Séminaire doctorant.e.s

Understanding the evolution of the genetic composition of cancer cell populations is of key interest for clinicians. In this talk we will study a toy model of carcinogenesis by considering 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). 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. In other words, we separate the birth-death typical time scale to the mutational one, but are interested in the natural time scale allowing mutations to be frequent.

Under non-increasing growth rate condition, namely the growth rate of any subpopulation is smaller than the growth rate of trait 0 (biologically meaning neutral, or deleterious, cancer evolution), we describe the time evolution of the first-order asymptotics 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}. Such results allow to characterize whose mutational paths along the edges of the graph are actually the evolutionary determining paths.

Without any condition on the growth rates, the analysis to get the first-order asymptotics of the mutant subpopulations is far more complex. We will motivate this increasing difficulty and give some insights about how one can deal with the understanding of such an evolutionary process (in a special restrictive case).