Accurate Phylogenetic Reconstruction Research Articles

Abstract PL05-03: Phylogenetic quantification of intra-tumor heterogeneity predicts time to relapse in high-grade serous ovarian cancer.

Abstract Intra-tumor genetic heterogeneity is the result of ongoing evolutionary change within each cancer. The expansion of genetically distinct sub-clonal populations may explain the emergence of drug resistance and if so would have prognostic and predictive utility. However, methods for objectively quantifying tumor heterogeneity have been missing and are particularly difficult to establish in cancers where predominant copy number variation prevents accurate phylogenetic reconstruction because of horizontal dependencies caused by long and cascading genomic rearrangements. To address these challenges we have developed MEDICC, a method for phylogenetic reconstruction and heterogeneity quantification, which determines optimal phasing of major and minor alleles, computes evolutionary distances between samples, and reconstructs ancestral genomes. Rigorous simulations and an extensive clinical study show the power of our method, which outperforms state-of-the-art competitors in reconstruction accuracy and additionally allows unbiased numerical quantification of tumor heterogeneity. Using MEDICC we investigated the relationship between tumor heterogeneity and patient outcome in high-grade serous ovarian cancer. We found that tumor heterogeneity in this cancer is driven by ongoing clonal evolution with fully branched evolutionary trajectories that do not have clock-like evolutionary rates. Quantitative measures of clonal expansion and temporal heterogeneity were the strongest predictors of progression-free survival. We show in 2 patients that clonal expansion of a minor subclone that was present prior to chemotherapy led to clinical relapse. Citation Information: Mol Cancer Ther 2013;12(11 Suppl):PL05-03. Citation Format: Florian M. Markowetz. Phylogenetic quantification of intra-tumor heterogeneity predicts time to relapse in high-grade serous ovarian cancer. [abstract]. In: Proceedings of the AACR-NCI-EORTC International Conference: Molecular Targets and Cancer Therapeutics; 2013 Oct 19-23; Boston, MA. Philadelphia (PA): AACR; Mol Cancer Ther 2013;12(11 Suppl):Abstract nr PL05-03.

PhyloBayes MPI: Phylogenetic Reconstruction with Infinite Mixtures of Profiles in a Parallel Environment

Modeling across site variation of the substitution process is increasingly recognized as important for obtaining more accurate phylogenetic reconstructions. Both finite and infinite mixture models have been proposed and have been shown to significantly improve on classical single-matrix models. Compared with their finite counterparts, infinite mixtures have a greater expressivity. However, they are computationally more challenging. This has resulted in practical compromises in the design of infinite mixture models. In particular, a fast but simplified version of a Dirichlet process model over equilibrium frequency profiles implemented in PhyloBayes has often been used in recent phylogenomics studies, while more refined model structures, more realistic and empirically more fit, have been practically out of reach. We introduce a message passing interface version of PhyloBayes, implementing the Dirichlet process mixture models as well as more classical empirical matrices and finite mixtures. The parallelization is made efficient thanks to the combination of two algorithmic strategies: a partial Gibbs sampling update of the tree topology and the use of a truncated stick-breaking representation for the Dirichlet process prior. The implementation shows close to linear gains in computational speed for up to 64 cores, thus allowing faster phylogenetic reconstruction under complex mixture models. PhyloBayes MPI is freely available from our website www.phylobayes.org.

Assessing statistical reliability of phylogenetic trees via a speedy double bootstrap method

