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Cited by 70Stochastic processes and statistical mechanicsEvolution and Genetic DynamicsMathematical and Theoretical Epidemiology and Ecology Models

A population birth-and-mutation process, I: explicit distributions for the number of mutants in an old culture of bacteria.

Mandelbrot, Benoit·1974·Journal of Applied Probability
DOI 10.2307/3212688
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Abstract

Luria and Delbrück (1943) have observed that, in old cultures of bacteria that have mutated at random, the distribution of the number of mutants is extremely long-tailed. In this note, this distribution will be derived (for the first time) exactly and explicitly. The rates of mutation will be allowed to be either positive or infinitesimal, and the rate of growth for mutants will be allowed to be either equal, greater or smaller than for non-mutants. Under the realistic limit condition of a very low mutation rate, the number of mutants is shown to be a stable-Lévy (sometimes called “Pareto Lévy”) random variable, of maximum skewness ß , whose exponent α is essentially the ratio of the growth rates of non-mutants and of mutants. Thus, the probability of the number of mutants exceeding the very large value m is proportional to m –α–1 (a behavior sometimes referred to as “asymptotically Paretian” or “hyperbolic”). The unequal growth rate cases α ≠ 1 are solved for the first time. In the α = 1 case, a result of Lea and Coulson is rederived, interpreted, and generalized. Various paradoxes involving divergent moments that were encountered in earlier approaches are either absent or fully explainable. The mathematical techniques used being standard, they will not be described in detail, so this note will be primarily a collection of results. However, the justification for deriving them lies in their use in biology, and the mathematically unexperienced biologists may be unfamiliar with the tools used. They may wish for more details of calculations, more explanations and Figures. To satisfy their needs, a report available from the author upon request has been prepared. It will be referred to as Part II.

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  • The second mutant distribution resulted from an attempt to relax assumption (A3) by Mandelbrot (1974).

    “The second mutant distribution resulted from an attempt to relax the assumption (A3) by Mandelbrot (1974).”

    From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Mandelbrot 1974

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Authors
Mandelbrot, B.
Venue
Journal of Applied Probability
Publisher
Cambridge University Press (CUP)
Published
1974-09
Volume
11
Issue
3
Pages
437-444
ISSN
  • 0021-9002 (ISSN-L)
  • 1475-6072
DOI
10.2307/3212688
OpenAlex
W2315323580
Cited by
70
References
15
Citations by year
2026: 1 · 2025: 2 · 2024: 1 · 2023: 3 · 2022: 3 · 2021: 1 · 2019: 3 · 2018: 3 · 2017: 3 · 2016: 3 · 2015: 3 · 2014: 4 · 2013: 6 · 2012: 3
FWCI
0.42
Type
journal article
Language
en
Open access
closed
Licence
www.cambridge.org/core/terms (unspecified, 15292 d)
Locations
Journal of Applied Probability (publishedVersion)
Topics
  • Physical Sciences › Mathematics › Mathematical Physics › Stochastic processes and statistical mechanics
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Genetics › Evolution and Genetic Dynamics
  • Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
Keywords
Pareto distribution, long-tailed distributions, mutation rate, bacterial growth rate, hyperbolic distribution, stochastic population growth

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