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.     

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.     

Testing the molecular clock: molecular and paleontological estimates of divergence times in the Echinoidea (Echinodermata)

The phylogenetic relationships of 46 echinoids, with representatives from 13 of the 14 ordinal-level clades and about 70% of extant families commonly recognized, have been established from 3 genes (3,226 alignable bases) and 119 morphological characters. Morphological and molecular estimates are similar enough to be considered suboptimal estimates of one another, and the combined data provide a tree that, when calibrated against the fossil record, provides paleontological estimates of divergence times and completeness of their fossil record. The order of branching on the cladogram largely agrees with the stratigraphic order of first occurrences and implies that their fossil record is more than 85% complete at family level and at a resolution of 5-Myr time intervals. Molecular estimates of divergence times derived from applying both molecular clock and relaxed molecular clock models are concordant with estimates based on the fossil record in up to 70% of cases, with most concordant results obtained using Sanderson's semiparametric penalized likelihood method and a logarithmic-penalty function. There are 3 regions of the tree where molecular and fossil estimates of divergence time consistently disagree. Comparison with results obtained when molecular divergence dates are estimated from the combined (morphology + gene) tree suggests that errors in phylogenetic reconstruction explain only one of these. In another region the error most likely lies with the paleontological estimates because taxa in this region are demonstrated to have a very poor fossil record. In the third case, morphological and paleontological evidence is much stronger, and the topology for this part of the molecular tree differs from that derived from the combined data. Here the cause of the mismatch is unclear but could be methodological, arising from marked inequality of molecular rates. Overall, the level of agreement reached between these different data and methodological approaches leads us to believe that careful application of likelihood and Bayesian methods to molecular data provides realistic divergence time estimates in the majority of cases (almost 80% in this specific example), thus providing a remarkably well-calibrated phylogeny of a character-rich clade of ubiquitous marine benthic invertebrates.     

Unraveling the evolutionary radiation of the thoracican barnacles using molecular and morphological evidence: a comparison of several divergence time estimation approaches

The Thoracica includes the ordinary barnacles found along the sea shore and is the most diverse and well-studied superorder of Cirripedia. However, although the literature abounds with scenarios explaining the evolution of these barnacles, very few studies have attempted to test these hypotheses in a phylogenetic context. The few attempts at phylogenetic analyses have suffered from a lack of phylogenetic signal and small numbers of taxa. We collected DNA sequences from the nuclear 18S, 28S, and histone H3 genes and the mitochondrial 12S and 16S genes (4,871 bp total) and data for 37 adult and 53 larval morphological characters from 43 taxa representing all the extant thoracican suborders (except the monospecific Brachylepadomorpha). Four Rhizocephala (highly modified parasitic barnacles) taxa and a Rhizocephala + Acrothoracica (burrowing barnacles) hypothetical ancestor were used as the outgroup for the molecular and morphological analyses, respectively. We analyzed these data separately and combined using maximum likelihood (ML) under "hill-climbing" and genetic algorithm heuristic searches, maximum parsimony procedures, and Bayesian inference coupled with Markov chain Monte Carlo techniques under mixed and homogeneous models of nucleotide substitution. The resulting phylogenetic trees answered key questions in barnacle evolution. The four-plated Iblomorpha were shown as the most primitive thoracican, and the plateless Heteralepadomorpha were placed as the sister group of the Lepadomorpha. These relationships suggest for the first time in an invertebrate that exoskeleton biomineralization may have evolved from phosphatic to calcitic. Sessilia (nonpedunculate) barnacles were depicted as monophyletic and appear to have evolved from a stalked (pedunculate) multiplated (5+) scalpelloidlike ancestor rather than a five-plated lepadomorphan ancestor. The Balanomorpha (symmetric sessile barnacles) appear to have the following relationship: (Chthamaloidea(Coronuloidea(Tetraclitoidea, Balanoidea))). Thoracican divergence times were estimated under ML-based local clock, Bayesian, and penalized likelihood approaches using an 18S data set and three calibration points: Heteralepadomorpha = 530 million years ago (MYA), Scalpellomorpha = 340 MYA, and Verrucomorpha = 120 MYA. Estimated dates varied considerably within and between approaches depending on the calibration point. Highly parameterized local clock models that assume independent rates (r > or = 15) for confamilial or congeneric species generated the most congruent estimates among calibrations and agreed more closely with the barnacle fossil record. Reasonable estimates were also obtained under the Bayesian procedure of Kishino et al. (2001, Mol. Biol. Evol. 18:352-361) but using multiple calibrations. Most of the dates estimated under the Bayesian procedure of Aris-Brosou and Yang (2002, Syst. Biol. 51:703-714) and the penalized likelihood method using single and/or multiple calibrations were inconsistent among calibrations and did not fit the fossil record.     

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.     

Approximate likelihood calculation on a phylogeny for Bayesian estimation of divergence times

The molecular clock provides a powerful way to estimate species divergence times. If information on some species divergence times is available from the fossil or geological record, it can be used to calibrate a phylogeny and estimate divergence times for all nodes in the tree. The Bayesian method provides a natural framework to incorporate different sources of information concerning divergence times, such as information in the fossil and molecular data. Current models of sequence evolution are intractable in a Bayesian setting, and Markov chain Monte Carlo (MCMC) is used to generate the posterior distribution of divergence times and evolutionary rates. This method is computationally expensive, as it involves the repeated calculation of the likelihood function. Here, we explore the use of Taylor expansion to approximate the likelihood during MCMC iteration. The approximation is much faster than conventional likelihood calculation. However, the approximation is expected to be poor when the proposed parameters are far from the likelihood peak. We explore the use of parameter transforms (square root, logarithm, and arcsine) to improve the approximation to the likelihood curve. We found that the new methods, particularly the arcsine-based transform, provided very good approximations under relaxed clock models and also under the global clock model when the global clock is not seriously violated. The approximation is poorer for analysis under the global clock when the global clock is seriously wrong and should thus not be used. The results suggest that the approximate method may be useful for Bayesian dating analysis using large data sets.     

