Gene genealogies in haploid populations evolving according to sweepstakes reproduction

TL;DR

The paper addresses how sweepstakes reproduction shapes gene genealogies in haploid populations by deriving continuous-time coalescents that are Kingman, Beta, or Poisson-Dirichlet in the domain of attraction of multiple-merger coalescents. It develops population models with heavy-tailed offspring distributions and random environments, yielding explicit limiting coalescents with time scales $N/\log N$ or $N$ and, when population size fluctuates, time-changed coalescents with a universal form $G(t)=\int_0^t v(s)^{-1}\,ds$. The work shows how upper bounds on offspring and environmental fluctuations affect the coalescent structure (including incomplete Beta and delta$_0$-coalescents) and demonstrates via simulations the extent to which functionals of the ancestral process approximate the limiting coalescent, both with and without conditioning on population ancestry. The findings are significant for inference in highly fecund populations and for understanding how sweepstakes-driven skew in offspring production interacts with demography to shape genetic diversity.

Abstract

Sweepstakes reproduction may be generated by chance matching of reproduction with favorable environmental conditions. Gene genealogies generated by sweepstakes reproduction are in the domain of attraction of multiple-merger coalescents where a random number of lineages merges at such times. We consider population genetic models of sweepstakes reproduction for haploid panmictic populations of both constant ($N$), and varying population size, and evolving in a random environment. We construct our models so that we can recover the observed number of new mutations in a given sample without requiring strong assumptions regarding the population size or the mutation rate. Our main results are {\it (i)} continuous-time coalescents that are either the Kingman coalescent or specific families of Beta- or Poisson-Dirichlet coalescents; when combining the results the parameter $α$ of the Beta-coalescent ranges from 0 to 2, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring an arbitrary individual may produce; {\it (ii)} in large populations we measure time in units proportional to either $ N/\log N$ or $N$ generations; {\it (iii)} incorporating fluctuations in population size leads to time-changed multiple-merger coalescents where the time-change does not depend on $α$; {\it (iv)} using simulations we show that in some cases approximations of functionals of a given coalescent do not match the ones of the ancestral process in the domain of attraction of the given coalescent; {\it (v)} approximations of functionals obtained by conditioning on the population ancestry (the ancestral relations of all gene copies at all times) are broadly similar (for the models considered here) to the approximations obtained without conditioning on the population ancestry.

Figures (10)

