On Haldane’s formulation of Luria and Delbrück’s mutation model.
Abstract
Two formulations of Luria and Delbrück's mutation model have been in common use since the 1940s. While mathematicians focused their attention on the formulation of Lea and Coulson that assumes asynchronous cell growth, biologists found more appealing the formulation of Haldane that assumes synchronous cell growth. This article attempts to solve several outstanding issues for the latter formulation. First, it provides an exact, closed-form expression for the mutant distribution by correcting a minor error in the literature. Second, it presents a novel algorithm for computing the mutant distribution, which leads to novel methods for computing point and interval estimates of mutation rates based on the maximum likelihood principle. Third, it critically examines existing methods based on the mean number of mutants. Finally, it compares the two formulations to underline their strengths and shortcomings.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In a study of the Haldane model, Zheng (2007) compared the Lea-Coulson model with the Haldane model and found that the two models produced comparable estimates of mutation rates.
“In a study of the Haldane model, Zheng (2007) compared the Lea-Coulson model with the Haldane model and found that the two models produced comparable estimates of mutation rates.”
From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Zheng 2007
rSalvador includes the Haldane model mainly for educational purposes, and it relies on algorithms developed by Zheng (2007).
“rSalvador includes the Haldane model mainly for educational purposes, and it relies on algorithms developed by Zheng (2007).”
From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Zheng 2007
Cites
Mutations of bacteria from virus sensitivity to virus resistance
Luria, S. E., & Delbrück, M. (1943) · Genetics · doi:10.1093/genetics/28.6.491 · Cited by 3,894 · see citing works
An Introduction to Difference Equations
Elaydi, S. (1999) · Undergraduate texts in mathematics · doi:10.1007/978-1-4757-3110-1 · Cited by 1,777 · see citing works
The distribution of the numbers of mutants in bacterial populations
Lea, D. E., & Coulson, C. A. (1949) · Journal of Genetics · doi:10.1007/bf02986080 · Cited by 1,450 · see citing works
Introduction to Genetic Analysis
Suzuki, D. T., & Griffiths, A. J. F. (1976) · Cited by 1,114 · see citing works
An Introduction to Genetic Analysis
Teplitz, R. R. (1987) · JAMA · doi:10.1001/jama.1987.03390160109041 · Cited by 1,070 · see citing works
Determining Mutation Rates in Bacterial Populations
Rosche, W. A., & Foster, P. L. (2000) · Methods · doi:10.1006/meth.1999.0901 · Cited by 482 · see citing works
Production of Staphylococcus Strains Resistant to Various Concentrations of Penicillin
Demerec, M. (1945) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.31.1.16 · Cited by 224 · see citing works
Measurements of rate of mutation of flagellar antigenic phase inSalmonella typhi-murium
Stocker, B. A. D. (1949) · Epidemiology and Infection · doi:10.1017/s002217240001473x · Cited by 160 · see citing works
Progress of a half century in the study of the Luria-Delbrück distribution
Zheng, Q. (1999) · Mathematical Biosciences · doi:10.1016/s0025-5564(99)00045-0 · Cited by 144 · see citing works
Self-citationDELAYED PHENOTYPIC EXPRESSION OF SPONTANEOUS MUTATIONS IN ESCHERICHIA COLI
Newcombe, H. B. (1948) · Genetics · doi:10.1093/genetics/33.5.447 · Cited by 140 · see citing works
The statistical theory of bacterial populations subject to mutation.
Armitage, P. (1952) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1952.tb00098.x · Cited by 139 · see citing works
Stewart, F. M., Gordon, D. M., & Levin, B. R. (1990) · Genetics · doi:10.1093/genetics/124.1.175 · Cited by 126 · see citing works
Analysis of the Luria–Delbrück distribution using discrete convolution powers
Ma, W. T., Sandri, G. V., & Sarkar, S. (1992) · Journal of Applied Probability · doi:10.1017/s0021900200043023 · Cited by 121 · see citing works
Birth-and-death processes, and the theory of carcinogenesis
Kendall, D. G. (1960) · Biometrika · doi:10.1093/biomet/47.1-2.13 · Cited by 107 · see citing works
Luria-Delbrück fluctuation experiments: design and analysis.
Jones, M. E., Thomas, S. M., & Rogers, A. (1994) · Genetics · doi:10.1093/genetics/136.3.1209 · Cited by 96 · see citing works
An Introduction to Stochastic Processes with Special Reference to Methods and Applications.
