Loss- and Gain-of-Function Mutations in Cancer: Mass-action, Spatial and Hierarchical Models

We study the stochastic dynamics of the two most common patterns in cancer initiation and progression: loss-of-function and gain-of-function mutations. We consider three stochastic models of cell populations with a constant size: a mass-action model, a spatial model and a hierarchical model. For gain-of-function mutations, we calculate the probability of mutant fixation starting from one mutant cell. For loss-of-function mutations, we calculate the rate of production of double-hit mutants. It turns out that the results are different in all models. This suggests that simple mass-action models are often misleading when studying cancer dynamics. Moreover, our results also allow us to think about various types of tissue architecture and its protective role against cancer. In particular, we show that hierarchical tissue organization lowers the risk of cancerous transformations. Also, cellular motility and long-range signaling can decrease the risk of cancer in solid tissues.