Evaluating the reliability of estimated phylogenetic trees is of critical importance in the field of molecular phylogenetics, and for other endeavors that depend on accurate phylogenetic reconstruction. The bootstrap method is a well-known computational approach to phylogenetic tree assessment, and more generally for assessing the reliability of statistical models. However, it is known to be biased under certain circumstances, calling into question the accuracy of the method. Several advanced bootstrap methods have been developed to achieve higher accuracy, one of which is the double bootstrap approach, but the computational burden of this method has precluded its application to practical problems of phylogenetic tree selection. We address this issue by proposing a simple method called the speedy double bootstrap, which circumvents the second-tier resampling step in the regular double bootstrap approach. We also develop an implementation of the regular double bootstrap for comparison with our speedy method. The speedy double bootstrap suffers no significant loss of accuracy compared with the regular double bootstrap, while performing calculations significantly more rapidly (at minimum around 371 times faster, based on analysis of mammalian mitochondrial amino acid sequences and 12S and 16S rRNA genes). Our method thus enables, for the first time, the practical application of the double bootstrap technique in the context of molecular phylogenetics. The approach can also be used more generally for model selection problems wherever the maximum likelihood criterion is used.

Staphylococcus aureus CC398: Host Adaptation and Emergence of Methicillin Resistance in Livestock

ABSTRACTSince its discovery in the early 2000s, methicillin-resistant Staphylococcus aureus (MRSA) clonal complex 398 (CC398) has become a rapidly emerging cause of human infections, most often associated with livestock exposure. We applied whole-genome sequence typing to characterize a diverse collection of CC398 isolates (n = 89), including MRSA and methicillin-susceptible S. aureus (MSSA) from animals and humans spanning 19 countries and four continents. We identified 4,238 single nucleotide polymorphisms (SNPs) among the 89 core genomes. Minimal homoplasy (consistency index = 0.9591) was detected among parsimony-informative SNPs, allowing for the generation of a highly accurate phylogenetic reconstruction of the CC398 clonal lineage. Phylogenetic analyses revealed that MSSA from humans formed the most ancestral clades. The most derived lineages were composed predominantly of livestock-associated MRSA possessing three different staphylococcal cassette chromosome mec element (SCCmec) types (IV, V, and VII-like) including nine subtypes. The human-associated isolates from the basal clades carried phages encoding human innate immune modulators that were largely missing among the livestock-associated isolates. Our results strongly suggest that livestock-associated MRSA CC398 originated in humans as MSSA. The lineage appears to have undergone a rapid radiation in conjunction with the jump from humans to livestock, where it subsequently acquired tetracycline and methicillin resistance. Further analyses are required to estimate the number of independent genetic events leading to the methicillin-resistant sublineages, but the diversity of SCCmec subtypes is suggestive of strong and diverse antimicrobial selection associated with food animal production.

Fast and Accurate Phylogenetic Reconstruction from High-Resolution Whole-Genome Data and a Novel Robustness Estimator

The rapid accumulation of whole-genome data has renewed interest in the study of genomic rearrangements. Comparative genomics, evolutionary biology, and cancer research all require models and algorithms to elucidate the mechanisms, history, and consequences of these rearrangements. However, even simple models lead to NP-hard problems, particularly in the area of phylogenetic analysis. Current approaches are limited to small collections of genomes and low-resolution data (typically a few hundred syntenic blocks). Moreover, whereas phylogenetic analyses from sequence data are deemed incomplete unless bootstrapping scores (a measure of confidence) are given for each tree edge, no equivalent to bootstrapping exists for rearrangement-based phylogenetic analysis. We describe a fast and accurate algorithm for rearrangement analysis that scales up, in both time and accuracy, to modern high-resolution genomic data. We also describe a novel approach to estimate the robustness of results-an equivalent to the bootstrapping analysis used in sequence-based phylogenetic reconstruction. We present the results of extensive testing on both simulated and real data showing that our algorithm returns very accurate results, while scaling linearly with the size of the genomes and cubically with their number. We also present extensive experimental results showing that our approach to robustness testing provides excellent estimates of confidence, which, moreover, can be tuned to trade off thresholds between false positives and false negatives. Together, these two novel approaches enable us to attack heretofore intractable problems, such as phylogenetic inference for high-resolution vertebrate genomes, as we demonstrate on a set of six vertebrate genomes with 8,380 syntenic blocks. A copy of the software is available on demand.

