A rapid heuristic algorithm for finding minimum evolution trees

The minimum sum of branch lengths (S), or the minimum evolution (ME) principle, has been shown to be a good optimization criterion in phylogenetic inference. Unfortunately, the number of topologies to be analyzed is computationally prohibitive when a large number of taxa are involved. Therefore, simplified, heuristic methods, such as the neighbor-joining (NJ) method, are usually employed instead. The NJ method analyzes only a small number of trees (compared with the size of the entire search space); so, the tree obtained may not be the ME tree (for which the S value is minimum over the entire search space). Different compromises between very restrictive and exhaustive search spaces have been proposed recently. In particular, the "stepwise algorithm" (SA) utilizes what is known in computer science as the "beam search," whereas the NJ method employs a "greedy search." SA is virtually guaranteed to find the ME trees while being much faster than exhaustive search algorithms. In this study we propose an even faster method for finding the ME tree. The new algorithm adjusts its search exhaustiveness (from greedy to complete) according to the statistical reliability of the tree node being reconstructed. It is also virtually guaranteed to find the ME tree. The performances and computational efficiencies of ME, SA, NJ, and our new method were compared in extensive simulation studies. The new algorithm was found to perform practically as well as the SA (and, therefore, ME) methods and slightly better than the NJ method. For searching for the globally optimal ME tree, the new algorithm is significantly faster than existing ones, thus making it relatively practical for obtaining all trees with an S value equal to or smaller than that of the NJ tree, even when a large number of taxa is involved.     

Efficiencies of different genes and different tree-building methods in recovering a known vertebrate phylogeny

The relative efficiencies of different protein-coding genes of the mitochondrial genome and different tree-building methods in recovering a known vertebrate phylogeny (two whale species, cow, rat, mouse, opossum, chicken, frog, and three bony fish species) was evaluated. The tree-building methods examined were the neighbor joining (NJ), minimum evolution (ME), maximum parsimony (MP), and maximum likelihood (ML), and both nucleotide sequences and deduced amino acid sequences were analyzed. Generally speaking, amino acid sequences were better than nucleotide sequences in obtaining the true tree (topology) or trees close to the true tree. However, when only first and second codon positions data were used, nucleotide sequences produced reasonably good trees. Among the 13 genes examined, Nd5 produced the true tree in all tree-building methods or algorithms for both amino acid and nucleotide sequence data. Genes Cytb and Nd4 also produced the correct tree in most tree-building algorithms when amino acid sequence data were used. By contrast, Co2, Nd1, and Nd41 showed a poor performance. In general, large genes produced better results, and when the entire set of genes was used, all tree-building methods generated the true tree. In each tree-building method, several distance measures or algorithms were used, but all these distance measures or algorithms produced essentially the same results. The ME method, in which many different topologies are examined, was no better than the NJ method, which generates a single final tree. Similarly, an ML method, in which many topologies are examined, was no better than the ML star decomposition algorithm that generates a single final tree. In ML the best substitution model chosen by using the Akaike information criterion produced no better results than simpler substitution models. These results question the utility of the currently used optimization principles in phylogenetic construction. Relatively simple methods such as the NJ and ML star decomposition algorithms seem to produce as good results as those obtained by more sophisticated methods. The efficiencies of the NJ, ME, MP, and ML methods in obtaining the correct tree were nearly the same when amino acid sequence data were used. The most important factor in constructing reliable phylogenetic trees seems to be the number of amino acids or nucleotides used.      

Efficiencies of fast algorithms of phylogenetic inference under the criteria of maximum parsimony, minimum evolution, and maximum likelihood when a large number of sequences are used

In phylogenetic inference by maximum-parsimony (MP), minimum-evolution (ME), and maximum-likelihood (ML) methods, it is customary to conduct extensive heuristic searches of MP, ME, and ML trees, examining a large number of different topologies. However, these extensive searches tend to give incorrect tree topologies. Here we show by extensive computer simulation that when the number of nucleotide sequences (m) is large and the number of nucleotides used (n) is relatively small, the simple MP or ML tree search algorithms such as the stepwise addition (SA) plus nearest neighbor interchange (NNI) search and the SA plus subtree pruning regrafting (SPR) search are as efficient as the extensive search algorithms such as the SA plus tree bisection-reconnection (TBR) search in inferring the true tree. In the case of ME methods, the simple neighbor-joining (NJ) algorithm is as efficient as or more efficient than the extensive NJ+TBR search. We show that when ME methods are used, the simple p distance generally gives better results in phylogenetic inference than more complicated distance measures such as the Hasegawa-Kishino-Yano (HKY) distance, even when nucleotide substitution follows the HKY model. When ML methods are used, the simple Jukes-Cantor (JC) model of phylogenetic inference generally shows a better performance than the HKY model even if the likelihood value for the HKY model is much higher than that for the JC model. This indicates that at least in the present case, selecting of a substitution model by using the likelihood ratio test or the AIC index is not appropriate. When n is small relative to m and the extent of sequence divergence is high, the NJ method with p distance often shows a better performance than ML methods with the JC model. However, when the level of sequence divergence is low, this is not the case.     

