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

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 2(2n-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/).

Figure 1

Bayesian inference of random local clocks on mammalian data. Most probable evolutionary tree relating three nuclear genes from 42 mammals [32]. The color of the branches in the tree indicate branch-specific relative rates from red (fast) to blue (slow). Regions with the same color signify local clocks. Branches with a posterior probability of a change in rate >0.1 are labeled with the estimated posterior probabilities from two independent runs. An arrow to the right signifies a rate increase on a branch (and its descendants), while an arrow to the left signifies a slow down.

Figure 2

Prior and posterior distributions of the number of rate changes for three molecular data sets. Comparison of posterior (red) to prior (blue) probability mass functions of the number of rate changes K for the (a) mammal, (b) primate and (c) influenza examples. In all examples, the prior probability of a global molecular clock (K = 0) is 50%. Greater posterior than prior probability for K = 0 supports the global clock hypothesis (primates); while small or negligible posterior probability for K = 0 strongly rejects the hypothesis (mammals and influenza).

Figure 3

Inferred mtDNA rates for primate phylogeny. Most probable evolutionary tree relating seven mtDNA sequences from primates [33]. Gray boxed regions depict 95% Bayesian credible intervals (BCIs) for relative divergence times (that is, in units of expected substitutions per site). Recorded for all branches are their relative rate parameter r_k 95% BCIs. All intervals cover 1, suggesting little or no rate variation across the tree.

Figure 4

Influenza A data analysis. (a) Most probable evolutionary tree relating 69 hemagglutinin sequences from human influenza A. Branch coloring indicates inferred rates of nucleotide substitution, with blue denoting the slowest rates and red the fastest. (b) Rate heterogeneity of hemagglutinin sequence evolution over time. The plot traces the marginal distribution of relative substitution rates across time. White indicates low posterior density, and yellow/red indicates high density. The estimated rates are higher towards the present, with a notable jump in rate approximately six and ten years before the last sequence sample.