Analysis of the Luria–Delbrück distribution using discrete convolution powers

T., W.; Sandri, G.; Sarkar, Sahotra

doi:10.1017/s0021900200043023

Cited by 122 publications

(87 citation statements)

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“…To analyze the data, we employed a program written by P. J. Gerrish (Los Alamos National Laboratory) that generates expected Luria-Delbrü ck distributions according to the method of Ma et al (1992) and then uses maximum likelihood to refine the mutation rate estimate. This procedure gave a mutation rate of 4.2 3 10…”

Section: Resultsmentioning

confidence: 99%

“…Fitness was then calculated as the ratio of the realized growth rates of the two strains during their direct competition, and t-tests were performed to evaluate whether the measured fitness values differed significantly from the null hypothetical value of 1. (Lea and Coulson 1949;Ma et al 1992). Evolutionary dynamics of malT substitutions: The approximate time of origin and the subsequent dynamics of the malT mutations that were substituted in each focal population were investigated using a PCR-RFLP strategy with clones isolated from samples frozen at various generations during the longterm evolution experiment.…”

Section: Methodsmentioning

confidence: 99%

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Parallel Changes in Global Protein Profiles During Long-Term Experimental Evolution in Escherichia coli

et al. 2006

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Twelve populations of Escherichia coli evolved in and adapted to a glucose-limited environment from a common ancestor. We used two-dimensional protein electrophoresis to compare two evolved clones, isolated from independently derived populations after 20,000 generations. Exceptional parallelism was detected. We compared the observed changes in protein expression profiles with previously characterized global transcription profiles of the same clones; this is the first time such a comparison has been made in an evolutionary context where these changes are often quite subtle. The two methodologies exhibited some remarkable similarities that highlighted two different levels of parallel regulatory changes that were beneficial during the evolution experiment. First, at the higher level, both methods revealed extensive parallel changes in the same global regulatory network, reflecting the involvement of beneficial mutations in genes that control the ppGpp regulon. Second, both methods detected expression changes of identical gene sets that reflected parallel changes at a lower level of gene regulation. The protein profiles led to the discovery of beneficial mutations affecting the malT gene, with strong genetic parallelism across independently evolved populations. Functional and evolutionary analyses of these mutations revealed parallel phenotypic decreases in the maltose regulon expression and a high level of polymorphism at this locus in the evolved populations.

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Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Parallel Changes in Global Protein Profiles During Long-Term Experimental Evolution in Escherichia coli

et al. 2006

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“…Put differently, quantile and zero-class methods have some advantage when there is some uncertainty about the underlying model or when there are known simplifying assumptions that can introduce error. The formula presented here derives from the simplest model for the LDD (Lea and Coulson 1949), which makes a number of simplifying assumptions, such as exponentially distributed times between replication events (a Markovian birth process) ( Jones et al 1999;Kepler and Oprea 2001), probabilities expressed in infinite (nontruncated) series thereby assigning nonzero (albeit miniscule) probabilities to unrealistically large mutant numbers (Armitage 1952;Bailey 1964;Stewart et al 1990;Ma et al 1992;Zheng 1999), and no difference in growth rates between wild-type and mutant subpopulations ( Jones 1994;Jaeger and Sarkar 1995;Zheng 1999). In light of these simplifying assumptions, whose impact can sometimes be significant, it might ultimately be desirable to use the less accurate but more robust quantile or zero-class methods to complement the more accurate but less robust likelihood methods.…”

Section: Discussionmentioning

confidence: 99%

“…Jones truncates the sum at 100, which is adequate for values L # 70, but thus limits applicability of his tabulated median method results to protocols for which plating efficiency is greater than $5%. A more elegant solution to the problem of infinite summation appears in Ma et al (1992) and Jones et al (1999).…”

