We study the fate of a newly-arising small beneficial sub-population in a large homogeneous asexual population, when  both populations accumulate deleterious mutations at the same rate.  For this purpose we build a reducible branching process with an infinite number of types, describing the composition of the population with respect to the number of acquired deleterious mutations. This allows us to describe how the speeds of accumulation of deleterious mutations in the two sub-populations compare (Muller's ratchet effect), as well as their respective fitnesses. This branching process model also enables us to compute the distribution of a neutral focal mutation in an expanding population (Luria-Delbrück distribution) when this population accumulate deleterious mutations.

branching process; hypermutable population; Muller's ratchet; Luria-Delbrück distribution