Determining evolutionary distances from highly diverged nucleic acid sequences: Operator metrics

Summary Operator metrics are explicity designed to measure evolutionary distances from nucleic acid sequences when substitution rates differ greatly among the organisms being compared, or when substitutions have been extensive. Unlike lengths calculated by the distance matrix and parsimony methods, in which substitutions in one branch of a tree can alter the measured length of another branch, lengths determined by operator metrics are not affected by substitutions outside the branch.In the method, lengths (operator metrics) corresponding to each of the branches of an unrooted tree are calculated. The metric length of a branch reconstructs the number of (transversion) differences between sequences at a tip and a node (or between nodes) of a tree. The theory is general and is fundamentally independent of differences in substitution rates among the organisms being compared. Mathematically, the independence has been obtained becuase the metrics are eigen vectors of fundamental equations which describe the evolution of all unrooted trees.Even under conditions when both the distance matrix method or a simple parsimony length method are show to indicate lengths than are an order of magnitude too large or too small, the operator metrics are accurate. Examples, using data calculated with evolutionary rates and branchings designed to confuse the measurement of branch lengths and to camouflage the topology of the true tree, demonstrate the validity of operator metrics. The method is robust. Operator metric distances are easy to calculated, can be extended to any number of taxa, and provide a statistical estimate of their variances.The utility of the method is demonstrated by using it to analyze the origins and evolutionary of chloroplasts, mitochondria, and eubacteria.     

Evolutionary change of restriction cleavage sites and phylogenetic inference for man and apes

A mathematical theory for the evolutionary change of restriction endonuclease cleavage sites is developed, and the probabilities of various types of restriction-site changes are evaluated. A computer simulation is also conducted to study properties of the evolutionary change of restriction sites. These studies indicate that parsimony methods of constructing phylogenetic trees often make erroneous inferences about evolutionary changes of restriction sites unless the number of nucleotide substitutions per site is less than 0.01 for all branches of the tree. This introduces a systematic error in estimating the number of mutational changes for each branch and, consequently, in constructing phylogenetic trees. Therefore, parsimony methods should be used only in cases where nucleotide sequences are closely related. Reexamination of Ferris et al.'s data on restriction-site differences of mitochondrial DNAs does not support Templeton's conclusions regarding the phylogenetic tree for man and apes and the molecular clock hypothesis. Templeton's claim that Nei and Li's method of estimating the number of nucleotide substitutions per site is seriously affected by parallel losses and loss-gains of restriction sites is also unsupported.      

On the best evolutionary rate for phylogenetic analysis

The effect of the evolutionary rate of a gene on the accuracy of phylogeny reconstruction was examined by computer stimulation. The evolutionary rate is measured by the tree length, that is, the expected total number of nucleotide substitutions per site on the phylogeny. DNA sequence data were simulated using both fixed trees with specified branch lengths and random trees with branch lengths generated from a model of cladogenesis. The parsimony and likelihood methods were used for phylogeny reconstruction, and the proportion of correctly recovered branch partitions by each method was estimated. Phylogenetic methods including parsimony appear quite tolerant of multiple substitutions at the same site. The optimum levels of sequence divergence were even higher than upper limits previously suggested for saturation of substitutions, indicating that the problem of saturation may have been exaggerated. Instead, the lack of information at low levels of divergence should be seriously considered in evaluation of a gene's phylogenetic utility, especially when the gene sequence is short. The performance of parsimony, relative to that of likelihood, does not necessarily decrease with the increase of the evolutionary rate.     

