How robust are the predictions of the W-F Model?

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1 How robust are the predictions of the W-F Model? As simplistic as the Wright-Fisher model may be, it accurately describes the behavior of many other models incorporating additional complexity. Many population genetical models have the property that they can be accurately approximated by the Wright-Fisher model provided that we replace the actual (or census) population size N by a different size N e. N e is said to be the effective population size of the population described by the original model. This result is often precise in the sense that the approximation becomes increasingly accurate as N increases. Perspective: The Wright-Fisher model captures some universal features of genetic drift and genealogies that hold in large neutral panmictic populations under very general conditions. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

2 Assuming that an alternative model can be approximated by the Wright-Fisher model, the effective population size can often be calculated from the variance in the change in allele frequencies from one generation to the next. Variance Effective Population Size (Haploid) A model is said to have variance effective population size N e if Var( p t) = where N e may depend on N but not p. p(1 p) N e, It follows that the effective population size determines the strength of genetic drift in such a model: the larger the value of N e, the weaker genetic drift will be. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

3 Many of the results derived using the Wright-Fisher model can be applied to other neutral models if we replace N by N e. Loss of genetic variation in a haploid population H t = ( 1 1 N e ) t H 0. Mean time to fixation of a neutral allele in a haploid population E[T fix ] 2N e { p ln(p) + (1 p) ln(1 p) }. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

4 Similarly, Kingman s coalescent describes the genealogy of a random sample of chromosomes for a large class of population genetical models provided that an appropriate timescale is chosen. Coalescent Effective Population Size (Haploid) A model is said to have coalescent effective population size N e if the rate of coalescence when there are k branches in the tree is equal to ( ) k 1 2 N e : ( ( ) ) k t P(τ k > t) exp. 2 N e It follows that the effective population size determines the rate of coalescence in such models: the larger the value of N e, the more slowly lineages will coalesce. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

5 Within-Generation Fecundity Variance We can illustrate these ideas by considering a large class of models in which each adult gives birth to a random number of offspring. Assumptions: 1 Non-overlapping generations with constant haploid population size N. 2 In each generation, each adult gives birth to a random number of offspring. These numbers are independent across individuals and generations and have mean m > 1 and variance σ 2 <. 3 N of the offspring are then sampled without replacement to form the next generation. 4 Reproduction and survival are independent of the individual s genotype (neutrality). The Wright-Fisher model corresponds to the limiting case where m = and σ 2 = 0. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

6 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

7 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

8 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

9 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

10 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

11 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

12 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

13 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

14 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

15 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

16 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

17 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

18 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

19 Two Examples p 1 = 0.9, p 2 = 0.1 p 0 = 0.75, p 5 = 0.25 gen gen gen gen gen gen gen gen gen gen gen gen Jay Taylor (ASU) Effective Population Size 24 Jan / 23

20 Under these assumptions, a lengthy calculation shows that where E[ p(t) p(t) = p] = 0 Var( p(t) p(t) = p) = p(1 p) N e Variance effective population size with variable offspring numbers N N e ( ) σ2 m m 2 In particular, the effective population size will be smaller and genetic drift stronger in populations in which the within-generation reproductive variance is larger. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

21 Furthermore, provided that N is large, the genealogy of a random sample from a population reproducing according to this model can still be approximated by Kingman s coalescent. However, the distribution of the time until the next coalescent event when there are k lineages is ( ( ) ) k t P(τ k > t) exp 2 where the coalescent effective population size is equal to the variance effective population size: N e Coalescent effective population size with variable offspring numbers N N e ( ) σ2 m m Thus, genealogies will tend to be more shallow in populations in which the within-generation reproductive variance is larger. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

22 Overlapping Generations: The Moran Model Effective population sizes can also be defined for models with overlapping generations. For example, the Moran model makes the following assumptions (Moran 1958): 1 The population contains N haploid individuals. 2 Each individual reproduces independently of the others at rate 1. 3 When an individual reproduces, they give birth to a single offspring. One of the remaining N 1 individuals is then chosen uniformly at random and removed from the population (i.e., dies). Since individuals reproduce independently, on average there are N births and N deaths per unit time, which is also the duration of a single generation under this model. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

23 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

24 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

25 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

26 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

27 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

28 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

29 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

30 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

31 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

32 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

33 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

34 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

35 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

36 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

37 Simulation of the Moran Model (N = 10) past present Jay Taylor (ASU) Effective Population Size 24 Jan / 23

38 Effective population size of the Moran model N e = N/2. As in the previous model, the reduction in the effective population size is due to an increased variance in reproductive success. However, now the variance has two components: 1 By chance, some individuals live longer than others and therefore have more opportunities to reproduce. 2 Since reproduction occurs at random times, even individuals that have the same lifespan can contribute different numbers of offspring to the population. 3 One can show that m = 1 and σ 2 = 2, which then gives the above result. More general results are available for age- and stage-structured models (Felsenstein 1971; Charlesworth 1980), but the important principle is that variance in reproductive success reduces N e. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

39 Changing Population Size In some cases, we can define an effective population size even when the census population size fluctuates from generation to generation. For example, consider the following modification of the W-F model: 1 Non-overlapping generations with N t haploid adults in generation t. 2 The population sizes N 1, N 2, are independent random variables with distribution P(N t = n) = q n. 3 Each individual alive in generation t + 1 independently chooses its parent uniformly at random and with replacement from the preceding generation. Here we are interested in the long-term effective population size, which will govern the amount of variation maintained in the population. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

