Phylogenetic Stochastic Mapping
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Phylogenetic stochastic mapping is a method for reconstructing the history of trait changes on a phylogenetic tree relating species/organisms carrying the trait. State-of-the-art methods assume that the trait evolves according to a continuous-time Markov chain (CTMC) and work well for small state spaces. The computations slow down considerably for larger state spaces (e.g. space of codons), because current methodology relies on exponentiating CTMC infinitesimal rate matrices --- an operation whose computational complexity grows as the size of the CTMC state space cubed. In this work, we introduce a new approach, based on a CTMC technique called uniformization, that does not use matrix exponentiation for phylogenetic stochastic mapping. Our method is based on a new Markov chain Monte Carlo (MCMC) algorithm that targets the distribution of trait histories conditional on the trait data observed at the tips of the tree. The computational complexity of our MCMC method grows as the size of the CTMC state space squared. Moreover, in contrast to competing matrix exponentiation methods, if the rate matrix is sparse, we can leverage this sparsity and increase the computational efficiency of our algorithm further. Using simulated data, we illustrate advantages of our MCMC algorithm and investigate how large the state space needs to be for our method to outperform matrix exponentiation approaches. We show that even on the moderately large state space of codons our MCMC method can be significantly faster than currently used matrix exponentiation methods. We apply our new stochastic mapping technique to two data sets. The first concerns the reproductive parity mode of squamates, and the second concerns the evolution of bioluminescent bacterial photophores in cephalopods. In both cases there were concerns that the standard CTMC model of trait evolution for the binary morphological traits was insufficient due to rate matrix heterogeneity across the phylogeny. To address these concerns we developed a Markov modulated Markov process model of trait evolution and integrated this hidden rates model with our matrix exponentiation free stochastic mapping technique. We found that the evidence supporting multiple gains of bioluminescence in cephalopods was mildly attenuated by accounting for potential rate matrix heterogeneity. Conversely, we found that accounting for rate matrix heterogeneity on the squamate phylogeny dramatically changed conclusions about the reproductive parity mode of the most recent common ancestor of squamates. The standard two state CTMC model of trait evolution found insufficient evidence to distinguish between oviparity and viviparity at the root of Squamata while a variety of hidden rates models found strong evidence that the most recent common ancestor of squamates was oviparous.
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