Abstract
Dating the tree of life is a task far more complicated than only determining the evolutionary relationships between species. It is therefore of interest to develop approaches apt to deal with undated phylogenetic trees. The main result of this work is a method to compute probabilities of undated phylogenetic trees under Markovian diversification models by constraining some of the divergence times to belong to given time intervals and by allowing diversification shifts on certain clades. If the diversification models considered are lineage-homogeneous, the time complexity of this computation is quadratic with the number of species of the phylogenetic tree and linear with the number of temporal constraints. The interest of this computation method is illustrated with three applications, namely, to compute the distribution of the divergence times of a tree topology with temporal constraints, to directly sample the divergence times of a tree topology, and to test for a diversification shift at a given clade.
Highlights
Peer Community Journal is a member of the Centre Mersenne for Open Scientific Publishing http:// www.centre-mersenne.org/
The first result is a method to compute the probability, under a Markovian diversification model, of a tree topology in which the divergence times are not exactly known but can be “constrained” to belong to given time intervals. This computation is performed by splitting the tree topology into small parts involving the times of the temporal constraints, referred to as patterns, and by combining their probabilities in order to get the probability of the whole tree topology
It can deal with phylogenetic trees with hundreds of tips on standard desktop computers. This computation can be used to obtain the divergence time distributions of a given undated phylogeny with temporal constraints, which can be applied to various questions. It can be used for dating phylogenetic trees from their topology only, like the method implemented in the function compute.brlen of the R-package APE (Grafen, 1989; Paradis et al, 2004)
Summary
The methods presented below apply to general diversification models. Namely, a diversification model Θ provides the parameters of a stochastic process which starts with a single lineage at time s and ends at time e, where e is usually the present time (both s and e are parameters of Θ). Under a birth-death-sampling model, the dynamics of speciation and extinction of species follows a birth-death process with constant rates λ and μ both through time and lineage, starting at origin time s and ending at time e which is generally the present time (Nee et al, 1994). By assuming that the diversification follows a simple birth-death process (i.e., with ρ = 1) with speciation rate λ and extinction rate μ, the probability pN (t) that a single lineage at time 0 has exactly N descendants at time t was given in Nee et al (1994). If one assumes that the diversification follows a birth-death-sampling process with speciation rate λ and extinction rate μ, the probability pN (t) that a single lineage at time 0 has exactly N descendants sampled with probability ρ at time t was given in Yang and Rannala (1997). Birth-death-sampling models are both Markovian and lineage-homogeneous
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