[Paper Review] Genealogical transition in the noisy $N$-Branching Random Walk. How stronger selection may promote genetic diversity

Emmanuel Schertzer, Alejandro H. Wences|arXiv (Cornell University)|Jan 18, 2023

Evolution and Genetic DynamicsBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper presents an exactly solvable model of a noisy $N$-branching random walk in a 1D fitness space, showing that stronger directional selection can paradoxically increase genetic diversity by reducing relative fitness differences within the population. The mechanism arises from a phase transition between weak (fully-pulled) and strong (semi-pulled) selection regimes, where higher selection strength boosts absolute fitness but lowers relative fitness, leading to non-monotonic diversity patterns contrary to classical intuition.

ABSTRACT

We consider an extension of the noisy $N$-Branching Random Walk that models the evolution of a population subject to natural selection. We show the existence of a critical value for the noise which separates the limiting genealogical structure into two regimes, which we respectively call the semi-pulled and the fully-pulled regimes. In the fully-pulled regime, the genealogy converges to a discrete time Poisson-Dirichlet coalescent. In the semi-pulled regime, the genealogy converges to the Bolthausen-Sznitman coalescent. We discuss some interesting biological consequences of this result. In particular, our model predicts a non-monotone relation between the selection strength and the effective population size.

Motivation & Objective

• To challenge the classical intuition that stronger directional selection reduces genetic diversity.
• To investigate the relationship between relative and absolute fitness in a population genetics model with selection and genetic drift.
• To analyze how selection strength affects genetic diversity in a stochastic, finite-population model.
• To identify a novel phase transition in front propagation dynamics in noisy FKPP-type equations.

Proposed method

• Modeling the population as a cloud of $N$ particles in a 1D fitness space evolving via discrete-time reproduction and selection steps.
• Using a Poisson point process with intensity $e^{-(s-x)}ds$ to model offspring fitness distribution after mutation.
• Implementing a two-step selection process: truncation to retain $N^\gamma$ rightmost particles ($\gamma > 1$), followed by fitness-proportional natural selection with parameter $\beta > 0$.
• Analyzing the system in the limit $N \to \infty$ using extreme value theory and point process convergence.
• Establishing a phase transition at a critical $\beta_c$ separating weak ($\beta < \beta_c$) and strong ($\beta > \beta_c$) selection regimes.
• Applying large deviation techniques and moment estimates to derive asymptotic behavior of fitness wave speed and diversity metrics.

(a) Weak selection regime ( $\beta=.3<\beta_{c}$ ).

Experimental results

Research questions

• RQ1 Does stronger directional selection always reduce genetic diversity, as traditionally believed?
• RQ2 How do relative and absolute fitness scale with selection strength in a finite, stochastic population?
• RQ3 What is the nature of the phase transition between weak and strong selection regimes in this model?
• RQ4 Can a higher selection strength lead to increased genetic diversity due to changes in relative fitness distribution?
• RQ5 How does the wave front propagation regime (pulled vs. semi-pulled) evolve with selection strength?

Key findings

• Genetic diversity is non-monotonic in selection strength and is typically higher in the strong selection regime, contradicting classical population genetics intuition.
• A phase transition occurs at a critical selection strength $\beta_c$, separating a weak selection regime (fully-pulled wave) from a strong selection regime (semi-pulled wave).
• In the strong selection regime, the wave's absolute fitness increases, but relative fitness differences between individuals decrease, promoting diversity.
• The speed of the fitness wave converges to $\chi \log N$ in the strong selection regime, with $\chi$ depending on $\beta$, and the front is semi-pulled.
• The expected number of particles at the front decays as $N^{-\epsilon\beta/2}$, indicating reduced dominance of top individuals under strong selection.
• The model reveals that inferring selection strength from genetic data may be conceptually flawed due to the decoupling of absolute and relative fitness effects.

(b) Strong selection regime ( $\beta=.8>\beta_{c}$ ).