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Bull Math Biol. 2016 Nov;78(11):2243-2276. Epub 2016 Oct 20.

Universal Asymptotic Clone Size Distribution for General Population Growth.

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SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, UK.
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK.


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.


Branching process; Cancer; Clone size; Luria–Delbrück

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