An approach of orthology detection from homologous sequences under minimum evolution

In the field of phylogenetics and comparative genomics, it is important to establish orthologous relationships when comparing homologous sequences. Due to the slight sequence dissimilarity between orthologs and paralogs, it is prone to regarding paralogs as orthologs. For this reason, several methods based on evolutionary distance, phylogeny and BLAST have tried to detect orthologs with more precision. Depending on their algorithmic implementations, each of these methods sometimes has increased false negative or false positive rates. Here, we developed a novel algorithm for orthology detection that uses a distance method based on the phylogenetic criterion of minimum evolution. Our algorithm assumes that sets of sequences exhibiting orthologous relationships are evolutionarily less costly than sets that include one or more paralogous relationships. Calculation of evolutionary cost requires the reconstruction of a neighbor-joining (NJ) tree, but calculations are unaffected by the topology of any given NJ tree. Unlike tree reconciliation, our algorithm appears free from the problem of incorrect topologies of species and gene trees. The reliability of the algorithm was tested in a comparative analysis with two other orthology detection methods using 95 manually curated KOG datasets and 21 experimentally verified EXProt datasets. Sensitivity and specificity estimates indicate that the concept of minimum evolution could be valuable for the detection of orthologs.     

Robustness of Phylogenetic Inference Based on Minimum Evolution

Minimum evolution is the guiding principle of an important class of distance-based phylogeny reconstruction methods, including neighbor-joining (NJ), which is the most cited tree inference algorithm to date. The minimum evolution principle involves searching for the tree with minimum length, where the length is estimated using various least-squares criteria. Since evolutionary distances cannot be known precisely but only estimated, it is important to investigate the robustness of phylogenetic reconstruction to imprecise estimates for these distances. The safety radius is a measure of this robustness: it consists of the maximum relative deviation that the input distances can have from the correct distances, without compromising the reconstruction of the correct tree structure. Answering some open questions, we here derive the safety radius of two popular minimum evolution criteria: balanced minimum evolution (BME) and minimum evolution based on ordinary least squares (OLS + ME). Whereas BME has a radius of \frac12\frac{1}{2}, which is the best achievable, OLS + ME has a radius tending to 0 as the number of taxa increases. This difference may explain the gap in reconstruction accuracy observed in practice between OLS + ME and BME (which forms the basis of popular programs such as NJ and FastME).     

Fast and accurate phylogeny reconstruction algorithms based on the minimum-evolution principle.

The Minimum Evolution (ME) approach to phylogeny estimation has been shown to be statistically consistent when it is used in conjunction with ordinary least-squares (OLS) fitting of a metric to a tree structure. The traditional approach to using ME has been to start with the Neighbor Joining (NJ) topology for a given matrix and then do a topological search from that starting point. The first stage requires O(n(3)) time, where n is the number of taxa, while the current implementations of the second are in O(p n(3)) or more, where p is the number of swaps performed by the program. In this paper, we examine a greedy approach to minimum evolution which produces a starting topology in O(n(2)) time. Moreover, we provide an algorithm that searches for the best topology using nearest neighbor interchanges (NNIs), where the cost of doing p NNIs is O(n(2) + p n), i.e., O(n(2)) in practice because p is always much smaller than n. The Greedy Minimum Evolution (GME) algorithm, when used in combination with NNIs, produces trees which are fairly close to NJ trees in terms of topological accuracy. We also examine ME under a balanced weighting scheme, where sibling subtrees have equal weight, as opposed to the standard "unweighted" OLS, where all taxa have the same weight so that the weight of a subtree is equal to the number of its taxa. The balanced minimum evolution scheme (BME) runs slower than the OLS version, requiring O(n(2) x diam(T)) operations to build the starting tree and O(p n x diam(T)) to perform the NNIs, where diam(T) is the topological diameter of the output tree. In the usual Yule-Harding distribution on phylogenetic trees, the diameter expectation is in log(n), so our algorithms are in practice faster that NJ. Moreover, this BME scheme yields a very significant improvement over NJ and other distance-based algorithms, especially with large trees, in terms of topological accuracy.     

