Branch Length Estimates Research Articles

Mitochondrial Genes and Mammalian Phylogenies: Increasing the Reliability of Branch Length Estimation

Since branch lengths provide important information about the timing and the extent of evolutionary divergence among taxa, accurate resolution of evolutionary history depends as much on branch length estimates as on recovery of the correct topology. However, the empirical relationship between the choice of genes to sequence and the quality of branch length estimation remains ill defined. To address this issue, we evaluated the accuracy of branch lengths estimated from subsets of the mitochondrial genome for a mammalian phylogeny with known subordinal relationships. Using maximum-likelihood methods, we estimated branch lengths from an 11-kb sequence of all 13 protein-coding genes and compared them with estimates from single genes (0.2-1.8 kb) and from 7 different combinations of genes (2-3.5 kb). For each sequence, we separated the component of the log-likelihood deviation due to branch length differences associated with alternative topologies from that due to those that are independent of the topology. Even among the sequences that recovered the same tree topology, some produced significantly better branch length estimates than others did. The combination of correct topology and significantly better branch length estimation suggests that these gene combinations may prove useful in estimating phylogenetic relationships for mammalian divergences below the ordinal level. Thus, the proper choice of genes to sequence is a critical factor for reliable estimation of evolutionary history from molecular data.

Complexity of the simplest phylogenetic estimation problem.

The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation of the tree topology is equivalent to partitioning the sample space (space of possible data outcomes) into subspaces, within each of which one of the three binary trees is the ML tree. Distance-based least squares and parsimony-like methods produce essentially the same estimate of the tree topology, although differences exist among methods even under this simple model. This seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogeny estimation. The solution to this real phylogeny estimation problem will be useful for studying the problem of significance evaluation.

War of the Iguanas: conflicting molecular and morphological phylogenies and long-branch attraction in iguanid lizards.

Recent studies based on different types of data (i.e., morphology, molecules) have found strongly conflicting phylogenies for the genera of iguanid lizards but have been unable to explain the basis for this incongruence. We reanalyze published data from morphology and from the mitochondrial ND4, cytochrome b, 12S, and 16S genes to explore the sources of incongruence and resolve these conflicts. Much of the incongruence centers on the genus Cyclura, which is the sister taxon of Iguana, according to parsimony analyses of the morphology and the ribosomal genes, but is the sister taxon of all other Iguanini, according to the protein-coding genes. Maximum likelihood analyses show that there has been an increase in the rate of nucleotide substitution in Cyclura in the two protein-coding genes (ND4 and cytochrome b), although this increase is not as clear when parsimony is used to estimate branch lengths. Parametric simulations suggest that Cyclura may be misplaced by the protein-coding genes as a result of long-branch attraction; even when Cyclura and Iguana are sister taxa in a simulated phylogeny, Cyclura is still placed as the basal member of the Iguanini by parsimony analysis in 55% of the replicates. A similar long-branch attraction problem may also exist in the morphological data with regard to the placement of Sauromalus with the Galápagos iguanas (Amblyrhynchus and Conolophus). The results have many implications for the analysis of diverse data sets, the impact of long branches on parsimony and likelihood methods, and the use of certain protein-coding genes in phylogeny reconstruction.

Branch length heterogeneity leads to nonindependent branch length estimates and can decrease the efficiency of methods of phylogenetic inference.

Branch length estimates play a central role in maximum-likelihood (ML) and minimum-evolution (ME) methods of phylogenetic inference. For various reasons, branch length estimates are not statistically independent under ML or ME. We studied the response of correlations among branch length estimates to the degree of among-branch length heterogeneity (BLH) in the model (true) tree. The frequency and magnitude of (especially negative) correlations among branch length estimates were both shown to increase as BLH increases under simulation and analytically. For ML, we used the correct model (Jukes-Cantor). For ME, we employed ordinary least-squares (OLS) branch lengths estimated under both simple p-distances and Jukes-Cantor distances, analyzed with and without an among-site rate heterogeneity parameter. The efficiency of ME and ML was also shown to decrease in response to increased BLH. We note that the shape of the true tree will in part determine BLH and represents a critical factor in the probability of recovering the correct topology. An important finding suggests that researchers cannot expect that different branches that were in fact the same length will have the same probability of being accurately reconstructed when BLH exists in the overall tree. We conclude that methods designed to minimize the interdependencies of branch length estimates (BLEs) may (1) reduce both the variance and the covariance associated with the estimates and (2) increase the efficiency of model-based optimality criteria. We speculate on possible ways to reduce the nonindependence of BLEs under OLS and ML.

