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Quantifying evolutionary dynamics of cancer initiation and progression can provide insights into more effective strategies of early detection and treatment. Here we develop a mathematical model of colorectal cancer initiation through inactivation of two tumor suppressor genes and activation of one oncogene, accounting for the well-known path to colorectal cancer through loss of tumor suppressors APC and TP53, and gain of the KRAS oncogene. In the model, we allow mutations to occur in any order, leading to a complex network of incomplete mutational genotypes on the way to colorectal cancer. We parametrize the model using experimentally measured parameter values, many of them only recently available, and compare its predictions to epidemiological data on colorectal cancer incidence. We find that the reported incidence of colorectal cancer can be recovered using a mathematical model of colorectal cancer initiation together with experimentally measured mutation rates in colorectal tissues and proliferation rates of premalignant lesions. We demonstrate that the order of driver events in colorectal cancer is determined by the combined effect of the rates at which driver genes are mutated and the fitness effects they provide. Our results imply that there may not be significant immune suppression of untreated benign and malignant colorectal lesions.

Recently available cancer sequencing data have revealed a complex view of the cancer genome containing a multitude of mutations, including drivers responsible for cancer progression and neutral passengers. Measuring selection in cancer and distinguishing drivers from passengers have important implications for development of novel treatment strategies. It has recently been argued that a third of cancers are evolving neutrally, as their mutational frequency spectrum follows a 1/f power law expected from neutral evolution in a particular intermediate frequency range. We study a stochastic model of cancer evolution and derive a formula for the probability distribution of the cancer cell frequency of a subclonal driver, demonstrating that driver frequency is biased towards 0 and 1. We show that it is difficult to capture a driver mutation at an intermediate frequency, and thus the calling of neutrality due to a lack of such driver will significantly overestimate the number of neutrally evolving tumors. Our approach provides quantification of the validity of the 1/f statistic across the entire range of relevant parameter values. We also show that our conclusions remain valid for non-exponential models: spatial 3d model and sigmoidal growth, relevant for early- and late stages of cancer growth., Author summary Darwinian evolution in cancer is responsible for the emergence of malignant traits in initially benign tumors. As tumor cells divide, they accumulate new mutations and while most of them are “passengers” which do not confer any selective growth advantage, “driver” mutations endow cells with traits that contribute to cancer spread. Identifying driver mutations that are under selection in cancer can point to new targets for cancer therapeutics and open new avenues for personalized cancer treatment. It has recently been argued that the presence or absence of selection in cancer can be deduced from deviation of mutant allele frequencies from 1/f power law in an intermediate frequency range. Using a stochastic mathematical model of cancer evolution we derive a formula for the frequency of a subclonal driver and show that frequencies of cancer drivers are biased towards 0 and 1; thus most mutations will inevitably appear to be either neutral (frequency ≈ 0) or clonal (frequency ≈ 1) despite very different levels of selection. Consequently, the proposed 1/f statistic will significantly overestimate the number of cancers deemed to be evolving neutrally. Our work quantifies the validity of the proposed neutral evolution statistic across the entire range of relevant parameter values.