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 ${\lambda }_1$. 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 $\mathcal{O}(1)$, (2) the intermediate frequency region occupied by clones of size $1\ll j(t)\ll \exp ({\lambda }_{1}t)$, and (3) the large frequency region occupied by clones of size at least $\exp ({\lambda }_{1}t).$ 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 $t\to \infty$. 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.

Changes from v2: added more description of proof strategy, shortened introduction