Matrix Representation With Parsimony Research Articles

Information content of trees: three-taxon statements, inference rules and dependency

Abstract The three-taxon statement is the fundamental unit of rooted trees in cladistics, stating that for three terminal taxa, two are more related to each other than to a third. Because of their fundamental role in phylogenetics, three-taxon statements are present in methodological research of various disciplines in evolutionary biology; for example consensus methods, supertree methods, species-tree methods, distance metrics and even phylogenetic reconstruction. However, three-taxon statement methods are subject to important flaws related to information redundancy. Here we aim to study the behaviour of three-taxon statements and the interactions among them in order to enhance their performance in evolutionary studies. We show how specific interactions between three-taxon statements are responsible for the emergence of redundancy and dependency within trees, and how they can be used for the improvement of weighting procedures. Our proposal is subsequently tested empirically in the supertree framework using simulations. We show that three-taxon statements using fractional weights perform much better than classical methods such as MRP (matrix representation with parsimony) or methods using unweighted statements. Our study shows that appropriate fractional weighting of three-taxon statements is of critical importance for removing redundancy in any method using them, such as in consensus, supertrees, distance metrics, and phylogenetic or biogeographical analyses.

Phylogenetic supertree reveals detailed evolution of SARS-CoV-2

Corona Virus Disease 2019 (COVID-19) caused by the emerged coronavirus SARS-CoV-2 is spreading globally. The origin of SARS-Cov-2 and its evolutionary relationship is still ambiguous. Several reports attempted to figure out this critical issue by genome-based phylogenetic analysis, yet limited progress was obtained, principally owing to the disability of these methods to reasonably integrate phylogenetic information from all genes of SARS-CoV-2. Supertree method based on multiple trees can produce the overall reasonable phylogenetic tree. However, the supertree method has been barely used for phylogenetic analysis of viruses. Here we applied the matrix representation with parsimony (MRP) pseudo-sequence supertree analysis to study the origin and evolution of SARS-CoV-2. Compared with other phylogenetic analysis methods, the supertree method showed more resolution power for phylogenetic analysis of coronaviruses. In particular, the MRP pseudo-sequence supertree analysis firmly disputes bat coronavirus RaTG13 be the last common ancestor of SARS-CoV-2, which was implied by other phylogenetic tree analysis based on viral genome sequences. Furthermore, the discovery of evolution and mutation in SARS-CoV-2 was achieved by MRP pseudo-sequence supertree analysis. Taken together, the MRP pseudo-sequence supertree provided more information on the SARS-CoV-2 evolution inference relative to the normal phylogenetic tree based on full-length genomic sequences.

Building combined MRP-matrices with BuM, an automated web-tool.

The most common methods for combining different phylogenetic trees with uneven but overlapping taxon sampling are the Matrix Representation with Parsimony (MRP) and consensus tree methods. Although straightforward, some steps of MRP are time-consuming and risky when manually performed, especially the preparation of the matrix representations from the original topologies, and the creation of the single matrix containing all the information of the individual trees. Here we present Building MRP-Matrices (BuM), a free online tool for generating a combined matrix, following Baum and Ragan coding scheme, from files containing phylogenetic trees in parenthetical format.

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.

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).

Implementing and testing Bayesian and maximum-likelihood supertree methods in phylogenetics.

Since their advent, supertrees have been increasingly used in large-scale evolutionary studies requiring a phylogenetic framework and substantial efforts have been devoted to developing a wide variety of supertree methods (SMs). Recent advances in supertree theory have allowed the implementation of maximum likelihood (ML) and Bayesian SMs, based on using an exponential distribution to model incongruence between input trees and the supertree. Such approaches are expected to have advantages over commonly used non-parametric SMs, e.g. matrix representation with parsimony (MRP). We investigated new implementations of ML and Bayesian SMs and compared these with some currently available alternative approaches. Comparisons include hypothetical examples previously used to investigate biases of SMs with respect to input tree shape and size, and empirical studies based either on trees harvested from the literature or on trees inferred from phylogenomic scale data. Our results provide no evidence of size or shape biases and demonstrate that the Bayesian method is a viable alternative to MRP and other non-parametric methods. Computation of input tree likelihoods allows the adoption of standard tests of tree topologies (e.g. the approximately unbiased test). The Bayesian approach is particularly useful in providing support values for supertree clades in the form of posterior probabilities.

