# Analysis of A Countable-Type Branching Process Model for the Tug-of-War Cancer Cell Dynamics

We consider a time-continuous Markov branching process of proliferating cells with a denumerable collection of types. Among-type transitions are inspired by the Tug-of-War process introduced in Mcfarland et al. (2014) as a mathematical model for competition of advantageous driver mutations and deleterious passenger mutations in cancer cells. We introduce a version of the model in which a driver mutation pushes the state of the cell L-units up, while a passenger mutation pulls it 1 unit down. The distribution of time to divisions depends on the type of cell, which is an integer. First, we analyze the probability of extinction of the process, using approach in Hautphenne (2013). Then, we consider the properties of the mean process, using theories in Seneta (2006). Finally, we consider the process in an infinitely long cell lineage of cells, using theory of difference equations in Bodine (2015), martingales, and random walk. The analysis leads to the result that under driver dominance, the process escapes to infinity, while under passenger dominance, it leads to a limit distribution. Our analysis reveals an asymmetric relationship between impacts of the two types of mutations. The process is driven to extinction with probability less than 1. Under passenger dominance regime (downward-trend) there exists with a positive probability a “reservoir” of cells with a wide range of types (fitnesses). In the context of cancer cell populations, this may mean that “indolent” cancer cell colonies may allow the biological process to rebound if conditions change, as in the “punctuated equilibria” theory of cancer evolution in Gao et al. (2016) and Davis et al. (2017).