Traveling waves of selective sweeps
Rick Durrett, John Mayberry

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 -fold mutant, , is approximately linear in and heuristics are used to obtain formulas for . Here, we consider the analogous problem for the Moran model and prove that as the mutation rate , , where the can be computed explicitly. In addition, we derive a limiting result on a log…
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