Discrete mugration model

Recap: Bayesian Phylogenetic Inference

The usual phylogenetic posterior is:

$$P(T,\mu,\theta|A,L) = \frac{1}{P(A)} P(A|T,\mu) P(T|\theta)P(\mu)P(\theta)$$

where

• $A$ is a sequence alignment,
• $T$ is the tree.

Inference: Modified tree likelihood

The standard phylogenetic posterior is modified:

\begin{align} P(T,\mu,\theta|A,L) =& \frac{1}{P(A)P(L)} P(A|T,\mu)P(L|T,M)\\ &\times P(T|\theta)P(\mu)P(\theta) \end{align}

where

• $L$ are the sampled locations, and
• $M$ is a matrix specifying the random walk.

Note the similarity between the two tree likelihood terms.

Mugration models treat location as just another trait/character.

Sampling assumption

A very important assumption made by the mugration model posterior:

Samples are assumed to be collected in a manner that is blind to their location.

• Mugration models use sample location as data.
• Just as for genetic data, non-random sampling procedures will bias results.

Equivalent population genetic model

A helpful way to visualise the mugration model is to imagine its effect on the population as a whole:

• Mugration => stochastically varying subpopulation sizes.
• A "neutral" model.

Continuous extensions

Use a contiunous diffusion process in place of the discrete random walk:

Lemey et al., MBE, 2010

Bouckaert, PeerJ, 2016

Essential features of mugration model remain, including sensitivity to sampling.

Structured Wright-Fisher Model

• Model as described by Notohara, 1990.
• Island populations are held constant by respective carrying capacities.

Structured Coalescent

• Backwards-in-time process that generates both the tree and ancestral locations.

• Parameterized by migration rates and (sub)population sizes.

Inference: Modified tree prior

Again, the standard phylogenetic posterior is modified:

\begin{align} P(T,\mu,\theta|A,L) &= \frac{1}{P(A)} P(A|T,\mu)\\ &\times P(T,C|\vec{N},\bar{M},L)P(\mu)P(\theta) \end{align}

where

• $L$ are the sampled locations,
• $\bar{M}$ is the migration rate matrix, and
• $C$ are the ancestral locations on the tree.

The sample locations and SC model affect the tree prior.

The shape of the tree is affected by structure.

Sampling assumption

• The coalescent tree prior is explicitly conditioned on the sample times
• Similarly, the structured coalescent tree prior is conditioned on sample locations.

The strucured coalescent makes no assumption about the manner in which samples are collected with respect to location.

• Sample distribution not used as data.
• Uneven sampling can reduce inference power, but will not bias results!

Birth-death Migration Model

• Introduced by Kühnert et al. (MBE 2016).
• A birth-death model of population dynamics in which individuals are permitted to change location due to discrete migration events.
• Migrations may be correlated with births, but not deaths.
• Sampling process explicitly modelled.
• Birth and death rates may be location-dependent: not "neutral"! (Tree shape affected by structure.)
• Inference is performed using modified tree prior.

Discrete Phylogeography

• Very well supported, BEAUti analysis setup.
• Tutorial on beast2.org/tutorials.
• Very fast, allows inference of which migrations are necessary to describe data.
• Prone to sampling biases.

Discrete Phylogeography

DensiTree output:

Continuous Spherical Phylogeography

• Also well supported and BEAUti analysis setup.
• Tutorial on beast2.org/tutorials.
• Output can be summarized using Spread and visualized using Google Earth.
• Prone to sampling biases.

Continuous Spherical Phylogeography

Google Earth visualization example:

Structured Coalescent (Full model)

• Newer analysis option, BEAUti setup.
• Tutorial at beast2.org/tutorials.
• No built-in assumptions regarding sampling procedure.
• More computationally demanding than mugration models, only smaller numbers of demes are feasible.

Structured Coalescent (Full model)

Structured Coalescent (Approximation)

• Very new analysis option, NO BEAUti setup.
• Temporary tutorial at gihub.com/tgvaughan/MultiTypeTree/wiki.
• Approximation cuts down on the computational demands of the SC model, allowing many more locations to be considered.
• Produces very similar results to MultiTypeTree, but only samples internal node locations (not mid-edge locations).

Structured Coalescent (Approximation)

Comparisson between mugration implementation and full and approx. SC models.

De Maio et al., PLoS Genetics, 2015

Structured Coalescent (Approximation)

Posterior comparisons:

De Maio et al., PLoS Genetics, 2015

Birth-death Migration Model

• While capable of performing inference, BDMM has not yet been officially released.
• Information can be found on the GitHub repository: github.com/denisekuehnert/bdmm.

Birth-death Migration Model

Teaser: time and space-dependent rate parameter estimation!

Kühnert et al., MBE, 2016