Unbiased estimation of mutation rates under fluctuating final counts

Efficient, robust, and versatile fluctuation data analysis using MLE MUtation Rate calculator (mlemur)

The fluctuation assay remains an important tool for analyzing the levels of mutagenesis in microbial populations. The mutant counts originating from some average number of mutations are usually assumed to obey the Luria-Delbrück distribution. While several tools for estimating mutation rates are available, they sometimes lack accuracy or versatility under non-standard conditions. In this work, extensions to the Luria-Delbrück protocol to account for phenotypic lag and cellular death with either perfect or partial plating were developed. Hence, the novel MLE MUtation Rate calculator, or mlemur, is the first tool that provides a user-friendly graphical interface allowing the researchers to model their data with consideration for partial plating, differential growth of mutants and non-mutants, phenotypic lag, cellular death, variability of the final number of cells, post-exponential-phase mutations, and the size of the inoculum. Additionally, mlemur allows the users to incorporate most of these special conditions at the same time to obtain highly accurate estimates of mutation rates and P values, confidence intervals for an arbitrary function of data (such as fold), and perform power analysis and sample size determination for the likelihood ratio test. The accuracy of point and interval estimates produced by mlemur against historical and simulated fluctuation experiments are assessed. Both mlemur and the analyses in this work might be of great help when evaluating fluctuation experiments and increase the awareness of the limitations of the widely-used Lea-Coulson formulation of the Luria-Delbrück distribution in the more realistic biological contexts.

The effect of bacterial mutation rate on the evolution of CRISPR-Cas adaptive immunity

CRISPR-Cas immune systems are present in around half of bacterial genomes. Given the specificity and adaptability of this immune mechanism, it is perhaps surprising that they are not more widespread. Recent insights into the requirement for specific host factors for the function of some CRISPR-Cas subtypes, as well as the negative epistasis between CRISPR-Cas and other host genes, have shed light on potential reasons for the partial distribution of this immune strategy in bacteria. In this study, we examined how mutations in the bacterial mismatch repair system, which are frequently observed in natural and clinical isolates and cause elevated host mutation rates, influence the evolution of CRISPR-Cas-mediated immunity. We found that hosts with a high mutation rate very rarely evolved CRISPR-based immunity to phage compared to wild-type hosts. We explored the reason for this effect and found that the higher frequency at which surface mutants pre-exist in the mutator host background causes them to rapidly become the dominant phenotype under phage infection. These findings suggest that natural variation in bacterial mutation rates may, therefore, influence the distribution of CRISPR-Cas adaptive immune systems. This article is part of a discussion meeting issue 'The ecology and evolution of prokaryotic CRISPR-Cas adaptive immune systems'.

Bacterial persistence promotes the evolution of antibiotic resistance by increasing survival and mutation rates

Persisters are transiently antibiotic-tolerant cells that complicate the treatment of bacterial infections. Both theory and experiments have suggested that persisters facilitate genetic resistance by constituting an evolutionary reservoir of viable cells. Here, we provide evidence for a strong positive correlation between persistence and the likelihood to become genetically resistant in natural and lab strains of E. coli. This correlation can be partly attributed to the increased availability of viable cells associated with persistence. However, our data additionally show that persistence is pleiotropically linked with mutation rates. Our theoretical model further demonstrates that increased survival and mutation rates jointly affect the likelihood of evolving clinical resistance. Overall, these results suggest that the battle against antibiotic resistance will benefit from incorporating anti-persister therapies.

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

Mycobacteria populations can undergo mutations in their DNA sequence during replication, which if not repaired would be transferred to future generations. Earlier studies have tackled the estimation of mutation rate in mycobacteria at fixed concentrations. However, in this study, in vitro spontaneous mutations in Mycobacterium smegmatis (Msm) mc² 155 (Msm) that confers resistance to some of the most important antitubercular drugs; isoniazid (INH^r ), rifampicin (RIF^r ), kanamycin (KAN^r ) and streptomycin (STR^r ) were first determined at several highly lethal concentrations, a few of which have not been previously investigated, in a fluctuation assay. Thereafter, mutation rate was estimated using the most commonly adopted Po method, and estimates were then compared concurrently with the Lea-Coulson method of the median and Ma-Sandri-Sarkar Maximum Likelihood Estimator method available on the Fluctuation AnaLysis CalculatOR (FALCOR). The mutation rates of RIF^r ranged from 9.24 × 10^-8 to 2.18 × 10^-10 , INH^r 1.2 × 10^-7 -1.20 × 10^-9 , STR^r 2.77 × 10^-8 -5.31 × 10^-8 and KAN^r 1.7 × 10^-8 mutations per cell division. Data obtained in this study provide mutation rate estimates to key antitubercular drugs at a range of concentrations while also validating a number of the frequent approaches for estimating mutation rates.

Opposing effects of final population density and stress on Escherichia coli mutation rate

Evolution depends on mutations. For an individual genotype, the rate at which mutations arise is known to increase with various stressors (stress-induced mutagenesis-SIM) and decrease at high final population density (density-associated mutation-rate plasticity-DAMP). We hypothesised that these two forms of mutation-rate plasticity would have opposing effects across a nutrient gradient. Here we test this hypothesis, culturing Escherichia coli in increasingly rich media. We distinguish an increase in mutation rate with added nutrients through SIM (dependent on error-prone polymerases Pol IV and Pol V) and an opposing effect of DAMP (dependent on MutT, which removes oxidised G nucleotides). The combination of DAMP and SIM results in a mutation rate minimum at intermediate nutrient levels (which can support 7 × 108 cells ml-1). These findings demonstrate a strikingly close and nuanced relationship of ecological factors-stress and population density-with mutation, the fuel of all evolution.