Optimizing phylogenetic supertrees using answer set programming

AbstractThe supertree construction problem is about combining several phylogenetic trees with possibly conflicting information into a single tree that has all the leaves of the source trees as its leaves and the relationships between the leaves are as consistent with the source trees as possible. This leads to an optimization problem that is computationally challenging and typically heuristic methods, such as matrix representation with parsimony (MRP), are used. In this paper we consider the use of answer set programming to solve the supertree construction problem in terms of two alternative encodings. The first is based on an existing encoding of trees using substructures known as quartets, while the other novel encoding captures the relationships present in trees through direct projections. We use these encodings to compute a genus-level supertree for the family of cats (Felidae). Furthermore, we compare our results to recent supertrees obtained by the MRP method.

MRL and SuperFine+MRL: new supertree methods

BackgroundSupertree 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.ResultsWe 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.ConclusionsSuperFine+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.

An experimental study of Quartets MaxCut and other supertree methods.

BackgroundSupertree methods represent one of the major ways by which the Tree of Life can be estimated, but despite many recent algorithmic innovations, matrix representation with parsimony (MRP) remains the main algorithmic supertree method.ResultsWe evaluated the performance of several supertree methods based upon the Quartets MaxCut (QMC) method of Snir and Rao and showed that two of these methods usually outperform MRP and five other supertree methods that we studied, under many realistic model conditions. However, the QMC-based methods have scalability issues that may limit their utility on large datasets. We also observed that taxon sampling impacted supertree accuracy, with poor results obtained when all of the source trees were only sparsely sampled. Finally, we showed that the popular optimality criterion of minimizing the total topological distance of the supertree to the source trees is only weakly correlated with supertree topological accuracy. Therefore evaluating supertree methods on biological datasets is problematic.ConclusionsOur results show that supertree methods that improve upon MRP are possible, and that an effort should be made to produce scalable and robust implementations of the most accurate supertree methods. Also, because topological accuracy depends upon taxon sampling strategies, attempts to construct very large phylogenetic trees using supertree methods should consider the selection of source tree datasets, as well as supertree methods. Finally, since supertree topological error is only weakly correlated with the supertree's topological distance to its source trees, development and testing of supertree methods presents methodological challenges.

Consequences of Different Null Models on the Tree Shape Bias of Supertree Methods

Supertree methods are widely applied and give rise to new conclusions about phylogenies (e.g., BinindaEmonds et al. 2007). Although several desiderata for supertree methods exist (Wilkinson, Thorley, et al. 2004), only few of them have been studied in greater detail, examples include shape bias (Wilkinson et al. 2005) or pareto properties (Wilkinson et al. 2007). Here I look more closely at two matrix representation methods, matrix representation with compatibility (MRC) and matrix representation with parsimony (MRP). Different null models of random data are studied and the resulting tree shapes are investigated. Thereby I consider unrooted trees and a bias in tree shape is determined by a tree balance measure. The measure for unrooted trees is a modification of a tree balance measure for rooted trees. I observe that depending on the underlying null model of random data, the methods may resolve conflict in favor of more balanced tree shapes. The analyses refer only to trees with the same taxon set, also known as the consensus setting (e.g., Wilkinson et al. 2007), but I will be able to draw conclusions on how to deal with missing data.

Performance of Matrix Representation with Parsimony for Inferring Species from Gene Trees

Phylogenomic datasets often contain sequence alignments on different subsets of taxa for different genes. A major goal of phylogenetics is often to combine estimated gene trees from many loci into an overall estimate of a species tree. When data are missing for some combinations of genes and taxa, supertree methods can be used to combine gene trees on different subsets of taxa into an overall tree. However, studies of the performance of supertree methods when gene tree conflict is due to incomplete lineage sorting are needed to understand their statistical properties in this setting.We find that Matrix Representation with Parsimony (MRP), the most commonly used supertree method, can in many cases infer the species tree in spite of high levels of conflict in the input gene trees. However, for some species trees with short branches, MRP can be increasingly likely to return a tree other than the species tree as the number of loci increases. In some cases, deleting taxa at random or using estimated (rather than known) gene trees can either improve or hinder MRP for recovering the species tree.Although MRP is able to handle large amounts of conflict in the input gene trees, MRP is not statistically consistent for estimating species trees when gene trees arise under the multispecies coalescent model. However, triplet MRP is statistically consistent in this setting.

