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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 μ0, T k c k log(1/μ), 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:
92C50Medical applications of mathematical biology
60J85Applications of branching processes
92C40Biochemistry, molecular biology
65C20Models (numerical methods)