Traveling waves of selective sweeps

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 μ → 0, T_k ∼ c_k log(1 / μ), 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.