Comparative Performance of Supertree Algorithms in Large Data Sets Using the Soapberry Family (Sapindaceae) as a Case Study

For the last 2 decades, supertree reconstruction has been an active field of research and has seen the development of a large number of major algorithms. Because of the growing popularity of the supertree methods, it has become necessary to evaluate the performance of these algorithms to determine which are the best options (especially with regard to the supermatrix approach that is widely used). In this study, seven of the most commonly used supertree methods are investigated by using a large empirical data set (in terms of number of taxa and molecular markers) from the worldwide flowering plant family Sapindaceae. Supertree methods were evaluated using several criteria: similarity of the supertrees with the input trees, similarity between the supertrees and the total evidence tree, level of resolution of the supertree and computational time required by the algorithm. Additional analyses were also conducted on a reduced data set to test if the performance levels were affected by the heuristic searches rather than the algorithms themselves. Based on our results, two main groups of supertree methods were identified: on one hand, the matrix representation with parsimony (MRP), MinFlip, and MinCut methods performed well according to our criteria, whereas the average consensus, split fit, and most similar supertree methods showed a poorer performance or at least did not behave the same way as the total evidence tree. Results for the super distance matrix, that is, the most recent approach tested here, were promising with at least one derived method performing as well as MRP, MinFlip, and MinCut. The output of each method was only slightly improved when applied to the reduced data set, suggesting a correct behavior of the heuristic searches and a relatively low sensitivity of the algorithms to data set sizes and missing data. Results also showed that the MRP analyses could reach a high level of quality even when using a simple heuristic search strategy, with the exception of MRP with Purvis coding scheme and reversible parsimony. The future of supertrees lies in the implementation of a standardized heuristic search for all methods and the increase in computing power to handle large data sets. The latter would prove to be particularly useful for promising approaches such as the maximum quartet fit method that yet requires substantial computing power.

Quartets MaxCut: A Divide and Conquer Quartets Algorithm

Accurate phylogenetic reconstruction methods are currently limited to a maximum of few dozens of taxa. Supertree methods construct a large tree over a large set of taxa, from a set of small trees over overlapping subsets of the complete taxa set. Hence, in order to construct the tree of life over a million and a half different species, the use of a supertree method over the product of accurate methods, is inevitable. Perhaps the simplest version of this task that is still widely applicable, yet quite challenging, is quartet-based reconstruction. This problem lies at the root of many tree reconstruction methods and theoretical as well as experimental results have been reported. Nevertheless, dealing with false, conflicting quartet trees remains problematic. In this paper, we describe an algorithm for constructing a tree from a set of input quartet trees even with a significant fraction of errors. We show empirically that conflicts in the inputs are handled satisfactorily and that it significantly outperforms and outraces the Matrix Representation with Parsimony (MRP) methods that have previously been most successful in dealing with supertrees. Our algorithm is based on a divide and conquer algorithm where our divide step uses a semidefinite programming (SDP) formulation of MaxCut. We remark that this builds on previous work of ours for piecing together trees from rooted triplet trees. The recursion for unrooted quartets, however, is more complicated in that even with completely consistent set of quartet trees the problem is NP-hard, as opposed to the problem for triples where there is a linear time algorithm. This complexity leads to several issues and some solutions of possible independent interest.

Robinson-Foulds supertrees.

BackgroundSupertree methods synthesize collections of small phylogenetic trees with incomplete taxon overlap into comprehensive trees, or supertrees, that include all taxa found in the input trees. Supertree methods based on the well established Robinson-Foulds (RF) distance have the potential to build supertrees that retain much information from the input trees. Specifically, the RF supertree problem seeks a binary supertree that minimizes the sum of the RF distances from the supertree to the input trees. Thus, an RF supertree is a supertree that is consistent with the largest number of clusters (or clades) from the input trees.ResultsWe introduce efficient, local search based, hill-climbing heuristics for the intrinsically hard RF supertree problem on rooted trees. These heuristics use novel non-trivial algorithms for the SPR and TBR local search problems which improve on the time complexity of the best known (naïve) solutions by a factor of Θ(n) and Θ(n2) respectively (where n is the number of taxa, or leaves, in the supertree). We use an implementation of our new algorithms to examine the performance of the RF supertree method and compare it to matrix representation with parsimony (MRP) and the triplet supertree method using four supertree data sets. Not only did our RF heuristic provide fast estimates of RF supertrees in all data sets, but the RF supertrees also retained more of the information from the input trees (based on the RF distance) than the other supertree methods.ConclusionsOur heuristics for the RF supertree problem, based on our new local search algorithms, make it possible for the first time to estimate large supertrees by directly optimizing the RF distance from rooted input trees to the supertrees. This provides a new and fast method to build accurate supertrees. RF supertrees may also be useful for estimating majority-rule(-) supertrees, which are a generalization of majority-rule consensus trees.

