The Fractional Birth Process with Power-Law Immigration

A semi-stochastic description of the mutation process with memory effects in a power-law growing cell colony is considered. Specifically, we give explicit expressions for the distribution of the number of mutants in a single clone, and of the total number of mutants. The investigation is performed under the assumption that clones grow according to a fractional linear birth process, characterized by a non-exponential, Mittag-Leffler waiting time distribution. Its slowly decaying long tail enables modeling bursty dynamics: very dense sequences of events are separated by long times of reduced activity. The probabilistic construction also allows for recovering the mean and the variance of the total number of mutants. We then give exact formulas for the higher-order moments of the fractional linear birth process and of the clone size, thus providing additional insight into this evolutionary process.