Fox, R. A., & Bartlett, M. S. (1962) · Journal of the Royal Statistical Society Series D (The Statistician) · doi:10.2307/2987233 · Cited by 92 · see citing works
An Introduction to Stochastic Processes with Special Reference to Methods and Applications.
Cox, D. R., & Bartlett, M. S. (1967) · Biometrika · doi:10.2307/2333887 · Cited by 87 · see citing works
Mutation and growth rates from Luria-Delbrück fluctuation tests.
Koch, A. L. (1982) · Mutation Research · doi:10.1016/0027-5107(82)90252-4 · Cited by 81 · see citing works
Statistical and algorithmic methods for fluctuation analysis with SALVADOR as an implementation
Zheng, Q. (2002) · Mathematical Biosciences · doi:10.1016/s0025-5564(02)00087-1 · Cited by 70 · see citing works
Self-citationMandelbrot, B. (1974) · Journal of Applied Probability · doi:10.2307/3212688 · Cited by 70 · see citing works
New algorithms for Luria-Delbruck fluctuation analysis
Zheng, Q. (2005) · Mathematical Biosciences · doi:10.1016/j.mbs.2005.03.011 · Cited by 57 · see citing works
Self-citationAn explicit representation of the Luria-Delbrück distribution
Angerer, W. P. (2001) · Journal of Mathematical Biology · doi:10.1007/s002850000053 · Cited by 55 · see citing works
Modeling and measurement of the spontaneous mutation rate in mammalian cells
Rossman, T. G., Goncharova, E. I., & Nadas, A. J. (1995) · Mutation research. Fundamental and molecular mechanisms of mutagenesis · doi:10.1016/0027-5107(94)00190-g · Cited by 49 · see citing works
Haldane's solution of the Luria-Delbrück distribution.
Sarkar, S. (1991) · Genetics · doi:10.1093/genetics/127.2.257 · Cited by 47 · see citing works
Mathematical models for estimating mutation rates in cell populations
CRUMP, K. S., & HOEL, D. G. (1974) · Biometrika · doi:10.1093/biomet/61.2.237 · Cited by 44 · see citing works
Remarks on the Luria–Delbrück distribution
Pakes, A. G. (1993) · Journal of Applied Probability · doi:10.2307/3214530 · Cited by 32 · see citing works
Fluctuation analysis of mutations to nalidixic acid resistance in Escherichia coli.
Boe, L., Tolker-Nielsen, T., Eegholm, K. M., Spliid, H., & Vrang, A. (1994) · Journal of Bacteriology · doi:10.1128/jb.176.10.2781-2787.1994 · Cited by 26 · see citing works
Fluctuation test for two-stage mutations: application to gene amplification
Kimmel, M., & Axelrod, D. (1994) · Mutation research. Fundamental and molecular mechanisms of mutagenesis · doi:10.1016/0027-5107(94)90166-x · Cited by 23 · see citing works
Estimation of Spontaneous Mutation Rates
Natarajan, L., Berry, C. C., & Gasché, C. (2003) · Biometrics · doi:10.1111/1541-0420.00065 · Cited by 19 · see citing works
Flagged first, then by citations
1 of the 30 references couldn’t be matched to a known paper.
Cited by
Sorted by citations
Hall, B. M., Ma, C.-X., Liang, P., & Singh, K. K. (2009) · Bioinformatics · doi:10.1093/bioinformatics/btp253 · Cited by 375 · see citing works
rSalvador: An R Package for the Fluctuation Experiment
Zheng, Q. (2017) · G3 Genes|Genomes|Genetics · doi:10.1534/g3.117.300120 · Cited by 98 · see citing works
“” In a study of the Haldane model, Zheng (2007) compared the Lea-Coulson model with the Haldane model and found that the two models produced comparable estimates of mutation rates.”
MethodResultSelf-citationReplicative DNA Polymerase δ but Not ε Proofreads Errors in Cis and in Trans
Flood, C. L., Rodriguez, G. P., Bao, G., Shockley, A. H., Kow, Y. W., & Crouse, G. F. (2015) · PLoS Genetics · doi:10.1371/journal.pgen.1005049 · Cited by 50 · see citing works
A new practical guide to the Luria–Delbrück protocol
Zheng, Q. (2015) · Mutation research. Fundamental and molecular mechanisms of mutagenesis · doi:10.1016/j.mrfmmm.2015.08.005 · Cited by 33 · see citing works
Self-citationUnbiased estimation of mutation rates under fluctuating final counts.
