Browsing by Author "Kurpas, Monika K."

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    Modes of Selection in Tumors as Reflected by Two Mathematical Models and Site Frequency Spectra
    (Frontiers Media S.A., 2022) Kurpas, Monika K.; Kimmel, Marek
    The tug-of-war model was developed in a series of papers of McFarland and co-authors to account for existence of mutually counteracting rare advantageous driver mutations and more frequent slightly deleterious passenger mutations in cancer. In its original version, it was a state-dependent branching process. Because of its formulation, the tug-of-war model is of importance for tackling the problem as to whether evolution of cancerous tumors is “Darwinian” or “non-Darwinian.” We define two Time-Continuous Markov Chain versions of the model, including identical mutation processes but adopting different drift and selection components. In Model A, drift and selection process preserves expected fitness whereas in Model B it leads to non-decreasing expected fitness. We investigate these properties using mathematical analysis and extensive simulations, which detect the effect of the so-called drift barrier in Model B but not in Model A. These effects are reflected in different structure of clone genealogies in the two models. Our work is related to the past theoretical work in the field of evolutionary genetics, concerning the interplay among mutation, drift and selection, in absence of recombination (asexual reproduction), where epistasis plays a major role. Finally, we use the statistics of mutation frequencies known as the Site Frequency Spectra (SFS), to compare the variant frequencies in DNA of sequenced HER2+ breast cancers, to those based on Model A and B simulations. The tumor-based SFS are better reproduced by Model A, pointing out a possible selection pattern of HER2+ tumor evolution. To put our models in context, we carried out an exploratory study of how publicly accessible data from breast, prostate, skin and ovarian cancers fit a range of models found in the literature.
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    Moran process version of the tug-of-war model: Behavior revealed by mathematical analysis and simulation studies
    (Amerian Institute of Mathematical Sciences, 2023) Bobrowski, Adam; Kimmel, Marek; Kurpas, Monika K.; Ratajczyk, Elżbieta
    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.