Mutation accumulation in exponentially growing populations

Stochastic models of mutation accumulation in exponentially growing cellular populations are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells exist with a given set of alterations, and how long will it take for these cells to appear. These questions have been tackled in special cases, often within a branching processes framework. However, the general situation of cells sequentially acquiring an arbitrary number of mutations which may be selectively advantageous, neutral, or disadvantageous remains unaddressed. Here, we consider this setting in the biologically relevant limiting regimes of large times and small mutation rates. We provide analytic expressions for the number, and arrival time, of cells with $n$ mutations. Universal probability distributions for both quantities are presented, and the consequences of our results on cancer driver mutation accumulation and bacterial fluctuation assays are highlighted