Continuous and tractable models of the variation of evolutionary rates. Math Biosci

Department of Mathematics and Statistics, McGill University, Montréal, Canada.
Mathematical Biosciences (Impact Factor: 1.3). 03/2006; 199(2):216-33. DOI: 10.1016/j.mbs.2005.11.002
Source: PubMed


We propose a continuous model for variation in the evolutionary rate across sites and over the phylogenetic tree. We derive exact transition probabilities of substitutions under this model. Changes in rate are modelled using the CIR process, a diffusion widely used in financial applications. The model directly extends the standard gamma distributed rates across site model, with one additional parameter governing changes in rate down the tree. The parameters of the model can be estimated directly from two well-known statistics: the index of dispersion and the gamma shape parameter of the rates across sites model. The CIR model can be readily incorporated into probabilistic models for sequence evolution. We provide here an exact formula for the likelihood of a three-taxon tree. The likelihoods of larger trees can be evaluated using Monte-Carlo methods.

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    • "Both of these processes have analytical transition probabilities in some special cases. Lepage et al. (2006) use the Cox-Ingersoll-Ross (CIR) process to model rate variation across a phylogeny. Like the OU process (but unlike Brownian motion ), the CIR process is ergodic. "
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    ABSTRACT: We present an efficient and flexible method for computing likelihoods of phenotypic traits on a phylogeny. The method does not resort to Monte-Carlo computation but instead blends Felsenstein's discrete character pruning algorithm with methods for numerical quadrature. It is not limited to Gaussian models and adapts readily to model uncertainty in the observed trait values. We demonstrate the framework by developing efficient algorithms for likelihood calculation and ancestral state reconstruction under Wright's threshold model, applying our methods to a dataset of trait data for extrafloral nectaries (EFNs) across a phylogeny of 839 Labales species.
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    • "Early studies achieved this by considering the evolutionary rate to be constant over time, that is, assuming a global molecular clock (also called a strict clock; Zuckerkandl and Pauling 1965). More recent methods allow the rate to vary over time under constraints specified by a relaxed-clock model, typically using a Bayesian inference framework (Hasegawa et al. 1989; Kishino et al. 1990; Thorne et al. 1998; Huelsenbeck et al. 2000; Yoder and Yang 2000; Kishino et al. 2001; Thorne and Kishino 2002; Aris-Brosou and Yang 2002; Yang and Yoder 2003; Drummond et al. 2006; Lepage et al. 2006; 2007; Rannala and Yang 2007; Drummond and Suchard 2010; Heath et al. 2012; Heath and Moore 2014). "
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    • "Under a 'relaxed-clock' model, substitution rates change over the tree in a constrained manner, thus separating the rate and time parameters associated with each branch and allowing inference of lineage divergence times. A considerable amount of effort has been directed at modeling lineage-specific substitution rate variation, with many different relaxed-clock models described in the literature [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]. When such models are coupled with a model on the distribution of speciation events over time (e.g., the Yule model [20] or birth-death process [21]), molecularsequence data can then inform the relative rates and node ages in a phylogenetic analysis. "
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