The Annals of Applied Probability

Traveling waves of selective sweeps

Rick Durrett and John Mayberry

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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, Tk, is approximately linear in k and heuristics are used to obtain formulas for ETk. Here, we consider the analogous problem for the Moran model and prove that as the mutation rate μ → 0, Tkck log(1 / μ), where the ck can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of Xk(t) = the number of cells with k mutations at time t.

Article information

Ann. Appl. Probab. Volume 21, Number 2 (2011), 699-744.

First available in Project Euclid: 22 March 2011

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Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 60J85, 92D25
Secondary: 92C50.

Moran model selective sweep rate of adaptation stochastic tunneling branching processes cancer models


Durrett, Rick; Mayberry, John. Traveling waves of selective sweeps. The Annals of Applied Probability 21 (2011), no. 2, 699--744. doi:10.1214/10-AAP721.

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