Traveling waves of selective sweeps

Details Traveling waves of selective sweeps Traveling waves of selective sweeps

2009-10-29

arxiv.org/abs/0910.5730v4

Annals of Applied Probability 2011, Vol. 21, No. 2, 699-744

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AAP721 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/10-AAP721

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 $t$.

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