• Figure 1: Definition \ref{['def:haplrandomalpha']} and the $\delta_{0}$-Beta$(\gamma,2-\alpha,\alpha)$-coalescent. Comparing $\overline\varrho_{i}(n)$ (black lines) and $\overline\varrho_{i}^{N}$ (recall \ref{['eq:functionals']}) when $N = 10^{3}$, $\kappa = 2$, $c = 1$, and $\gamma$ as shown; black lines are $\overline \varrho_{i}(n)$ for sample size $n$ as shown with rates as in \ref{['eq:ratesarandall']} in Theorem \ref{['thm:haplrandomalpha']}, coloured lines are $\overline \varrho_{i}^{N}(n)$ for a sample from a population of finite size $N$ evolving according to Definition \ref{['hschwpop']} and Definition \ref{['def:haplrandomalpha']} with the potential offspring distributed as in \ref{['eq:40']} and with $\varepsilon_{N} = cN^{\alpha - 2}\log N$ as in \ref{['eq:33']} in Lemma \ref{['lm:cNhaplrandomall']}; the case $\gamma = 1$ is compared to ${\zeta(N)} = N\log N$, and $\gamma = 1/(1 + \text{\sf m})$ to ${\zeta(N)} = N$ with $\text{\sf m}$ as in \ref{['eq:57']} approximating $m_{\infty}$; $\overline \varrho_{i}^{N}(n)$ from $10^{4}$ experiments
• Figure 2: Definition \ref{['def:alpha-random-one']} and the $\delta_{0}$-Beta$(\gamma,2-\alpha,\alpha)$-coalescent. Comparing $\overline\varrho_{i}(n)$ (black lines) and $\overline \varrho_{i}^{N}(n)$ (recall \ref{['eq:functionals']}) when $N=10^{3}$, $\kappa = 2$, $c = 1$, and with $\alpha$, $\gamma$, and sampled size $n$ as shown; black lines are $\overline \varrho_{i}(n)$ with rates as in \ref{['eq:ratesarandall']}, coloured lines are estimates of $\mathds{E}\left[ R_{i}^{N}(n) \right]$ for a sample from a population evolving according to Definition \ref{['hschwpop']} and Definition \ref{['def:alpha-random-one']} with the potential offspring distributed as in \ref{['eq:40']} and with $\overline\varepsilon_{N} = cN^{\alpha - 1}\log N$ as in \ref{['eq:51']} in Lemma \ref{['lm:cNrandomalphaone']}; the case $\gamma = 1$ is compared to ${\zeta(N)} = N\log N$, and the case $\gamma = 1/(1 + \text{\sf m})$ compared to ${\zeta(N)} = N$ with $\text{\sf m}$ as in \ref{['eq:57']} approximating $m_{\infty}$; $\overline \varrho_{i}^{N}(n)$ from $10^{4}$ experiments
• Figure 3: The $\delta_{0}$-Poisson-Dirichlet$(\alpha,0)$-coalescent. Approximations $(\overline \varrho_{i}(n))$ of $\mathds{E}\left[ R_{i}(n) \right]$ (lines) predicted by the $\delta_{0}$-Poisson-Dirichlet$(\alpha,0)$-coalescent compared to $\overline\varrho_{i}^{N}(n)$ (circles in c and d) when the population evolves according to Definition \ref{['def:haplrandomalpha']} with ${\zeta(N)} = \infty$, $\varepsilon = c(\log N)/N$ for $c$ as shown, $N=3000$, $\alpha$ as shown and $\kappa = 2$. The scale of the ordinate (y-axis) may vary between the graphs. see Appendix § \ref{['sec:samplingKPD']} for an algorithm for sampling from the $\delta_{0}$-Poisson-Dirichlet$(\alpha,0)$-coalescent
• Figure 4: Quenched vs. annealed. Comparing $\overline\rho_{i}(n)$ and $\overline \varrho_{i}^{N}(n)$ (blue lines; recall \ref{['eq:functionals']}) when the population evolves according to Definition \ref{['hschwpop']} and Definition \ref{['def:haplrandomalpha']} (a,b,c,d) and Definition \ref{['def:alpha-random-one']} (e,f); $N=10^{3}$, $\alpha$ as shown, $\kappa = 2$, ${\zeta(N)} = N^{1/\alpha}\log N$ (a,b) and ${\zeta(N)} = N$ (c,d,e,f); $\varepsilon_{N} = \overline\varepsilon_{N} = 0.1$; the approximations are from $10^{4}$ experiments. Appendices \ref{['sec:sampl-discr-trees']} ($\overline \varrho_{i}^{N}(n)$) and \ref{['sec:estimatequenched']} ($\overline\rho_{i}^{N}(n)$) contain brief descriptions of the sampling algorithms
• Figure 5: Approximations $\overline \varrho_{i}(n)$ (recall \ref{['eq:functionals']}) for the time-changed $\delta_{0}$-Beta$(\gamma,2-\alpha,\alpha)$-coalescent with time-change $G(t) = \int_{0}^{t}e^{\rho s}{\rm d}s = \mathds{1}_{\left\{ \rho > 0 \right\}}\left( e^{\rho t} - 1 \right)/\rho + \mathds{1}_{\left\{ \rho = 0 \right\}}t$ with $\rho$ as shown, $\kappa = 2$, $c=1$, $n=100$. The scale of the ordinate ($y$-axis) may vary between the graphs; results from $10^{5}$ experiments

Theorems & Definitions (60)

• Definition 2.1: Standard notation
• Definition 2.2: The coalescence probability
• Proposition 2.3: schweinsberg03, Proposition 3; conditions for convergence to a $\Lambda$-coalescent
• Definition 2.4: The Beta$(\gamma, 2-\alpha,\alpha)$-coalescent
• Proposition 2.5: schweinsberg03 Proposition 1; convergence to a $\Xi$-coalescent
• Definition 2.6: schweinsberg03; The Poisson-Dirichlet$(\alpha,0)$-coalescent
• Remark 2.7: The transition probability of the Poisson-Dirichlet coalescent
• Definition 2.8: Evolution of the population
• Theorem 2.9: schweinsberg03; Theorem 4, Lemmas 6, 13, and 16
• Definition 2.10: The $\delta_{0}$-Beta$(\gamma,2-\alpha,\alpha)$-coalescent