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

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Trees & Topologies Chapter 3, Part 2

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  1. Trees & TopologiesChapter 3, Part 2

  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? COMP 790 Trees & Topologies

  3. Single Lineage • How many events pass before it coalesces with another gene? COMP 790 Trees & Topologies

  4. 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. COMP 790 Trees & Topologies

  5. 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: COMP 790 Trees & Topologies

  6. Disjoint Subsamples (cont’d) COMP 790 Trees & Topologies

  7. 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. COMP 790 Trees & Topologies

  8. Disjoint Subsamples Example Gene tree of the PHDA1 gene from a sample of Africans and non-Africans. COMP 790 Trees & Topologies

  9. 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. COMP 790 Trees & Topologies

  10. 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) COMP 790 Trees & Topologies

  11. 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) COMP 790 Trees & Topologies

  12. The age of the MRCA for two sequences Now consider the situation of two sequences with S2 = k segregating sites. 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. COMP 790 Trees & Topologies

  14. 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. COMP 790 Trees & Topologies

  15. Probability of going from n ancestors to k ancestors Probability that a sample of three genes have two ancestors at time r. COMP 790 Trees & Topologies

  16. Questions? • Slides are available on the Wiki at:http://compgen.unc.edu/Courses/index.php/Comp_790-087 COMP 790 Trees & Topologies

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