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Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics

Reinhard Bürger

TL;DR

The paper develops explicit, sharp bounds for survival probabilities $S^{(n)}$ of a single advantageous mutation in supercritical Galton-Watson processes by leveraging fractional-linear probability generating functions that match the ultimate extinction probability $P^ ^\infty$ and the convergence rate $' (P^ ^\infty)$. It proves such bounds exist for classical offspring laws (Poisson, binomial, negative binomial) and fully characterizes three-offspring cases, while offering series expansions in the selective advantage $s$ to approximate $S^ ^\infty$ and $'$, with connections to Wright–Fisher diffusion results. The work also provides a comprehensive comparison with Pollak's and Agresti's bounds, and develops a suite of applications to population-genetic questions, including convergence times $T_()$, survival up to generation $n$, and the response of a quantitative trait under directional selection. Overall, the results yield analytically tractable tools to bound and approximate time-dependent allele-frequency dynamics in finite populations, facilitating multi-mutation adaptive analyses in population genetics.

Abstract

Population genetic processes, such as the adaptation of a quantitative trait to directional selection, may occur on longer time scales than the sweep of a single advantageous mutation. To study such processes in finite populations, approximations for the time course of the distribution of a beneficial mutation were derived previously by branching process methods. The application to the evolution of a quantitative trait requires bounds for the probability of survival $S^{(n)}$ up to generation $n$ of a single beneficial mutation. Here, we present a method to obtain a simple, analytically explicit, either upper or lower, bound for $S^{(n)}$ in a supercritical Galton-Watson process. We prove the existence of an upper bound for offspring distributions including Poisson, binomial, and negative binomial. They are constructed by bounding the given generating function, $\varphi$, by a fractional linear one that has the same survival probability $S^\infty$ and yields the same rate of convergence of $S^{(n)}$ to $S^\infty$ as $\varphi$. For distributions with at most three offspring, we characterize when this method yields an upper bound, a lower bound, or only an approximation. Because for many distributions it is difficult to get a handle on $S^\infty$, we derive an approximation by series expansion in $s$, where $s$ is the selective advantage of the mutant. We briefly review well-known asymptotic results that generalize Haldane's approximation $2s$ for $S^\infty$, as well as less well-known results on sharp bounds for $S^\infty$. We apply them to explore when bounds for $S^{(n)}$ exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for $S^\infty$ and $S^{(n)}$. Finally, as an application we determine the response of a quantitative trait caused by new beneficial mutations to prolonged directional selection.

Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics

TL;DR

The paper develops explicit, sharp bounds for survival probabilities of a single advantageous mutation in supercritical Galton-Watson processes by leveraging fractional-linear probability generating functions that match the ultimate extinction probability and the convergence rate . It proves such bounds exist for classical offspring laws (Poisson, binomial, negative binomial) and fully characterizes three-offspring cases, while offering series expansions in the selective advantage to approximate and , with connections to Wright–Fisher diffusion results. The work also provides a comprehensive comparison with Pollak's and Agresti's bounds, and develops a suite of applications to population-genetic questions, including convergence times , survival up to generation , and the response of a quantitative trait under directional selection. Overall, the results yield analytically tractable tools to bound and approximate time-dependent allele-frequency dynamics in finite populations, facilitating multi-mutation adaptive analyses in population genetics.

Abstract

Population genetic processes, such as the adaptation of a quantitative trait to directional selection, may occur on longer time scales than the sweep of a single advantageous mutation. To study such processes in finite populations, approximations for the time course of the distribution of a beneficial mutation were derived previously by branching process methods. The application to the evolution of a quantitative trait requires bounds for the probability of survival up to generation of a single beneficial mutation. Here, we present a method to obtain a simple, analytically explicit, either upper or lower, bound for in a supercritical Galton-Watson process. We prove the existence of an upper bound for offspring distributions including Poisson, binomial, and negative binomial. They are constructed by bounding the given generating function, , by a fractional linear one that has the same survival probability and yields the same rate of convergence of to as . For distributions with at most three offspring, we characterize when this method yields an upper bound, a lower bound, or only an approximation. Because for many distributions it is difficult to get a handle on , we derive an approximation by series expansion in , where is the selective advantage of the mutant. We briefly review well-known asymptotic results that generalize Haldane's approximation for , as well as less well-known results on sharp bounds for . We apply them to explore when bounds for exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for and . Finally, as an application we determine the response of a quantitative trait caused by new beneficial mutations to prolonged directional selection.

Paper Structure

This paper contains 30 sections, 11 theorems, 188 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Basic notation and assumptions
  4. Seneta's method of bounding the extinction probabilities $P_{\varphi}^{(n)}$
  5. Fractional linear generating functions
  6. The basic result and alternative methods for deriving bounds for $S_{\varphi}^{(n)}$
  7. Basic result
  8. Alternative approaches
  9. Bounds for the survival probabilities $S_{\varphi}^{(n)}$ for common families of offspring distributions
  10. The Poisson distribution
  11. Comparison of Pollak's and Agresti's bounds with our and the simple bound
  12. The binomial distribution
  13. The negative binomial distribution
  14. Offspring distributions with at most three offspring
  15. Bounds and approximations for $S_{\varphi}^\infty$
  16. ...and 15 more sections

Key Result

Proposition 3.1

Let $\varphi(x)$ be a pgf satisfying our general assumptions stated in Section Sec:basics, so that $m>1$ and $0<P_{\varphi}^\infty<1$. Let $\varphi_{\rm{FL}}(x;\pi_\varphi,\rho_\varphi)$ denote the uniquely determined fractional linear pgf that satisfies cond_Pinf_gainf. If then the probability of extinction by generation $n$ satisfies Equivalently, the probability $S_{\varphi}^{(n)}$ of surviva

