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Zbl 1219.92037
Durrett, Rick; Mayberry, John
Traveling waves of selective sweeps. (English)
[J] Ann. Appl. Probab. 21, No. 2, 699-744 (2011). ISSN 1050-5164

Summary: 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, {\it N. Beerenwinkel} et al. [PLoS Comput. Biol. 3, 2239--2246 (2007)] considered 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 $ET_k$. We consider the analogous problem for the Moran model and prove that as the mutation rate $\mu \rightarrow 0$, $T_k \sim c_k \log (1 / \mu )$, 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 $t$.

MSC 2000:: *92C50 Medical appl. of mathematical biology
60J85 Appl. of branching processes
92C40 Biochemistry, etc.
65C20 Models (numerical methods)

Keywords: Moran model; rate of adaptation; stochastic tunneling; branching processes; cancer models

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