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.     

Catarrhine primate divergence dates estimated from complete mitochondrial genomes: concordance with fossil and nuclear DNA evidence

Accurate divergence date estimates improve scenarios of primate evolutionary history and aid in interpretation of the natural history of disease-causing agents. While molecule-based estimates of divergence dates of taxa within the superfamily Hominoidea (apes and humans) are common in the literature, few such estimates are available for the Cercopithecoidea (Old World monkeys), the sister taxon of the hominoids in the primate infraorder Catarrhini. To help fill this gap, we have sequenced the entire mitochondrial DNA (mtDNA) genomes from a representative of three cercopithecoid tribes, Cercopithecini (Chlorocebus aethiops), Colobini (Colobus guereza), and Presbytini (Trachypithecus obscurus), and analyzed these new data together with other catarrhine mtDNA genomes available in public databases. Molecular divergence date estimates are dependent on calibration points gleaned from the paleontological record. We defined criteria for the selection of good calibration points and identified three points meeting these criteria: Homo-Pan, 6.0 Ma; Pongo-hominines, 14.0 Ma; hominoid/cercopithecoid, 23.0 Ma. Because a uniform molecular clock does not fit the catarrhine mtDNA data, we estimated divergence dates using a penalized likelihood and a Bayesian method, both of which take into account the effects of rate differences on lineages, phylogenetic tree structure, and multiple calibration points. The penalized likelihood method applied to the coding regions of the mtDNA genome yielded the following divergence date estimates, with approximate 95% confidence intervals: cercopithecine-colobine, 16.2 (14.4-17.9) Ma; colobin-presbytin, 10.9 (9.6-12.3) Ma; cercopithecin-papionin, 11.6 (10.3-12.9) Ma; and Macaca-Papio, 9.8 (8.6-10.9) Ma. Within the hominoids, the following dates were inferred: hylobatid-hominid, 16.8 (15.0-18.5) Ma; Gorilla-Homo+Pan, 8.1 (7.1-9.0) Ma; Pongo pygmaeus pygmaeus-P. p. abelii, 4.1 (3.5-4.7) Ma; and Pan troglodytes-P. paniscus, 2.4 (2.0-2.7) Ma. These dates were similar to those found using penalized likelihood on other subsets of the data, but slightly younger than several of the Bayesian estimates.     

The first mistletoes: origins of aerial parasitism in Santalales

Past molecular phylogenetic work has shown that aerial parasites have evolved five times independently in the sandalwood order (Santalales), but the absolute timing of these diversifications was not addressed. DNA sequences from nuclear SSU and LSU rDNA, and chloroplast rbcL, matK and trnL-F from 39 santalalean taxa were obtained. Separate and combined data partitions were analyzed with maximum parsimony and Bayesian inference. Time estimates were performed with Bayesian relaxed molecular clock and penalized likelihood methods using published fossil data. Both methods gave comparable divergence dates for the major clades. These data confirm five origins of aerial parasitism, first in Misodendraceae ca. 80 Mya and subsequently in Viscaceae (72 Mya), "Eremolepidaceae" (53 Mya), tribe Amphorogyneae in Santalaceae (46 Mya), and Loranthaceae (28 Mya). The rapid adaptive radiation and speciation in Loranthaceae coincides with the appearance of savanna biomes during the Oligocene. In all clades except Misodendraceae, it appears that aerial parasites evolved from ancestors that were polymorphic for either root or stem parasitism-a condition here termed amphiphagous. Convergences in morphological features associated with the mistletoe habit have occurred such as the squamate habit, seed attachment structures, unisexual flowers, and loss of chlorophyll.     

Rooting and dating maples (Acer) with an uncorrelated-rates molecular clock: implications for north American/Asian disjunctions

Simulations suggest that molecular clock analyses can correctly identify the root of a tree even when the clock assumption is severely violated. Clock-based rooting of phylogenies may be particularly useful when outgroup rooting is problematic. Here, we explore relaxed-clock rooting in the Acer/Dipteronia clade of Sapindaceae, which comprises genera of highly uneven species richness and problematic mutual monophyly. Using an approach that does not presuppose rate autocorrelation between ancestral and descendant branches and hence does not require a rooted a priori topology, we analyzed data from up to seven chloroplast loci for some 50 ingroup species. For comparison, we used midpoint and outgroup rooting and dating methods that rely on rooted input trees, namely penalized likelihood, a Bayesian autocorrelated-rates model, and a strict clock. The chloroplast sequences used here reject a single global substitution rate, and the assumption of autocorrelated rates was also rejected. The root was placed between Acer and Dipteronia by all three rooting methods, albeit with low statistical support. Analyses of Acer diversification with a lineage-through-time plot and different survival models, although sensitive to missing data, suggest a gradual decrease in the average diversification rate. The nine North American species of Acer diverged from their nearest relatives at widely different times: eastern American Acer diverged in the Oligocene and Late Miocene; western American species in the Late Eocene and Mid Miocene; and the Acer core clade, including A. saccharum, dates to the Miocene. Recent diversification in North America is strikingly rare compared to diversification in eastern Asia.