### Trees & Topologies Chapter 3, Part 2

```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?
2.5
2
1.5
1
0.5
0
1
2/24/2009
2
3
4
COMP 790 Trees & Topologies
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
2
Single Lineage
• How many events pass before it coalesces
with another gene?
8
7
6
5
4
3
2
1
0
1
2/24/2009
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
COMP 790 Trees & Topologies
3
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.
2/24/2009
COMP 790 Trees & Topologies
4
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:
2/24/2009
COMP 790 Trees & Topologies
5
Disjoint Subsamples (cont’d)
2/24/2009
COMP 790 Trees & Topologies
6
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.
2/24/2009
COMP 790 Trees & Topologies
7
Disjoint Subsamples Example
Gene tree of the PHDA1 gene from a sample of
Africans and non-Africans.
2/24/2009
COMP 790 Trees & Topologies
8
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.
2/24/2009
COMP 790 Trees & Topologies
9
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
COMP 790 Trees & Topologies
10
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)
2/24/2009
COMP 790 Trees & Topologies
11
The age of the MRCA for two
sequences
Now consider the situation of two sequences
with S2 = k segregating sites.
2/24/2009
COMP 790 Trees & Topologies
12
Probability of going from n
ancestors to k ancestors
• Probability of different number of ancestors
starting with seven ancestors at time 0.
2/24/2009
COMP 790 Trees & Topologies
13
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.
2/24/2009
COMP 790 Trees & Topologies
14
Probability of going from n
ancestors to k ancestors
Probability that a sample of three genes have
two ancestors at time r.
2/24/2009
COMP 790 Trees & Topologies
15
Questions?
• Slides are available on the Wiki at:
http://compgen.unc.edu/Courses/index.php/C
omp_790-087
2/24/2009
COMP 790 Trees & Topologies
16
```