The number of males in primate social groups: a comparative test of the socioecological model

As applied to polygynous mammals, the socioecological model assumes that environmental risks and resources determine the spatial and temporal distribution of females, which then sets male strategies for monopolizing fertile matings. The effects of female spatial distribution (i.e., female number) and temporal overlap (female mating synchrony) have been examined in comparative studies of primates, but the relative influence of these two factors on male monopolization potential (the number of males) remains unclear. One particular problem is that female synchrony is more difficult to estimate than female number. This paper uses multivariate statistical methods and three independent estimates of female synchrony to assess the roles of spatial and temporal effects in the context of a phylogenetically corrected dataset. These analyses are based on sensitivity analyses involving a total of four phylogenies, with two sets of branch length estimates for each tree, and one nonphylogenetic analysis in which species values are used (because male behavior may represent a facultative response to the distribution of females). The results show: (1) that breeding seasonality predicts male number (statistically significant in six out of nine sensitivity tests); (2) that expected female overlap, after controlling for female group size using residuals, also accounts for the number of males in primate groups (significant in eight out of nine tests), and (3) that actual estimates of female mating synchrony predict male number, again after correcting for female group size (significant in five out of nine tests). Nonsignificant results are in the predicted direction, and female group size is significant in all statistical tests. These analyses therefore demonstrate an independent influence of female temporal overlap on male monopolization strategies in mammalian social systems.

Nucleotide substitution models and estimation of phylogeny.

The nucleotide substitution matrix inferred from avian data sets using cytochrome b differs considerably from the models commonly used in phylogenetic analyses. To analyze the possible effects of this particular pattern of change in phylogeny estimation we performed a computer simulation in which we started with a real sequence and used the inferred model of change to produce a tree of 10 species. Maximum parsimony (MP), maximum likelihood (ML), and various distance methods were then used to recover the topology and the branch lengths. We used two kinds of data with varying levels of variation. In addition, we tested with the removal of third positions and different weighting schemes. At low levels of variation, MP was outstanding in recovering the topology (90% correct), while unweighted pair-group method, arithmetic average (UPGMA), regardless of distances used, was poor (40%). At the higher level, most methods had a chance of around 40%-58% of finding the true tree. However, in most cases, the trees found were only slightly wrong, with only one or a few branches misplaced. On the other hand, the use of a "wrong" model had serious effects on the estimation of branch lengths (distances). Although precision was high, accuracy was poor with most methods, giving branch lengths that were biased downward. When seeded with the true distance matrix, Fitch and NJ always found the true tree, while UPGMA frequently failed to do so. The effect of removing third positions was dramatic at low levels of variation, because only one MP program was able to find a true tree at all, albeit rarely, while none of the others ever did so. At higher levels, the situation was better, but still much worse than with the whole data set.

