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.     

Relaxed phylogenetics and dating with confidence

In phylogenetics, the unrooted model of phylogeny and the strict molecular clock model are two extremes of a continuum. Despite their dominance in phylogenetic inference, it is evident that both are biologically unrealistic and that the real evolutionary process lies between these two extremes. Fortunately, intermediate models employing relaxed molecular clocks have been described. These models open the gate to a new field of “relaxed phylogenetics.” Here we introduce a new approach to performing relaxed phylogenetic analysis. We describe how it can be used to estimate phylogenies and divergence times in the face of uncertainty in evolutionary rates and calibration times. Our approach also provides a means for measuring the clocklikeness of datasets and comparing this measure between different genes and phylogenies. We find no significant rate autocorrelation among branches in three large datasets, suggesting that autocorrelated models are not necessarily suitable for these data. In addition, we place these datasets on the continuum of clocklikeness between a strict molecular clock and the alternative unrooted extreme. Finally, we present analyses of 102 bacterial, 106 yeast, 61 plant, 99 metazoan, and 500 primate alignments. From these we conclude that our method is phylogenetically more accurate and precise than the traditional unrooted model while adding the ability to infer a timescale to evolution.     

Simulating and detecting autocorrelation of molecular evolutionary rates among lineages

Evolutionary timescales can be estimated from genetic data using phylogenetic methods based on the molecular clock. To account for molecular rate variation among lineages, a number of relaxed‐clock models have been developed. Some of these models assume that rates vary among lineages in an autocorrelated manner, so that closely related species share similar rates. In contrast, uncorrelated relaxed clocks allow all of the branch‐specific rates to be drawn from a single distribution, without assuming any correlation between rates along neighbouring branches. There is uncertainty about which of these two classes of relaxed‐clock models are more appropriate for biological data. We present an R package, NELSI, that allows the evolution of DNA sequences to be simulated according to a range of clock models. Using data generated by this package, we assessed the ability of two Bayesian phylogenetic methods to distinguish among different relaxed‐clock models and to quantify rate variation among lineages. The results of our analyses show that rate autocorrelation is typically difficult to detect, even when there is complete taxon sampling. This provides a potential explanation for past failures to detect rate autocorrelation in a range of data sets.     

A dirichlet process prior for estimating lineage-specific substitution rates

We introduce a new model for relaxing the assumption of a strict molecular clock for use as a prior in Bayesian methods for divergence time estimation. Lineage-specific rates of substitution are modeled using a Dirichlet process prior (DPP), a type of stochastic process that assumes lineages of a phylogenetic tree are distributed into distinct rate classes. Under the Dirichlet process, the number of rate classes, assignment of branches to rate classes, and the rate value associated with each class are treated as random variables. The performance of this model was evaluated by conducting analyses on data sets simulated under a range of different models. We compared the Dirichlet process model with two alternative models for rate variation: the strict molecular clock and the independent rates model. Our results show that divergence time estimation under the DPP provides robust estimates of node ages and branch rates without significantly reducing power. Further analyses were conducted on a biological data set, and we provide examples of ways to summarize Markov chain Monte Carlo samples under this model.     

A compound poisson process for relaxing the molecular clock

The molecular clock hypothesis remains an important conceptual and analytical tool in evolutionary biology despite the repeated observation that the clock hypothesis does not perfectly explain observed DNA sequence variation. We introduce a parametric model that relaxes the molecular clock by allowing rates to vary across lineages according to a compound Poisson process. Events of substitution rate change are placed onto a phylogenetic tree according to a Poisson process. When an event of substitution rate change occurs, the current rate of substitution is modified by a gamma-distributed random variable. Parameters of the model can be estimated using Bayesian inference. We use Markov chain Monte Carlo integration to evaluate the posterior probability distribution because the posterior probability involves high dimensional integrals and summations. Specifically, we use the Metropolis-Hastings-Green algorithm with 11 different move types to evaluate the posterior distribution. We demonstrate the method by analyzing a complete mtDNA sequence data set from 23 mammals. The model presented here has several potential advantages over other models that have been proposed to relax the clock because it is parametric and does not assume that rates change only at speciation events. This model should prove useful for estimating divergence times when substitution rates vary across lineages.     

