Statistics for the Luria-Delbrück distribution

Abstract

The Luria-Delbrück distribution is a classical model of mutations in cell kinetics. It is obtained as a limit when the probability of mutation tends to zero and the number of divisions to infinity. It can be interpreted as a compound Poisson distribution (for the number of mutations) of exponential mixtures (for the developing time of mutant clones) of geometric distributions (for the number of cells produced by a mutant clone in a given time). The probabilistic interpretation, and a rigourous proof of convergence in the general case, are deduced from classical results on Bellman-Harris branching processes. The two parameters of the Luria-Delbrück distribution are the expected number of mutations, which is the parameter of interest, and the relative fitness of normal cells compared to mutants, which is the heavy tail exponent. Both can be simultaneously estimated by the maximum likehood method. However, the computation becomes numerically unstable when the maximal value of the sample is large, which occurs frequently due to the heavy tail property. Based on the empirical probability generating function, robust estimators are proposed and their asymptotic variance is given. They are comparable in precision to maximum likelihood estimators, with a much broader range of calculability, a better numerical stability, and a negligible computing time.

Spontaneous mutations conferring antibiotic resistance to antitubercular drugs at a range of concentrations in Mycobacterium smegmatis

“…Thus, unlike the Po method, this approach is recommended in situations when most of cultures exhibit mutant colonies and ideally should be avoided when over half of the tubes have no mutations (Pope, O'Sullivan, McHugh, & Gillespie, 2008). The Ma-Sarka-Sandri Maximum Likelihood (ML) estimator, devised by Sarkar, Ma, and Sandri (1992), is currently the benchmark for estimating m and is the only technique able to correct for inaccuracies that arise from partial plaiting (Hamon & Ycart, 2012 (Lea & Coulson, 1949). The MSS algorithm is represented as:…”

Bacterial defenses against a natural antibiotic promote collateral resilience to clinical antibiotics

“…That PYO increases µapp by promoting the growth of pre-existing partially-resistant mutants was further supported by the alternative approach of evaluating the fit of our data to different mathematical models. Specifically, Pearson's chi-square test indicated that our data closely fit the Hamon and Ycart model 32 (Fig. S9, Table S2), which allows for differential fitness of mutants compared to WT cells, but assumes that all mutants arise pre-plating.…”

“…Specifically, for the Hamon and Ycart version of the LD model 32 , we estimated m and w (relative fitness of mutants compared to the parent strain in the non-selective pre-plating liquid growth medium) using the function GF.est from the R script available at http://ljk.imag.fr/membres/Bernard.Ycart/LD/ (version 1.0; note that in the script, m is called alpha and 1/w is called rho); then, we used the function pLD from the same script to generate the theoretical distribution. For the mixed LD-Poisson and basic LD models, we wrote and used an R translation of the MATLAB code written by Lang et al 33 ; the original code is available at https://github.com/AWMurrayLab/FluctuationTest_GregLang.…”

PynA is a pyrimidine 5′‐nucleotidase that functions as an antimutator protein inStreptococcus pneumoniae

“…The mutation rate (l) determinations and the statistical analysis from the fluctuation assays were carried out using the bz rates Web tool (http://www.lcqb.upmc.fr/bzrates) with the default settings [51]. For each strain, 50 independent 100-lL cultures in C+Y medium or C+Y plus 5-BrdU (10 lM) were started with an inoculum of approximately 1 9 10 2 cfu (N 0 ) in a microtitre plate.…”

“…The remaining 100-lL cultures were spread onto separate agar plates supplemented with rifampicin (2 lgÁmL À1 ; 29 MIC), and the total number of resistant mutants was determined after a 24-h incubation period. The mutation rate (l) determinations and the statistical analysis from the fluctuation assays were carried out using the bz rates Web tool (http://www.lcqb.upmc.fr/bzrates) with the default settings [51].…”