title: Universal Asymptotic Clone Size Distribution for General Population Growth.
abstract: 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 the Luria-Delbrück or Lea-Coulson model, is often assumed but seldom realistic. In this article, we generalise this 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-that is the number of progeny of a founder mutant-which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. Additionally, for a large class of population growth, we prove that the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. Considering metastases in cancer as the mutant clones, upon analysing a data-set of their size distribution, we indeed find that a power-law tail is more likely than an exponential one.
authors.affiliation: SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, UK. Michael.Nicholson@ed.ac.uk.
authors.forename: Michael D
authors.identifier: 0000-0001-8476-4072
authors.initials: MD
authors.lastname: Nicholson
authors.affiliation: School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK.
authors.forename: Tibor
authors.initials: T
authors.lastname: Antal
mesh_terms: D006801:Humans; D016013:Likelihood Functions; D016015:Logistic Models; D055641:Mathematical Concepts; D008954:Models, Biological; D008957:Models, Genetic; D009154:Mutation; D009369:Neoplasms / Q000235:genetics / Q000473:pathology; D016012:Poisson Distribution; D011158:Population Growth; D013269:Stochastic Processes; D013997:Time Factors
keywords: Branching process; Cancer; Clone size; Luria–Delbrück