Quartet MaxCut: A fast algorithm for amalgamating quartet trees

Accurate phylogenetic reconstruction methods are inherently computationally heavy and therefore are limited to relatively small numbers of taxa. Supertree construction is the task of amalgamating small trees over partial sets into a big tree over the complete taxa set. The need for fast and accurate supertree methods has become crucial due to the enormous number of new genomic sequences generated by modern technology and the desire to use them for classification purposes. In particular, the Assembling the Tree of Life (ATOL) program aims at constructing the evolutionary history of all living organisms on Earth. When dealing with unrooted trees, a quartet – an unrooted tree over four taxa – is the most basic piece of phylogenetic information. Therefore, quartet amalgamation stands at the heart of any supertree problem as it concerns combining many minimal pieces of information into a single, coherent, and more comprehensive piece of information.We have devised an extremely fast algorithm for quartet amalgamation and implemented it in a very efficient code. The new code can handle over a hundred millions of quartet trees over several hundreds of taxa with very high accuracy.

LifePrint: a novel k-tuple distance method for construction of phylogenetic trees

PurposeHere we describe LifePrint, a sequence alignment-independent k-tuple distance method to estimate relatedness between complete genomes.MethodsWe designed a representative sample of all possible DNA tuples of length 9 (9-tuples). The final sample comprises 1878 tuples (called the LifePrint set of 9-tuples; LPS9) that are distinct from each other by at least two internal and noncontiguous nucleotide differences. For validation of our k-tuple distance method, we analyzed several real and simulated viroid genomes. Using different distance metrics, we scrutinized diverse viroid genomes to estimate the k-tuple distances between these genomic sequences. Then we used the estimated genomic k-tuple distances to construct phylogenetic trees using the neighbor-joining algorithm. A comparison of the accuracy of LPS9 and the previously reported 5-tuple method was made using symmetric differences between the trees estimated from each method and a simulated “true” phylogenetic tree.ResultsThe identified optimal search scheme for LPS9 allows only up to two nucleotide differences between each 9-tuple and the scrutinized genome. Similarity search results of simulated viroid genomes indicate that, in most cases, LPS9 is able to detect single-base substitutions between genomes efficiently. Analysis of simulated genomic variants with a high proportion of base substitutions indicates that LPS9 is able to discern relationships between genomic variants with up to 40% of nucleotide substitution.ConclusionOur LPS9 method generates more accurate phylogenetic reconstructions than the previously proposed 5-tuples strategy. LPS9-reconstructed trees show higher bootstrap proportion values than distance trees derived from the 5-tuple method.

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.

Modelling the morphology and evolution of the linea sinuosa (crown–root junction) in arvicolid rodents; a test with Pliocene Ogmodontomys from Kansas, USA

The linea sinuosa of 39 specimens of Pliocene Ogmodontomys sawrockensis and Ogmodontomys poaphagus from the Meade Basin of southwestern Kansas, USA, was deconstructed into eight enamel segments, representing the major peaks and troughs that characterized Ogmodontomys. Coordinates for the positions of the major peaks and troughs were determined from standard Cartesian coordinate drawings, and then scaled for uniformity. Descriptive statistics for the amplitude (vertical) and drift (lateral) variation were calculated for each coordinate. A modified cubic spline analysis showed that accurate representations of the linea sinuosa, both of individuals and species, could be generated given these coordinate data. Comparisons of individual coordinates by analyses of variance distinguished O. sawrockensis from O. poaphagus, and the same coordinates were statistically distinguishable when the samples of O. sawrockensis were combined and tested against the combined samples of O. poaphagus by Student's t-tests. A mosaic of evolutionary trends, predominantly in peak morphology, was documented for Ogmodontomys from early through middle Pliocene time. Principal components analysis of coordinate data distinguished the linea sinuosa pattern of Ogmodontomys from a small sample of Pliomys lenki. This combined statistical approach offers a new and relatively simple quantitative method to model the form and evolution of the arvicolid linea sinuosa, thereby leading to more accurate phylogenetic and biostratigraphic reconstructions.

