Traveling waves of selective sweeps

TL;DR

This paper analyzes the timing and growth patterns of advantageous mutations in exponentially growing populations, providing explicit formulas and limits for mutation times and sizes in a Moran model relevant to cancer evolution.

Contribution

It extends previous Wright-Fisher model results to the Moran model, deriving explicit asymptotic formulas for mutation times and sizes as mutation rate approaches zero.

Findings

Mutation times scale as c_k log(1/μ) for small μ.

Explicit formulas for the constants c_k are derived.

Log-scale limits for the number of k-mutant cells are established.

Abstract

The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponentially growing population. To better understand this process, Beerenwinkel et al. [PLoS Comput. Biol. 3 (2007) 2239--2246] consider a Wright--Fisher model in which cells from an exponentially growing population accumulate advantageous mutations. Simulations show a traveling wave in which the time of the first $k$-fold mutant, $T_k$, is approximately linear in $k$ and heuristics are used to obtain formulas for $ET_k$. Here, we consider the analogous problem for the Moran model and prove that as the mutation rate $\mu\to0$, $T_k\sim c_k\log(1/\mu)$, where the $c_k$ can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of $X_k(t)={}$the number of cells with $k$ mutations at time $t$.