Phylogenetic Analysis of Reptilian Hemoglobins: Trees, Rates, and Divergences

Phylogenetic relationships among reptiles were examined using previously published and newly determined hemoglobin sequences. Trees reconstructed from these sequences using maximum-parsimony, neighbor-joining, and maximum-likelihood algorithms were compared with a phylogenetic tree of Amniota, which was assembled on the basis of published morphological data. All analyses differentiated alpha chains into alphaA and alphaD types, which are present in all reptiles except crocodiles, where only alphaA chains are expressed. The occurrence of the alphaD chain in squamates (lizards and snakes only in this study) appears to be a general characteristic of these species. Lizards and snakes also express two types of beta chains (betaI and betaII), while only one type of beta chain is present in birds and crocodiles. Reconstructed hemoglobin trees for both alpha and beta sequences did not yield the monophyletic Archosauria (i.e., crocodilians + birds) and Lepidosauria (i.e., Sphenodon + squamates) groups defined by the morphology tree. This discrepancy, as well as some other poorly resolved nodes, might be due to substantial heterogeneity in evolutionary rates among single hemoglobin lineages. Estimation of branch lengths based on uncorrected amino acid substitutions and on distances corrected for multiple substitutions (PAM distances) revealed that relative rates for squamate alphaA and alphaD chains and crocodilian beta chains are at least twice as high as those of the rest of the chains considered. In contrast to these rate inequalities between reptilian orders, little variation was found within squamates, which allowed determination of absolute evolutionary rates for this subset of hemoglobins. Rate estimates for hemoglobins of lizards and snakes yielded 1.7 (alphaA) and 3.3 (beta) million years/PAM when calibrated with published divergence time vs. PAM distance correlates for several speciation events within snakes and for the squamate left and right arrow sphenodontid split. This suggests that hemoglobin chains of squamate reptiles evolved approximately 3.5 (alphaA) or approximately 1.7 times (beta) faster than their mammalian equivalents. These data also were used to obtain a first estimate of some intrasquamate divergence times.

Maximum-Likelihood Models for Combined Analyses of Multiple Sequence Data

Abstract. Models of nucleotide substitution were constructed for combined analyses of heterogeneous sequence data (such as those of multiple genes) from the same set of species. The models account for different aspects of the heterogeneity in the evolutionary process of different genes, such as differences in nucleotide frequencies, in substitution rate bias (for example, the transition/transversion rate bias), and in the extent of rate variation across sites. Model parameters were estimated by maximum likelihood and the likelihood ratio test was used to test hypotheses concerning sequence evolution, such as rate constancy among lineages (the assumption of a molecular clock) and proportionality of branch lengths for different genes. The example data from a segment of the mitochondrial genome of six hominoid species (human, common and pygmy chimpanzees, gorilla, orangutan, and siamang) were analyzed. Nucleotides at the three codon positions in the protein-coding regions and from the tRNA-coding regions were considered heterogeneous data sets. Statistical tests showed that the amount of evolution in the sequence data reflected in the estimated branch lengths can be explained by the codon-position effect and lineage effect of substitution rates. The assumption of a molecular clock could not be rejected when the data were analyzed separately or when the rate variation among sites was ignored. However, significant differences in substitution rate among lineages were found when the data sets were combined and when the rate variation among sites was accounted for in the models. Under the assumption that the orangutan and African apes diverged 13 million years ago, the combined analysis of the sequence data estimated the times for the human-chimpanzee separation and for the separation of the gorilla as 4.3 and 6.8 million years ago, respectively.

A Hidden Markov Model approach to variation among sites in rate of evolution.

The method of Hidden Markov Models is used to allow for unequal and unknown evolutionary rates at different sites in molecular sequences. Rates of evolution at different sites are assumed to be drawn from a set of possible rates, with a finite number of possibilities. The overall likelihood of phylogeny is calculated as a sum of terms, each term being the probability of the data given a particular assignment of rates to sites, times the prior probability of that particular combination of rates. The probabilities of different rate combinations are specified by a stationary Markov chain that assigns rate categories to sites. While there will be a very large number of possible ways of assigning rates to sites, a simple recursive algorithm allows the contributions to the likelihood from all possible combinations of rates to be summed, in a time proportional to the number of different rates at a single site. Thus with three rates, the effort involved is no greater than three times that for a single rate. This "Hidden Markov Model" method allows for rates to differ between sites and for correlations between the rates of neighboring sites. By summing over all possibilities it does not require us to know the rates at individual sites. However, it does not allow for correlation of rates at nonadjacent sites, nor does it allow for a continuous distribution of rates over sites. It is shown how to use the Newton-Raphson method to estimate branch lengths of a phylogeny and to infer from a phylogeny what assignment of rates to sites has the largest posterior probability. An example is given using beta-hemoglobin DNA sequences in eight mammal species; the regions of high and low evolutionary rates are inferred and also the average length of patches of similar rates.