Modeling body size evolution in Felidae under alternative phylogenetic hypotheses

The use of phylogenetic comparative methods in ecological research has advanced during the last twenty years, mainly due to accurate phylogenetic reconstructions based on molecular data and computational and statistical advances. We used phylogenetic correlograms and phylogenetic eigenvector regression (PVR) to model body size evolution in 35 worldwide Felidae (Mammalia, Carnivora) species using two alternative phylogenies and published body size data. The purpose was not to contrast the phylogenetic hypotheses but to evaluate how analyses of body size evolution patterns can be affected by the phylogeny used for comparative analyses (CA). Both phylogenies produced a strong phylogenetic pattern, with closely related species having similar body sizes and the similarity decreasing with increasing distances in time. The PVR explained 65% to 67% of body size variation and all Moran's I values for the PVR residuals were non-significant, indicating that both these models explained phylogenetic structures in trait variation. Even though our results did not suggest that any phylogeny can be used for CA with the same power, or that “good” phylogenies are unnecessary for the correct interpretation of the evolutionary dynamics of ecological, biogeographical, physiological or behavioral patterns, it does suggest that developments in CA can, and indeed should, proceed without waiting for perfect and fully resolved phylogenies.

Gene content differences across strains of Streptococcus uberis identified using oligonucleotide microarray comparative genomic hybridization

Streptococcus uberis is one of the principal causative agents of bovine mastitis. The organism is typically considered an environmental pathogen. In this study, two multilocus sequence typing (MLST) schemes and whole genome DNA microarrays were used to evaluate the degree and nature of genome flexibility between S. uberis strains. The 21 isolates examined in this study arise from a collection of 232 international isolates for which previous epidemiological and preliminary genotyping data existed. The microarray analysis resulted in an estimate of the core genome for S. uberis, consisting of 1530 ORFs, among 1855 tested, representing 82.5% of the S. uberis 0140J genome. The remaining ORFs were variable in gene content across the 21 tested strains. A total of 26 regions of difference (RDs), consisting of three or more contiguous ORFs, were identified among the variable genes. Core genes mainly encoded housekeeping functions, while the variable genes primarily fell within categories such as protection responses, degradation of small molecules, laterally acquired elements, and two component systems. Recombination detection procedures involving the MLST loci suggested S. uberis is a highly recombinant species, precluding accurate phylogenetic reconstructions involving these data. On the other hand, the microarray data did provide limited support for an association of gene content with strains found in multiple cows and/or multiple herds, suggesting the possibility of genes related to bovine transmissibility or host-adaptation.

Estimating true evolutionary distances under the DCJ model

Motivation: Modern techniques can yield the ordering and strandedness of genes on each chromosome of a genome; such data already exists for hundreds of organisms. The evolutionary mechanisms through which the set of the genes of an organism is altered and reordered are of great interest to systematists, evolutionary biologists, comparative genomicists and biomedical researchers. Perhaps the most basic concept in this area is that of evolutionary distance between two genomes: under a given model of genomic evolution, how many events most likely took place to account for the difference between the two genomes?Results: We present a method to estimate the true evolutionary distance between two genomes under the ‘double-cut-and-join’ (DCJ) model of genome rearrangement, a model under which a single multichromosomal operation accounts for all genomic rearrangement events: inversion, transposition, translocation, block interchange and chromosomal fusion and fission. Our method relies on a simple structural characterization of a genome pair and is both analytically and computationally tractable. We provide analytical results to describe the asymptotic behavior of genomes under the DCJ model, as well as experimental results on a wide variety of genome structures to exemplify the very high accuracy (and low variance) of our estimator. Our results provide a tool for accurate phylogenetic reconstruction from multichromosomal gene rearrangement data as well as a theoretical basis for refinements of the DCJ model to account for biological constraints.Availability: All of our software is available in source form under GPL at http://lcbb.epfl.chContact: bernard.moret@epfl.ch

Accurate gene-tree reconstruction by learning gene- and species-specific substitution rates across multiple complete genomes.