Estimating species trees from unrooted gene trees

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

Consistency of Topological Moves Based on the Balanced Minimum Evolution Principle of Phylogenetic Inference

Many phylogenetic algorithms search the space of possible trees using topological rearrangements and some optimality criterion. FastME is such an approach that uses the {em balanced minimum evolution (BME)} principle, which computer studies have demonstrated to have high accuracy. FastME includes two variants: {em balanced subtree prune and regraft (BSPR)} and {em balanced nearest neighbor interchange (BNNI)}. These algorithms take as input a distance matrix and a putative phylogenetic tree. The tree is modified using SPR or NNI operations, respectively, to reduce the BME length relative to the distance matrix, until a tree with (locally) shortest BME length is found. Following computer simulations, it has been conjectured that BSPR and BNNI are consistent, i.e. for an input distance that is a tree-metric, they converge to the corresponding tree. We prove that the BSPR algorithm is consistent. Moreover, even if the input contains small errors relative to a tree-metric, we show that the BSPR algorithm still returns the corresponding tree. Whether BNNI is consistent remains open.     

The relative contribution of band number to phylogenetic accuracy in AFLP data sets

We examined the effect of increasing the number of sampled amplified fragment length polymorphism (AFLP) bands to reconstruct an accurate and well-supported AFLP-based phylogeny. In silico AFLP was performed using simulated DNA sequences evolving along balanced and unbalanced model trees with recent, uniform and ancient radiations and average branch lengths (from the most internal node to the tip) ranging from 0.02 to 0.05 substitutions per site. Trees were estimated by minimum evolution (ME) and maximum parsimony (MP) methods from both DNA sequences and virtual AFLP fingerprints. The comparison of the true tree with the estimated AFLP trees suggests that moderate numbers of AFLP bands are necessary to recover the correct topology with high bootstrap support values (i.e. >70%). Fewer numbers of bands are necessary for shorter tree lengths and for balanced than for unbalanced tree topologies. However, branch length estimation was rather unreliable and did not improve substantially after a certain number of bands were sampled. These results hold for different levels of genome coverage and number of taxa analysed. In silico AFLP using bacterial genomic DNA sequences recovered a well-supported tree topology that mirrored an empirical phylogeny based on a set of 31 orthologous gene sequences when as few as 263 AFLP bands were scored. These results suggest that AFLPs may be an efficient alternative to traditional DNA sequencing for accurate topology reconstruction of shallow trees when not very short ancestral branches exist.     

EVALUATION OF THE RESTRICTED MAXIMUM-LIKELIHOOD METHOD FOR ESTIMATING PHYLOGENETIC TREES USING SIMULATED ALLELE-FREQUENCY DATA

Comparisons are made of the accuracy of the restricted maximum-likelihood, Wagner parsimony, and UPGMA (unweighted pair-group method using arithmetic averages) clustering methods to estimate phylogenetic trees. Data matrices were generated by constructing simulated stochastic evolution in a multidimensional gene-frequency space using a simple genetic-drift model (Brownian-motion, random-walk) with constant rates of divergence in all lineages. Ten differentphylogenetic tree topologies of 20 operational taxonomic units (OTU's), representing a range of tree shapes, were used. Felsenstein's restricted maximum-likelihood method, Wagner parsimony, and UPGMA clustering were used to construct trees from the resulting data matrices. The computations for the restricted maximum-likelihood method were performed on a Cray-1 supercomputer since the required calculations (especially when optimized for the vector hardware) are performed substantially faster than on more conventional computing systems. The overall level of accuracy of tree reconstruction depends on the topology of the true phylogenetic tree. The UPGMA clustering method, especially when genetic-distance coefficients are used, gives the most accurate estimates of the true phylogeny (for our model with constant evolutionary rates). For large numbers of loci, all methods give similar results, but trends in the results imply that the restricted maximum-likelihood method would produce the most accurate trees if sample sizes were large enough.     

DO PHYLOGENETIC METHODS PRODUCE TREES WITH BIASED SHAPES?