40 Effective population size with changing population size The effective population size of the Wright-Fisher model with fluctuating population size is equal to the weighted harmonic mean of the census population size ( ) 1 q n N e =. n n Because the harmonic mean is very sensitive to small values, occasionally severe population contractions can greatly reduce the effective population size. This result also holds for more realistic population dynamical models, e.g., when N t+1 depends on N t according to a Markov chain. For the Wright-Fisher approximation to be accurate, we must also require that the minimal census population size is not too small (N t 100). Jay Taylor (ASU) Effective Population Size 24 Jan / 23

41 Example: Suppose that a haploid population contains 1000 adults in good years, but is reduced to 50 adults when environmental conditions deteriorate. If, on average, poor years occur once per decade, then the effective population size is N e = ( ) If, instead, good years occur only once per decade, then the effective population size is N e = ( ) Moral: Population crashes and declines have disproportionately strong impact on effective population size and therefore on genetic variation. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

42 Sex Ratio and Diploid Populations We can easily extend the Wright-Fisher model to a randomly mating diploid populations with two sexes. 1 Overlapping generations with N diploid individuals: N f females and N m males. 2 Here we consider an autosomal locus, so each individual carries two alleles, one inherited from their mother and one from their father, chosen as follows. 3 Each individual, male or female, alive in generation t + 1 chooses an allele uniformly at random and with replacement from a female alive in the previous generation. Independently, they also choose an allele uniformly at random and with replacement from a male alive in the previous generation. Although the frequency of A 1 can differ between males and females, biparental reproduction will prevent this difference from becoming large. Thus it suffices to consider the population-wide frequency. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

43 The effective population size is defined slightly differently for diploid populations: Var(δ(t) p(t) = p) = p(1 p) 2N e. The factor of two which appears in the denominator accounts for the fact that each individual contains two copies of an autosomal locus. Accordingly, we have to use 2N e rather than N e in all of the results derived from the haploid Wright-Fisher model. Effective population size of a two-sex model N e = 4N f N m N f + N m With a 1:1 sex ratio, N e = N. However, skewed sex ratios will reduce the effective population size. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

44 Ploidy and Mode of Transmission All else being equal, sex chromosomes and organelles (mtdna, ptdna) will have smaller effective population sizes than the autosomes and therefore be more strongly affected by genetic drift. For X -linked loci For Y -linked loci N e = 9N f N m 2N f + 4N m N e = Nm 2. Caveat: In many cases, there will be other sex-specific forces acting that change these predictions, e.g., differences in reproductive success variance (sexual selection) between the sexes, as well as differences in selection and recombination between different regions of the genome. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

45 The Effective Size of a Metapopulation Under some conditions, we can define the effective size of a subdivided population. Consider the following version of Wright s island model (Wright 1940): 1 The population is subdivided into D subpopulations, each containing N haploid individuals. 2 Each individual reproduces at rate 1, independently of all others, and gives birth to a single offspring. 3 With probability (1 m), the offspring remains within the subpopulation where it was born, in which case one of the N adults in that subpopulation is chosen at random and dies. 4 Otherwise, with probability m, the offspring migrates to another subpopulation, which is chosen uniformly at random, where it replaces a random chosen adult. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

46 Although structured models usually cannot be simplified to the Wright-Fisher model, if D N, then we can exploit the separation of timescales between the fluctuations of the global frequency of A 1 (slow) and the fluctuations within demes (fast) to obtain: Effective population size in the island model N e = D ( N + 1 m ). m Effective Population Size of the Island Model (D =1000) The magnitude of N e depends on whether Nm is less than or greater than 1. When Nm 1, the population is nearly panmictic and N e DN. When Nm 1, the demes are isolated and N e D/m DN. N e isolation N =100 N =50 N =10 panmixis migration rate (m) Jay Taylor (ASU) Effective Population Size 24 Jan / 23

47 Example: Genetic drift at an eye color locus in experimental populations of D. melanogaster Source: P. Buri (1956) Gene frequency in small populations of mutant Drosophila. Evolution 10: Jay Taylor (ASU) Effective Population Size 24 Jan / 23

48 Estimation of Effective Population Size The effective population size can be estimated from genetic data using several different approaches: If time series data on allele frequencies is available, then the variance effective population size can be estimated directly using ˆN e = p(1 p) Var( p). Effective population size can also be estimated from the observed amount of genetic variation in a sample using the expected amount of variation at mutation-drift balance, e.g., under the infinite sites model of mutation, ˆN e = π 4µ, where π is the nucleotide diversity and µ is the mutation rate per site per generation. If sequence data is available, the coalescent effective population size can be estimated jointly with the gene tree using Bayesian methods (e.g., BEAST). Jay Taylor (ASU) Effective Population Size 24 Jan / 23

49 As the following table makes clear, for many species the estimated effective population size is much smaller than the census population size. Source: B. Charlesworth (2009) Effective population size and patterns of molecular evolution and variation. Nat. Genet. Rev. 10: Jay Taylor (ASU) Effective Population Size 24 Jan / 23

50 Caveats It is important to distinguish between short-term and long-term averages of N e. The variance method, for example, gives a short-term estimate, while estimates based on mutation-drift balance usually reflect long-term averages. If the short-term and long-term averages are very different, then we should be careful in using N e to make predictions about evolution. Different parts of the genome may have very different estimates of N e. If this is due to selection, either direct or at linked sites, then again we need to be cautious in how we use N e. Jay Taylor (ASU) Effective Population Size 24 Jan / 23

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