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Analysis of the Luria–Delbrück distribution using discrete convolution powers

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

W. T. Ma ,
G. vH. Sandri  and
S. Sarkar
W. T. Ma
Affiliation:
Boston University
G. vH. Sandri
Affiliation:
Boston University
S. Sarkar*
Affiliation:
Boston University
*
∗Postal address: Boston Theoretical Biology Group, Center for the Philosophy and History of Science, Boston University, 745 Commonwealth Avenue, Boston, MA 02215, USA.
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Abstract

The Luria–Delbrück distribution arises in birth-and-mutation processes in population genetics that have been systematically studied for the last fifty years. The central result reported in this paper is a new recursion relation for computing this distribution which supersedes all past results in simplicity and computational efficiency: p0 = e–m; where m is the expected number of mutations. A new relation for the asymptotic behavior of pn (≈ c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for the z-transform of the distribution is also reported.

Keywords

BACTERIAL MUTAGENESISDIRECTED MUTATIONS

MSC classification

Primary: 60E05: Distributions
Secondary: 92D10: Genetics
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 255 - 267
Copyright
Copyright © Applied Probability Trust 1992 

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References

Armitage, P. (1952) The statistical theory of bacterial populations subject to mutation. J.R. Statist. Soc. B14, 1–40.Google Scholar
Armitage, P. (1953) Statistical concepts in the theory of bacterial mutation. J. Hygiene 51, 162–184.CrossRefGoogle ScholarPubMed
Bartlett, M. S. (1978) An Introduction to Stochastic Processes, 3rd edn. Cambridge University Press.Google Scholar
Cairns, J., Overbaugh, J. and Miller, S. (1988) The origin of mutants. Nature 335, 142–145.CrossRefGoogle ScholarPubMed
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Vol. 1. 3rd edn. Wiley, New York.Google Scholar
Fu, J., Li, I.C. and Chu, E. H. Y. (1982) The parameters for quantitative analysis of mutation rates with cultured mammalian somatic cells. Mutat. Res. 105, 363–370.CrossRefGoogle ScholarPubMed
Hilderbrand, F. B. (1916) Advanced Calculus for Applications, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Koch, A. L. (1982) Mutation and growth rates from Luria-Delbrück fluctuation tests. Mutat. Res. 95, 129–143.Google Scholar
Lea, D. E. and Coulson, C. A. (1949) The distribution of the number of mutants in bacterial populations. J. Genetics 49, 264–285.Google Scholar
Li, I.-C., Wu, S.-C. H., Fu, J. and Chu, E. H. Y. (1985) A determinist approach for the estimation of mutation rates in cultured mammalian cells. Mutation Res. 149, 127–132.Google Scholar
Luria, S. E. and Delbrück, M. (1943) Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28, 491–511.CrossRefGoogle ScholarPubMed
Ma, W. T., Sandri, G. vH. and Sarkar, S. (1991) Novel representation of power series that arise in statistical mechanics and population genetics. Phys. Lett.Google Scholar
Mandelbrot, B. (1974) A population birth-and-mutation process, I: explicit distributions for the number of mutants in an old culture of bacteria. J. Appl. Prob. 11, 437–444.Google Scholar
Mayer, J. E. and Mayer, M. G. (1940) Statistical Mechanics, Wiley, New York.Google Scholar
Sarkar, S. (1990) On the possibility of directed mutations in bacteria: statistical analysis and reductionist strategies. In PSA 1990; ed. Fine, A. et al., Vol. I. Philosophy of Science Association, East Lansing, 111–124.Google Scholar
Sarkar, S. (1991a) Lamarck contre Darwin, reduction versus statistics: conceptual issues in the controversy over directed mutagenesis in bacteria. In Organism and the Origin of Self ed. Tauber, A. I., pp. 235–271. Kluwer, Dordrecht.CrossRefGoogle Scholar
Sarkar, S. (1991b) Haldane's solution of the Luria-Delbrück distribution. Genetics 127, 257–261.CrossRefGoogle ScholarPubMed
Stewart, F. M., Gordon, D. M. and Levin, B. R. (1990) Fluctuation analysis: the probability distribution of the number of mutants under different conditions. Genetics 124, 175–185.Google Scholar