Probability Distribution of Tree Age for the Simple Birth–Death Process, with Applications to Distributions of Number of Ancestral Lineages and Divergence Times for Pairs of Taxa in a Yule Tree

Abstract

In this contribution, a general expression is derived for the probability density of the time to the most recent common ancestor (TMRCA) of a simple birth–death tree, a widely used stochastic null-model of biological speciation and extinction, conditioned on the constant birth and death rates and number of extant lineages. This density is contrasted with a previous result which was obtained using a uniform prior for the time of origin. The new distribution is applied to two problems of phylogenetic interest. First, that of the probability of the number of taxa existing at any time in the past in a tree of a known number of extant species, and given birth and death rates, and second, that of determining the TMRCA of two randomly selected taxa in an unobserved tree that is produced by a simple birth-only, or Yule, process. In the latter case, it is assumed that only the rate of bifurcation (speciation) and the size, or number of tips, are known. This is shown to lead to a closed-form analytical expression for the probability distribution of this parameter, which is arrived at based on the known mathematical form of the age distribution of Yule trees of a given size and branching rate, which is derived here de novo, and a similar distribution which additionally is conditioned on tree age. The new distribution is the exact Yule prior for divergence times of pairs of taxa under the stated conditions and is potentially useful in statistical (Bayesian) inference studies of phylogenies.

Appendix: Comparison with Age Distribution Based on an Improper Uniform Prior for the Time of Origin

In a previous study on the density of the times of speciation events, Gernhard (2008) considers the distribution of the time of origin of a BD tree having a known number of descendants at the present time. This earlier work takes a somewhat different approach where it is assumed that the moment at which a tree of unknown size and birth and death rates emerged is entirely unknown and is equally likely to have occurred at any time in the past. This amounts to assuming the age τ of any tree to follow an improper uniform distribution on [0, ∞) which is thus taken to be the prior (for a more recent application of this model assumption, see Ignatieva et al 2020). If the parameters n, λ and μ are known, application of Bayes’ theorem gives rise to the posterior distribution for tree age density conditioned on n, λ, μ which, in the notation used by Gernhard (2008; theorem 3.2), is found to be

$$ q_{or} (\tau |n,\lambda ,\mu ) = n\lambda \left( {1 - \frac{\mu }{\lambda }} \right)^{2} e^{ - (\lambda - \mu )\tau } \frac{{\left( {1 - e^{ - (\lambda - \mu )\tau } } \right)^{n - 1} }}{{\left( {1 - \frac{\mu }{\lambda }e^{ - (\lambda - \mu )\tau } } \right)^{n + 1} }}. $$

(A.1)

In this case, the tree starts with a single lineage which may split after some time.

This result cannot be compared directly with Eq. (8) of the present study which defines the time of origin of a BD tree as that of the first bifurcation. To make a comparison between the two approaches requires a slight modification of the argument presented in subSect. 2.1, where we now interpret the MRCA of the subtree highlighted in Fig. 1 to mark the “birth” of either one of its daughter trees.

Thus, the probability that a tree that starts with a single lineage at time T – τ will have grown to size n at time T is proportional to

$$ 2p_{n}^{2} (\tau ) + 2p_{n} (\tau )(1 - p_{n} (\tau )) = 2p_{n} (\tau ). $$

(A.2)

After normalising with $2\sum\nolimits_{n = 0}^{\infty } {p_{n} (\tau )} = 2$, this probability is found to be simply equal to $p_{n} (\tau )$, which should then replace $\hat{p}_{n} (\tau )$ in Eq. (3). Otherwise, the procedure for calculating the new distribution, which will be denoted as $\tilde{P}(\tau |n,\lambda ,\mu )$, is exactly the same as that followed in deriving Eq. (8).

The analogue of Eq. (5) is

$$ \tilde{P}(\tau |T,n,\lambda ,\mu )d\tau = \tilde{C}_{n} (T,\lambda ,\mu )e^{ - 2(\lambda - \mu )\tau } \frac{{\left( {1 - e^{ - (\lambda - \mu )\tau } } \right)^{n - 1} }}{{\left( {1 - \frac{\mu }{\lambda }e^{ - (\lambda - \mu )\tau } } \right)^{n + 1} }}d\tau , $$

(A.3)

where $\tilde{C}_{n} (T,\lambda ,\mu )$ is the normalisation factor which, in the limit T → ∞, follows from

$$ \begin{aligned} \mathop {\lim }\limits_{T \to \infty } \tilde{C}_{n} (T,\lambda ,\mu )^{ - 1} & = \int_{0}^{\infty } {d\tau e^{ - 2(\lambda - \mu )\tau } } \frac{{\left( {1 - e^{ - (\lambda - \mu )\tau } } \right)^{n - 1} }}{{\left( {1 - \frac{\mu }{\lambda }e^{ - (\lambda - \mu )\tau } } \right)^{n + 1} }} \\ & = \frac{1}{\lambda - \mu }\left( {\frac{1}{n(1 - \mu /\lambda )} - \sum\limits_{k = 0}^{\infty } {\frac{{(\mu /\lambda )^{k} }}{n + k + 1}} } \right). \\ \end{aligned} $$

(A.4)

With tree age τ thus redefined, its distribution now becomes

$$ \tilde{P}(\tau |n,\lambda ,\mu ) = \frac{{\lambda (1 - \mu /\lambda )e^{ - 2(\lambda - \mu )\tau } \left( {1 - e^{ - (\lambda - \mu )\tau } } \right)^{n - 1} }}{{\left( {\frac{1}{n(1 - \mu /\lambda )} - \sum\limits_{k = 0}^{\infty } {\frac{{(\mu /\lambda )^{k} }}{n + k + 1}} } \right)\left( {1 - \frac{\mu }{\lambda }e^{ - (\lambda - \mu )\tau } } \right)^{n + 1} }}, $$

(A.5)

which differs from Eq. (A.1) by an extra factor $e^{ - (\lambda - \mu )\tau }$ (and hence, the normalisation constant will also be different). It should be noted that this is precisely the factor by which the average total population from which the clade is sampled would have shrunk when returning to the point where the MRCA emerged.

That the results are not identical is therefore not surprising based on the different perspectives, viz. a uniform prior on time of origin vs. a picture based on a subtree embedded in an exponentially increasing population (assuming λ and μ known).