We examine whether phylogenetic methods provide biased estimates of tree shape with respect to the random branching model. We investigate the performance of five commonly used phylogenetic methods using computer simulation: (1) maximum parsimony; (2) neighbor joining; (3) UPGMA with an outgroup taxon; (4) UPGMA without an outgroup taxon; and (5) maximum likelihood. All methods provide estimates of tree shape that are, on average, more asymmetrical than the true tree, especially when rates of evolution are high. We suggest a simple explanation for the bias and propose a modified test of tree shape that corrects for it.     

A multi-neighbor-joining approach for phylogenetic tree reconstruction and visualization

The computationally challenging problem of reconstructing the phylogeny of a set of contemporary data, such as DNA sequences or morphological attributes, was treated by an extended version of the neighbor-joining (NJ) algorithm. The original NJ algorithm provides a single-tree topology, after a cascade of greedy pairing decisions that tries to simultaneously optimize the minimum evolution and the least squares criteria. Given that some sub-trees are more stable than others, and that the minimum evolution tree may not be achieved by the original NJ algorithm, we propose a multi-neighbor-joining (MNJ) algorithm capable of performing multiple pairing decisions at each level of the tree reconstruction, keeping various partial solutions along the recursive execution of the NJ algorithm. The main advantages of the new reconstruction procedure are: 1) as is the case for the original NJ algorithm, the MNJ algorithm is still a low-cost reconstruction method; 2) a further investigation of the alternative topologies may reveal stable and unstable sub-trees; 3) the chance of achieving the minimum evolution tree is greater; 4) tree topologies with very similar performances will be simultaneously presented at the output. When there are multiple unrooted tree topologies to be compared, a visualization tool is also proposed, using a radial layout to uniformly distribute the branches with the help of well-known metaheuristics used in computer science.     

A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates [published erratum appears in Mol Biol Evol 1995 May;12(3):525]

Using simulated data, we compared five methods of phylogenetic tree estimation: parsimony, compatibility, maximum likelihood, Fitch- Margoliash, and neighbor joining. For each combination of substitution rates and sequence length, 100 data sets were generated for each of 50 trees, for a total of 5,000 replications per condition. Accuracy was measured by two measures of the distance between the true tree and the estimate of the tree, one measure sensitive to accuracy of branch lengths and the other not. The distance-matrix methods (Fitch- Margoliash and neighbor joining) performed best when they were constrained from estimating negative branch lengths; all comparisons with other methods used this constraint. Parsimony and compatibility had similar results, with compatibility generally inferior; Fitch- Margoliash and neighbor joining had similar results, with neighbor joining generally slightly inferior. Maximum likelihood was the most successful method overall, although for short sequences Fitch- Margoliash and neighbor joining were sometimes better. Bias of the estimates was inferred by measuring whether the independent estimates of a tree for different data sets were closer to the true tree than to each other. Parsimony and compatibility had particular difficulty with inaccuracy and bias when substitution rates varied among different branches. When rates of evolution varied among different sites, all methods showed signs of inaccuracy and bias.      

Phylogenetic relationships within the genus Pimephales as inferred from ND4 and ND4L nucleotide sequences

In phylogenetic analyses, conducted on the ND4L gene and part of the ND4 gene from species of the genus Pimephales , maximum parsimony yielded four trees, with the strict consensus providing no resolution of relationships among species. Maximum likelihood and minimum evolution methods yielded identical tree topologies, which differed from previous hypotheses of relationships for these species. If this topology is correct, it implies independent evolution of morphological characters, possibly associated with convergent trophic specialization.     

Property and efficiency of the maximum likelihood method for molecular phylogeny

Summary The maximum likelihood (ML) method for constructing phylogenetic trees (both rooted and unrooted trees) from DNA sequence data was studied. Although there is some theoretical problem in the comparison of ML values conditional for each topology, it is possible to make a heuristic argument to justify the method. Based on this argument, a new algorithm for estimating the ML tree is presented. It is shown that under the assumption of a constant rate of evolution, the ML method and UPGMA always give the same rooted tree for the case of three operational taxonomic units (OTUs). This also seems to hold approximately for the case with four OTUs. When we consider unrooted trees with the assumption of a varying rate of nucleotide substitution, the efficiency of the ML method in obtaining the correct tree is similar to those of the maximum parsimony method and distance methods. The ML method was applied to Brown et al.'s data, and the tree topology obtained was the same as that found by the maximum parsimony method, but it was different from those obtained by distance methods.     