Comparative genomics provides a general methodology for discovering functional DNA elements and understanding their evolution. The availability of many related genomes enables more powerful analyses, but requires rigorous phylogenetic methods to resolve orthologous genes and regions. Here, we use 12 recently sequenced Drosophila genomes and nine fungal genomes to address the problem of accurate gene-tree reconstruction across many complete genomes. We show that existing phylogenetic methods that treat each gene tree in isolation show large-scale inaccuracies, largely due to insufficient phylogenetic information in individual genes. However, we find that gene trees exhibit common properties that can be exploited for evolutionary studies and accurate phylogenetic reconstruction. Evolutionary rates can be decoupled into gene-specific and species-specific components, which can be learned across complete genomes. We develop a phylogenetic reconstruction methodology that exploits these properties and achieves significantly higher accuracy, addressing the species-level heterotachy and enabling studies of gene evolution in the context of species evolution.

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.

Exploring the Relationship between Sequence Similarity and Accurate Phylogenetic Trees

We have characterized the relationship between accurate phylogenetic reconstruction and sequence similarity, testing whether high levels of sequence similarity can consistently produce accurate evolutionary trees. We generated protein families with known phylogenies using a modified version of the PAML/EVOLVER program that produces insertions and deletions as well as substitutions. Protein families were evolved over a range of 100-400 point accepted mutations; at these distances 63% of the families shared significant sequence similarity. Protein families were evolved using balanced and unbalanced trees, with ancient or recent radiations. In families sharing statistically significant similarity, about 60% of multiple sequence alignments were 95% identical to true alignments. To compare recovered topologies with true topologies, we used a score that reflects the fraction of clades that were correctly clustered. As expected, the accuracy of the phylogenies was greatest in the least divergent families. About 88% of phylogenies clustered over 80% of clades in families that shared significant sequence similarity, using Bayesian, parsimony, distance, and maximum likelihood methods. However, for protein families with short ancient branches (ancient radiation), only 30% of the most divergent (but statistically significant) families produced accurate phylogenies, and only about 70% of the second most highly conserved families, with median expectation values better than 10(-60), produced accurate trees. These values represent upper bounds on expected tree accuracy for sequences with a simple divergence history; proteins from 700 Giardia families, with a similar range of sequence similarities but considerably more gaps, produced much less accurate trees. For our simulated insertions and deletions, correct multiple sequence alignments did not perform much better than those produced by T-COFFEE, and including sequences with expressed sequence tag-like sequencing errors did not significantly decrease phylogenetic accuracy. In general, although less-divergent sequence families produce more accurate trees, the likelihood of estimating an accurate tree is most dependent on whether radiation in the family was ancient or recent. Accuracy can be improved by combining genes from the same organism when creating species trees or by selecting protein families with the best bootstrap values in comprehensive studies.

Measuring genome conservation across taxa: divided strains and united kingdoms

Species evolutionary relationships have traditionally been defined by sequence similarities of phylogenetic marker molecules, recently followed by whole-genome phylogenies based on gene order, average ortholog similarity or gene content. Here, we introduce genome conservation—a novel metric of evolutionary distances between species that simultaneously takes into account, both gene content and sequence similarity at the whole-genome level. Genome conservation represents a robust distance measure, as demonstrated by accurate phylogenetic reconstructions. The genome conservation matrix for all presently sequenced organisms exhibits a remarkable ability to define evolutionary relationships across all taxonomic ranges. An assessment of taxonomic ranks with genome conservation shows that certain ranks are inadequately described and raises the possibility for a more precise and quantitative taxonomy in the future. All phylogenetic reconstructions are available at the genome phylogeny server: <>.

