On Haldane’s formulation of Luria and Delbrück’s mutation model

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.

A cautionary note on the mutation frequency in microbial research

The mutation frequency, also known as the mutant frequency, is an unnormalized quantity, and its normalized counterpart is the mutation rate. Due to historical reasons, the mutation frequency has been a predominant yardstick of microbial mutability in the field of mutator identification. While the mean mutation frequency is infamously erratic, replacing it with the median mutation frequency is not an effective remedy. By encouraging investigators to substitute mutation rates for mutation frequencies in microbial research, this paper directs attention to substantial open problems such as false positive control and massive nonmutant cell death.

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.

Luria-Delbrück, revisited: the classic experiment does not rule out Lamarckian evolution

We re-examined data from the classic Luria-Delbrück fluctuation experiment, which is often credited with establishing a Darwinian basis for evolution. We argue that, for the Lamarckian model of evolution to be ruled out by the experiment, the experiment must favor pure Darwinian evolution over both the Lamarckian model and a model that allows both Darwinian and Lamarckian mechanisms (as would happen for bacteria with CRISPR-Cas immunity). Analysis of the combined model was not performed in the original 1943 paper. The Luria-Delbrück paper also did not consider the possibility of neither model fitting the experiment. Using Bayesian model selection, we find that the Luria-Delbrück experiment, indeed, favors the Darwinian evolution over purely Lamarckian. However, our analysis does not rule out the combined model, and hence cannot rule out Lamarckian contributions to the evolutionary dynamics.

The Luria-Delbrück protocol is still the most practical

It is theoretically appealing to determine microbial mutation rates by counting mutant cells in two successive generations. However, major experimental difficulties have been largely ignored, causing unreliable estimates of mutation rates to be unwittingly accepted. Counting mutant cells twice in the same liquid culture incurs appreciable errors that are difficult to quantify; maintaining synchronous cell growth for 25 or more generations is an equally daunting laboratory feat.

A new practical guide to the Luria-Delbrück protocol

Since 2000 several review papers have been published about the analysis of experimental data obtained using the Luria-Delbrück protocol. These timely papers cleared much of the confusion surrounding various methods for estimating or comparing mutation rates. As a result, today the fluctuation test is more widely applied with much improved accuracy. The present paper provides guidelines on a few remaining problems that continue to baffle mutation researchers. Among the issues addressed are incomplete plating, relative fitness, and comparison of experiments where average final cell population sizes differ. It also offers a fresh view on the estimation methods that are based on the sample median.

Replicative DNA polymerase δ but not ε proofreads errors in Cis and in Trans

It is now well established that in yeast, and likely most eukaryotic organisms, initial DNA replication of the leading strand is by DNA polymerase ε and of the lagging strand by DNA polymerase δ. However, the role of Pol δ in replication of the leading strand is uncertain. In this work, we use a reporter system in Saccharomyces cerevisiae to measure mutation rates at specific base pairs in order to determine the effect of heterozygous or homozygous proofreading-defective mutants of either Pol ε or Pol δ in diploid strains. We find that wild-type Pol ε molecules cannot proofread errors created by proofreading-defective Pol ε molecules, whereas Pol δ can not only proofread errors created by proofreading-defective Pol δ molecules, but can also proofread errors created by Pol ε-defective molecules. These results suggest that any interruption in DNA synthesis on the leading strand is likely to result in completion by Pol δ and also explain the higher mutation rates observed in Pol δ-proofreading mutants compared to Pol ε-proofreading defective mutants. For strains reverting via AT→GC, TA→GC, CG→AT, and GC→AT mutations, we find in addition a strong effect of gene orientation on mutation rate in proofreading-defective strains and demonstrate that much of this orientation dependence is due to differential efficiencies of mispair elongation. We also find that a 3′-terminal 8 oxoG, unlike a 3′-terminal G, is efficiently extended opposite an A and is not subject to proofreading. Proofreading mutations have been shown to result in tumor formation in both mice and humans; the results presented here can help explain the properties exhibited by those proofreading mutants.

