[Paper Review] Rigorous results for a population model with selection II: genealogy of the population

Jason Schweinsberg|arXiv (Cornell University)|Jul 1, 2015

Evolution and Genetic Dynamics7 citations

TL;DR

This paper rigorously establishes that under specific scaling limits of mutation and selection rates, the genealogy of a large, fixed-size population under weak positive selection converges to the Bolthausen-Sznitman coalescent. Using a diffusion approximation and coupling techniques, it confirms nonrigorous predictions that multiple lineages merge simultaneously due to rapid fixation of beneficial mutations, a hallmark of populations under strong selection.

ABSTRACT

We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $μ_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. Under certain conditions on the parameters $μ_N$ and $s_N$, we show that the genealogy of the population can be described by the Bolthausen-Sznitman coalescent. This result confirms predictions of Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).

Motivation & Objective

• To provide a mathematically rigorous justification for the nonrigorous prediction that the genealogy of a population under selection converges to the Bolthausen-Sznitman coalescent.
• To extend previous results on the rate of adaptation and fitness distribution by analyzing the ancestral process of the population.
• To establish conditions on the mutation rate $\mu_N$ and selection strength $s_N$ under which the coalescent limit is the Bolthausen-Sznitman process.
• To confirm the role of multiple mergers in genealogies due to rapid fixation of beneficial mutations in large populations.

Proposed method

• The analysis uses a diffusion scaling with time rescaled by $a_N = \frac{\log(s_N/\mu_N)}{s_N}$, which captures the natural time scale of lineage mergers.
• The genealogical process is approximated via a sequence of coalescent processes $\Pi_N$, which are coupled to a Poisson point process $\Phi$ modeling merger events.
• Key technical tools include coupling arguments between the finite-population process $\Pi_N$ and the limiting Bolthausen-Sznitman coalescent $\Pi$, controlling discrepancies via small error terms.
• The proof relies on bounding the probability of mismatched coalescence events using moment bounds and Poisson tail estimates, particularly for rare, simultaneous mergers.
• The construction leverages results from prior work [27] on the dynamics of fitness and mutation accumulation, especially the typical number of mutations $k_N = \frac{\log N}{\log(s_N/\mu_N)}$.
• A time-translation argument ensures that the finite-time genealogical processes $\Pi_N(t)$ converge weakly to $\Pi(t)$ at fixed time points, under the assumptions A1–A3.

Experimental results

Research questions

• RQ1 Under what conditions on $\mu_N$ and $s_N$ does the genealogy of a large population under selection converge to the Bolthausen-Sznitman coalescent?
• RQ2 How do the scaling limits of mutation and selection rates affect the frequency and structure of multiple-merger events in the ancestral process?
• RQ3 Can the nonrigorous prediction of Desai, Walczak, and Fisher (2013) and Neher and Hallatschek (2013) be formally justified using stochastic processes and coupling techniques?
• RQ4 What role does the typical number of mutations $k_N$ play in determining the time scale and structure of coalescent events?

Key findings

• Under assumptions A1–A3, the genealogy of the population converges weakly to the Bolthausen-Sznitman coalescent as $N \to \infty$.
• The time scale $a_N = \frac{\log(s_N/\mu_N)}{s_N}$ is shown to be the correct scaling for observing the limiting coalescent behavior.
• The number of mutations $k_N = \frac{\log N}{\log(s_N/\mu_N)}$ characterizes the typical fitness difference between fittest and average individuals, and controls the resolution of the coalescent process.
• The probability of mismatched coalescence events between the finite process $\Pi_N$ and the limiting $\Pi$ is bounded by terms that vanish as $N \to \infty$, ensuring weak convergence.
• The proof confirms that multiple-merger events—where more than two lineages merge simultaneously—are the dominant feature of the genealogy under the given selection regime.
• The convergence holds for any fixed time horizon $T$, with error bounds that vanish as $N \to \infty$ under the stated parameter scaling.