A simulation study comparing supertree and combined analysis methods using SMIDGen

BackgroundSupertree methods comprise one approach to reconstructing large molecular phylogenies given multi-marker datasets: trees are estimated on each marker and then combined into a tree (the "supertree") on the entire set of taxa. Supertrees can be constructed using various algorithmic techniques, with the most common being matrix representation with parsimony (MRP). When the data allow, the competing approach is a combined analysis (also known as a "supermatrix" or "total evidence" approach) whereby the different sequence data matrices for each of the different subsets of taxa are concatenated into a single supermatrix, and a tree is estimated on that supermatrix.ResultsIn this paper, we describe an extensive simulation study we performed comparing two supertree methods, MRP and weighted MRP, to combined analysis methods on large model trees. A key contribution of this study is our novel simulation methodology (Super-Method Input Data Generator, or SMIDGen) that better reflects biological processes and the practices of systematists than earlier simulations. We show that combined analysis based upon maximum likelihood outperforms MRP and weighted MRP, giving especially big improvements when the largest subtree does not contain most of the taxa.ConclusionsThis study demonstrates that MRP and weighted MRP produce distinctly less accurate trees than combined analyses for a given base method (maximum parsimony or maximum likelihood). Since there are situations in which combined analyses are not feasible, there is a clear need for better supertree methods. The source tree and combined datasets used in this study can be used to test other supertree and combined analysis methods.

Complete Generic-Level Phylogenetic Analyses of Palms (Arecaceae) with Comparisons of Supertree and Supermatrix Approaches

Supertree and supermatrix methods have great potential in the quest to build the tree of life and yet they remain controversial, with most workers opting for one approach or the other, but rarely both. Here, we employed both methods to construct phylogenetic trees of all genera of palms (Arecaceae/Palmae), an iconic angiosperm family of great economic importance. We assembled a supermatrix consisting of 16 partitions, comprising DNA sequence data, plastid restriction fragment length polymorphism data, and morphological data for all genera, from which a highly resolved and well-supported phylogenetic tree was built despite abundant missing data. To construct supertrees, we used variants of matrix representation with parsimony (MRP) analysis based on input trees generated directly from subsamples of the supermatrix. All supertrees were highly resolved. Standard MRP with bootstrap-weighted matrix elements performed most effectively in this case, generating trees with the greatest congruence with the supermatrix tree and fewest clades unsupported by any input tree. Nonindependence due to input trees based on combinations of data partitions was an acceptable trade-off for improvements in supertree performance. Irreversible MRP and the use of strictly independent input trees only provided no obvious benefits. Contrary to previous claims, we found that unsupported clades are not infrequent under some MRP implementations, with up to 13% of clades lacking support from any input tree in some irreversible MRP supertrees. To build a formal synthesis, we assessed the cross-corroboration between supermatrix trees and the variant supertrees using semistrict consensus, enumerating shared clades and compatible clades. The semistrict consensus of the supermatrix tree and the most congruent supertree contained 160 clades (of a maximum of 204), 137 of which were present in both trees. The relationships recovered by these trees strongly support the current phylogenetic classification of palms. We evaluate 2 composite supertree support measures (rQS and V) and conclude that it is more informative to report numbers of input trees that support or conflict with a given supertree clade. This study demonstrates that supertree and supermatrix methods can provide effective, explicit, and complimentary mechanisms for synthesizing disjointed phylogenetic evidence while emphasizing the need for further refinement of supertree methods.

Supertrees and the Tree of Life: generating a metaphylogeny for a diverse invertebrate family (Insecta:Diptera:Therevidae) using constraint trees and the parsimony ratchet to overcome low taxon overlap

