-based methods can reconcile trees with hundreds of taxa and>50 transfers in fractions of a second, which enables repeated calculations during the course of an iterative search. Our approach can accommodate trees in which uncertain relationships have been collapsed to multifurcating nodes. Using a series of benchmark datasets simulated under plausible rates of LGT, we show that SPR supertrees are more similar to correct species histories than supertrees based on parsimony or Robinson-Foulds distance criteria. We successfully constructed an SPR supertree from a phylogenomic dataset of 40,631 gene trees that covered 244 genomes representing several major bacterial phyla. Our SPR-based approach also allowed direct inference of highways of gene transfer between bacterial classes and genera. A Small number of these highways connect genera in different phyla and can highlight specific genes implicated in long-distance LGT. [Lateral gene transfer; matrix representation with parsimony; phylogenomics; prokaryotic phylogeny; Robinson-Foulds; subtree prune-and-regraft; supertrees.].

An empirical evaluation of two-stage species tree inference strategies using a multilocus dataset from North American pines

Background

As it becomes increasingly possible to obtain DNA sequences of orthologous genes from diverse sets of taxa, species trees are frequently being inferred from multilocus data. However, the behavior of many methods for performing this inference has remained largely unexplored. Some methods have been proven to be consistent given certain evolutionary models, whereas others rely on criteria that, although appropriate for many parameter values, have peculiar zones of the parameter space in which they fail to converge on the correct estimate as data sets increase in size.

Results

Here, using North American pines, we empirically evaluate the behavior of 24 strategies for species tree inference using three alternative outgroups (72 strategies total). The data consist of 120 individuals sampled in eight ingroup species from subsection Strobus and three outgroup species from subsection Gerardianae, spanning ∼47 kilobases of sequence at 121 loci. Each “strategy” for inferring species trees consists of three features: a species tree construction method, a gene tree inference method, and a choice of outgroup. We use multivariate analysis techniques such as principal components analysis and hierarchical clustering to identify tree characteristics that are robustly observed across strategies, as well as to identify groups of strategies that produce trees with similar features. We find that strategies that construct species trees using only topological information cluster together and that strategies that use additional non-topological information (e.g., branch lengths) also cluster together. Strategies that utilize more than one individual within a species to infer gene trees tend to produce estimates of species trees that contain clades present in trees estimated by other strategies. Strategies that use the minimize-deep-coalescences criterion to construct species trees tend to produce species tree estimates that contain clades that are not present in trees estimated by the Concatenation, RTC, SMRT, STAR, and STEAC methods, and that in general are more balanced than those inferred by these other strategies.

Conclusions

When constructing a species tree from a multilocus set of sequences, our observations provide a basis for interpreting differences in species tree estimates obtained via different approaches that have a two-stage structure in common, one step for gene tree estimation and a second step for species tree estimation. The methods explored here employ a number of distinct features of the data, and our analysis suggests that recovery of the same results from multiple methods that tend to differ in their patterns of inference can be a valuable tool for obtaining reliable estimates.

Fast local search for unrooted Robinson-Foulds supertrees

A Robinson-Foulds (RF) supertree for a collection of input trees is a tree containing all the species in the input trees that is at minimum total RF distance to the input trees. Thus, an RF supertree is consistent with the maximum number of splits in the input trees. Constructing RF supertrees for rooted and unrooted data is NP-hard. Nevertheless, effective local search heuristics have been developed for the restricted case where the input trees and the supertree are rooted. We describe new heuristics, based on the Edge Contract and Refine (ECR) operation, that remove this restriction, thereby expanding the utility of RF supertrees. Our experimental results on simulated and empirical data sets show that our unrooted local search algorithms yield better supertrees than those obtained from MRP and rooted RF heuristics in terms of total RF distance to the input trees and, for simulated data, in terms of RF distance to the true tree.

MRL and SuperFine+MRL: new supertree methods

Background

Supertree methods combine trees on subsets of the full taxon set together to produce a tree on the entire set of taxa. Of the many supertree methods, the most popular is MRP (Matrix Representation with Parsimony), a method that operates by first encoding the input set of source trees by a large matrix (the "MRP matrix") over {0,1, ?}, and then running maximum parsimony heuristics on the MRP matrix. Experimental studies evaluating MRP in comparison to other supertree methods have established that for large datasets, MRP generally produces trees of equal or greater accuracy than other methods, and can run on larger datasets. A recent development in supertree methods is SuperFine+MRP, a method that combines MRP with a divide-and-conquer approach, and produces more accurate trees in less time than MRP. In this paper we consider a new approach for supertree estimation, called MRL (Matrix Representation with Likelihood). MRL begins with the same MRP matrix, but then analyzes the MRP matrix using heuristics (such as RAxML) for 2-state Maximum Likelihood.

Results

We compared MRP and SuperFine+MRP with MRL and SuperFine+MRL on simulated and biological datasets. We examined the MRP and MRL scores of each method on a wide range of datasets, as well as the resulting topological accuracy of the trees. Our experimental results show that MRL, coupled with a very good ML heuristic such as RAxML, produced more accurate trees than MRP, and MRL scores were more strongly correlated with topological accuracy than MRP scores.

Conclusions

SuperFine+MRP, when based upon a good MP heuristic, such as TNT, produces among the best scores for both MRP and MRL, and is generally faster and more topologically accurate than other supertree methods we tested.

Polynomial supertree methods revisited

Supertree methods allow to reconstruct large phylogenetic trees by combining smaller trees with overlapping leaf sets into one, more comprehensive supertree. The most commonly used supertree method, matrix representation with parsimony (MRP), produces accurate supertrees but is rather slow due to the underlying hard optimization problem. In this paper, we present an extensive simulation study comparing the performance of MRP and the polynomial supertree methods MinCut Supertree, Modified MinCut Supertree, Build-with-distances, PhySIC, PhySIC_IST, and super distance matrix. We consider both quality and resolution of the reconstructed supertrees. Our findings illustrate the tradeoff between accuracy and running time in supertree construction, as well as the pros and cons of voting- and veto-based supertree approaches. Based on our results, we make some general suggestions for supertree methods yet to come.