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Models and their benefits.

Models and their benefits. Models + Data probability of data (statistics...) probability of individual histories hypothesis testing parameter estimation. Wright-Fisher Model of Population Reproduction. Haploid Model.

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Models and their benefits.

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  1. Models and their benefits. • Models+ Data • probability of data (statistics...) • probability of individual histories • hypothesis testing • parameter estimation

  2. Wright-Fisher Model of Population Reproduction Haploid Model i. Individuals are made by sampling with replacement in the previous generation. ii. The probability that 2 alleles have same ancestor in previous generation is 1/2N Assumptions Constant population size No geography No Selection No recombination Diploid Model Individuals are made by sampling a chromosome from the female and one from the male previous generation with replacement

  3. 10 Alleles’ Ancestry for 15 generations

  4. Waiting for most recent common ancestor - MRCA Distribution until 2 alleles had a common ancestor, X2?: P(X2 > j) = (1-(1/2N))j P(X2 > 1) = (2N-1)/2N = 1-(1/2N) P(X2 = j) = (1-(1/2N))j-1 (1/2N) j j 2 2 1 1 1 1 1 1 2N 2N 2N Mean, E(X2) = 2N. Ex.: 2N = 20.000, Generation time 30 years, E(X2) = 600000 years.

  5. P(k):=P{k alleles had k distinct parents} 1 1 2N Ancestor choices: k -> k k -> any k -> k-1 k -> j (2N)k 2N *(2N-1) *..* (2N-(k-1)) =: (2N)[k] Sk,j - the number of ways to group k labelled objects into j groups.(Stirling Numbers of second kind. For k << 2N:

  6. Expected Height and Total Branch Length Branch Lengths Time Epoch 1 2 1 2 1 1/3 3 2/(k-1) k Expected Total height of tree: Hk= 2(1-1/k) i.Infinitely many alleles finds 1 allele in finite time. ii. In takes less than twice as long for k alleles to find 1 ancestors as it does for 2 alleles. Expected Total branch length in tree, Lk: 2*(1 + 1/2 + 1/3 +..+ 1/(k-1)) ca= 2*ln(k-1)

  7. 6 Realisations with 25 leaves Observations: Variation great close to root. Trees are unbalanced.

  8. Sampling more sequences The probability that the ancestor of the sample of size n is in a sub-sample of size k is Letting n go to infinity gives (k-1)/(k+1), i.e. even for quite small samples it is quite large.

  9. Infinite Site Model Final Aligned Data Set:

  10. Recombination-Coalescence Illustration Copied from Hudson 1991 Intensities Coales.Recomb. 0  1 (1+b) b 3 (2+b) 6 2 3 2 1 2

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