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confidence: 99%

A Simple Formula for Obtaining Markedly Improved Mutation Rate Estimates

Gerrish

2008

Genetics

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In previous work by M. E. Jones and colleagues, it was shown that mutation rate estimates can be improved and corresponding confidence intervals tightened by following a very easy modification of the standard fluctuation assay: cultures are grown to a larger-than-usual final density, and mutants are screened for in only a fraction of the culture. Surprisingly, this very promising development has received limited attention, perhaps because there has been no efficient way to generate the predicted mutant distribution to obtain nonmoment-based estimates of the mutation rate. Here, the improved fluctuation assay discovered by Jones and colleagues is made amenable to quantile-based, likelihood, and other Bayesian methods by a simple recursion formula that efficiently generates the entire mutant distribution after growth and dilution. This formula makes possible a further protocol improvement: grow cultures as large as is experimentally possible and severely dilute before plating to obtain easily countable numbers of mutants. A preliminary look at likelihood surfaces suggests that this easy protocol adjustment gives markedly improved mutation rate estimates and confidence intervals.A common method for estimating mutation rates is the ''fluctuation assay,'' in which several bacterial cultures are grown from small, presumably mutant-free, inoculums to high-density cultures and subsequently screened for mutants by plating on selective media (Rosche and Foster 2000;Pope et al. 2008). On selective media, only selected mutants are able to form colonies, and the numbers of mutants counted on plates are an indicator of the mutation rate. To make an estimate of the mutation rate from these numbers, a priori knowledge of their distribution is required. The fundamental assumption upon which this distribution is derived is that no Lamarckian-like processes exist: mutations are assumed to occur spontaneously during cell division in a way that is completely independent of any detriment or benefit that the mutations might subsequently incur. Such mutations can occur at any time during the growth of a culture. If a mutation happens to occur early in the growth of a culture, it will leave a large number of mutant descendants in the culture at the time of plating. The nonnegligible probability of a mutation occurring early thus translates to nonnegligible probabilities in the right tail of the mutant distribution. The resulting heavy-tailed mutant distribution was first proposed by Luria and Delbrü ck (1943) and has since been dubbed the ''Luria-Delbrü ck distribution'' (LDD) in their honor. Its first rigorous derivation was given by Lea and Coulson (1949), and many subsequent derivations account for different biological and experimental details (Armitage 1952;Bailey 1964;Mandelbrot 1974;Bartlett 1978;Stewart et al. 1990;Stewart 1991;Pakes 1993;Zheng 1999;Kepler and Oprea 2001;Oprea and Kepler 2001;Dewanji et al. 2005).Each of these derivations culminates in a version of the LDD expressed in probability-generating function (PGF...

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“…Among the later contributions that enhanced our ability to analyze fluctuation experiments are those made by Lea and Coulson (1949), Armitage (1952), Crump and Hoel (1974), Mandelbrot (1974), Koch (1982), Stewart et al (1990), Ma et al (1992), Jones et al (1994), and many others found in a recent review (Zheng 1999). Rosche and Foster's (2000) critical comparison of the then existing methods is a useful guide for biologists.…”

Section: T He Fluctuation Test Protocol Devised Bymentioning

confidence: 99%

Update on Estimation of Mutation Rates Using Data From Fluctuation Experiments

Zheng

2005

Genetics

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This note discusses a minor mathematical error and a problematic mathematical assumption in Luria and Delbrü ck 's (1943) classic article on fluctuation analysis. In addition to suggesting remedial measures, the note provides information on the latest development of techniques for estimating mutation rates using data from fluctuation experiments. T HE fluctuation test protocol devised by Luria andDelbrü ck (1943) still serves as the basis for estimating microbial mutation rates today, although later developments have resulted in much improved methods. Among the later contributions that enhanced our ability to analyze fluctuation experiments are those made by Lea and Coulson (1949), Armitage (1952), Crump and Hoel (1974), Mandelbrot (1974), Koch (1982), Stewart et al. (1990), Ma et al. (1992), Jones et al. (1994), and many others found in a recent review (Zheng 1999). Rosche and Foster's (2000) critical comparison of the then existing methods is a useful guide for biologists. One goal of this note is to notify the reader of the latest developments that can further help biologists improve their ability to measure mutation rates. Another goal is to discuss a minor mathematical error and a problematic mathematical assumption in Luria and Delbrü ck's (1943) article that have caused lingering confusion. Previous attempts to clarify the confusion were scarce and to a large extent failed to resolve some relevant practical issues. As a result, the genetics literature is increasingly fraught with mutation rates that were computed using either incorrect or unreliable methods. It appears helpful that the minor error and the problematic assumption are explained and remedial measures are provided. I begin with a paradox that has puzzled many.In a fluctuation experiment, each of n parallel cultures is seeded at time zero with N 0 nonmutant cells for incubation. At a later time T each culture has about N T nonmutant cells and the contents of each culture are plated to facilitate counting of mutants existing at time T in the n cultures. This process results in experimental data in the form of X 1 , X 2 , . . . , X n , the numbers of mutants existing in the n cultures immediately before plating. If z of the n cultures still remain devoid of mutant cells at time T, Luria and Delbrü ck's (1943) P 0 method estimates the mutation rate bŷwithp 0 ¼ z=n. Although Luria and Delbrü ck did not give the above equation, their numerical example on page 507 clearly indicates that they used (1) to estimate mutation rates defined as ''mutations per bacterium per division cycle.'' This definition of mutation rates has been widely accepted, and throughout this note the term ''mutation rate'' is used in that sense. In other words, a mutation rate is the probability that a cell undergoes a mutation during the cell's life cycle. Using the same definition, Lederberg (1951, p. 99) argued that mutation rates should be estimated bŷLederberg's reasoning runs as follows. In each culture N T ÿ N 0 cellular divisions have happened. If m 0 is the mut...

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Analysis of the Luria–Delbrück distribution using discrete convolution powers

Cited by 122 publications

References 10 publications

Parallel Changes in Global Protein Profiles During Long-Term Experimental Evolution in Escherichia coli

Parallel Changes in Global Protein Profiles During Long-Term Experimental Evolution in Escherichia coli

A Simple Formula for Obtaining Markedly Improved Mutation Rate Estimates

Update on Estimation of Mutation Rates Using Data From Fluctuation Experiments

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