Many DNA polymerases are able to proofread their errors: after incorporation of a wrong base, the resulting mispair invokes an exonuclease activity of the polymerase that removes the mispaired base and allows replication to continue. Elimination of the proofreading activity thus results in much higher mutation rates. We demonstrate that the two major replicative DNA polymerases in yeast, Pol δ and Pol ε, have different proofreading abilities. In diploid cells, Pol ε is not able to proofread errors created by other Pol ε molecules, whereas Pol δ can proofread not only errors created by other Pol δ molecules but also errors created by Pol ε molecules. We also find that mispaired bases not corrected by proofreading have much different likelihoods of being extended, depending on the particular base-base mismatch. In humans, defects in Pol δ or Pol ε proofreading can lead to cancer, and these results help explain the formation of those tumors and the finding that Pol ε mutants seem to be found as frequently, or more so, in human tumors as Pol δ mutants.

Unbiased estimation of mutation rates under fluctuating final counts

Estimation methods for mutation rates (or probabilities) in Luria-Delbrück fluctuation analysis usually assume that the final number of cells remains constant from one culture to another. We show that this leads to systematically underestimate the mutation rate. Two levels of information on final numbers are considered: either the coefficient of variation has been independently estimated, or the final number of cells in each culture is known. In both cases, unbiased estimation methods are proposed. Their statistical properties are assessed both theoretically and through Monte-Carlo simulation. As an application, the data from two well known fluctuation analysis studies on Mycobacterium tuberculosis are reexamined.

Fluctuation analysis: can estimates be trusted?

The estimation of mutation rates and relative fitnesses in fluctuation analysis is based on the unrealistic hypothesis that the single-cell times to division are exponentially distributed. Using the classical Luria-Delbrück distribution outside its modelling hypotheses induces an important bias on the estimation of the relative fitness. The model is extended here to any division time distribution. Mutant counts follow a generalization of the Luria-Delbrück distribution, which depends on the mean number of mutations, the relative fitness of normal cells compared to mutants, and the division time distribution of mutant cells. Empirical probability generating function techniques yield precise estimates both of the mean number of mutations and the relative fitness of normal cells compared to mutants. In the case where no information is available on the division time distribution, it is shown that the estimation procedure using constant division times yields more reliable results. Numerical results both on observed and simulated data are reported.

Oxidative damage and mutagenesis in Saccharomyces cerevisiae: genetic studies of pathways affecting replication fidelity of 8-oxoguanine

Oxidative damage to DNA constitutes a major threat to the faithful replication of DNA in all organisms and it is therefore important to understand the various mechanisms that are responsible for repair of such damage and the consequences of unrepaired damage. In these experiments, we make use of a reporter system in Saccharomyces cerevisiae that can measure the specific increase of each type of base pair mutation by measuring reversion to a Trp+ phenotype. We demonstrate that increased oxidative damage due to the absence of the superoxide dismutase gene, SOD1, increases all types of base pair mutations and that mismatch repair (MMR) reduces some, but not all, types of mutations. By analyzing various strains that can revert only via a specific CG → AT transversion in backgrounds deficient in Ogg1 (encoding an 8-oxoG glycosylase), we can study mutagenesis due to a known 8-oxoG base. We show as expected that MMR helps prevent mutagenesis due to this damaged base and that Pol η is important for its accurate replication. In addition we find that its accurate replication is facilitated by template switching, as loss of either RAD5 or MMS2 leads to a significant decrease in accurate replication. We observe that these ogg1 strains accumulate revertants during prolonged incubation on plates, in a process most likely due to retromutagenesis.

Discussion on research methods of bacterial resistant mutation mechanisms under selective culture--uncertainty analysis of data from the Luria-Delbrück fluctuation experiment

Whether bacterial drug-resistance is drug-induced or results from rapid propagation of random spontaneous mutations in the flora prior to exposure, remains a long-term key issue concerned and debated in both genetics and medicinal fields. In a pioneering study, Luria and Delbrück exposed E. coli to T1 phage, to investigate whether the number of resistant colonies followed the Poisson distribution. They deduced that the development of resistant colonies is independent of phage presence. Similar results have since been obtained on solid medium containing antibacterial agents. Luria and Delbrück's conclusions were long considered a gold standard for analyzing drug resistance mutations. More recently, the concept of adaptive mutation has triggered controversy over this approach. Microbiological observation shows that, following exposure to drugs of various concentrations, drug-resistant cells emerge and multiply depending on the time course, and show a process function, inconsistent with the definition of Poisson distribution (which assumes not only that resistance is independent of drug quantity but follows no specific time course). At the same time, since cells tend to aggregate after division rather than separating, colonies growing on drug plates arise from the multiplication of resistant bacteria cells of various initial population sizes. Thus, statistical analysis based on equivalence of initial populations will yield erroneous results. In this paper, 310 data from the Luria-Delbrück fluctuation experiment were reanalyzed from this perspective. In most cases, a high-end abnormal value, resulting from the non-synchronous variation of the two above-mentioned time variables, was observed. Therefore, the mean value cannot be regarded as an unbiased expectation estimate. The ratio between mean value and variance was similarly incomparable, because two different sampling methods were used. In fact, the Luria-Delbrück data appear to follow an aggregated, rather than Poisson distribution. In summary, the statistical analysis of Luria and Delbrück is insufficient to describe rules of resistant mutant development and multiplication. Correction of this historical misunderstanding will enable new insight into bacterial resistance mechanisms.

