Logarithmic scaling of selective sweep curves: from tents to houses

Abstract

One of the classical results of mathematical population genetics states that the frequency of a beneficial mutant's offspring, on its way to fixation in a large population, looks like a logistic curve. A logarithmic scaling (both in height and time) of these selective sweep curves leads (in the case of strong selection) to a tent-like shape in the large population limit: First the logarithmic frequency of the mutant increases linearly from 0 to 1, then that of the former resident decreases from 1 to 0. For moderate selection the logarithmic frequencies develop (in the large population limit) a jump at the beginning/the end of the sweep, which takes the shape of the tent into that of a house. Our main result (proved for the Moran model) assesses the regularity of this convergence in the large population limit: It is uniform in the house's roof (phases of linear growth and decline) and "Skorokhod $M_1$" in the house's walls (closely around the jumps). Apart from interest in its own right, we anticipate that this result and the proof techniques will be instrumental for extending the description of clonal interference by Poissonian interacting trajectories (as it was done in Hermann et al. (2024) for strong selection) also to moderate selection.