Traveling waves of selective sweeps. (English) Zbl 1219.92037

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, 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 \(t\).


92C50 Medical applications (general)
60J85 Applications of branching processes
92C40 Biochemistry, molecular biology
65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text: DOI arXiv


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