"Genetic composition of supercritical branching populations under power law mutation rates"Brouard, VianneyWe aim at understanding the evolution of the genetic composition of cancer cell populations. To this aim, we consider a branching individual based model representing a cell population where cells divide, die and mutate along the edges of a finite directed graph (V,E). The process starts with only one cell of trait 0. Following typical parameter values in cancer cell populations we study the model under large population and power law mutation rates limit, in the sense that the mutation probabilities are parameterized by negative powers of n and the typical sizes of the population of our interest are positive powers of n. Under non-increasing growth rate condition (namely the growth rate of any subpopulation is smaller than the growth rate of trait 0), we describe the time evolution of the first-order asymptotics of the size of each subpopulation on the log(n) time scale, as well as in the random time scale at which the initial population, resp. the total population, reaches the size n^{t}. In particular, such results allow to characterize whose mutational paths along the edges of the graph are actually contributing to the size order of each subpopulation. Without any condition on the growth rates, we describe the time evolution of the order of magnitude of each subpopulation. Adapting techniques from Durrett and Mayberry 2011, we show that these converge to positive deterministic non-decreasing piecewise linear continuous functions, whose slopes are given by an algorithm. From now on, we are trying to relax the non-increasing growth rate condition in order to obtain the first-order asymptotics when allowing selective mutations. We hope that this generalisation would be obtained up to Jully 2024. In this case, it would be part of the talk. |
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