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.
Highlights
Cancerous tumours spawning metastases, bacterial colonies developing antibiotic resistance or pathogens kickstarting the immune system are examples in which events in a primary population initiate a distinct, secondary population
While we focus on clone sizes, we demonstrate that the distribution for the total number of mutants follows as a consequence, and results hold in that setting
We make the following remarks on the above. (i) The mutation rate μ does not appear in the density for initiation times in (6); mutant clone sizes are independent of the mutation rate and all following results regarding clone sizes will be . (ii)
Summary
Bacterial colonies developing antibiotic resistance or pathogens kickstarting the immune system are examples in which events in a primary population initiate a distinct, secondary population. In the large time small mutation rate limit, the clone size distribution at a fixed wild-type population size coincides for stochastic and deterministic exponential wild-type growth (Kessler and Levine 2015; Keller and Antal 2015). We present some general results which are valid for a large class of growth functions This extends the classic results found in Kendall (1948), Athreya and Ney (2004), Karlin and Taylor (1981), Tavare (1987) and recent work in Tomasetti (2012), Houchmandzadeh (2015) who considered the wild-type population growth rate to be time-dependent but coupled with the mutant growth rate.
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