Haldane’s formula in Cannings models: the case of moderately strong selection

Boenkost, Florin; Casanova, Adrián González; Pokalyuk, Cornelia; Wakolbinger, Anton

doi:10.1007/s00285-021-01698-9

Cited by 8 publications

(5 citation statements)

References 26 publications

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“…For mutants with substantial selective advantage, namely for st ≫ 1 /K here, extinction events happen when mutants are still rare, due to stochastic fluctuations associated to sampling. Indeed, in a well-mixed population, if their fraction reaches a given threshold, beneficial mutants are very likely to fix in the end [78, 79]. Therefore, the branching process approach yields accurate results on extinction probabilities and extinction times provided that K ≫ 1 and Kst ≫ 1.…”

Section: Methodsmentioning

confidence: 99%

Frequent asymmetric migrations suppress natural selection in spatially structured populations

Abbara

Bitbol

2023

Preprint

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Natural microbial populations often have complex spatial structures. This can impact their evolution, in particular the ability of mutants to take over. While mutant fixation probabilities are known to be unaffected by sufficiently symmetric structures, evolutionary graph theory has shown that some graphs can amplify or suppress natural selection, in a way that depends on microscopic update rules. We propose a model of spatially structured populations on graphs directly inspired by batch culture experiments, alternating within-deme growth on nodes and migration-dilution steps, and yielding successive bottlenecks. This setting bridges models from evolutionary graph theory with Wright-Fisher models. Using a branching process approach, we show that spatial structure with frequent migrations can only yield suppression of natural selection. More precisely, in this regime, circulation graphs, where the total incoming migration flow equals the total outgoing one in each deme, do not impact fixation probability, while all other graphs strictly suppress selection. Suppression becomes stronger as the asymmetry between incoming and outgoing migrations grows. Amplification of natural selection can nevertheless exist in a restricted regime of rare migrations and very small fitness advantages, where we recover the predictions of evolutionary graph theory for the star graph.

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Section: Methodsmentioning

confidence: 99%

Frequent asymmetric migrations suppress natural selection in spatially structured populations

Abbara

Bitbol

2023

Preprint

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“…We note that the scaling Assumption A1 for s and N reflects the fact that, compared to diffusion approximations when K = 1, in our model selection is stronger than random genetic drift. In Boenkost et al (2021), Assumption A1 with K > 2 is called moderately strong selection and used to prove that the fixation probability of a single mutant in Cannings models is asymptotically equivalent to 2( σ − 1) /v (in our notation) as N → ∞. The first requirement in Assumption A2 implies that (see Remark 3.3).…”

Section: Evolution Of the Phenotypic Mean And Variancementioning

confidence: 99%

“…In Boenkost et al (2021) We denote the values of Ḡ(τ ) and V G (τ ) in the quasi-stationary phase by Ḡ * (τ ) and V * G , respectively. For notational simplicity, we set ν = ν(s, α, N ) = N P sur (e sα ) .…”

Section: Approximations For the Quasi-stationary Phasementioning

confidence: 99%

Polygenic dynamics underlying the response of quantitative traits to directional selection

Götsch¹,

Bürger²

2023

Preprint

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We study the response of a quantitative trait to exponential directional selection in a finite haploid population, both on the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean Θ) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a slightly supercritical Galton-Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation n, conditioned on survival until n, is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution corresponds to the absolutely continuous part W+ of the random variable, typically denoted W, that characterizes the stochasticity accumulating during the mutant's sweep. In bypassing, we prove the folklore result that W+ is approximately exponentially distributed, but in general not exactly so by deriving its second and third moment. A proper transformation yields the approximate mutant frequency dynamics in a Wright-Fisher population of size N. Then we derive explicitly the (approximate) time dependence of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate expressions even for the quasi-stationary phase, when the expected per-generation response has equilibrated. These refine classical results, and they are derived from first principles. We find that Θ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the response is mainly due to sweeps or to small allele-frequency shifts, and the number of segregating sites is an appropriate indicator for it. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested for a Poisson offspring distribution by comprehensive simulations in a Wright-Fisher framework.

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“…where c N ∼ N (a N )/a N ∼ (a N )/ * (a N ) → 0 as N → ∞. Thanks to the monotonicity property (5) this formula is only needed for j ∈ {1, 2} in the previous proof.…”

Section: Proof Of Theorem 1 (Vi) Letmentioning

confidence: 99%

“…The same multinomial scheme with an additional dormancy mechanism is considered in the recent work by Cordero et al [8]. A class of Dirichlet models in the domain of attraction of the Kingman coalescent is also studied in two recent works by Boenkost et al [4,5] with an emphasis on Haldane's formula [14]. We refer the reader to Athreya [1] for some more information on Haldane's formula.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic genealogies for a class of generalized Wright–Fisher models

Huillet¹,

Möhle²

2021

Modern Stochastics: Theory and Applications

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A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities ${W_{1}},\dots ,{W_{N}}$ induced by independent and identically distributed positive random variables ${X_{1}},{X_{2}},\dots $ via ${W_{i}}:={X_{i}}/{S_{N}}$, $i\in \{1,\dots ,N\}$, where ${S_{N}}:={X_{1}}+\cdots +{X_{N}}$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths ${W_{1}},\dots ,{W_{N}}$. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of ${X_{1}}$ is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727–761] for the case when ${X_{1}}$ is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107–139] for models where sampling is performed without replacement from a supercritical branching process.

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Haldane’s formula in Cannings models: the case of moderately strong selection

Abstract: For a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N … Show more

Cited by 8 publications

References 26 publications

Frequent asymmetric migrations suppress natural selection in spatially structured populations

Frequent asymmetric migrations suppress natural selection in spatially structured populations

Polygenic dynamics underlying the response of quantitative traits to directional selection

Asymptotic genealogies for a class of generalized Wright–Fisher models

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