Moran Process Version of the Tug-of-War Model: Complex Behavior Revealed by Mathematical Analysis and Simulation Studies
In a series of publications McFarland and co-authors introduced the tug-of-warmodel of evolution of cancer cell populations. The model is explaining the joint effect ofrare advantageous and frequent slightly deleterious mutations, which may be identifiable withdriver and passenger mutations in cancer. In this paper, we put the Tug-of-War model inthe framework of a denumerable-type Moran process and use mathematics and simulationsto understand its behavior. The model is associated with a time-continuous Markov Chain(MC), with a generator that can be split into a sum of the drift and selection process partand of the mutation process part. Operator semigroup theory is then employed to prove thatthe MC does not explode, as well as to characterize a strong-drift limit version of the MCwhich displays instant fixation effect, which was an assumption in the original McFarlandsmodel. Mathematical results are fully confirmed by simulations of the complete and limitversions. They also visualize complex stochastic transients and genealogies of clones arising inthe model.