Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 45 / 4 / iv

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- Solution iv

Solution

An ancestral recombination graph, or ARG, is the directed ancestry of a sample of chromosomal segments when ancestral lineages can both coalesce and undergo genetic recombination. It records which ancestral material each lineage carries, the times of events and the recombination breakpoints. At any fixed genomic position, following only the ancestral material at that position gives a marginal phylogenetic tree. Different positions can have different trees, while sharing substantial parts of their history. The graph is acyclic in time, although its underlying undirected graph can have loops because a lineage splits backwards and its ancestors can later rejoin.

For the standard neutral, constant-size, randomly mating diploid model, measure time in

2 N_{e}

generations and set

ρ = 4 N_{e} r

, where

r

is the recombination rate across the whole segment. The backward process is a continuous-time Markov jump process on collections of lineages carrying ancestral material:

Each unordered pair of current lineages coalesces at rate one. The parent lineage carries the union of their ancestral material, with common ancestry recorded at overlapping positions. With $k$ lineages the total coalescence rate is $(2 k)$ . The full ARG permits a pair to merge even when its carried material is disjoint; such an event creates no merger in a local tree at that instant.
A lineage splits into two parental lineages at a recombination breakpoint. With a uniform recombination map on a segment of normalized length one, the intensity of breakpoints along a fully ancestral lineage is $ρ /2$ per unit length. If a lineage carries only part of the segment, only cuts leaving ancestral material on both sides need be retained, giving rate $(ρ /2) b_{i}$ , where $b_{i}$ is the span between its leftmost and rightmost ancestral positions. Material to the left goes to one parent and material to the right to the other. A gap within that span still permits a cut that separates two carried pieces.

For a state with

k

lineages, independent exponential event clocks give total rate

R = (2 k) + \frac{ρ}{2} \sum_{i = 1}^{k} b_{i} .

(1)

The next waiting time has exponential distribution with rate

R

; choose its event type in proportion to the corresponding rate. A coalescing pair is uniform among unordered pairs; a recombining lineage is selected with weight

b_{i}

and its breakpoint is uniform over its active span. With a nonuniform genetic map, use the corresponding integrated intensity and map-weighted breakpoint distribution. Initially all sample lineages span the whole segment, so the initial rate is

(2 n) + n ρ /2

. Ancestral material no longer needed after its sample copies have found their local common ancestor can be pruned.

Mutations are placed independently on the resulting ancestral branches, with local rate

θ /2

in coalescent units for a locus having scaled mutation parameter

θ

. Without recombination, the graph reduces to a single Kingman's coalescent tree. With recombination, each individual locus still has that marginal coalescent distribution, but trees at different loci are dependent. The ARG combines backward pairwise mergers with backward recombination splits; local trees are its position-specific projections. The neutral two-locus construction is developed in Hudson's two-locus model.

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 45 / 4 / iv

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Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 45 / 4 / iv

Solution

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