Universal asymptotic clone size distribution for general population growth
Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed as the Luria-Delbruck or Lea-Coulson model, is often assumed but seldom realistic. In this article we generalise the model to different types of wild-type population growth, with mutants evolving as a birth-death branching process. Our focus is on the size distribution of clones after some time, which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. We prove that the large time limit of the clone size distribution has a general two-parameter form for a large class of population growth. The large time clone size distribution always has a power-law tail, and for subexponential wild-type growth the probability of a given clone size is inversely proportional to the clone size. We support our results by analysing a data-set on tumour metastasis sizes, and we find that a power-law tail is more likely than an exponential one, in agreement with our predictions.