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The tournament ratchet's clicktime process, and metastability in a Moran model

Jan Lukas Igelbrink, Charline Smadi, Anton Wakolbinger

TL;DR

This work analyzes Muller's ratchet under tournament selection within a Moran-model framework, focusing on slow-click regimes where $N^{-1}\ll m_N< s_N\ll1$ and $\mathfrak c_N\ll\mathfrak a_N$. By exploiting hierarchical autonomy of fitness classes, the authors study a two-type Moran subsystem for the fittest class and establish metastable behaviour with Ornstein–Uhlenbeck fluctuations around a center of attraction, as well as exponential extinction times. The central result is that the rescaled clicktime process converges to a rate-1 Poisson process with inter-click times of order $e_N$, where $e_N$ is explicitly given in terms of $\mathfrak a_N$, $\mathfrak c_N$, and $\mathfrak u_N$ and involves an explicit function $\eta(m,\rho)$. The analysis combines forward-in-time metastability arguments, Green-function-type estimates, Boltzmann–Gibbs insights, and an OU limit, yielding a precise probabilistic description of click dynamics and the conditions under which a new fittest class successfully arises after each click.

Abstract

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate $\ll 1/N$) we prove that the rescaled process of click times of the tournament ratchet converges as $N\to \infty$ to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.

The tournament ratchet's clicktime process, and metastability in a Moran model

TL;DR

This work analyzes Muller's ratchet under tournament selection within a Moran-model framework, focusing on slow-click regimes where and . By exploiting hierarchical autonomy of fitness classes, the authors study a two-type Moran subsystem for the fittest class and establish metastable behaviour with Ornstein–Uhlenbeck fluctuations around a center of attraction, as well as exponential extinction times. The central result is that the rescaled clicktime process converges to a rate-1 Poisson process with inter-click times of order , where is explicitly given in terms of , , and and involves an explicit function . The analysis combines forward-in-time metastability arguments, Green-function-type estimates, Boltzmann–Gibbs insights, and an OU limit, yielding a precise probabilistic description of click dynamics and the conditions under which a new fittest class successfully arises after each click.

Abstract

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate ) we prove that the rescaled process of click times of the tournament ratchet converges as to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.
Paper Structure (16 sections, 14 theorems, 182 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and main results
  3. The tournament ratchet and its clicktime process
  4. Metastability in a two-type Moran model with selection and deleterious mutations
  5. Proof of Theorem \ref{['thmY0']}
  6. Properties of the potential function
  7. Expected hitting times: Proof of Theorem \ref{['thmY0']}.\ref{['thmY0:a']}
  8. Ornstein-Uhlenbeck fluctuations: Proof of Theorem \ref{['thmY0']}.\ref{['thmY0:b']}
  9. Asymptotic normality of the Boltzmann-Gibbs distribution
  10. Asymptotic normality of the quasi-equilibria: Proof of Theorem \ref{['thmY0']}.\ref{['thmY0:c']}
  11. Asymptotic exponentiality of the extinction time: Proof of Theorem \ref{['thmY0']}.\ref{['thmY0:d']}
  12. On the way to extinction: Proof of Theorem \ref{['thmY0']}.\ref{['thmY0:e']}
  13. Proof of Theorem \ref{['th:clicktimepois']}
  14. Lower-bounding the size of the new fittest class at a clicktime
  15. From one click to the next
  16. ...and 1 more sections

Key Result

Theorem 2.3

Assume the conditionssmallmut andexpreg. Then, with $e_N$ as ineN, and with the initial condition the sequence of $\mathbb{N}_0$-valued processes $(K_N^\star(e_N^{}t))_{t\ge 0}$ converges in distribution as $N\to \infty$ to a rate 1 Poisson counting process. In particular, the sequence of time-rescaled clicktime processes converges in distribution as $N\rightarrow \infty$ to a rate 1 Poisson poi

Figures (1)

  • Figure 1: A plot of the potential function $U(n) = \sum_{\ell=1}^n \log\frac{\mu_\ell}{\lambda_\ell}$, $n\in [N]$, for $N=10^5$, $m = N^{-0.6}$, $\rho = 1- N^{-0.1}=0.68$. The left panel shows the full domain and range of$U$, the right panel restricts to $n\le 6.5\cdot 10^4$. The quantities $\mathfrak a$, $\mathfrak c$ and$\sigma$ are defined and explained in\ref{['def_a']}, \ref{['defh']} and\ref{['defb']}.

Theorems & Definitions (36)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 26 more