TRAVELING WAVES OF SELECTIVE SWEEPS

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 -> 0, T(k) similar to 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.