Ribosomal ITS sequences and plant phylogenetic inference

One of the most popular sequences for phylogenetic inference at the generic and infrageneric levels in plants is the internal transcribed spacer (ITS) region of the 18S–5.8S–26S nuclear ribosomal cistron. The prominence of this source of nuclear DNA sequence data is underscored by a survey of phylogenetic publications involving comparisons at the genus level or below, which reveals that of 244 papers published over the last five years, 66% included ITS sequence data. Perhaps even more striking is the fact that 34% of all published phylogenetic hypothesis have been based exclusively on ITS sequences. Notwithstanding the many important contributions of ITS sequence data to phylogenetic understanding and knowledge of genome relationships, a number of molecular genetic processes impact ITS sequences in ways that may mislead phylogenetic inference. These molecular genetic processes are reviewed here, drawing attention to both underlying mechanism and phylogenetic implications. Among the most prevalent complications for phylogenetic inference is the existence in many plant genomes of extensive sequence variation, arising from ancient or recent array duplication events, genomic harboring of pseudogenes in various states of decay, and/or incomplete intra- or inter-array homogenization. These phenomena separately and collectively create a network of paralogous sequence relationships potentially confounding accurate phylogenetic reconstruction. Homoplasy is shown to be higher in ITS than in other DNA sequence data sets, most likely because of orthology/paralogy conflation, compensatory base changes, problems in alignment due to indel accumulation, sequencing errors, or some combination of these phenomena. Despite the near-universal usage of ITS sequence data in plant phylogenetic studies, its complex and unpredictable evolutionary behavior reduce its utility for phylogenetic analysis. It is suggested that more robust insights are likely to emerge from the use of single-copy or low-copy nuclear genes.

Molecular Phylogeny of Arthropods and the Significance of the Cambrian “Explosion” for Molecular Systematics

SYNOPSIS. Accurate phylogenetic reconstruction requires character systems that have evolved fast enough to have kept pace with cladogenesis but slowly enough to have conveyed the resulting phylogenetic signal to the present. Because stratigraphic evidence suggests that basal arthropod lineages arose rapidly during an ancient (Cambrian) phylogenetic radiation, the discovery of molecular sequences capable of resolving arthropod phylogeny may be a significant challenge for molecular systematists. This challenge is exemplified by our attempt to resolve arthropod phylogeny using the amino acid sequence of elongation factor-1α (EF-1α). Our fossil-based assessment of evolutionary rates indicates that EF-la should be capable of resolving Cambrian-age divergences. However, phylogenetic analysis using EF-1α fails to establish relationships among most higher-level groups, although it does recover more recently derived clades. Here we propose two models to explain this incongruity. The Rapid Radiation Model maintains that fossil-based estimates of arthropod diversification are essentially accurate and that diversification occurred so rapidly during the Cambrian that few phylogenetically significant changes occurred in the slowly evolving EF-1α sequence. The Enhanced Preservation Model maintains that fossil-based estimates of Cambrian-age divergences reflect enhanced preservation of pre-existing lineages and that arthropod diversification occurred before the Cambrian. This model attributes lack of resolution to degradation of phylogenetic signal within EF-1α by subsequent evolution. Current evidence is more consistent with the Enhanced Preservation Model, which implies that fossil-based methods can be very misleading when attempting to gauge the phylogenetic information content of molecular sequences for Cambrian- and Precambrian-age divergences.

Least Squares Estimation of Molecular Distance - Noise Abatement in Phylogenetic Reconstruction

Zuckerkandl and Pauling (1962, "Horizons in Biochemistry," pp. 189-225, Academic Press, New York) first noticed that the degree of sequence similarity between the proteins of different species could be used to estimate their phylogenetic relationship. Since then models have been developed to improve the accuracy of phylogenetic inferences based on amino acid or DNA sequences. Most of these models were designed to yield distance measures that are linear with time, on average. The reliability of phylogenetic reconstruction, however, depends on the variance of the distance measure in addition to its expectation. In this paper we show how the method of generalized least squares can be used to combine data types, each most informative at different points in time, into a single distance measure. This measure reconstructs phylogenies more accurately than existing non-likelihood distance measures. We illustrate the approach for a two-rate mutation model and demonstrate that its application provides more accurate phylogenetic reconstruction than do currently available analytical distance measures.