The genealogical history of a population The coalescent process. Identity by descent Distribution of pairwise coalescence times

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1 The coalescent The genealogical history of a population The coalescent process Identity by descent Distribution of pairwise coalescence times Adding mutations Expected pairwise differences Evolutionary and mutational variance The n-coalescent The number of segregating sites Testing neutrality Tajima s D statistic Other tests of neutrality Factors affecting test power Simple deviations from the neutral model Population growth Population structure Simulation and inference in the coalescent Limitations of the coalescent Copyright: Gilean McVean, 00

2 Sample variance and evolutionary stochasticity Parameters Population Sample N e u Parameter estimation Assuming that the population conforms to the Fisher- Wright model, what can I deduce about the underlying evolutionary parameters from the sample? Hypothesis testing Can I reject the hypothesis that the sample is taken from a population for which the neutral model is an accurate description of the underlying evolutionary process? Copyright: Gilean McVean, 00

3 Genes in populations Ancestry of current population Copyright: Gilean McVean, 00 3

4 Samples in populations The genealogical history of a sample Copyright: Gilean McVean, 00 4

5 The coalescent process Most recent common ancestor (MRCA) coalescence Ancestral lineages time Copyright: Gilean McVean, 00 5

6 Identity by descent and the coalescent Probability from same parent (coalescence) N e Probability from different parents N e Probability of coalescence t generations ago t N e N e Not coalesced for first t- generations Coalesce in next generation Copyright: Gilean McVean, 00 6

7 The time to the MRCA for two alleles E[ T MRCA ] N e N e N e 4N e 6N e 63% of outcomes have T MRCA < N e Adding mutations Genealogy DNA sequences E[ π] u E[ T MRCA ] 4N e u Copyright: Gilean McVean, 00 7

8 Variance in pairwise differences Variance due to A) Distribution of coalescence times (geometric) B) Randomness of the mutation process (Poisson) Pr{ k + θ θ 4N u mutations} e θ + θ k Probability θ 0. θ θ Number of differences Copyright: Gilean McVean, 00 8

9 The n-coalescent time Genealogy Sample Assumptions A) Lineages coalesce independently B) No more than one coalescence can occur in a single generation C) The time-scale is so large, that it can be represented as continuous In effect N e Copyright: Gilean McVean, 00 9

10 The time to coalescence with n sequences Pr{coalescence given n lineages} n( n ) N e Number of pairs of lineages Probability of a given pair coalescing E[ T co ] 4Ne n( n ) E[no. mutations] 4Ne n( n ) n u Time Total mutation rate θ n Copyright: Gilean McVean, 00 0

11 The number of segregating sites The coalescent is a Markovian process The times between coalescent events are independent of each other The expected number of segregating sites in the entire sample is the sum over coalescent events [ S] n θ i i E Watterson (975) Relative value E[S] E[ T MRCA ] Sample size Copyright: Gilean McVean, 00

12 The variance in the number of segregating sites The number of segregating sites is a compound distribution Var( S ) ue[ T ] + u Var[ tot T tot Due to the independence of successive coalescent events, the variances in coalescence times are additive Var[ T tot ] Var[ S] θ n i n i i i Var[ T The full distribution can be calculated by a simple recursion (Tavaré, 984) + θ co n i E[ S] 7.7 ( i)] i ] θ n S Copyright: Gilean McVean, 00

13 Testing neutrality Tajima s D statistic E[ π] E[ S] θ n θ i i D π S / a n a n Var( π S / a ) n Tajima (989) n i i π < S / a n π > S / an D < 0 D > 0 Copyright: Gilean McVean, 00 3

14 An example: human mtdna Ingman et al. (000) 5 complete molecules from a worldwide sample (linguistic groups) 5 segregating sites excluding D-loop π 44. a5 4.5 S / a5 5.3 Vˆ ( d) D Probability of observing such an extreme value under neutrality 0.0 Human mtdna have an excess of low-frequency variants Population growth, selection, or sampling? Copyright: Gilean McVean, 00 4

15 Other tests of neutrality Using frequency spectrum information Expected number of external mutations η ] E[ e θ Fu and Li (993) Outgroup Expected number of derived alleles at frequency i in n θ E[ξ ] i i Fu (995) Copyright: Gilean McVean, 00 5

16 Human mtdna is consistently non-neutral Fu and Li D* statistic ( n ) n S anηs P < 0.0 Var( D*) Frequency spectrum of segregating sites Observed Expected Frequency rare allele Copyright: Gilean McVean, 00 6

17 Factors affecting test power The number of mutations in the sample is of critical importance In general, sequencing a large region is more important than sequencing many individuals Recombination reduces the possibility of drawing trees from sequences, but evens out evolutionary stochasticity T tot 4N e a n Sequence position Copyright: Gilean McVean, 00 7

18 Simple deviations from neutrality Population growth N e Exponential growth of % per generation N t Nt + α Generations before present Pairwise coalescence times Pr{coalesce at t} t Ne( t) i 0 Ne( i) Probability of coalescence t generations ago Growing population Constant population Copyright: Gilean McVean, 00 8

19 The effect of population growth on genealogies Long external branches Most segregating sites singletons Low pairwise diversity Negative Tajima D statistic Growth rate of 0.% per generation in 00,000 years 0 Frequency in sample of 5 sequences Copyright: Gilean McVean, 00 9

20 Simulation in the coalescent Joint simulation of mutations and genealogies Active lineages Coalescence Mutation n( n ) nu N e A) Rescale time τ t N e B) Time until next event exponentially distributed with rate equal to the sum of all possibilities φ( τ) λe λτ n( n ) λ + nθ C) Probability next event is a mutation probability of mutation divided by total probability of events Pr{mutation} θ θ + n Copyright: Gilean McVean, 00 0

21 Population structure m m migration Pr{next event is a migration} M M + θ + n - M 4N e m Copyright: Gilean McVean, 00

22 Likelihood inference in the coalescent θ k 5 ASK What is distribution of times to the MRCA given the data? Pr{ t k, θ} Pr{ t & k θ} Pr{ k θ} ( θτ) e + θ τ(+θ) θ + θ / where τ t N e Conditional on k, θ 5 Unconditional distribution τ Copyright: Gilean McVean, 00

23 Conditional simulation A B C C M M C C M C C M Pr{ A} Pr{ B} 3 ( θ + 3) ( 3 6) ( 3 6) ( 3θ + 6) ( θ + ) θ θ + θ θ + 3 θ θ + ( 3) ( θ + 6) ( 3θ + 6) ( ) ( θ + ) θ θ + Pr{A data} < Pr{B data} Want to sample histories (genealogy plus mutations) in proportion to their likelihood given the data e.g. E[ T MRCA data] G T N MRCA i T ( G) Pr{G}Pr{data G} MRCA ( i) Copyright: Gilean McVean, 00 3

24 Strengths and weaknesses of coalescent theory Very flexible, simulations are easy to implement irrespective of population and mutational models Deals explicitly with basic unit of empirical population genetics research Full likelihood analysis within the coalescent framework uses all possible information BUT Natural selection difficult to incorporate Coalescence depends on allelic state and rest of population Full likelihood inference is computationally intensive and may be unnecessary Copyright: Gilean McVean, 00 4