Limitations of the evolutionary parsimony method of phylogenetic analysis [published erratum appears in Mol Biol Evol 1990 Mar;7(2):201]

Lake's evolutionary parsimony (EP) method of constructing a phylogenetic tree is primarily applied to four DNA sequences. In this method, three quantities--X, Y, and Z--that correspond to three possible unrooted trees are computed, and an invariance property of these quantities is used for choosing the best tree. However, Lake's method depends on a number of unrealistic assumptions. We therefore examined the theoretical basis of his method and reached the following conclusions: (1) When the rates of two transversional changes from a nucleotide are unequal, his invariance property breaks down. (2) Even if the rates of two transversional changes are equal, the invariance property requires some additional conditions. (3) When Kimura's two- parameter model of nucleotide substitution applies and the rate of nucleotide substitution varies greatly with branch, the EP method is generally better than the standard maximum-parsimony (MP) method in recovering the correct tree but is inferior to the neighbor-joining (NJ) and a few other distance matrix methods. (4) When the rate of nucleotide substitution is the same or nearly the same for all branches, the EP method is inferior to the MP method even if the proportion of transitional changes is high. (5) When Lake's assumptions fail, his chi2 test may identify an erroneous tree as the correct tree. This happens because the test is not for comparing different trees. (6) As long as a proper distance measure is used, the NJ method is better than the EP and MP methods whether there is a transition/transversion bias or whether there is variation in substitution rate among different nucleotide sites.      

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.     

Additive Uncorrelated Relaxed Clock Models for the Dating of Genomic Epidemiology Phylogenies

Phylogenetic dating is one of the most powerful and commonly used methods of drawing epidemiological interpretations from pathogen genomic data. Building such trees requires considering a molecular clock model which represents the rate at which substitutions accumulate on genomes. When the molecular clock rate is constant throughout the tree then the clock is said to be strict, but this is often not an acceptable assumption. Alternatively, relaxed clock models consider variations in the clock rate, often based on a distribution of rates for each branch. However, we show here that the distributions of rates across branches in commonly used relaxed clock models are incompatible with the biological expectation that the sum of the numbers of substitutions on two neighboring branches should be distributed as the substitution number on a single branch of equivalent length. We call this expectation the additivity property. We further show how assumptions of commonly used relaxed clock models can lead to estimates of evolutionary rates and dates with low precision and biased confidence intervals. We therefore propose a new additive relaxed clock model where the additivity property is satisfied. We illustrate the use of our new additive relaxed clock model on a range of simulated and real data sets, and we show that using this new model leads to more accurate estimates of mean evolutionary rates and ancestral dates.     

Rate and mode differences between nuclear and mitochondrial small-subunit rRNA genes in mushrooms.

Sequences from homologous regions of the nuclear and mitochondrial small-subunit rRNA genes from 10 members of the mushroom order Boletales were used to construct evolutionary trees and to compare the rates and modes of evolution. Trees constructed independently for each gene by parsimony and tested by bootstrap analysis have identical topologies in all statistically significant branches. Examination of base substitutions revealed that the nuclear gene is biased toward C-T transitions and that the distribution of transversions in the mitochondrial gene is strongly effected by an A-T bias. When only homologous regions of the two genes were compared, base substitutions per nucleotide were roughly 16-fold greater in the mitochondrial gene. The difference in the frequency of length mutations was at least as great but was impossible to estimate accurately because of their absence in the nuclear gene. Maximum likelihood was used to show that base-substitution rates vary dramatically among the branches. A significant part of the rate inconstancy was caused by an accelerated nuclear rate in one branch and a retarded mitochondrial rate in a different branch. A second part of the rate variability involved a consistent inconstancy: short branches exhibit ratios of mitochondrial to nuclear divergences of less than 1, while longer branches had ratios of approximately 4:1-8:1. This pattern suggests a systematic error in the branch length calculation. The error may be related to the simplicity of the divergence estimates, which assumes that all base positions have an equal probability of change.     

The consistency of several phylogeny-inference methods under varying evolutionary rates.

A phylogenetic method is a consistent estimator of phylogeny if and only if it is guaranteed to give the correct tree, given that sufficient (possibly infinite) independent data are examined. The following methods are examined for consistency: UPGMA (unweighted pair-group method, averages), NJ (neighbor joining), MF (modified Farris), and P (parsimony). A two-parameter model of nucleotide sequence substitution is used, and the expected distribution of character states is calculated. Without perfect correction for superimposed substitutions, all four methods may be inconsistent if there is but one branch evolving at a faster rate than the other branches. Partial correction of observed distances improves the robustness of the NJ method to rate variation, and perfect correction makes the NJ method a consistent estimator for all combinations of rates that were examined. The sensitivity of all the methods to unequal rates varies over a wide range, so relative-rate tests are unlikely to be a reliable guide for accepting or rejecting phylogenies based on parsimony analysis.     