Statistical Tests of DNA Phylogenies

In this article, we review (1) statistical tests of DNA phylogenies inferred by the maximum- parsimony method, including tests that take into account the effect of different base compositions among lineages, (2) statistical tests based on the minimum-evolution criterion, i.e., the best tree is the tree with the smallest sum of branch-length estimates, and (3) the bootstrap technique for estimating the confidence level of a phylogenetic hypothesis based on either the maximum-parsimony or the neighbor-joining method. We explain why the bootstrap technique usually gives biased estimates and how to correct the bias. (DNA phylogenies; parsimony; neighbor joining; minimum evolution; statis? tical tests; bootstrap; bias correction.)

Estimating the Rate of Phenotypic Evolution from Comparative Data

This study presents a method to estimate rates of evolutionary change in continuous characters from comparative data. The technique is similar to those introduced previously in which between-species divergence is estimated as a function of time since divergence but also takes into account the possible statistical nonindependence of trait values measured from phylogenetically related species in an approach similar to the independent contrasts methods used in interspecific data analysis. The use of Ornstein-Uhlenbeck processes is also proposed to extend the standard Brownian motion model of evolutionary change and to allow for tests of neutral evolution versus evolution under stabilizing selection. Applications of the method to test specific hypotheses of phenotypic evolution (including the adequacy of Brownian motion to describe real data), to compare rates of change of different types of characters or different groups of organisms, and to estimate branch lengths in units of expected variance of change as required by most comparative-method data analysis techniques are discussed.

Theoretical foundation of the minimum-evolution method of phylogenetic inference.

The minimum-evolution (ME) method of phylogenetic inference is based on the assumption that the tree with the smallest sum of branch length estimates is most likely to be the true one. In the past this assumption has been used without mathematical proof. Here we present the theoretical basis of this method by showing that the expectation of the sum of branch length estimates for the true tree is smallest among all possible trees, provided that the evolutionary distances used are statistically unbiased and that the branch lengths are estimated by the ordinary least-squares method. We also present simple mathematical formulas for computing branch length estimates and their standard errors for any unrooted bifurcating tree, with the least-squares approach. As a numerical example, we have analyzed mtDNA sequence data obtained by Vigilant et al. and have found the ME tree for 95 human and 1 chimpanzee (outgroup) sequences. The tree was somewhat different from the neighbor-joining tree constructed by Tamura and Nei, but there was no statistically significant difference between them.

A Simple Method for Estimating and Testing Minimum-Evolution Trees

A simple method for estimating and testing phylogenetic trees under the principle of minimum evolution (ME) is presented. The basic procedure of this method is first to obtain the neighbor-joining (NJ) tree by Saitou and Nei’s method and then to search for a tree with the minimum value of the sum (S) of branch lengths by examining all trees that are closely related to the NJ tree. Once the ME tree is identified, a statistical test is conducted for the difference in S between this tree and other closely related trees. The mathematical method required for conducting this test is developed by using the least-squares approach. Computer simulation has shown that this method identifies the correct tree with a high probability, as long as the number of nucleotides examined is sufficiently large. It has also been shown that the topology of the NJ tree is almost always identical with that of the ME tree. A method for obtaining least-squares estimates (and their standard errors) of branch lengths for a given topology is also presented. This method can be used for testing the reliability of the branching pattern of the ME tree. However, the statistical test of S values is more powerful in rejecting incorrect trees than is the branch-length test or bootstrapping. Furthermore, both a mathematical method for computing the number of trees with a given value of topological difference from the NJ tree and a computer algorithm for identifying all the topologies are developed.