Fluctuation analysis on mutation models with birth-date dependence

The classic Luria-Delbrück model can be interpreted as a Poisson compound (number of mutations) of exponential mixtures (developing time of mutant clones) of geometric distributions (size of a clone in a given time). This "three-ingredients" approach is generalized in this paper to the case where the split instant distributions of cells are not i.i.d. : the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. Previous results on the convergence of the distribution of mutant counts are recovered. The particular case where the instantaneous division rates of normal and mutant cells are proportional is studied. The classic Luria-Delbrück and Haldane models are recovered. Probability computations and simulation algorithms are provided. Robust estimation methods developed for the classic mutation models are adapted to the new model: their properties of consistency and asymptotic normality remain true; their asymptotic variances are computed. Finally, the estimation biases induced by considering classic mutation models instead of an inhomogeneous model are studied with simulation experiments.

Identification of Rifampicin Resistance Mutations in Escherichia coli, Including an Unusual Deletion Mutation

Rifampicin is an effective antibiotic against mycobacterial and other bacterial infections, but resistance readily emerges in laboratory and clinical settings. We screened Escherichia coli for rifampicin resistance and identified numerous mutations to the gene encoding the β-chain of RNA polymerase (rpoB), including an unusual 9-nucleotide deletion mutation. Structural modeling of the deletion mutant indicates locations of potential steric clashes with rifampicin. Sequence conservation in the region near the deletion mutation suggests a similar mutation may also confer resistance during the treatment of tuberculosis.

rSalvador: An R Package for the Fluctuation Experiment

The past few years have seen a surge of novel applications of the Luria-Delbrück fluctuation assay protocol in bacterial research. Appropriate analysis of fluctuation assay data often requires computational methods that are unavailable in the popular web tool FALCOR. This paper introduces an R package named rSalvador to bring improvements to the field. The paper focuses on rSalvador’s capabilities to alleviate three kinds of problems found in recent investigations: (i) resorting to partial plating without properly accounting for the effects of partial plating; (ii) conducting attendant fitness assays without incorporating mutants’ relative fitness in subsequent data analysis; and (iii) comparing mutation rates using methods that are in general inapplicable to fluctuation assay data. In addition, the paper touches on rSalvador’s capabilities to estimate sample size and the difficulties related to parameter nonidentifiability.

Time Inhomogeneous Mutation Models with Birth Date Dependence

The classic Luria-Delbrück model for fluctuation analysis is extended to the case where the split instant distributions of cells are not i.i.d.: the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. In particular, it is possible to consider subprobability distributions and to model non-exponential growth. The extended model leads to a family of probability distributions which depend on the expected number of mutations, the death probability of mutant cells, and the split instant distributions of normal and mutant cells. This is deduced from the Bellman-Harris integral equation, written for the birth date inhomogeneous case. A new theorem of convergence for the final mutant counts is proved, using an analytic method. Particular examples like the Haldane model or the case where hazard functions of the split-instant distributions are proportional are studied. The Luria-Delbrück distribution with cell deaths is recovered. A computation algorithm for the probabilities is provided.

A second look at the final number of cells in a fluctuation experiment

In a fluctuation experiment, the number of cells existing in a culture immediately before plating (commonly known as Nt) varies across the parallel cultures. However, most existing mathematical models for fluctuation assay data do not recognize the variation in Nt. Despite repeated attempts in the past to integrate this source of variability in the estimation of microbial mutation rates, several questions of practical importance remain unanswered. The present investigation finds that the variation needs accounting for only when the coefficient of variation for Nt is large, and experimental data suggest that the coefficient of variation is often moderate or small. Moreover, an increase in the inoculum size can reduce the coefficient of variation. Through extensive simulation, several existing methods that accommodate the variation in Nt are compared. It was found that a newly devised likelihood method based on the existing gamma mixture model outperforms other existing methods. The investigation focuses on the estimation of mutation rates using the Lea-Coulson model, under which mutation is selectively neutral; however, the paper also explores the major findings' implications for the comparison of mutation rates using the likelihood ratio test, and for the estimation of mutation rates using the Mandelbrot-Koch model that allows for non-neutral mutations.

bz-rates: A Web Tool to Estimate Mutation Rates from Fluctuation Analysis

Fluctuation analysis is the standard experimental method for measuring mutation rates in micro-organisms. The appearance of mutants is classically described by a Luria-Delbrück distribution composed of two parameters: the number of mutations per culture (m) and the differential growth rate between mutant and wild-type cells (b). A precise estimation of these two parameters is a prerequisite to the calculation of the mutation rate. Here, we developed bz-rates, a Web tool to calculate mutation rates that provides three useful advances over existing Web tools. First, it allows taking into account b, the differential growth rate between mutant and wild-type cells, in the estimation of m with the generating function. Second, bz-rates allows the user to take into account a deviation from the Luria-Delbrück distribution called z, the plating efficiency, in the estimation of m. Finally, the Web site provides a graphical visualization of the goodness-of-fit between the experimental data and the model. bz-rates is accessible at http://www.lcqb.upmc.fr/bzrates.

General formulation of Luria-Delbrück distribution of the number of mutants

The Luria-Delbrück experiment is a cornerstone of evolutionary theory, demonstrating the randomness of mutations before selection. The distribution of the number of mutants in this experiment has been the subject of intense investigation during the past 70 years. Despite this considerable effort, most of the results have been obtained under the assumption of constant growth rate, which is far from the experimental condition. We derive here the properties of this distribution for arbitrary growth function for both the deterministic and stochastic growth of the mutants. The derivation we propose uses the number of wild-type bacteria as the independent variable instead of time. The derivation is surprisingly simple and versatile, allowing many generalizations to be taken easily into account.