Electron. J. Differential Equations, Vol. 2021 (2021), No. 56, pp. 1-22.

Age-dependent branching processes and applications to the Luria-Delbruck experiment

Stephen J. Montgomery-Smith, Hesam Oveys

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbrück that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution have spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by Lea, Coulson, and Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.

Submitted May 26, 2021. Published June 23, 2021.
Math Subject Classifications: 92D15.
Key Words: Probability generating function; fluctuation analysis; asymmetric cell division; Laplace transform.

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Stephen J. Montgomery-Smith
Department of Mathematics
University of Missouri
Columbia MO 65211, USA
email: stephen@missouri.edu
Hesam Oveys
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012, USA
email: hoveys@cims.nyu.edu

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