Constructing minimal ancestral recombination graphs.

By viewing the ancestral recombination graph as defining a sequence of trees, we show how possible evolutionary histories consistent with given data can be constructed using the minimum number of recombination events. In contrast to previously known methods, which yield only estimated lower bounds, our method of detecting recombination always gives the minimum number of recombination events if the right kind of rooted trees are used in our algorithm. A new lower bound can be defined if rooted trees with fewer constraints are used. As well as studying how often it actually is equal to the minimum, we test how this new lower bound performs in comparison to some other lower bounds. Our study indicates that the new lower bound is an improvement on earlier bounds. Also, using simulated data, we investigate how well our method can recover the actual site-specific evolutionary relationships. In the presence of recombination, using a single tree to describe the evolution of the entire locus clearly leads to lower average recovery percentages than does our method. Our study shows that recovering the actual local tree topologies can be done more accurately than estimating the actual number of recombination events.     

The problem of rooting rapid radiations

There are many examples of groups (such as birds, bees, mammals, multicellular animals, and flowering plants) that have undergone a rapid radiation. In such cases, where there is a combination of short internal and long external branches, correctly estimating and rooting phylogenetic trees is known to be a difficult problem. In this simulation study, we tested the performances of different phylogenetic methods at estimating a tree that models a rapid radiation. We found that maximum likelihood, corrected and uncorrected neighbor-joining, and corrected and uncorrected parsimony, all suffer from biases toward specific tree topologies. In addition, we found that using a single-taxon outgroup to root a tree frequently disrupts an otherwise correct ingroup phylogeny. Moreover, for uncorrected parsimony, we found cases where several individual trees (in which the outgroup was placed incorrectly) were selected more frequently than the correct tree. Even for parameter settings where the correct tree was selected most frequently when using extremely long sequences, for sequences of up to 60,000 nucleotides the incorrectly rooted trees were each selected more frequently than the correct tree. For all the cases tested here, tree estimation using a two taxon outgroup was more accurate than when using a single-taxon outgroup. However, the ingroup was most accurately recovered when no outgroup was used.     

Polyhedral Geometry of Phylogenetic Rogue Taxa

It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this “rogue taxon” effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a “rogue taxon” such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we computationally construct polyhedral cones that give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation.     

Bayes estimation of species divergence times and ancestral population sizes using DNA sequences from multiple loci

The effective population sizes of ancestral as well as modern species are important parameters in models of population genetics and human evolution. The commonly used method for estimating ancestral population sizes, based on counting mismatches between the species tree and the inferred gene trees, is highly biased as it ignores uncertainties in gene tree reconstruction. In this article, we develop a Bayes method for simultaneous estimation of the species divergence times and current and ancestral population sizes. The method uses DNA sequence data from multiple loci and extracts information about conflicts among gene tree topologies and coalescent times to estimate ancestral population sizes. The topology of the species tree is assumed known. A Markov chain Monte Carlo algorithm is implemented to integrate over uncertain gene trees and branch lengths (or coalescence times) at each locus as well as species divergence times. The method can handle any species tree and allows different numbers of sequences at different loci. We apply the method to published noncoding DNA sequences from the human and the great apes. There are strong correlations between posterior estimates of speciation times and ancestral population sizes. With the use of an informative prior for the human-chimpanzee divergence date, the population size of the common ancestor of the two species is estimated to be approximately 20,000, with a 95% credibility interval (8000, 40,000). Our estimates, however, are affected by model assumptions as well as data quality. We suggest that reliable estimates have yet to await more data and more realistic models.     

New Weighting Methods for Phylogenetic Tree Reconstruction Using Multiple Loci

Efficient determination of evolutionary distances is important for the correct reconstruction of phylogenetic trees. The performance of the pooled distance required for reconstructing a phylogenetic tree can be improved by applying large weights to appropriate distances for reconstructing phylogenetic trees and small weights to inappropriate distances. We developed two weighting methods, the modified Tajima–Takezaki method and the modified least-squares method, for reconstructing phylogenetic trees from multiple loci. By computer simulations, we found that both of the new methods were more efficient in reconstructing correct topologies than the no-weight method. Hence, we reconstructed hominoid phylogenetic trees from mitochondrial DNA using our new methods, and found that the levels of bootstrap support were significantly increased by the modified Tajima–Takezaki and by the modified least-squares method.