Procedures for the Analysis of Comparative Data Using Phylogenetically Independent Contrasts

We discuss and clarify several aspects of applying Felsenstein's (1985, Am. Nat. 125: 1n15) procedures to test for correlated evolution of continuous traits. This is one of several available comparative methods that maps data for phenotypic traits onto an existing phylogenetic tree (derived from independent information). Application of Felsenstein's method does not require an entirely dichotomous topology. It also does not require an assumption of gradual, clocklike character evolution, as might be modeled by Brownian motion. Almost any available information can be used to estimate branch lengths (e.g., genetic distances, divergence times estimated from the fossil record or from molecular clocks, numbers of character changes from a cladistic analysis). However, the adequacy for statistical purposes of any proposed branch lengths must be verified empirically for each phytogeny and for each character. We suggest a simple way of doing this, based on graphical analysis of plots of standardized independent contrasts versus their standard deviations (i.e., the square roots of the sums of their branch lengths). In some cases, the branch lengths and/or the values of traits being studied will require transformation. An example involving the scaling of mammalian home range area is presented. Once adequately standardized, sets of independent contrasts can be analyzed using either linear or nonlinear (multiple) regression. In all cases, however, regressions (or correlations) must be computed through the origin. We also discuss ways of correcting for body size effects and how this relates to making graphical representations of relationships of standardized independent contrasts. We close with a consideration of the types of traits that can be analyzed with independent contrasts procedures and conclude that any (continuous) trait that is inherited from ancestors is appropriate for analysis, regardless of the mechanism of inheritance (e.g., genetic or cultural).

Systematic Underestimation of Tree Branch Lengths by Lake's Operator Metrics: An Effect of Position-dependent Substitution Rates

Lake has proposed a method, which he calls “operator metrics,” for estimating branch lengths (as transversion differences) on an unrooted phylogenetic tree of nucleotide sequences. Examination of published applications of operator metrics reveals that it often assigns branch lengths that sum to less than the observed sequence differences, indicating a substantial systematic bias. The problem occurs in all sets of sequence data examined. It is shown analytically that operator metrics does not yield the claimed estimate of transversion sequence differences when sequence positions differ in their nucleotide substitution rate, in which case the method underestimates tree branch lengths. The site-to-site variations in substitution rate that have been characterized by previous authors are of sufficient magnitude to explain the problems observed in the operator-metrics branch length estimates. Transversion substitutions estimated using Kimura’s two-parameter (transition / transversion) model are less subject to this problem and are more consistent with directly observed differences.

Robustness of maximum likelihood tree estimation against different patterns of base substitutions

In the maximum likelihood (ML) method for estimating a molecular phylogenetic tree, the pattern of nucleotide substitutions for computing likelihood values is assumed to be simpler than that of the actual evolutionary process, simply because the process, considered to be quite devious, is unknown. The problem, however, is that there has been no guarantee to endorse the simplification. To study this problem, we first evaluated the robustness of the ML method in the estimation of molecular trees against different nucleotide substitution patterns, including Jukes and Cantor's, the simplest ever proposed. Namely, we conducted computer simulations in which we could set up various evolutionary models of a hypothetical gene, and define a true tree to which an estimated tree by the ML method was to be compared. The results show that topology estimation by the ML method is considerably robust against different ratios of transitions to transversions and different GC contents, but branch length estimation is not so. The ML tree estimation based on Jukes and Cantor's model is also revealed to be resistant to GC content, but rather sensitive to the ratio of transitions to transversions. We then applied the ML method with different substitution patterns to nucleotide sequence data on tax gene from T-cell leukemia viruses whose evolutionary process must have been more complicated than that of the hypothetical gene. The results are in accordance with those from the simulation study, showing that Jukes and Cantor's model is as useful as a more complicated one for making inferences about molecular phylogeny of the viruses.

A Theoretical Study of the Underestimation of Branch Lengths by the Maximum Parsimony Principle

