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Bayesian Random Local Clocks, Or One Rate to Rule Them All Alexei J Drummond1,2*, Marc a Suchard3,4*

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Bayesian Random Local Clocks, Or One Rate to Rule Them All Alexei J Drummond1,2*, Marc a Suchard3,4* Drummond and Suchard BMC Biology 2010, 8:114 http://www.biomedcentral.com/1741-7007/8/114 RESEARCH ARTICLE Open Access Bayesian random local clocks, or one rate to rule them all Alexei J Drummond1,2*, Marc A Suchard3,4* Abstract 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 22n-2 possible rate models available for averaging with 1, 2, ..., 2n-2 different rate categories. Results: We propose an efficient method to sample this model space while simultaneously estimating the phylogeny. The new method conveniently allows a direct test of the strict molecular clock, in which one rate rules them all, against a large array of alternative local molecular clock models. We illustrate the method’s utility on three example data sets involving mammal, primate and influenza evolution. Finally, we explore methods to visualize the complex posterior distribution that results from inference under such models. Conclusions: The examples suggest that large sequence datasets may only require a small number of local molecular clocks to reconcile their branch lengths with a time scale. All of the analyses described here are implemented in the open access software package BEAST 1.5.4 (http://beast-mcmc.googlecode.com/). Background been a slowdown of the rate of albumin evolution in In 1967, Allan Wilson and his then doctoral student African apes and humans to reconcile their great simi- Vincent Sarich described an “evolutionary clock” for larity with the presumed antiquity of their common albumin proteins and exploited the clock to date the ancestor. common ancestor of humans and chimpanzees to five Researchers have grappled with the tension between million years ago [1]. Given the limited informative- molecular and non-molecular evidence for evolutionary ness of these immunological data, this estimate has time scales ever since. Recently, a number of authors survived the intervening years remarkably well. This [3-7], have advanced “relaxed molecular clock” methods. work was the first prominent application of the con- These methods accommodate variation in the rate of cept of a molecular clock [2] and, at the time, the molecular evolution from lineage to lineage. In addition result raised extreme controversy, as the commonly to allowing non-clock-like relationships among held belief advocated that the common ancestor of sequences related by a phylogeny, modeling rate varia- humans and African apes was much more ancient. In tion among lineages in a gene tree also enable research- fact, previous authors had argued that there must have ers to incorporate multiple calibration points that may not be consistent with a strict molecular clock. These * Correspondence: [email protected]; [email protected] calibration points can be associated either with the 1Allan Wilson Centre for Molecular Ecology and Evolution, University of Auckland, Private Bag 92019, Auckland, New Zealand internal nodes of the tree or the sampled sequences 3Departments of Biomathematics and Human Genetics, David Geffen School themselves. Furthermore, relaxed molecular clock mod- of Medicine at UCLA, Los Angeles, CA 90095, USA els appear to fit real data better than either a strict Full list of author information is available at the end of the article © 2010 Drummond and Suchard; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Drummond and Suchard BMC Biology 2010, 8:114 Page 2 of 12 http://www.biomedcentral.com/1741-7007/8/114 molecular clock or the other extreme of no clock Methods assumption [6]. In spite of these successes, controversy Basic evolutionary model still remains around the particular assumptions underly- We begin by considering data Y, consisting of aligned ing some of the popular relaxed molecular clock models molecular sequences of length S from n taxa. We orien- currently employed. A number of authors [8-10], argue tate these data such that we may write Y =(Y1, ..., YS), that changes in the rate of evolution do not necessarily where Ys for s = 1, ..., S are the n homologous charac- occur smoothly nor on every branch of a gene tree. The ters at each site s of the sequence alignment. To model alternative expounds that large subtrees share the same this homology, we follow standard likelihood-based phy- underlying rate of evolution and that any variation can logenetic reconstruction practice [13] and assume the be described entirely by the stochastic nature of the evo- data arise from an underlying continuous-time Markov lutionary process. These phylogenetic regions or sub- chain (CTMC) process [14] along an unobserved tree τ. trees of rate homogeneity are separated by changes in The tree τ consists of a rooted, bifurcating topology that the rate of evolution. This alternative model may be characterizes the relatedness among the taxa, the gener- especially important for gene trees that have dense ally unknown historical times when lineages diverge in taxon sampling, in which case there are potentially the topology and up to 2n − 2 rate parameters rk that many short closely related lineages amongst which there relate historical time and expected number of substitu- is not reason a priori to assume differences in the tions on each branch k. The CTMC process describes underlying rate of substitution. the relative rates at which different sequence characters Local molecular clocks are another alternative to the change along the independent branches in τ. We restrict global molecular clock [11]. A local molecular clock per- our attention in this paper to nucleotide substitution mits different regions in the tree to have different rates, processes characterized by either the HKY85 [15] or but within each region the rate must be the same. Up GTR [16] infinitesimal rate matrices Λ and discrete- until now these models have been difficult to employ Gamma distributed across-site rate variation [17] with because their implementations did not permit the mod- shape parameter a. However, our approach admits any eling of uncertainty in (1) the phylogenetic tree topology standardly used CTMC for nucleotides, codons or or (2) the phylogenetic positions of the rate changes amino acids. between the local clock regions. For a model that allows Letting F =(Λ,a), we write the data sampling density one rate change on a rooted tree there are 2n − 2 of site s as P(Ys|τ, F). Felsenstein’s peeling/pruning branches on which the rate change can occur. To con- algorithm [18] enables computational efficient calcula- sider two rate changes, one must consider (2n − 2) × tions of P(Ys|τ,F). Assuming that sites are independent (2n − 3) possible rate placements. If each branch can and identically distributed given (τ,F) yields the com- have 0 or 1 rate changes then the total number of local plete data likelihood n− clock models that might be considered is 22 2,where n is the number of sequences under study. For even S n = moderate this number of local clock models can not PP()YY|,∏ ()s |, . (1) be evaluated exhaustively. s=1 In this paper we employ Markov chain Monte Carlo (MCMC) to investigate a Bayesian random local clock (RLC) model, in which all possible local clock configura- Branch-specific rate variation tions are nested. We implement our method in the We take the opinion that variation in the rate of mole- BEAST 1.x [12] and BEAST 2 (http://code.google.com/ cular evolution is widespread [5,6], but, following Yoder p/beast2/) open software frameworks. The resulting andYang[11],weassumedthatinanygiventreethere method co-estimates from the sequence data both the exist a small number of rate changes. This contrasts phylogenetic tree and the number, magnitude and loca- with most previous Bayesian MCMC relaxed clock mod- tion of rate changes along the tree. Our method samples els that favor many small or smoothly changing events n− a state space that includes the product of all 22 2 pos- [3,7,19,20]. In general, the numerous small changes arise sible local clock models on all possible rooted trees. as a modeling consequence, and are not necessarily Because the RLC model includes the possibility of zero data-driven. Apart from the induced smoothing, some rate changes, it also serves to test whether one rate is structure remains quite useful; at certain time scales one sufficient to rule all the gene sequences at hand, as was expects rate changes to be heritable and persist for Wilson and Sarich’s view of the African primate some time down the subtree extending from the albumins. change-point. Drummond and Suchard BMC Biology 2010, 8:114 Page 3 of 12 http://www.biomedcentral.com/1741-7007/8/114 Model parameterization Recent work in BSSVS [22,23] efficiently performs the We introduce the RLC model that allows for sparse, exploration in two steps. In the first step, the approach possibly large-scale changes while maintaining spatial augments the model state-space with a set of P binary correlation along the tree.
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