The Equilibrium Allele Frequency Distribution for a Population with Reproductive Skew

Abstract

We study the population genetics of two neutral alleles under reversible mutation in a model that features a skewed offspring distribution, called the L-Fleming-Viot process. We describe the shape of the equilibrium allele frequency distribution as a function of the model parameters. We show that the mutation rates can be uniquely identified from this equilibrium distribution, but the form of the offspring distribution cannot itself always be so identified. We introduce an estimator for the mutation rate that is consistent, independent of the form of reproductive skew. We also introduce a two-allele infinite-sites version of the L-Fleming-Viot process, and we use it to study how reproductive skew influences standing genetic diversity in a population. We derive asymptotic formulas for the expected number of segregating sites as a function of sample size and offspring distribution. We find that the WrightFisher model minimizes the equilibrium genetic diversity, for a given mutation rate and variance effective population size, compared to all other L-processes. MANY questions in population genetics concern the role of demographic stochasticity and its interaction with mutation and selection in determining the fates of allelic types. The foundational work of Fisher, Wright, Haldane, Kimura (Wright 1931;Haldane 1932;Fisher 1958;Kimura 1994), and others has been instrumental in shaping our intuition about the powerful role that genetic drift plays in evolution and especially its role in maintaining diversity. This classical theory, which views genetic drift as a strong force, emanates from the Wright-Fisher model of replication and its large-population limit, the Kimura diffusion (Kimura 1955). The diffusion approximation has been particularly well studied, not only because it is mathematically tractable, but also because it is robust to variation in many of the underlying model details. Many discrete population-genetic models, including a large number of Karlin-Taylor and Cannings processes (Karlin and McGregor 1964;Cannings 1974;Ewens 2004), share the same diffusion limit as the Wright-Fisher model, and they therefore exhibit qualitatively similar behavior.Nevertheless, Kimura's classical diffusion is not appropriate in every circumstance. Its central assumption is the absence of skew in the reproduction process-that is, the assumption that no single individual can contribute a sizable proportion to the composition of the population in a single generation. Recent studies have suggested that this assumption is violated in several species, especially in marine taxa but also including many types of plants (Beckenbach 1994;Hedgecock 1994), whose mode of reproduction involves a heavy-tailed offspring distribution.While the number of empirical studies on heavy-tailed offspring distributions is limited, there is a rich mathematical theory to describe the dynamics of populations with heavy reproductive skew. Beginning with Cannings' (1974) paper on neutral exchangeable reproduction processes, this literature has le...

Selection-Like Biases Emerge in Population Models with Recurrent Jackpot Events

“…One approach to account for these jackpot events is to consider an effective model where, in each generation, the population is sampled from an effective offspring number distribution with a broad, power-law tail. While such extensions of the Wright-Fisher diffusion process to skewed offspring distributions have been formally constructed (Donnelly and Kurtz 1999;Pitman 1999;Bertoin and Le Gall 2003;Berestycki 2009;Griffiths 2014), also including selection and mutations (Etheridge et al 2010;Der et al 2012;Foucart 2013;Der and Plotkin 2014;Baake et al 2016), we still lack explicit finite time predictions for the probability distribution of allele frequency trajectories. My goal here is to fill this gap for the particular case of the Luria-Delbrück jackpot distribution, by characterizing the allele frequency process with and without selection in such a way that it can be generalized, intuitively understood, and integrated in time.…”

Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

“…length. This does not contradict the unidentifiability claim of Der and Plotkin [2014], because the authors only consider independent draws from π Λ . In contrast, in our setting it is information about transition densities p Λ ∆ (x, y) which facilitates posterior consistency.…”

“…The fact that π δ 0 (x) = π δ 1 (x) when θ = 1 in Example 1 was pointed out by Der and Plotkin [2014] as proof of the fact that Λ-measures cannot in general be uniquely identified from independent draws from π Λ (x). Our calculations illustrate that inference suffers from low power and poor stability even when θ = 1 if all observations are contemporaneous.…”

“…. The stationary law π Λ (x) is known in the parent-independent, two-allele case for both of these atoms [Der and Plotkin, 2014]:…”

Selection-like biases emerge in population models with recurrent jackpot events

“…While extensions of the Wright-Fisher diffusion process to capture skewed offspring numbers have been formally constructed [2,3,14,20], also including selection and mutations [1,10,11,16,18], we still lack explicit finite time predictions for the probability distribution of allele frequency trajectories. Our goal here is to fill this gap for the particular case of the Luria-Delbrück jackpot distribution, by characterizing the allele frequency process in such a way that it can be easily generalized, intuitively understood and integrated in time.…”