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.      

Constructing and counting phylogenetic invariants.

The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of m taxa using nucleotide sequence data. Models for the respective probabilities of the 4m possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution occurs along the branches of a putative phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. If the invariant is typically non-zero for another phylogenetic tree, then estimates of the invariant can be used as evidence to support one phylogeny over another. Previous work of Evans and Speed showed that, for certain commonly used substitution models, the problem of finding a minimal generating set for the ideal of invariants can be reduced to the linear algebra problem of finding a basis for a certain lattice (that is, a free Z-module). They also conjectured that the cardinality of such a generating set can be computed using a simple "degrees of freedom" formula. We verify this conjecture. Along the way, we explain in detail how the observations of Evans and Speed lead to a simple, computationally feasible algorithm for constructing a minimal generating set.     

Construction of linear invariants in phylogenetic inference.

An analytical method is presented for constructing linear invariants. All linear invariants of a k-species tree can be derived from those of (k-1)-species trees using this method. The new method is simpler than that of Cavender, which relies on numerical computations. Moreover, the new method provides a convenient tool to study the relationships between linear invariants of the same tree or of different trees. All linear invariants of trees of up to five species are derived in this study. For four species, there are 16 independent linear invariants for each of the three possible unrooted trees, 14 of which are shared by two unrooted trees and 12 of these are shared by all three unrooted trees; the last types of linear invariants can be used to construct tests on the assumptions about nucleotide substitutions. The number of linear invariants for a tree is found to increase rapidly with the number of species.     

Relative efficiencies of the maximum parsimony and distance-matrix methods in obtaining the correct phylogenetic tree

The relative efficiencies of the maximum parsimony (MP) and distance-matrix methods in obtaining the correct tree (topology) were studied by using computer simulation. The distance-matrix methods examined are the neighbor-joining, distance-Wagner, Tateno et al. modified Farris, Faith, and Li methods. In the computer simulation, six or eight DNA sequences were assumed to evolve following a given model tree, and the evolutionary changes of the sequences were followed. Both constant and varying rates of nucleotide substitution were considered. From the sequences thus obtained, phylogenetic trees were constructed using the six tree-making methods and compared with the model (true) tree. This process was repeated 300 times for each different set of parameters. The results obtained indicate that when the number of nucleotide substitutions per site is small and a relatively small number of nucleotides are used, the probability of obtaining the correct topology (P1) is generally lower in the MP method than in the distance-matrix methods. The P1 value for the MP method increases with increasing number of nucleotides but is still generally lower than the value for the NJ or DW method. Essentially the same conclusion was obtained whether or not the rate of nucleotide substitution was constant or whether or not a transition bias in nucleotide substitution existed. The relatively poor performance of the MP method for these cases is due to the fact that information from singular sites is not used in this method. The MP method also showed a relatively low P1 value when the model of varying rate of nucleotide substitution was used and the number of substitutions per site was large. However, the MP method often produced cases in which the correct tree was one of several equally parsimonious trees. When these cases were included in the class of "success," the MP method performed better than the other methods, provided that the number of nucleotide substitutions per site was small.     

Markov invariants, plethysms, and phylogenetics

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log–Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.     

When is it safe to use an oversimplified substitution model in tree- making?

