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Comments on the Luria–Delbrück distribution

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Adrienne W. Kemp
Adrienne W. Kemp*
Affiliation:
University of St Andrews
*
∗ Postal address: Mathematical Institute, North Haugh, St Andrews KY16 9SS UK.
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Abstract

The long-tailed Luria–Delbrück distribution arises in connection with the ‘random mutation’ hypothesis (whereas the ‘directed adaptation' hypothesis is thought to give a Poisson distribution). At time t the distribution depends on the parameter m = gNt/(a + g) where Nt is the current population size and g/(a + g) is the relative mutation rate (assumed constant). The paper identifies three models for the distribution in the existing literature and gives a fourth model. Ma et al. (1992) recently proved that there is a remarkably simple recursion relation for the Luria–Delbrück probabilities pn and found that asymptotically pn ≈ c/n2; their numerical studies suggested that c = 1 when the parameter m is unity. Cairns et al. (1988) had previously argued and shown numerically that Pn = Σj ≧ n Pj ≈ m/n. Here we prove that n(n + 1)pn < m(1 + 11m/30) for n = 1, 2, ···, and hence prove that as n becomes large n(n + 1)pn, ≈ m; the result mPn ≈ m follows immediately.

Keywords

RANDOM MUTATIONINFINITE DIVISIBILITYPOISSON-STOPPED SUM DISTRIBUTION

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 822 - 828
Copyright
Copyright © Applied Probability Trust 1994 

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References

Armitage, P. (1952) The statistical theory of bacterial populations subject to mutation. J. R. Statist. Soc. B14, 1–44.Google Scholar
Bartlett, M. S. (1952) Contribution to discussion on ‘The statistical theory of bacterial populations subject to mutation’. J. R. Statist. Soc. B14, 37–38.Google Scholar
Bartlett, M. S. (1966) An Introduction to Stochastic Processes, 2nd. 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. (1957) An Introduction to Probability Theory, Vol. 1, 2nd ed. Wiley, New York.Google Scholar
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992) Univariate Discrete Distributions, 2nd ed. Wiley, New York.Google Scholar
Kemp, C. D. (1967) On a contagious distribution suggested for accident data. Biometrics 23, 241–255.CrossRefGoogle ScholarPubMed
Lea, D. E. and Coulson, C. A. (1949) The distribution of the numbers of mutants in bacterial populations. J. Genetics 49, 264–285.CrossRefGoogle ScholarPubMed
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. A155, 103–106.CrossRefGoogle Scholar
Ma, W. T., Sandri, G. Vh. and Sarkar, S. (1992) Analysis of the Luria-Delbrück distribution using discrete convolution powers. J. Appl. Prob. 29, 255–267.CrossRefGoogle Scholar
Ryan, F. J. (1952) Contribution to discussion on ‘The statistical theory of bacterial populations subject to mutation’. J. R. Statist. Soc. B14, 39–40.Google Scholar