Large population solution of the stochastic luria-delbruck evolution model

Luria and Delbruck introduced a very useful and subsequently widely adopted framework for quantitatively understanding the emergence of new cellular lineages. Here, we provide an analytical treatment of the fully stochastic version of the model, enabled by the fact that population sizes at the time of measurement are invariably very large and mutation rates are low. We show that the Lea-Coulson generating function describes the 'inner solution,' where the number of mutants is much smaller than the total population. We find that the corresponding distribution function interpolates between a monotonic 1=m m+1 decrease at relatively small populations, (compared with the inverse of the mutation probability), whereas it goes over to a Lévy a-stable distribution in the very large population limit. The moments are completely determined by the outer solution, and so are devoid of practical significance. The key to our solution is focusing on the fixed population size ensemble, which we show is very different from the fixed time ensemble due to the extreme variability in the evolutionary process.