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Abstract Cancers are typically fueled by sequential accumulation of driver mutations in a previously healthy cell. Some of these mutations, such as inactivation of the first copy of a tumor suppressor gene, can be neutral, and some, like those resulting in activation of oncogenes, may provide cells with a selective growth advantage. We study a multi-type branching process that starts with healthy tissue in homeostasis and models accumulation of neutral and advantageous mutations on the way to cancer. We provide results regarding the sizes of premalignant populations and the waiting times to the first cell with a particular combination of mutations, including the waiting time to malignancy. Finally, we apply our results to two specific biological settings: initiation of colorectal cancer and age incidence of chronic myeloid leukemia. Our model allows for any order of neutral and advantageous mutations and can be applied to other evolutionary settings. 

The questions of how healthy colonic crypts maintain their size, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies. 

We study a multi-stage model for the development of colorectal cancer from initially healthy tissue. The model incorporates a complex sequence of driver gene alterations, some of which result in immediate growth advantage, while others have initially neutral effects. We derive analytic estimates for the sizes of premalignant subpopulations, and use these results to compute the waiting times to premalignant and malignant genotypes. This work contributes to the quantitative understanding of colorectal tumor evolution and the lifetime risk of colorectal cancer. 

To increase our basic understanding of the ecology and evolution of conjugative plasmids, we need reliable estimates of their rate of transfer between bacterial cells. Current assays to measure transfer rate are based on deterministic modeling frameworks. However, some cell numbers in these assays can be very small, making estimates that rely on these numbers prone to noise. Here, we take a different approach to estimate plasmid transfer rate, which explicitly embraces this noise. Inspired by the classic fluctuation analysis of Luria and Delbrück, our method is grounded in a stochastic modeling framework. In addition to capturing the random nature of plasmid conjugation, our new methodology, the Luria–Delbrück method (“LDM”), can be used on a diverse set of bacterial systems, including cases for which current approaches are inaccurate. A notable example involves plasmid transfer between different strains or species where the rate that one type of cell donates the plasmid is not equal to the rate at which the other cell type donates. Asymmetry in these rates has the potential to bias or constrain current transfer estimates, thereby limiting our capabilities for estimating transfer in microbial communities. In contrast, the LDM overcomes obstacles of traditional methods by avoiding restrictive assumptions about growth and transfer rates for each population within the assay. Using stochastic simulations and experiments, we show that the LDM has high accuracy and precision for estimation of transfer rates compared to the most widely used methods, which can produce estimates that differ from the LDM estimate by orders of magnitude.