The choice of an "optimal" mathematical model for computing evolutionary distances from real sequences is not currently supported by easy-to-use software applicable to large data sets, and an investigator frequently selects one of the simplest models available. Here we study properties of the observed proportion of differences (p- distance) between sequences as an estimator of evolutionary distance for tree-making. We show that p-distances allow for consistent tree- making with any of the popular methods working with evolutionary distances if evolution of sequences obeys a "molecular clock" (more precisely, if it follows a stationary time-reversible Markov model of nucleotide substitution). Next, we show that p-distances seem to be efficient in recovering the correct tree topology under a "molecular clock," but produce "statistically supported" wrong trees when substitutions rates vary among evolutionary lineages. Finally, we outline a practical approach for selecting an "optimal" model of nucleotide substitution in a real data analysis, and obtain a crude estimate of a "prior" distribution of the expected tree branch lengths under the Jukes-Cantor model. We conclude that the use of a model that is obviously oversimplified is inadvisable unless it is justified by a preliminary analysis of the real sequences.      

Phylogenetic signal in nucleotide data from seed plants: implications for resolving the seed plant tree of life

Effects of taxonomic sampling and conflicting signal on the inference of seed plant trees supported in previous molecular analyses were explored using 13 single-locus data sets. Changing the number of taxa in single-locus analyses had limited effects on log likelihood differences between the gnepine (Gnetales plus Pinaceae) and gnetifer (Gnetales plus conifers) trees. Distinguishing among these trees also was little affected by the use of different substitution parameters. The 13-locus combined data set was partitioned into nine classes based on substitution rates. Sites evolving at intermediate rates had the best likelihood and parsimony scores on gnepine trees, and those evolving at the fastest rates had the best parsimony scores on Gnetales-sister trees (Gnetales plus other seed plants). When the fastest evolving sites were excluded from parsimony analyses, well-supported gnepine trees were inferred from the combined data and from each genomic partition. When all sites were included, Gnetales-sister trees were inferred from the combined data, whereas a different tree was inferred from each genomic partition. Maximum likelihood trees from the combined data and from each genomic partition were well-supported gnepine trees. A preliminary stratigraphic test highlights the poor fit of Gnetales-sister trees to the fossil data.     

Discordance of species trees with their most likely gene trees: the case of five taxa

Under a coalescent model for within-species evolution, gene trees may differ from species trees to such an extent that the gene tree topology most likely to evolve along the branches of a species tree can disagree with the species tree topology. Gene tree topologies that are more likely to be produced than the topology that matches that of the species tree are termed anomalous, and the region of branch-length space that gives rise to anomalous gene trees (AGTs) is the anomaly zone. We examine the occurrence of anomalous gene trees for the case of five taxa, the smallest number of taxa for which every species tree topology has a nonempty anomaly zone. Considering all sets of branch lengths that give rise to anomalous gene trees, the largest value possible for the smallest branch length in the species tree is greater in the five-taxon case (0.1934 coalescent time units) than in the previously studied case of four taxa (0.1568). The five-taxon case demonstrates the existence of three phenomena that do not occur in the four-taxon case. First, anomalous gene trees can have the same unlabeled topology as the species tree. Second, the anomaly zone does not necessarily enclose a ball centered at the origin in branch-length space, in which all branches are short. Third, as a branch length increases, it is possible for the number of AGTs to increase rather than decrease or remain constant. These results, which help to describe how the properties of anomalous gene trees increase in complexity as the number of taxa increases, will be useful in formulating strategies for evading the problem of anomalous gene trees during species tree inference from multilocus data.     

Evolution of the Diatoms (Bacillariophyta): IV. A Reconstruction of Their Age from Small Subunit rRNA Coding Regions and the Fossil Record

