The goal of cancer genome sequencing projects is to determine the geneticalterations that cause common cancers. Many malignancies arise during theclonal expansion of a benign tumor which motivates the study of recurrentselective sweeps in an exponentially growing population. To better understandthis process, Beerenwinkel et al. [PLoS Comput. Biol. 3 (2007) 2239--2246]consider a Wright--Fisher model in which cells from an exponentially growingpopulation accumulate advantageous mutations. Simulations show a traveling wavein which the time of the first $k$-fold mutant, $T_k$, is approximately linearin $k$ and heuristics are used to obtain formulas for $ET_k$. Here, we considerthe 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 computedexplicitly. In addition, we derive a limiting result on a log scale for thesize of $X_k(t)={}$the number of cells with $k$ mutations at time $t$.
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机译：癌症基因组测序项目的目标是确定导致常见癌症的遗传变异。在良性肿瘤的克隆扩增过程中会出现许多恶性肿瘤，这促使人们对呈指数增长的人群进行反复选择性扫描的研究成为可能。为了更好地理解这个过程，Beerenwinkel等人。 [PLoS计算。生物学3（2007）2239--2246]考虑了Wright-Fisher模型，其中来自指数增长的种群的细胞积累了有利的突变。模拟显示了行进波动，第一个$ k $倍突变体$ T_k $的时间大约是线性k $ k $，启发式算法用于获得$ ET_k $的公式。在这里，我们考虑了Moran模型的类似问题，并证明了当突变率$ \ mu \ to0 $时，$ T_k \ sim c_k \ log（1 / \ mu）$，其中$ c_k $可以被精确地计算出来。另外，我们得出了对数尺度上的限制结果，其大小为$ X_k（t）= {} $在时间$ t $处具有$ k $突变的细胞数量。
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