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Background

Pesto is an acronym for "Phylogenetic Estimation of Shifts in the Tempo of Origination". Broadly speaking, it is a method for detecting shifts in the process of diversification that led to the biodiversity present today. In order to study diversification, we are interested in two events: speciation and extinction events. Since these events are difficult to observe, we use a simple model, the birth-death process, to model what happened in the past. On this page, there is some background material that introduces the mathematical models that we use.

The birth-death model

Under the standard birth-death model, the tempo at which species speciate and go extinct are controlled by two parameters (Nee et al. 1994):

  • The speciation rate ($\lambda$)
  • The extinction rate ($\mu$)

In its simplest model, the rates of the birth-death process are the same across different lineages in the tree, and across time. By simulating under the birth-death process, and pruning the extinct lineages from the tree, one gets a reconstructed phylogenetic tree as a result (i.e. it is ultrametric, all tips end at the same time point).

The birth-death-shift model

The question we are interested in, is whether the process of diversification changed throughout the phylogenetic tree. In other words, was there a shift or not, and if so, how large was the shift? To do so, we are employing a variant of the state-dependent birth-death model (first presented by Maddison et al. 2007). This is also called the birth-death-shift model, or the lineage-specific birth-death model (Höhna et al. 2019). The birth-death-shift model has three parameters:

  • The state-dependent speciation rate ($\lambda_i$)
  • The state-dependent extinction rate ($\mu_i$)
  • The common shift rate ($\eta$).
bdshift

When a shift event occurs, the speciation and extinction rate shifts from the previous state (say $\lambda_1,\mu_1$) to a new state with different rates ($\lambda_2,\mu_2$). A rate shift to any other rate category occurs with rate $\eta$, and a rate shift from state $i$ to a specific other state $j$ occurs with rate $\eta/(K-1)$. The figure above depicts a three-state model. However, depending on how the model is set up, there can by any $K$ number of states.

The likelihood

The probability of observing the tree is the same as for the binary-state speciation and extinction model (BiSSE, Maddison et al. 2007), and the multi-state speciation and extinction model (MuSSE, FitzJohn 2012). We first calculate the probability of going extinct before the present, if a lineage was alive at some age t in the past:

\[ \frac{dE_{i}(t)}{dt} = \mu_i - (\lambda_i + \mu_i + \eta) E_{i}(t) + \lambda_i E_{i}(t)^2 + \frac{\eta}{K-1} \sum_{j \neq i}^K E_{j}(t)\]

The initial state for $E_i(t)$ is equal to $1-\rho$ for all states, where $\rho$ is the taxon sampling fraction (we assume uniform taxon sampling probability). Next, we calculate the probability that a lineage alive at age t was observed in the reconstructed tree:

\[\frac{dD_{M,i}(t)}{dt} = - (\lambda_i + \mu_i + \eta) D_{M,i}(t) + 2 \lambda_i D_{M,i}(t) E_i(t) + \frac{\eta}{K-1} \sum_{j \neq i}^K D_{M,j}(t)\]

We solve $D_{M,i}(t)$ for each branch M and each state i, in a postorder tree traversal. At the tips, the initial state is $D_{M,i}(t)=\rho$ for all states. At the branching events, the initial state for the parent branch P is assigned $D_{P,i}(t) := \lambda_i \times D_{L,i}(t) \times D_{R,i}(t)$ where L and R are the left and right descendant branches. Continuing this toward the root of the tree, we calculate the likelihood as follows:

\[L = \frac{1}{K}\sum_{i=1}^K \Big [ \frac{D_{L,i}(t) \times D_{R,i}(t)}{(1 - E_k(t))^2} \Big ]\]

where t is the age of the most recent common ancestor.

References

  • Nee, S., May, R. M., & Harvey, P. H. (1994). The reconstructed evolutionary process. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 344(1309), 305-311.
  • Maddison, W. P., Midford, P. E., & Otto, S. P. (2007). Estimating a binary character's effect on speciation and extinction. Systematic biology, 56(5), 701-710.
  • FitzJohn, R. G. (2012). Diversitree: comparative phylogenetic analyses of diversification in R. Methods in Ecology and Evolution, 3(6), 1084-1092.
  • Höhna, S., Freyman, W. A., Nolen, Z., Huelsenbeck, J. P., May, M. R., & Moore, B. R. (2019). A Bayesian approach for estimating branch-specific speciation and extinction rates. BioRxiv, 555805.