Figures (5)

  • Figure 4.1: The graph of the function $f_{\rm{Poi}}(x)$ with $m=1.5$. Then $\pi_m\approx0.506$, $\rho_m\approx0.211$, $P_{\rm{Poi}}^\infty\approx0.4172$. As $m$ decreases to 1, $P_{\rm{Poi}}^\infty$ increases to 1, and $(\pi_m,\rho_m)$ approaches $(\tfrac{1}{3},\tfrac{1}{3})$.
  • Figure 4.2: Possible shapes of graphs of $f_{\rm{F3}}(x)=\varphi_{\rm{F3}}(x)-\varphi_{\rm{FL}}(x)$. All possible cases are obtained by choosing $p_0=p_2$, $p_3=\tfrac{1}{2}p_2$, and varying $p_2$. Then the relations between $p_0$, $p_2$, and $p_3$ are retained as $p_2$ or $p_1=1-\tfrac{5}{2}p_2$ varies (e.g., $p_1=0.625$ in panel A). In the degenerate case of panel D, we have $p_0=p_0^{(+)}<p_0^{(r)}$, so that $f_{\rm{F3}}"(P_{\rm{F3}}^\infty)=0$ and $f_{\rm{F3}}"'(P_{\rm{F3}}^\infty)>0$. In the degenerate case of panel F, we have $f_{\rm{F3}}'(1)=0$ and $f_{\rm{F3}}"(1)<0$. In addition to the indicated relations, $p_0^{(r)}>p_0^{(+)}>0$ holds in A and B, $p_0^{(+)}>p_0^{(r)}$ in F and G, and $p_0>p_0^{(\gamma)}$ in A -- E. In all cases, $P_{\rm{F3}}^\infty=\tfrac{1}{2}(\sqrt{17}-3)\approx0.56155$, and $m_{\rm{F3}}=1+p_2$. Figure A applies if $0.4\ge p_2>\tfrac{1}{2}(1-3/\sqrt{17})$, and the lower bound yields the critical case B. The critical case D occurs if $p_2=\tfrac{2}{17}$, F applies if $p_2=-\tfrac{1}{2}+\tfrac{5}{34}\sqrt{17}$, and G applies for all smaller values of $p_2$. The values of $f_{\rm{F3}}(0)$ are $\approx 0.00081$, $-0.00222$, $-0.002959$, $-0.003273$, and $-0.00376$ in panels C, D, E, F, and G, respectively. Note that the vertical scales in A and B differ from those in the other panels.
  • Figure 4.3: The three regions defined in Corollary \ref{['cor:main_result_F3']} shown from two angles in panels A and B. The region defined by \ref{['lower_bound']} is shown in shades of yellow and brown. Here, the extinction probability $P_{\rm{F3}}^{(n)}$ can be bounded from below by the fractional linear extinction probability $P_{\rm{FL}}^{(n)}$ obtained from \ref{['rFthpFth']}. The yellow plane in A is the boundary $p_0+p_2+p_3=1$ ($p_1=0$). The region defined by \ref{['no_bound']} is shown in shades of red. Here, $P_{\rm{F3}}^{(n)}$ cannot be bounded by $P_{\rm{FL}}^{(n)}$ from one side. The region defined by \ref{['upper_bound']} is shown in shades of green. Here, $P_{\rm{F3}}^{(n)}$ is bounded from below by $P_{\rm{FL}}^{(n)}$. The boundary plane $p_0=p_2+2p_3$ ($m_{\rm{F3}}=1$) is visible in A, close to the bottom of the cube.
  • Figure 5.1: Possible shapes of graphs of $f_{\rm{GP}}(x;s,\lambda)$. We chose $s=0.3$ for good visibility. Then $P_{\rm{GP}}^\infty\approx0.7435$ if $\lambda=0.30$, and $P_{\rm{GP}}^\infty\approx0.7515$ if $\lambda=0.3145$. At the critical value $\lambda_{c_1}\approx0.30160$, $f_{\rm{GP}}'(1)$ changes sign; at $\lambda_{c_2}\approx0.30596$, $f_{\rm{GP}}"(P_{\rm{GP}}^\infty)$ changes sign; at $\lambda_{c_0}\approx0.31433$, $f_{\rm{GP}}(0)$ changes sign.
  • Figure 6.1: Relative errors of survival probabilities by generation $n$, $(S^{(n)}_{\rm{app}}-S_{\rm{GP}}^{(n)})/S_{\rm{GP}}^{(n)}$, for the generalized Poisson distribution. In all cases, $s=0.1$. The values of $\lambda$ are given in the legend; $\lambda=0$ yields the Poisson distribution. If $\lambda=0.276$, then $S^{(n)}_{\rm{app}}-S_{\rm{GP}}^{(n)}$ changes sign between $n=3$ and $n=4$; cf. Fig. \ref{['fig_F2']}. We note that for given $m=1+s$, $\varphi_{\rm{GP}}$ and $\varphi_{\rm{NB}}$ have the same variance if $\lambda=1-\sqrt{r/(r+1+s)}$. With $r=5$ this yields $\lambda\approx0.095$, $S_{\rm{GP}}^\infty\approx0.14841$, $S_{\rm{NB}}^\infty\approx0.14834$. Thus, on this scale of resolution, the blue curve would be almost indistinguishable from the corresponding curve for the negative binomial with $r=5$.

Theorems & Definitions (24)

  • Proposition 3.1
  • Remark 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Lemma 4.3
  • proof
  • proof : Proof of Theorem \ref{['thm:Poi_ge_FL']}
  • Theorem 4.4
  • Corollary 4.5
  • Theorem 4.6
  • ...and 14 more