Testing a molecular clock without an outgroup: derivations of induced priors on branch-length restrictions in a Bayesian framework

We propose a Bayesian method for testing molecular clock hypotheses for use with aligned sequence data from multiple taxa. Our method utilizes a nonreversible nucleotide substitution model to avoid the necessity of specifying either a known tree relating the taxa or an outgroup for rooting the tree. We employ reversible jump Markov chain Monte Carlo to sample from the posterior distribution of the phylogenetic model parameters and conduct hypothesis testing using Bayes factors, the ratio of the posterior to prior odds of competing models. Here, the Bayes factors reflect the relative support of the sequence data for equal rates of evolutionary change between taxa versus unequal rates, averaged over all possible phylogenetic parameters, including the tree and root position. As the molecular clock model is a restriction of the more general unequal rates model, we use the Savage-Dickey ratio to estimate the Bayes factors. The Savage-Dickey ratio provides a convenient approach to calculating Bayes factors in favor of sharp hypotheses. Critical to calculating the Savage-Dickey ratio is a determination of the prior induced on the modeling restrictions. We demonstrate our method on a well-studied mtDNA sequence data set consisting of nine primates. We find strong support against a global molecular clock, but do find support for a local clock among the anthropoids. We provide mathematical derivations of the induced priors on branch length restrictions assuming equally likely trees. These derivations also have more general applicability to the examination of prior assumptions in Bayesian phylogenetics.     

Bayesian Phylogenetic Inference using Relaxed-clocks and the Multispecies Coalescent

The multispecies coalescent (MSC) model accommodates both species divergences and within-species coalescent and provides a natural framework for phylogenetic analysis of genomic data when the gene trees vary across the genome. The MSC model implemented in the program bpp assumes a molecular clock and the Jukes–Cantor model, and is suitable for analyzing genomic data from closely related species. Here we extend our implementation to more general substitution models and relaxed clocks to allow the rate to vary among species. The MSC-with-relaxed-clock model allows the estimation of species divergence times and ancestral population sizes using genomic sequences sampled from contemporary species when the strict clock assumption is violated, and provides a simulation framework for evaluating species tree estimation methods. We conducted simulations and analyzed two real datasets to evaluate the utility of the new models. We confirm that the clock-JC model is adequate for inference of shallow trees with closely related species, but it is important to account for clock violation for distant species. Our simulation suggests that there is valuable phylogenetic information in the gene-tree branch lengths even if the molecular clock assumption is seriously violated, and the relaxed-clock models implemented in bpp are able to extract such information. Our Markov chain Monte Carlo algorithms suffer from mixing problems when used for species tree estimation under the relaxed clock and we discuss possible improvements. We conclude that the new models are currently most effective for estimating population parameters such as species divergence times when the species tree is fixed.     

ACCOUNTING FOR RATE VARIATION AMONG LINEAGES IN COMPARATIVE DEMOGRAPHIC ANALYSES

Genetic analyses of contemporary populations can be used to estimate the demographic histories of species within an ecological community. Comparison of these demographic histories can shed light on community responses to past climatic events. However, species experience different rates of molecular evolution, and this presents a major obstacle to comparative demographic analyses. We address this problem by using a Bayesian relaxed‐clock method to estimate the relative evolutionary rates of 22 small mammal taxa distributed across northwestern North America. We found that estimates of the relative molecular substitution rate for each taxon were consistent across the range of sampling schemes that we compared. Using three different reference rates, we rescaled the relative rates so that they could be used to estimate absolute evolutionary timescales. Accounting for rate variation among taxa led to temporal shifts in our skyline‐plot estimates of demographic history, highlighting both uniform and idiosyncratic evolutionary responses to directional climate trends for distinct ecological subsets of the small mammal community. Our approach can be used in evolutionary analyses of populations from multiple species, including comparative demographic studies.     

Bayesian random local clocks,or one rate to rule them all

Background  

Relaxed molecular clock models allow divergence time dating and "relaxed phylogenetic" inference, in which a time tree is estimated in the face of unequal rates across lineages. We present a new method for relaxing the assumption of a strict molecular clock using Markov chain Monte Carlo to implement Bayesian modeling averaging over random local molecular clocks. The new method approaches the problem of rate variation among lineages by proposing a series of local molecular clocks, each extending over a subregion of the full phylogeny. Each branch in a phylogeny (subtending a clade) is a possible location for a change of rate from one local clock to a new one. Thus, including both the global molecular clock and the unconstrained model results, there are a total of 2^2n-2 possible rate models available for averaging with 1, 2, ..., 2n - 2 different rate categories.     

