Muller's ratchet in a near-critical regime: tournament versus fitness proportional selection

TL;DR

The paper extends Muller's ratchet to a near-critical regime and compares tournament versus fitness-proportional selection within a Moran framework. It develops a detailed, Green-function–based analysis to derive sharp interclick-time asymptotics, revealing polynomial and exponential regimes depending on the near-critical scaling parameters, and connects these results to Poisson-profile approximations and ancestral-selection-graph inspired dualities. The findings provide strong evidence for the near-critical tournament ratchet's rate behavior and establish a rigorous bridge to the classical ratchet's Poisson-profile description. Together, these results sharpen understanding of how mutation, selection, and drift interact near criticality in non-recombining populations and illuminate the dynamics of the best-class lineage over long times.

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. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as $N\to \infty$). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as $N\to \infty$.

Figures (7)

• Figure 1: This is an illustration of the Rule of Thumb (RTC) predicting the order of magnitude of the interclick times of the classical ratchet. Each data point was obtained by pooling the interclick times no. 50 to 150 from 100 simulations of the (classical) ratchet for the corresponding parameter configuration $(N, \beta, \delta)$ in the $(\beta, \delta)$-scaling \ref{['bdclassi']}. In the exponential regime, (RTC) predicts an order of magnitude $\exp ( c \, N^{1-\beta-\delta})$ for the interclick times. In panel (\ref{['Fig1a']}), we see that the constant $c$ is difficult to estimate from simulations up to $N=10^4$, but $c =2.3$ as chosen there gives a reasonable fit. For the polynomial regime, (RTC) predicts the order of magnitude $N^{1-\delta}$, which fits very well the data in the situation of panel (\ref{['Fig1b']}).
• Figure 2: This is an illustration of the Rule of Thumb for the tournament ratchet (RTT) in the light of Theorem \ref{['mainth']}. Each data point was obtained by pooling the interclick times no. 50 to 150 from 100 simulations of the tournament ratchet for the corresponding value of $N$. Here, in panel (\ref{['subfig2b']}) $(\beta,\delta) = (0.6,0.28)$, which belongs to the polynomial regime$\mathcal{P}$, and in panel (\ref{['subfig2a']}) $(\beta,\delta) = (0.8,0.08)$, which belongs to the exponential regime$\mathcal{E}$. Each panel shows two predictions based on the asymptotics of Theorem\ref{['mainth']}, using the initial values $\mathfrak a = N^{1-\delta}$ and $\mathfrak b = N^{1-\delta/2}$, respectively. In the exponential regime the predictions using $\mathfrak a$ and $\mathfrak b$, respectively, are virtually indistinguishable, while in the polynomial regime the prediction using$\mathfrak b$ is by far better than the one using$\mathfrak a$.
• Figure 3: The empirical occupation times of the size of the best class in a simulation of the tournament ratchet are compared to the Green functions $G(\mathfrak a, \cdot)$, $G(\mathfrak b, \cdot)$, which are computed numerically using formula \ref{['generalgreen1']}. Panels (\ref{['fig7f:a']}) and (\ref{['fig7f:b']}) feature the exponential and the polynomial regime, respectively, with $\gamma = 0.2$ in panel (\ref{['fig7f:a']}) and $\gamma = \tfrac23$ in panel (\ref{['fig7f:b']}). In panel (\ref{['fig7f:b']}) the population size is $N=500$ and simulations were run up to the first $10^4 +1$ clicks, where the first click was ignored. In (\ref{['fig7f:a']}), 101 clicks were observed and the first one ignored. Here the population size was $N=100$. See EPW09 for similar plots concerning the classical ratchet.
• Figure 4: For $N=10^5$ we compare the size of the "new best class" of the classical ratchet immediately after a click (observed in simulations) with the two theoretical predictions $N\pi_0= N^{1-\delta}$ and $N\pi_1= N^{1-\delta}\delta \log N$, cf. Remark \ref{['remcentatt']}.\ref{['rem:remcentatt_b']}. For various values of $\gamma= \delta/(1-\beta)$, we consider (the logarithms of) these observed and predicted quantities as functions of $\beta$. Each data point was obtained by pooling the interclick times no. 5 to 30 from 20 simulations of the classical ratchet for the corresponding parameter configuration. Roughly, the average of the observed logarithmic sizes of the new best class seems to wander away from $N\pi_0$ towards $N\pi_1$ (and beyond) as $\gamma$ increases.
• Figure 5: For $N=10^5$ we compare the size of the "new best class" of the tournament ratchet immediately after a click (as observed by simulations) with $\mathfrak a = N^{1-\delta}$ and $\mathfrak b = N^{1-\delta/2}$, which are the centers of attraction of the best and the second best class of the tournament ratchet(cf. Remark\ref{['remcentatt']}.\ref{['rem:remcentatt_b']}). For various values of $\gamma= \delta/(1-\beta)$, we consider (the logarithms of) these quantities as functions of $\beta$. Each data point was obtained by pooling the interclick times no. 5 to 30 from 20 simulations of the tournament ratchet for the corresponding parameter configuration. For a wide range of parameters with $\gamma$ between $1/2$ and $1$, $\mathfrak b$ is a better fit for the size of the new best class than $\mathfrak a$.

Theorems & Definitions (23)

• Definition 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.4
• Remark 3.5
• Remark 3.6
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3