The fixed-size Luria-Delbruck model with a nonzero death rate

What is the expected number of mutants in a stochastically growing colony once it reaches a given size, N? This is a variant of the famous Luria-Delbruck model which studies the distribution of mutants after a given time-lapse. Instead of fixing the time-lapse, we assume that the colony size is a measurable quantity, which is the case in many in-vivo oncological and other applications. We study the mean number of mutants for an arbitrary cell death rate, and give partial results for the variance. For a restricted set of parameters we provide analytical results; we also design a very efficient computational method to calculate the mean, which works for most of the parameter values, and any colony size, no matter how large. We find that a cellular population with a higher death rate will contain a larger number of mutants than a population of equal size with a smaller death rate. Also, a very large population will contain a larger percentage of mutants; that is, irreversible mutations act like a force of selection, even though here the mutants are assumed to have no selective advantage. Finally, we investigate the applicability of the traditional, 'fixed-time' approach and find that it approximates the 'fixed-size' problem whenever stochastic effects are negligible.