A Bayesian evaluation of human mitochondrial substitution rates

Accurate estimates of mitochondrial substitution rates are central to molecular studies of human evolution, but meaningful comparisons of published studies are problematic because of the wide range of methodologies and data sets employed. These differences are nowhere more pronounced than among rates estimated from phylogenies, genealogies, and pedigrees. By using a data set comprising mitochondrial genomes from 177 humans, we estimate substitution rates for various data partitions by using Bayesian phylogenetic analysis with a relaxed molecular clock. We compare the effect of multiple internal calibrations with the customary human-chimpanzee split. The analyses reveal wide variation among estimated substitution rates and divergence times made with different partitions and calibrations, with evidence of substitutional saturation, natural selection, and significant rate heterogeneity among lineages and among sites. Collectively, the results support dates for migration out of Africa and the common mitochondrial ancestor of humans that are considerably more recent than most previous estimates. Our results also demonstrate that human mitochondrial genomes exhibit a number of molecular evolutionary complexities that necessitate the use of sophisticated analytical models for genetic analyses.     

Relaxed clocks and inferences of heterogeneous patterns of nucleotide substitution and divergence time estimates across whales and dolphins (Mammalia: Cetacea)

Various nucleotide substitution models have been developed to accommodate among lineage rate heterogeneity, thereby relaxing the assumptions of the strict molecular clock. Recently developed "uncorrelated relaxed clock" and "random local clock" (RLC) models allow decoupling of nucleotide substitution rates between descendant lineages and are thus predicted to perform better in the presence of lineage-specific rate heterogeneity. However, it is uncertain how these models perform in the presence of punctuated shifts in substitution rate, especially between closely related clades. Using cetaceans (whales and dolphins) as a case study, we test the performance of these two substitution models in estimating both molecular rates and divergence times in the presence of substantial lineage-specific rate heterogeneity. Our RLC analyses of whole mitochondrial genome alignments find evidence for up to ten clade-specific nucleotide substitution rate shifts in cetaceans. We provide evidence that in the uncorrelated relaxed clock framework, a punctuated shift in the rate of molecular evolution within a subclade results in posterior rate estimates that are either misled or intermediate between the disparate rate classes present in baleen and toothed whales. Using simulations, we demonstrate abrupt changes in rate isolated to one or a few lineages in the phylogeny can mislead rate and age estimation, even when the node of interest is calibrated. We further demonstrate how increasing prior age uncertainty can bias rate and age estimates, even while the 95% highest posterior density around age estimates decreases; in other words, increased precision for an inaccurate estimate. We interpret the use of external calibrations in divergence time studies in light of these results, suggesting that rate shifts at deep time scales may mislead inferences of absolute molecular rates and ages.     

Effects of models of rate evolution on estimation of divergence dates with special reference to the metazoan 18S ribosomal RNA phylogeny

The molecular clock, i.e., constancy of the rate of evolution over time, is commonly assumed in estimating divergence dates. However, this assumption is often violated and has drastic effects on date estimation. Recently, a number of attempts have been made to relax the clock assumption. One approach is to use maximum likelihood, which assigns rates to branches and allows the estimation of both rates and times. An alternative is the Bayes approach, which models the change of the rate over time. A number of models of rate change have been proposed. We have extended and evaluated models of rate evolution, i.e., the lognormal and its recent variant, along with the gamma, the exponential, and the Ornstein-Uhlenbeck processes. These models were first applied to a small hominoid data set, where an empirical Bayes approach was used to estimate the hyperparameters that measure the amount of rate variation. Estimation of divergence times was sensitive to these hyperparameters, especially when the assumed model is close to the clock assumption. The rate and date estimates varied little from model to model, although the posterior Bayes factor indicated the Ornstein-Uhlenbeck process outperformed the other models. To demonstrate the importance of allowing for rate change across lineages, this general approach was used to analyze a larger data set consisting of the 18S ribosomal RNA gene of 39 metazoan species. We obtained date estimates consistent with paleontological records, the deepest split within the group being about 560 million years ago. Estimates of the rates were in accordance with the Cambrian explosion hypothesis and suggested some more recent lineage-specific bursts of evolution.     

Molecular‐clock methods for estimating evolutionary rates and timescales

The molecular clock presents a means of estimating evolutionary rates and timescales using genetic data. These estimates can lead to important insights into evolutionary processes and mechanisms, as well as providing a framework for further biological analyses. To deal with rate variation among genes and among lineages, a diverse range of molecular‐clock methods have been developed. These methods have been implemented in various software packages and differ in their statistical properties, ability to handle different models of rate variation, capacity to incorporate various forms of calibrating information and tractability for analysing large data sets. Choosing a suitable molecular‐clock model can be a challenging exercise, but a number of model‐selection techniques are available. In this review, we describe the different forms of evolutionary rate heterogeneity and explain how they can be accommodated in molecular‐clock analyses. We provide an outline of the various clock methods and models that are available, including the strict clock, local clocks, discrete clocks and relaxed clocks. Techniques for calibration and clock‐model selection are also described, along with methods for handling multilocus data sets. We conclude our review with some comments about the future of molecular clocks.     

