From clonal interference to Poissonian interacting trajectories

TL;DR

This paper models the dynamics of competing beneficial mutations in a fixed-size population, introducing a Poissonian system of interacting trajectories to analyze the speed of adaptation and its fluctuations.

Contribution

It introduces the Poissonian system of interacting trajectories (PIT) as a new large population limit model for clonal interference in evolution.

Findings

PIT accurately describes the sizes of competing clonal subpopulations.

The speed of adaptation is positive and finite if fitness increments have finite expectation.

A functional central limit theorem characterizes fluctuations in the population's fitness.

Abstract

We consider a population whose size $N$ is fixed over the generations, and in which random beneficial mutations arrive at a rate of order $1/\log N$ per generation. In this so-called Gerrish--Lenski regime, typically a finite number of contending mutations are present together with one resident type. These mutations compete for fixation, a phenomenon addressed as clonal interference. We introduce and study a Poissonian system of interacting trajectories (PIT), and prove that it arises as a large population scaling limit of the logarithmic sizes of the contending clonal subpopulations in a continuous-time Moran model with strong selection. We show that the PIT exhibits an almost surely positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation. We relate this speed to heuristic predictions from the literature. Furthermore, we derive a functional central limit theorem for the fitness of the resident population in the PIT.