The Site Frequency Spectrum in an Exponentially Growing Population with Selection
Abstract
We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate λ1. In this setting, we investigate a component of the site frequency spectrum (SFS), describing the number of driver mutations present at any given frequency in the population. We establish asymptotic convergence results at large times and detection sizes. We examine three distinct regions of the SFS: (1) the small frequency region occupied by clones of size 𝒪(1), (2) the intermediate frequency region occupied by clones of size 1≪ j(t)≪ exp(λ1 t), and (3) the large frequency region occupied by clones of size at least exp(λ1 t). We obtain strong laws of large numbers for the driver SFS in the small- and large-frequency regions by constructing suitable L2-approximations. In the intermediate frequency region, we identify a cutoff frequency j(t) growing with time at which there are an order-one number of mutant clones. At this frequency, we obtain a Poisson convergence theorem as t→∞. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters. As an example of this, we propose and show asymptotic consistency of a new estimator of the fitness increase.
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BibTeX
@misc{ahmed2026site,
title = {{The Site Frequency Spectrum in an Exponentially Growing Population with Selection}},
author = {Armaan Ahmed and Jasmine Foo and Einar Gunnarsson and Kevin Leder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.16479},
eprint = {2607.16479},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16479},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Site Frequency Spectrum in an Exponentially Growing Population with Selection
AU - Armaan Ahmed
AU - Jasmine Foo
AU - Einar Gunnarsson
AU - Kevin Leder
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2607.16479
UR - https://arxiv.org/abs/2607.16479
AB - We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate λ_1. In this setting, we investigate a component of the site frequency spectrum (SFS), describing the number of driver mutations present at any given frequency in the population. We establish asymptotic convergence results at large times and detection sizes. We examine three distinct regions of the SFS: (1) the small frequency region occupied by clones of size 𝒪(1), (2) the intermediate frequency region occupied by clones of size 1≪ j(t)≪ exp(λ_1 t), and (3) the large frequency region occupied by clones of size at least exp(λ_1 t). We obtain strong laws of large numbers for the driver SFS in the small- and large-frequency regions by constructing suitable L^2-approximations. In the intermediate frequency region, we identify a cutoff frequency j(t) growing with time at which there are an order-one number of mutant clones. At this frequency, we obtain a Poisson convergence theorem as t→∞. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters. As an example of this, we propose and show asymptotic consistency of a new estimator of the fitness increase.
ER -