Comparison of likelihood and Bayesian methods for estimating divergence times using multiple gene Loci and calibration points, with application to a radiation of cute-looking mouse lemur species

Divergence time and substitution rate are seriously confounded in phylogenetic analysis, making it difficult to estimate divergence times when the molecular clock (rate constancy among lineages) is violated. This problem can be alleviated to some extent by analyzing multiple gene loci simultaneously and by using multiple calibration points. While different genes may have different patterns of evolutionary rate change, they share the same divergence times. Indeed, the fact that each gene may violate the molecular clock differently leads to the advantage of simultaneous analysis of multiple loci. Multiple calibration points provide the means for characterizing the local evolutionary rates on the phylogeny. In this paper, we extend previous likelihood models of local molecular clock for estimating species divergence times to accommodate multiple calibration points and multiple genes. Heterogeneity among different genes in evolutionary rate and in substitution process is accounted for by the models. We apply the likelihood models to analyze two mitochondrial protein-coding genes, cytochrome oxidase II and cytochrome b, to estimate divergence times of Malagasy mouse lemurs and related outgroups. The likelihood method is compared with the Bayes method of Thorne et al. (1998, Mol. Biol. Evol. 15:1647-1657), which uses a probabilistic model to describe the change in evolutionary rate over time and uses the Markov chain Monte Carlo procedure to derive the posterior distribution of rates and times. Our likelihood implementation has the drawbacks of failing to accommodate uncertainties in fossil calibrations and of requiring the researcher to classify branches on the tree into different rate groups. Both problems are avoided in the Bayes method. Despite the differences in the two methods, however, data partitions and model assumptions had the greatest impact on date estimation. The three codon positions have very different substitution rates and evolutionary dynamics, and assumptions in the substitution model affect date estimation in both likelihood and Bayes analyses. The results demonstrate that the separate analysis is unreliable, with dates variable among codon positions and between methods, and that the combined analysis is much more reliable. When the three codon positions were analyzed simultaneously under the most realistic models using all available calibration information, the two methods produced similar results. The divergence of the mouse lemurs is dated to be around 7-10 million years ago, indicating a surprisingly early species radiation for such a morphologically uniform group of primates.     

Molecular dating for phylogenies containing a mix of populations and species by using Bayesian and RelTime approaches

Simultaneous molecular dating of population and species divergences is essential in many biological investigations, including phylogeography, phylodynamics and species delimitation studies. In these investigations, multiple sequence alignments consist of both intra‐ and interspecies samples (mixed samples). As a result, the phylogenetic trees contain interspecies, interpopulation and within‐population divergences. Bayesian relaxed clock methods are often employed in these analyses, but they assume the same tree prior for both inter‐ and intraspecies branching processes and require specification of a clock model for branch rates (independent vs. autocorrelated rates models). We evaluated the impact of a single tree prior on Bayesian divergence time estimates by analysing computer‐simulated data sets. We also examined the effect of the assumption of independence of evolutionary rate variation among branches when the branch rates are autocorrelated. Bayesian approach with coalescent tree priors generally produced excellent molecular dates and highest posterior densities with high coverage probabilities. We also evaluated the performance of a non‐Bayesian method, RelTime, which does not require the specification of a tree prior or a clock model. RelTime's performance was similar to that of the Bayesian approach, suggesting that it is also suitable to analyse data sets containing both populations and species variation when its computational efficiency is needed.     

Efficient selection of branch-specific models of sequence evolution

The analysis of extant sequences shows that molecular evolution has been heterogeneous through time and among lineages. However, for a given sequence alignment, it is often difficult to uncover what factors caused this heterogeneity. In fact, identifying and characterizing heterogeneous patterns of molecular evolution along a phylogenetic tree is very challenging, for lack of appropriate methods. Users either have to a priori define groups of branches along which they believe molecular evolution has been similar or have to allow each branch to have its own pattern of molecular evolution. The first approach assumes prior knowledge that is seldom available, and the second requires estimating an unreasonably large number of parameters. Here we propose a convenient and reliable approach where branches get clustered by their pattern of molecular evolution alone, with no need for prior knowledge about the data set under study. Model selection is achieved in a statistical framework and therefore avoids overparameterization. We rely on substitution mapping for efficiency and present two clustering approaches, depending on whether or not we expect neighbouring branches to share more similar patterns of sequence evolution than distant branches. We validate our method on simulations and test it on four previously published data sets. We find that our method correctly groups branches sharing similar equilibrium GC contents in a data set of ribosomal RNAs and recovers expected footprints of selection through dN/dS. Importantly, it also uncovers a new pattern of relaxed selection in a phylogeny of Mantellid frogs, which we are able to correlate to life-history traits. This shows that our programs should be very useful to study patterns of molecular evolution and reveal new correlations between sequence and species evolution. Our programs can run on DNA, RNA, codon, or amino acid sequences with a large set of possible models of substitutions and are available at http://biopp.univ-montp2.fr/forge/testnh.     

