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AB was supported by the National Science Centre (Poland) grant 2017/25/B/ST1/01804. MK was partially supported by the National Science Centre (Poland) grant 2018/29/B/ST7/02550 and by NIH Grant P01 CA265748
(Margaret Goodell, PI). MKK was supported by the National Science Centre (Poland) grant 2021/41/B/NZ2/04134.
In a series of publications McFarland and co-authors introduced the tug-of-war model of evolution of cancer cell populations. The model is explaining the joint effect of rare advantageous and frequent slightly deleterious mutations, which may be identifiable with driver and passenger mutations in cancer. In this paper, we put the tug-of-war model in the framework of a denumerable-type Moran process and use mathematics and simulations to 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 part and of the mutation process part. Operator semigroup theory is then employed to prove that the MC does not explode, as well as to characterize a strong-drift limit version of the MC which displays "instant fixation" effect, which was an assumption in the original McFarland's model. Mathematical results are confirmed by simulations of the complete and limit versions. Simulations also visualize complex stochastic transients and genealogies of clones arising in the model.