Small subunit ribosomal RNA (ssu rRNA) coding regions from 30 diatoms, 3 oomycetes, and 6 pelagophytes were used to construct linearized trees, maximum-likelihood trees, and neighbor-joining trees inferred from both unweighted and weighted distances. Stochastic accumulation of sequence substitutions among the diatoms was assessed with relative rate tests. Pennate diatoms evolved relatively slowly but within the limits set by a stochastic model; centric diatoms exceeded those limits. A rate distribution test was devised to identify those taxa showing an aberrant distribution of base substitutions within the ssu rRNA coding region. First appearance dates of diatom taxa from the fossil record were regressed against their corresponding branch lengths to infer the average and earliest possible age for the origin of the diatoms, the pennate diatoms, and the centric diatom order Thalassiosirales. Our most lenient age estimate (based on the median-evolving diatom taxon in the maximum-likelihood tree or on the average branch length in a linearized tree) suggests that their average age is approximately 164–166 Ma, which is close to their earliest fossil record. Both calculations suggest that it is unlikely that diatoms existed prior to 238–266 Ma. Rate variation among the diatoms' ssu rRNA coding regions and uncertainties associated with the origin of extant taxa in the fossil record contribute significantly to the variation in age estimates obtained. Different evolutionary models and the exclusion of fast or slow evolving taxa did not significantly affect age estimates; however, the inclusion of aberrantly fast evolving taxa did. Our molecular clock calibrations indicate that the rRNA coding regions in the diatoms are evolving at approximately 1% per 18 to 26 Ma, which is the fastest substitution rate reported in any pro- or eukaryotic group of organisms to date.     

Log Transformation Improves Dating of Phylogenies

Phylogenetic trees inferred from sequence data often have branch lengths measured in the expected number of substitutions and therefore, do not have divergence times estimated. These trees give an incomplete view of evolutionary histories since many applications of phylogenies require time trees. Many methods have been developed to convert the inferred branch lengths from substitution unit to time unit using calibration points, but none is universally accepted as they are challenged in both scalability and accuracy under complex models. Here, we introduce a new method that formulates dating as a nonconvex optimization problem where the variance of log-transformed rate multipliers is minimized across the tree. On simulated and real data, we show that our method, wLogDate, is often more accurate than alternatives and is more robust to various model assumptions.     

The information content of a character under a Markov model of evolution

The rate of evolutionary change associated with a character determines its utility for the reconstruction of phylogenetic history. For a given age of lineage splits, we examine the information content of a character to assess the magnitude and range of an optimal rate of substitution. On the one hand an optimal transition rate must provide sufficiently many character changes to distinguish subclades, whereas on the other hand changes must be sufficiently rare that reversals on a single branch (and hence homoplasy) are uncommon. In this study, we evolve binary characters over three tree topologies with fixed branch lengths, while varying transition rate as a parameter. We use the character state distribution obtained to measure the "information content" of a character given a transition rate. This is done with respect to several criteria-the probability of obtaining the correct tree using parsimony, the probability of infering the correct ancestral state, and Shannon-Weaver and Fisher information measures on the configuration of probability distributions. All of the information measures suggest the intuitive result of the existence of optimal rates for phylogeny reconstruction. This nonzero optimum is less pronounced if one conditions on there having been a change, in which case the parsimony-based results of minimum change being the most informative tends to hold.     

Rapid evolutionary divergences in reef fishes of the family Acanthuridae (Perciformes: Teleostei)

A phylogenetic analysis of the sugeonfish family Acanthuridae was conducted to investigate: (a) the pattern of divergences among outgroup and basal ingroup taxa, (b) the pattern of species divergences within acanthurid genera, (c) monophyly in the genus Acanthurus, and (d) the evolution of thick-walled stomach morphology in the genera Acanthurus and Ctenochaetus. Fragments of the 12S, 16S, t-Pro, and control region mitochondrial genes were sequenced for 21 acanthurid taxa (representing all extant genera) and four outgroup taxa. Unweighted parsimony analysis produced two optimal trees. Both of these were highly incongruent with a previous morphological phylogeny, especially with regard to the placement of the monotypic outgroups Zanclus and Luvarus. The maximum likelihood tree and the morphological phylogeny were not significantly different and the conflicting branches were very short. Split decomposition analysis identified conflict in the placement of long basal branches separated by short internodes, providing further evidence that long branch attraction is an important cause of disagreement between molecular and morphological trees. Parametric bootstrapping rejected hypotheses of monophyly of: (a) the genus Acanthurus and (b) a group containing representatives of Acanthurus/Ctenochaetus with thick-walled stomachs. The branching pattern of the likelihood and split decomposition trees indicates that evolution in the acanthurid clade has involved at least three periods of intense speciation.