Tree Space

Display the topologies of 3-taxa tree sampled from MCMC under Kingman’s coalescent model.

3 Taxa

There are 3 different topologies in this tree space:

1. T=0, where the tree is (A,B),C;
2. T=1, where the tree is (A,C),B;
3. T=2, where the tree is (B,C),A.

We define the tree interval k₁ is the root height minus internal node height, and k₂ is the internal node height. Both k₁ and k₂ are simulated by Kingman’s coalescent model using MCMC, where the probability of genealogies in a population is:

P(k₁, k₂, T | Ne) = (1/Ne) * e^(-k₁/Ne) * (1/Ne) * e^(-3*k₂/Ne)

Assuming Ne = 1, then the above equation is simplified to:

P(k₁, k₂, T) = e^-k₁ * e^(-3*k₂)

The proposal during MCMC sampling makes the equal probability change k₁ and k₂ using the random walk algorithm, where the jump size w is default to 1. When k₁ is smaller than 0, there will be an equal probability to change the tree from this topology to any of another two topologies.

The animation of 10,000 samples is available from below.

More animations with different jump size w are available under the folder note in the repository.

Code

The python 3 source code is available from the Jupyter Notebook file TreeSpace3Taxa.ipynb

Diagnose

• MovieWriter ffmpeg unavailable:

brew install ffmpeg
ffmpeg