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Fixed Parameters: Population Structure, Mutation, Selection, Recombination,...

Coalescent Theory in Biology www. coalescent .dk. Fixed Parameters: Population Structure, Mutation, Selection, Recombination,. Reproductive Structure. Genealogies of non-sequenced data. Genealogies of sequenced data. CATAGT. CGTTAT. TGTTGT. Parameter Estimation Model Testing.

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Fixed Parameters: Population Structure, Mutation, Selection, Recombination,...

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  1. Coalescent Theory in Biology www. coalescent.dk Fixed Parameters: Population Structure, Mutation, Selection, Recombination,... Reproductive Structure Genealogies of non-sequenced data Genealogies of sequenced data CATAGT CGTTAT TGTTGT Parameter Estimation Model Testing

  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. 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:

  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. 10 Alleles’ Ancestry for 15 generations

  6. Multiple and Simultaneous Coalescents 1. Simultaneous Events 2. Multifurcations. 3. Underestimation of Coalescent Rates

  7. Discrete  Continuous Time tc:=td/2Ne 6 6/2Ne 0 2N 0 1 4 1.0 corresponds to 2N generations 1.0 0.0 2 6 5 3

  8. 1 2 3 4 5 The Standard Coalescent Two independent Processes Continuous: Exponential Waiting Times Discrete: Choosing Pairs to Coalesce. Waiting Coalescing {1,2,3,4,5} (1,2)--(3,(4,5)) {1,2}{3,4,5} 1--2 {1}{2}{3,4,5} 3--(4,5) {1}{2}{3}{4,5} 4--5 {1}{2}{3}{4}{5}

  9. 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)

  10. Effective Populations Size, Ne. In an idealised Wright-Fisher model: i. loss of variation per generation is 1-1/(2N). ii. Waiting time for random alleles to find a common ancestor is 2N. Factors that influences Ne: i.Variance in offspring. WF: 1. If variance is higher, then effective population size is smaller. ii.Population size variation - example k cycle: N1, N2,..,Nk. k/Ne= 1/N1+..+ 1/Nk. N1 = 10 N2= 1000 => Ne= 50.5 iii.Two sexesNe = 4NfNm/(Nf+Nm)I.e. Nf-10Nm -1000 Ne - 40

  11. 6 Realisations with 25 leaves Observations: Variation great close to root. Trees are unbalanced.

  12. 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.

  13. Adding Mutations m mutation pr. nucleotide pr.generation. L: seq. length µ = m*L Mutation pr. allele pr.generation. 2Ne - allele number. Q := 4N*µ -- Mutation intensity in scaled process. Continuous time Continuous sequence Discrete time Discrete sequence 1/L time 1/(2Ne) time sequence sequence mutation mutation coalescence Probability for two genes being identical: P(Coalescence < Mutation) = 1/(1+Q). 1 Q/2 Q/2 Note: Mutation rate and population size usually appear together as a product, making separate estimation difficult.

  14. Three Models of Alleles and Mutations. Finite Site Infinite Allele Infinite Site acgtgctt acgtgcgt acctgcat tcctgcat tcctgcat Q Q Q acgtgctt acgtgcgt acctgcat tcctggct tcctgcat i. Allele is represented by a sequence. ii. A mutation changes nucleotide at chosen position. i. Only identity, non-identity is determinable ii. A mutation creates a new type. i. Allele is represented by a line. ii. A mutation always hits a new position.

  15. Infinite Allele Model 4 5 1 2 3

  16. Infinite Site Model Final Aligned Data Set:

  17. Labelling and unlabelling:positions and sequences 1 2 3 4 5 Ignoring mutation position Ignoring sequence label 1 2 3 5 4 Ignoring mutation position Ignoring sequence label { , , } The forward-backward argument 4 classes of mutation events incompatible with data 9 coalescence events incompatible with data

  18. Infinite Site Model: An example Theta=2.12 2 3 2 3 4 5 5 9 5 10 14 19 33

  19. Impossible Ancestral States

  20. Finite Site Model acgtgctt acgtgcgt acctgcat tcctgcat tcctgcat s s s Final Aligned Data Set:

  21. Diploid Model with Recombination An individual is made by: The paternal chromosome is taken by picking random father. Making that father’s chromosomes recombine to create the individuals paternal chromosome. Similarly for maternal chromosome.

  22. The Diploid Model Back in Time. A recombinant sequence will have have two different ancestor sequences in the grandparent.

  23. 1- recombination histories I:Branch length change 1 2 4 3 2 1 4 3 2 1 4 3

  24. 1- recombination histories II:Topology change 1 2 4 3 2 1 4 3 2 1 4 3

  25. 1- recombination histories III:Same tree 1 2 4 3 2 1 4 3 2 1 4 3

  26. 1- recombination histories IV:Coalescent time must be further back in time than recombination time. c r 1 2 4 3

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

  28. From Wiuf and Hein, 1999 Genetics Age to oldest most recent common ancestor Scaled recombination rate - r 0 kb 250 kb Age to oldest most recent common ancestor

  29. Number of genetic ancestors to the Human Genome Sr– number of Segments E(Sr) = 1 + r time C C C R R R sequence Simulations Statements about number of ancestors are much harder to make.

  30. Applications to Human Genome (Wiuf and Hein,97) 0 260 Mb 0 52.000 *35 0 7.5 Mb 6890 8360 *250 30kb 0 Parameters used 4Ne 20.000 Chromos. 1: 263 Mb. 263 cM Chromosome 1: Segments 52.000 Ancestors 6.800 All chromosomes Ancestors 86.000 Physical Population. 1.3-5.0 Mill. A randomly picked ancestor: (ancestral material comes in batteries!)

  31. 1 2 3 4 1 2 4 3 Ignoring recombination in phylogenetic analysis General Practice in Analysis of Viral Evolution!!! Recombination Assuming No Recombination Mimics decelerations/accelerations of evolutionary rates. No & Infinite recombination implies molecular clock.

  32. Simulated Example

  33. Genotype and Phenotype Covariation: Gene Mapping Decay of local dependency Time Reich et al. (2001) Genetype -->Phenotype Function Dominant/Recessive. Penetrance Spurious Occurrence Heterogeneity genotype phenotype Genotype  Phenotype Sampling Genotypes and Phenotypes Result:The Mapping Function A set of characters. Binary decision (0,1). Quantitative Character.

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