Ycart, B., & Veziris, N. (2014) · PLoS ONE · doi:10.1371/journal.pone.0101434 · Cited by 19 · see citing works
Mean field mutation dynamics and the continuous Luria–Delbrück distribution
Kashdan, E., & Pareschi, L. (2012) · Mathematical Biosciences · doi:10.1016/j.mbs.2012.08.001 · Cited by 17 · see citing works
On Bartlett’s formulation of the Luria–Delbrück mutation model
Zheng, Q. (2008) · Mathematical Biosciences · doi:10.1016/j.mbs.2008.05.005 · Cited by 16 · see citing works
Self-citationFluctuation analysis: can estimates be trusted?
Ycart, B. (2013) · PLoS ONE · doi:10.1371/journal.pone.0080958 · Cited by 15 · see citing works
Luria–Delbrück, revisited: the classic experiment does not rule out Lamarckian evolution
Holmes, C. M., Ghafari, M., Abbas, A., Saravanan, V., & Nemenman, I. M. (2017) · Physical Biology · doi:10.1088/1478-3975/aa8230 · Cited by 16 · see citing works
Shockley, A. H., Doo, D. W., Rodriguez, G. P., & Crouse, G. F. (2013) · Genetics · doi:10.1534/genetics.113.153874 · Cited by 14 · see citing works
Numbers of Mutations within Multicellular Bodies: Why It Matters
Frank, S. A. (2022) · Axioms · doi:10.3390/axioms12010012 · Cited by 7 · see citing works
Carvajal‐Rodríguez, A. (2012) · Biochemistry and Molecular Biology Education · doi:10.1002/bmb.20615 · Cited by 7 · see citing works
A cautionary note on the mutation frequency in microbial research
Zheng, Q. (2018) · Mutation research. Fundamental and molecular mechanisms of mutagenesis · doi:10.1016/j.mrfmmm.2018.04.001 · Cited by 6 · see citing works
Self-citationMultiscale Estimation of Cell Kinetics
Jean, L. W., Suchorolski, M. T., Jeon, J., & Luebeck, E. G. (2009) · Computational and Mathematical Methods in Medicine · doi:10.1080/17486700903535922 · Cited by 3 · see citing works
Jin, J., Wei, G., Yang, W., Zhang, H., & Gao, P. (2012) · Science China Life Sciences · doi:10.1007/s11427-012-4395-7 · Cited by 1 · see citing works
Fluctuation analysis on mutation models with birth-date dependence
Mazoyer, A. (2018) · Mathematical Biosciences · doi:10.1016/j.mbs.2018.06.006 · Cited by 1 · see citing works
Fascination with Fluctuation: Luria and Delbrück’s Legacy
Robeva, R., & Jungck, J. R. (2023) · Axioms · doi:10.3390/axioms12030280 · Cited by 1 · see citing works
The Luria–Delbrück protocol is still the most practical
Zheng, Q. (2015) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2015.09.003 · Cited by 1 · see citing works
Self-citationMutant Number Laws and Infinite Divisibility
Pakes, A. G. (2022) · Axioms · doi:10.3390/axioms11110584 · Cited by 1 · see citing works
- Authors
- Zheng, Q.
- Affiliations
- Texas A&M Health Science Center (US)
- Corresponding author
- Zheng, Qi
- Venue
- Mathematical Biosciences
- Publisher
- Elsevier BV
- Published
- 2007-10
- Volume
- 209
- Issue
- 2
- Pages
- 500-513
- ISSN
- 0025-5564 (ISSN-L)
- OpenAlex
- W1985173348
- Cited by
- 19
- References
- 30
- Citations by year
- 2023: 1 · 2022: 2 · 2018: 2 · 2017: 2 · 2015: 3 · 2014: 1 · 2013: 2 · 2012: 3
- FWCI
- 0.54
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Mathematical Biosciences (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Genetics › Evolution and Genetic Dynamics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Mathematical Physics › Stochastic processes and statistical mechanics
- Keywords
- analytical solution, mutation rate estimation, maximum likelihood estimation, bacterial mutation
- MeSH terms
- Algorithms, Markov Chains, Mathematics, Models, Genetic, Mutation, Confidence Intervals, Likelihood Functions, Cell Proliferation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
