Trees & Topologies Chapter 3, Part 2

Trees & TopologiesChapter 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?

2/24/2009 2COMP 790 Trees & Topologies

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)2/24/2009 10COMP 790 Trees & Topologies

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 S2 = 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 different number of ancestors starting with seven ancestors at time 0 and ending with 4 ancestors at a different time.



Probability that a sample of three genes have two ancestors at time r.


Questions?

• Slides are available on the Wiki at: http://compgen.unc.edu/Courses/index.php/Comp_790-087


http://compgen.unc.edu/Courses/index.php/Comp_790-087

http://compgen.unc.edu/Courses/index.php/Comp_790-087

Trees & Topologies Chapter 3, Part 2

Documents

Transcript of Trees & Topologies Chapter 3, Part 2