On the distribution of bacterial mutants: The effects of differential fitness of mutants and non-mutants

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.

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.

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.

The probability of evolutionary rescue: towards a quantitative comparison between theory and evolution experiments

Evolutionary rescue occurs when a population genetically adapts to a new stressful environment that would otherwise cause its extinction. Forecasting the probability of persistence under stress, including emergence of drug resistance as a special case of interest, requires experimentally validated quantitative predictions. Here, we propose general analytical predictions, based on diffusion approximations, for the probability of evolutionary rescue. We assume a narrow genetic basis for adaptation to stress, as is often the case for drug resistance. First, we extend the rescue model of Orr & Unckless (Am. Nat. 2008 172, 160-169) to a broader demographic and genetic context, allowing the model to apply to empirical systems with variation among mutation effects on demography, overlapping generations and bottlenecks, all common features of microbial populations. Second, we confront our predictions of rescue probability with two datasets from experiments with Saccharomyces cerevisiae (yeast) and Pseudomonas fluorescens (bacterium). The tests show the qualitative agreement between the model and observed patterns, and illustrate how biologically relevant quantities, such as the per capita rate of rescue, can be estimated from fits of empirical data. Finally, we use the results of the model to suggest further, more quantitative, tests of evolutionary rescue theory.

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).

A simple formula for obtaining markedly improved mutation rate estimates

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 non-moment-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.