The dipteran family Therevidae (stiletto flies) is cosmopolitan and has been the focus of many taxonomic and phylogenetic studies over the last 25 years. Despite this work, questions remain concerning the relationships between subfamilies, genera and generic groups and membership of those groups. We use the supertree method to produce an inclusive phylogeny for the family Therevidae from 24 phylogenetic studies using matrix representation with parsimony (MRP) analysis. The supertree method, one of the most common approaches to calculating globally inclusive phylogenies from smaller more exclusive analyses, produced the therevid metaphylogeny despite only 34% of the terminal taxa being found in more than one source tree. We describe a method for handling low taxon overlap in supertree analyses, in combination with the parsimony ratchet and constraint tree techniques. The supertree presented here is an overarching phylogenetic hypothesis of the Therevidae, incorporating extensive sampling of major lineages and summarising past phylogenetic work on the family. The inclusive metaphylogeny for 362 therevid taxa robustly retrieves the subfamilies Agapophytinae, Phycinae, Therevinae and Xestomyzinae, and the tribes Cyclotelini and Therevini. The Phycinae and Xestomyzinae form a clade, sister to the remaining Therevidae. The Australasian and South American Taenogera Kröber genus-group is monophyletic and sister to a clade of Therevinae and the Australian endemic Agapophytinae. The Therevinae consists of the Anabarhynchus Macquart genus-group of Australian, South American, New Caledonian and New Zealand taxa as sister to the non-Australasian ‘higher Therevinae’, which contains the tribes Cyclotelini and Therevini. The Therevini includes the Hoplosathe Lyneborg & Zaitzev, Litolinga Irwin & Lyneborg, Baryphora Loew, Pandivirilia Irwin & Lyneborg and Thereva Latreille generic-groups. MRP supertree methods can be used to produce inclusive metaphylogenies in situations where source trees have poor data overlap and low taxon overlap, and are therefore valuable in species-rich groups such as arthropods. These methods may be necessary for constructing the ‘Tree of Life’, representing phylogenetic relationships among the millions of known species. However, our analyses show that in situations of source tree conflict, MRP supertree analyses present only the majority signal. We also show that conflict between source trees can be hidden in MRP supertrees, thus our results emphasise the need to evaluate the resulting clades with reference to the source trees.

Properties of Supertree Methods in the Consensus Setting

Supertree methods (SMs) are techniques for inferring (super)trees from sets of (input) trees. Classical consensus methods are SMs that were designed for the special case where input trees have identical leaf sets. The need for methods that can also combine information from input trees with nonidentical leaf sets has led to many alternative SMs. Some of these SMs are generalizations from conservative consensus methods (strict and semistrict) that do not resolve input tree conflicts (e.g., Gordon, 1986; Goloboff and Pol, 2002). Our focus here is on more liberal SMs, those capable of resolving conflicts among input trees. Liberal SMs comprise the majority of described methods and have been the most used in practice by biologists seeking well-resolved phylogenies. However, today's practitioners are confronted with choosing among a potentially bewildering array of liberal SM(s). Wilkinson et al. (2004) argued that nonarbitrary, rational choice among liberal SMs would best be guided by knowledge of the comparative accuracy of alternative methods. However, there have been few comparisons of accuracy using simulations (Bininda-Emonds and Sanderson, 2001; Chen et al., 2003; Eulenstein et al., 2004; Lapointe and Levasseur, 2004; Ross and Rodrigo, 2004) over a restricted range of conditions. Thus, Wilkinson et al. (2004) also discussed a number of properties that they suggested provide surrogates for accuracy and might therefore be expected of any SM. One of these, that input tree conflicts should be resolved independently of input tree shape, was investigated by Wilkinson et al. (2005a), who used a simple example (Fig. 1) and simulations to demonstrate input tree shape effects with 8 of the 14 methods they investigated, including the widely used matrix representation with parsimony (MRP) of Baum (1992) and Ragan (1992). Here we introduce a class of sub-Pareto properties that we argue constitute particularly weak expectations of how accurate SMs should handle consensus problems. We then use the same example to substantiate and extend results reported in Wilkinson et al. (2004) and to demonstrate that seven of the liberal SMs that are sensitive to input tree shape also lack some seemingly reasonable consensus properties. Lastly, we consider the relevance of these properties to choice and PRELIMINARIES

Imputing Supertrees and Supernetworks from Quartets

Inferring species phylogenies is an important part of understanding molecular evolution. Even so, it is well known that an accurate phylogenetic tree reconstruction for a single gene does not always necessarily correspond to the species phylogeny. One commonly accepted strategy to cope with this problem is to sequence many genes; the way in which to analyze the resulting collection of genes is somewhat more contentious. Supermatrix and supertree methods can be used, although these can suppress conflicts arising from true differences in the gene trees caused by processes such as lineage sorting, horizontal gene transfer, or gene duplication and loss. In 2004, Huson et al. (IEEE/ACM Trans. Comput. Biol. Bioinformatics 1:151-158) presented the Z-closure method that can circumvent this problem by generating a supernetwork as opposed to a supertree. Here we present an alternative way for generating supernetworks called Q-imputation. In particular, we describe a method that uses quartet information to add missing taxa into gene trees. The resulting trees are subsequently used to generate consensus networks, networks that generalize strict and majority-rule consensus trees. Through simulations and application to real data sets, we compare Q-imputation to the matrix representation with parsimony (MRP) supertree method and Z-closure, and demonstrate that it provides a useful complementary tool.