-The degree of underestimation of branch lengths by the maximum parsimony principle is studied. The expected number of nucleotide changes per site under the maximum parsimony principle is computed, and it is compared with the expected number of nucleotide substitutions. A tree topology with no hierarchical structure is considered for mathematical simplicity. It is shown that as long as the evolutionary distance is less than 0.2, the maximum parsimony principle gives good estimates of nucleotide substitutions. When the evolutionary distance is greater than 0.2, however, the method gives gross underestimates of nucleotide substitutions. [Branching; parsimony; phylogenetics; topology.] There are two major problems in constructing a phylogenetic tree from molecular data. One is the determination of the topology of a tree and the other is the estimation of branch lengths. For the first problem, the maximum parsimony method (Camin and Sokal, 1965; Fitch, 1977) has been extensively used for amino acid or nucleotide sequence data. For the estimation of branch lengths, however, this method is expected to underestimate the number of amino acid or nucleotide substitutions. This property comes from the principle of the method itself: minimize the number of changes required. Thus each branch length (estimated by Fitch's [1971] method) is expected to be smaller than the real length on average. In spite of this known shortcoming, the maximum parsimony method seems to be quite appropriate for the estimation of branch length in terms of amino acid or nucleotide substitutions when closely related sequences are compared, since the probability of backward and parallel substitutions is negligible in this situation. But how close should sequences be? The number of sequences compared is also related to this problem, because we expect to extract more and more changes as the number of sequences is increased. So far, there seems to be no theoretical study on these subjects. In this paper, I show the effect of the amount of divergence and the number of nucleotide sequences on the estimates of branch lengths by the maximum parsimony principle. For simplicity, I consider the model of random nucleotide substitution (Jukes and Cantor, 1969). A constant rate of evolution, or the molecular clock, is also assumed. Further, a tree topology with no hierarchical structure is considered. Under these assumptions, the expected number of required nucleotide changes per site estimated by the maximum parsimony principle is computed.

Nonuniformity of nucleotide substitution rates in molecular evolution: computer simulation and analysis of 5S ribosomal RNA sequences.

The effects of temporal (among different branches of a phylogeny) and spatial (among different nucleotide sites within a gene) nonuniformities of nucleotide substitution rates on the construction of phylogenetic trees from nucleotide sequences are addressed. Spatial nonuniformity may be estimated by using Shannon's (1948) entropy formula to measure the Relative Nucleotide Variability (RNV) at each nucleotide site in an aligned set of sequences; this is demonstrated by a comparative analysis of 5S rRNAs. New methods of constructing phylogenetic trees are proposed that augment the Unweighted Pair-Group Using Arithmetic Averages (UPGMA) algorithm by estimating and compensating for both spatial and temporal nonuniformity in substitution rates. These methods are evaluated by computer simulations of 5S rRNA evolution that include both kinds of nonuniformities. It was found that the proposed Reference Ratio Method improved both the ability to reconstruct the correct topology of a tree and also the estimation of branch lengths as compared to UPGMA. A previous method (Farris et al. 1970; Klotz et al. 1979; Li 1981) was found to be less successful in reconstructing topologies when there is high probability of multiple mutations at some sites. Phylogenetic analyses of 5S rRNA sequences support the endosymbiotic origins of both chloroplasts and mitochondria, even though the latter exhibit an accelerated rate of nucleotide substitution. Phylogenetic trees also reveal an adaptive radiation within the eubacteria and another within the eukaryotes for the origins of most major phyla within each group during the Precambrian era.

Simple method for constructing phylogenetic trees from distance matrices.

A simple method is proposed for constructing phylogenetic trees from distance matrices. The procedure for constructing tree topologies is similar to that of the unweighted pair-group method (UPG method) but makes corrections for unequal rates of evolution among lineages. The procedure for estimating branch lengths is the same as that of the Fitch and Margoliash method (F-M method) except that it allows no negative branch lengths. The performance of the present procedure for the construction of tree topologies is compared with that of the UPG method, the F-M method, Farris' method, and the modified Farris method by using Tateno's simulation outputs for nucleotide sequence divergence and his results for the performances of the latter four methods [Tateno, Y. (1978) Dissertation (Univ. Texas, Houston, TX). In this limited comparison, the present method performs considerably better than the UPG method and the F-M method and about equally well as the last two methods. The present method appears to be preferable to the UPG method for analysis of data from populations that have not differentiated much. Indeed, an application of the present method to gene frequency data from some Amerindian populations gives a tree topology far more reasonable than that obtained by the UPG method.

A non-sequential method for constructing trees and hierarchical classifications.

A procedure is presented that forms an unrooted tree-like structure from a matrix of pairwise differences. The tree is not formed a portion at a time, as methods now in use generally do, but is formed en toto without intervening estimates of branch lengths. The method is based on a relaxed additivity (four-point metric) constraint. From the tree, a classification may be formed.