Stochastic Tunnels in Evolutionary Dynamics

We study a situation that arises in the somatic evolution of cancer. Consider a finite population of replicating cells and a sequence of mutations: type 0 can mutate to type 1, which can mutate to type 2. There is no back mutation. We start with a homogeneous population of type 0. Mutants of type 1 emerge and either become extinct or reach fixation. In both cases, they can generate type 2, which also can become extinct or reach fixation. If mutation rates are small compared to the inverse of the population size, then the stochastic dynamics can be described by transitions between homogeneous populations. A "stochastic tunnel" arises, when the population moves from all 0 to all 2 without ever being all 1. We calculate the exact rate of stochastic tunneling for the case when type I is as fit as type 0 or less fit. Type 2 has the highest fitness. We discuss implications for the elimination of tumor suppressor genes and the activation of genetic instability. Although our theory is developed for cancer genetics, stochastic tunnels are general phenomena that could arise in many circumstances.