Universal asymptotic clone size distribution for general population growth

TL;DR

This paper generalizes the clone size distribution model for populations with various growth dynamics, deriving exact formulas and showing that the long-term distribution tail follows a power-law, with implications for cancer metastasis analysis.

Contribution

It introduces a generalized model for clone size distribution under different population growth types, extending beyond the exponential case and proving a universal power-law tail behavior.

Findings

Exact clone size distributions for exponential, power-law, and logistic growth.

Long-term clone size distribution has a universal power-law tail.

Cancer metastasis data supports power-law tail over exponential.

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-Delbr\"uck 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.