Bayesian selection of continuous-time Markov chain evolutionary models

We develop a reversible jump Markov chain Monte Carlo approach to estimating the posterior distribution of phylogenies based on aligned DNA/RNA sequences under several hierarchical evolutionary models. Using a proper, yet nontruncated and uninformative prior, we demonstrate the advantages of the Bayesian approach to hypothesis testing and estimation in phylogenetics by comparing different models for the infinitesimal rates of change among nucleotides, for the number of rate classes, and for the relationships among branch lengths. We compare the relative probabilities of these models and the appropriateness of a molecular clock using Bayes factors. Our most general model, first proposed by Tamura and Nei, parameterizes the infinitesimal change probabilities among nucleotides (A, G, C, T/U) into six parameters, consisting of three parameters for the nucleotide stationary distribution, two rate parameters for nucleotide transitions, and another parameter for nucleotide transversions. Nested models include the Hasegawa, Kishino, and Yano model with equal transition rates and the Kimura model with a uniform stationary distribution and equal transition rates. To illustrate our methods, we examine simulated data, 16S rRNA sequences from 15 contemporary eubacteria, halobacteria, eocytes, and eukaryotes, 9 primates, and the entire HIV genome of 11 isolates. We find that the Kimura model is too restrictive, that the Hasegawa, Kishino, and Yano model can be rejected for some data sets, that there is evidence for more than one rate class and a molecular clock among similar taxa, and that a molecular clock can be rejected for more distantly related taxa.     

Efficient inference on known phylogenetic trees using Poisson regression

MOTIVATION: We suggest the use of Poisson regression for time inference and hypothesis testing on a bifurcating Phylogenetic tree with known topology. This method is computationally simple and naturally accommodates variable substitution rates across different sites, without requiring the estimation of these rates. We identify the assumptions under which this is a maximum-likelihood inference approach and show that in some realistic situations--in particular, when the probability of repeated mutation within each branch of the tree is small--these assumptions hold with high probability. RESULTS: Our motivating domain is human mitochondrial DNA trees, and we illustrate our method on a problem of estimating the time to most recent common ancestor of all non-African mtDNA, using publicly available data. We test for molecular clock violations using multiple comparisons, and conclude that the global molecular clock hypothesis cannot be rejected based on these data.     

Inconsistencies in estimating the age of HIV-1 subtypes due to heterotachy

Rate heterogeneity among lineages is a common feature of molecular evolution, and it has long impeded our ability to accurately estimate the age of evolutionary divergence events. The development of relaxed molecular clocks, which model variable substitution rates among lineages, was intended to rectify this problem. Major subtypes of pandemic HIV-1 group M are thought to exemplify closely related lineages with different substitution rates. Here, we report that inferring the time of most recent common ancestor of all these subtypes in a single phylogeny under a single (relaxed) molecular clock produces significantly different dates for many of the subtypes than does analysis of each subtype on its own. We explore various methods to ameliorate this problem. We conclude that current molecular dating methods are inadequate for dealing with this type of substitution rate variation in HIV-1. Through simulation, we show that heterotachy causes root ages to be overestimated.     

Coestimation of recombination,substitution and molecular adaptation rates by approximate Bayesian computation

The estimation of parameters in molecular evolution may be biased when some processes are not considered. For example, the estimation of selection at the molecular level using codon-substitution models can have an upward bias when recombination is ignored. Here we address the joint estimation of recombination, molecular adaptation and substitution rates from coding sequences using approximate Bayesian computation (ABC). We describe the implementation of a regression-based strategy for choosing subsets of summary statistics for coding data, and show that this approach can accurately infer recombination allowing for intracodon recombination breakpoints, molecular adaptation and codon substitution rates. We demonstrate that our ABC approach can outperform other analytical methods under a variety of evolutionary scenarios. We also show that although the choice of the codon-substitution model is important, our inferences are robust to a moderate degree of model misspecification. In addition, we demonstrate that our approach can accurately choose the evolutionary model that best fits the data, providing an alternative for when the use of full-likelihood methods is impracticable. Finally, we applied our ABC method to co-estimate recombination, substitution and molecular adaptation rates from 24 published human immunodeficiency virus 1 coding data sets.