Mean field mutation dynamics and the continuous Luria-Delbrück distribution

The Luria-Delbrück mutation model has a long history and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using some mathematical tools from nonlinear statistical physics. Starting from the classical formulations we derive the corresponding differential models and show that under a suitable mean field scaling they correspond to generalized Fokker-Planck equations for the mutants distribution whose solutions are given by the corresponding Luria-Delbrück distribution. Numerical results confirming the theoretical analysis are also presented.

Teaching the fluctuation test in silico by using mutate: a program to distinguish between the adaptive and spontaneous mutation hypotheses

Mutate is a program developed for teaching purposes to impart a virtual laboratory class for undergraduate students of Genetics in Biology. The program emulates the so-called fluctuation test whose aim is to distinguish between spontaneous and adaptive mutation hypotheses in bacteria. The plan is to train students in certain key multidisciplinary aspects of current genetics such as sequence databases, DNA mutations, and hypothesis testing, while introducing the fluctuation test. This seminal experiment was originally performed studying Escherichia coli resistance to the infection by bacteriophage T1. The fluctuation test initiated the modern bacterial genetics that 25 years later ushered in the era of the recombinant DNA. Nowadays we know that some deletions in fhuA, the gene responsible for E. coli membrane receptor of T1, could cause the E. coli resistance to this phage. For the sake of simplicity, we will introduce the assumption that a single mutation generates the resistance to T1. During the practical, the students use the program to download some fhuA gene sequences, manually introduce some stop codon mutations, and design a fluctuation test to obtain data for distinguishing between preadaptative (spontaneous) and induced (adaptive) mutation hypotheses. The program can be launched from a browser or, if preferred, its executable file can be downloaded from http://webs.uvigo.es/acraaj/MutateWeb/Mutate.html. It requires the Java 5.0 (or higher) Runtime Environment (freely available at http://www.java.com).

Multiscale estimation of cell kinetics

We introduce a methodology based on the Luria-Delbruck fluctuation model for estimating the cell kinetics of clonally expanding populations. In particular, this approach allows estimation of the net cell proliferation rate, the extinction coefficient and the initial (viable) population size. We present a systematic approach based on spatial partitioning, which captures the local fluctuations of the number and sizes of individual clones. However, partitioning introduces measurement error by inflating the number of clones, which is dependent on time and the degree of cell migration. We perform various in silico experiments to explore the properties of the estimators and we show that there exists a direct relationship between precision and observation time. We also explore the trade-off between the measurement error and the estimation accuracy. By exploring different scales of cellular fluctuations, from the entire population down to those of individual clones, we show that this methodology is useful for inferring important parameters in neoplastic progression.

On Bartlett's formulation of the Luria-Delbrück mutation model

Current knowledge of microbial mutation rates was accumulated largely by means of fluctuation experiments. A mathematical model describing the cell dynamics in a fluctuation experiment is indispensable to the estimation of mutation rates through fluctuation experiments. In almost six decades the model formulated by Lea and Coulson dominated in research and application, although the model formulated by Bartlett is generally believed to describe the cell dynamics more faithfully. The neglect of the Bartlett formulation was mainly due to mathematical difficulties. The present investigation overcomes some of these difficulties, thereby paving the way for the application of the Bartlett formulation in estimating mutation rates. Specifically, the article offers an algorithm for computing the distribution function of the number of mutants under the Bartlett formulation. The article also provides algorithms for computing point and interval estimates of mutation rates that are based on the maximum-likelihood principle. In addition, the article examines and compares the asymptotic behavior of the distributions of the number of mutants under the two formulations.