Trees & Topologies Chapter 3, Part 2

A simple lineage Consider a given gene of sample size n. How long does it take before this gene coalesces with another gene in the sample?

Single Lineage How many events pass before it coalesces with another gene?

Disjoint subsamples Consider a sample of size n that is divided into two disjoint subsamples, A and B of sizes k and n-k, respectively.

Disjoint Subsamples (cont’d) The probability that all genes in A find a MRCA coalescing with any gene in B is: The probability that one of the two samples finds a MRCA before coalescing with members of the other sample is:

Disjoint Subsamples (cont’d)

Jump Process of Disjoint Subsamples Jump processes: – (i,j) -> (i-1, j) with probability (i+1)/(i+j) – (i,j) -> (i,j-1) with probability (j-1)/(i+j) Process starts in (k, n-k) and continues until (1,j) for some j. Eventually jumps to (0,j) for some j and finally reaches (0,1), where 0 denotes that sample A has been fully absorbed into B.

Disjoint Subsamples Example Gene tree of the PHDA1 gene from a sample of Africans and non-Africans.

A sample partitioned by a mutation Now, consider a sample of size n where a polymorphism divides the sample into two disjoint subsamples, A and B, of size k and n-k, respectively.

Comparing the mean values Jump processes: (i,j) -> (i-1,j) with probability i/(i+j-1) (i,j) -> (i, j-1) with probability (j-1)/(i+j-1)

Unknown ancestral state If we do not know which of the two alleles is older, we have a slightly different situation. Probability that an allele found in frequency k out of n genes is the oldest is k/n. Probability that A carries the mutant allele is 1-k/n = (n-k)/n. Jump processes become: – (i,j) -> (i-1,j) with probability i/(i+j) – (i,j) -> (i, j-1) with probability j/(i+j)

The age of the MRCA for two sequences Now consider the situation of two sequences with S 2 = k segregating sites.

Probability of going from n ancestors to k ancestors Probability of different number of ancestors starting with seven ancestors at time 0.

Probability of going from n ancestors to k ancestors Probability of different number of ancestors starting with seven ancestors at time 0 and ending with 4 ancestors at a different time.

Probability of going from n ancestors to k ancestors Probability that a sample of three genes have two ancestors at time r.

Questions? Slides are available on the Wiki at: omp_ omp_