The Coalescent with Recombination
Lecturer: Alexei Drummond

Recombination

  • Occurs in eukaryotes, bacteria and viruses.
  • Eukaryotic recombination occurs during meiosis via chromosomal crossover.
  • Bacterial recombination occurs via
    • phage-mediated transduction,
    • natural transformation, and
    • conjugation.
  • Viral recombination occurs when multiple strains infect a single cell. (Example: reassortment in influenza.)
Extremely important for phylogenetics: affects the validity of the "tree" assumption!

Wright-Fisher model with recombination

  • Consider an WF model with male and female diploid individuals.
  • Focus on a small segment of the genome.
  • Each child selects 1 male and 1 female parent uniformly at random.
  • With probability $r$ (which depends on the segment size) the two chromosome pairs of a parent are recombined before being passed on.

Wright-Fisher model with recombination

  • Since the specific pairing of chromosomes only matters over a single generation, in the long term the haploid approximation is very good:
  • Each child in $i+1$ selects a parent uniformly at random from generation $i$.
  • With probability $r$ an additional parent is selected.
  • In this case, a breakpoint chosen uniformly on the chromosome, and everything to either the left or right replaced by the homologous section of the second parent's chromosome.
  • (Equivalent to selecting 2 parents for each child and letting the resulting sequences recombine with some probability.)

Coalescent with recombination

For fixed recombination rate $\rho=r/g$ in the limit $r\ll 1$, $g\ll 1$ and $N\gg 1$, the genealogical process is the coalescent with recombination (Hudson, 1983):

  • Coalescence rate: $T_c(k)=\binom{k}{2}\frac{1}{Ng}$
  • Recombination rate: $T_r(k)=\rho k$.
  • Recombination break points chosen uniformly along sequence: everything to the left descends from one parent, everything else to the other.
  • Each site possess a local tree.
  • Local trees may find MRCAs (grey sites) before grand (G)MRCA of the process.

CwR as a birth-death process

The number of surviving ancestral lineages $k$ evolves under the birth-death process corresponding to the following pair of reactions:

\begin{align*} 2L &\overset{\chi}{\longrightarrow} L\\ L &\overset{\rho}{\longrightarrow} 2L \end{align*}

where $\chi=1/Ng$. The deterministic approximation for the evolution of $\langle k\rangle=\langle k\rangle$ is \begin{align*} \partial_t \langle k\rangle &\simeq -\frac{1}{2N}\langle k\rangle(\langle k\rangle-1) + \rho\langle k\rangle\\ &\propto \langle k\rangle(1-\langle k\rangle/k_c) \end{align*} where $k_c=1+2N\rho$.

$\langle k\rangle$ stabilizes at $k_c$, but noise eventually drives $k$ to 1 (GMRCA).

CwR example ARG

(Simulated using this MASTER script.)

Coalescent with gene conversion

A gene conversion event refers to the replacement of a single contiguous stretch of sequence with a homologous stretch from a different parent. Incorporated into a modified CwR process by Wiuf and Hein (2000).

  • Allows similar patterns of site ancestry but with fewer events.
  • Model applicable to prokaryotic recombination.
  • Coalescence and recombination rates as for CwR.
  • Conversions initiate at randomly chosen starting sites and extend for $d$ sites where $P(d)=(1-\delta^{-1})^{d-1}\delta^{-1}$ (geometric distribution).

Sequentially Markov Coalescent (SMC)

McVean and Cardin (2005)
  • Sites between breakpoints exponentially distributed with rate $\rho T$ where $T$ is the total length of the current local tree.
  • Neglects some possible recombinations, e.g. those that do not affect the data.

Population inference using SMC

  • Li &l; Durbin (2011) developed an SMC-based HMM on pairs of alignments.
  • Baum-Welch used to estimate parameters and hidden states (local TMRCAs).
  • TMRCA distribution used to produce estimates of population size dynamics.
Li and Durbin (2011)

Bayesian inference of ARGs

Can easily write down an expression for the posterior distribution for the ARG given an alignment:

$$P(G,\rho,N,\mu|A) = \frac{1}{P(A)}P(A|G,\mu)P(G|\rho,N)P(\rho,N,\mu)$$
  • $G$ is the recombination graph,
  • $\rho$ is the recombination rate.

In practice, this is non-trivial since:

  1. the likelihood for $G$, $P(A|G,\mu)$, is invariant under many features of $G$,
  2. the likelihood surface often contains many distinct peaks, and
  3. the state space of ARGs is huge: much larger than for trees.

Despite this, many approximate algorithms exist.

Algorithms for Bayesian ARG inference

Summary

  • Recombination is an ever-present feature of real evolution.
  • Ignoring recombination when it is present can lead to biased phylogenetic inferences.
  • Coalescent models for recombination produce Ancestral Recombination Graphs (ARGs).
  • Coalescent models are derivable from modified Wright-Fisher process.
  • Sequentially Markovian approximation can be used to construct an HMM where TMRCAs are hidden states.
  • Full Bayesian inference of ARGs is an ongoing research topic.

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