How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime
Abstract
We determine statistics of allele-frequency trajectories across replicate populations. Defining R = 2Ne|s| where Ne is the effective population size and s is the additive selection coefficient of a focal allele, we focus on strong selection which corresponds to R ≫ 1. Strong selection does not require large selection coefficients: in populations of realistic size, it can readily arise from small selection coefficients. Our analysis concerns a biallelic locus over timescales on which mutation can be neglected. Every allele-frequency trajectory belongs to one of two absorbing classes: fixation or loss. When R is large, individual allele-frequency trajectories can appear nearly deterministic. This does not imply that an approximation of allele-frequency statistics can be based on a single deterministic trajectory. For a beneficial allele, the deterministic trajectory approaches fixation, yet a non-negligible fraction of stochastic trajectories may be lost. Ignoring trajectories that achieve loss can produce severely erroneous statistics. We address this by conditioning separately on the eventual loss and eventual fixation of the focal allele. This yields an outcome-conditioned representation of a statistic. In the strong-selection regime, we apply a small-noise expansion that reasonably captures transient behaviours of statistics while recovering their exact long-time limits. We illustrate results with numerical examples for the mean allele frequency and its variance. The biological interpretation of this work is that, even under strong selection, both trajectory types are of importance. We provide a simple framework for calculating time-dependent allele-frequency statistics relevant to experimental evolution, longitudinal population-genomic data, and the dynamics of alleles driving selective sweeps.
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BibTeX
@misc{waxman2026deterministic,
title = {{How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime}},
author = {David Waxman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14549},
eprint = {2609.14549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14549},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - How deterministic trajectories and their fluctuations underlie allele-frequency statistics in the strong-selection regime
AU - David Waxman
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.14549
UR - https://arxiv.org/abs/2609.14549
AB - We determine statistics of allele-frequency trajectories across replicate populations. Defining R = 2N_e|s| where N_e is the effective population size and s is the additive selection coefficient of a focal allele, we focus on strong selection which corresponds to R ≫ 1. Strong selection does not require large selection coefficients: in populations of realistic size, it can readily arise from small selection coefficients. Our analysis concerns a biallelic locus over timescales on which mutation can be neglected. Every allele-frequency trajectory belongs to one of two absorbing classes: fixation or loss. When R is large, individual allele-frequency trajectories can appear nearly deterministic. This does not imply that an approximation of allele-frequency statistics can be based on a single deterministic trajectory. For a beneficial allele, the deterministic trajectory approaches fixation, yet a non-negligible fraction of stochastic trajectories may be lost. Ignoring trajectories that achieve loss can produce severely erroneous statistics. We address this by conditioning separately on the eventual loss and eventual fixation of the focal allele. This yields an outcome-conditioned representation of a statistic. In the strong-selection regime, we apply a small-noise expansion that reasonably captures transient behaviours of statistics while recovering their exact long-time limits. We illustrate results with numerical examples for the mean allele frequency and its variance. The biological interpretation of this work is that, even under strong selection, both trajectory types are of importance. We provide a simple framework for calculating time-dependent allele-frequency statistics relevant to experimental evolution, longitudinal population-genomic data, and the dynamics of alleles driving selective sweeps.
ER -