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.     

Toward a Unique Definition of the Mutation Rate

In around 1943, while writing a classic paper with Luria, Delbrück envisioned two kinds of mutation rates: One was expressed as mutations per bacterium per unit time, the other as mutations per bacterium per division cycle. Due to minor mathematical errors, the precise connection between the two concepts remained elusive for Delbrück. As a result, researchers and educators alike are still grappling with the definition of the mutation rate. Within the context of microbial mutation, the current author proposes an idealized model to bring new clarity to the distinction between the two forms of the mutation rate that Delbrück once attempted to define and characterize. The paper also critiques two incorrect estimators of mutation rates and brings to light two important yet unexplored “invariance” hypotheses about mutation rates.     

A deterministic approach for the estimation of mutation rates in cultured mammalian cells

Unequal growth rates between mutant and wild-type cells in a large population constitute a problem for the estimation of mutation rate. Over a period of cell growth, a selective advantage of one cell type over the other might lead to considerable error in the estimation of mutation rate if equal growth rates are assumed. In this study, we propose a formula and apply it to the estimation of spontaneous mutation rate in a growing population of Chinese hamster V79 cells in which ouabain-resistant mutant cells exhibit a slower growth rate than the wild-type cells. The formula is a generalization of that previously presented by Armitage (1953), and this is the first attempt to apply the deterministic approach for mutation rate estimation to cultured mammalian cells. The value of the estimated rate is compared with that derived from a parallel experiment using the fluctuation test of Luria and Delbrück (1943). The limitations and advantages of taking the deterministic approach to mutation rate estimation in mammalian cell systems are discussed.     

Mutations of Chinese Hamster Somatic Cells from 2-Deoxygalactose Sensitivity to Resistance

In Chinese hamster somatic cells, the spontaneous change of phenotype from 2-deoxygalactose sensitivity to resistance was studied using fluctuation test experiments à la Luria and Delbrück (1943) for four Chinese hamster cell strains derived from V79. The results are consistent with true mutational events. The mutation rates are in the range of 1 to 3.5 X 10(-5) per cell per generation. The relationship between the 2-deoxyglactose resistance and the galactokinase markers is discussed.     

Evolution of resistance during clonal expansion

Acquired drug resistance is a major limitation for cancer therapy. Often, one genetic alteration suffices to confer resistance to an otherwise successful therapy. However, little is known about the dynamics of the emergence of resistant tumor cells. In this article, we consider an exponentially growing population starting from one cancer cell that is sensitive to therapy. Sensitive cancer cells can mutate into resistant ones, which have relative fitness alpha prior to therapy. In the special case of no cell death, our model converges to the one investigated by Luria and Delbrück. We calculate the probability of resistance and the mean number of resistant cells once the cancer has reached detection size M. The probability of resistance is an increasing function of the detection size M times the mutation rate u. If Mu < 1, then the expected number of resistant cells in cancers with resistance is independent of the mutation rate u and increases with M in proportion to M(1-1/alpha) for advantageous mutants with relative fitness alpha>1, to l nM for neutral mutants (alpha = 1), but converges to an upper limit for deleterious mutants (alpha<1). Further, the probability of resistance and the average number of resistant cells increase with the number of cell divisions in the history of the tumor. Hence a tumor subject to high rates of apoptosis will show a higher incidence of resistance than expected on its detection size only.     

Transformation of temperature-sensitive growth mutant of BHK21 cell line to wild-type phenotype with hamster and mouse DNA

A temperature-sensitive (ts) mutant, tsBN2, which was derived from BHK21 and is defective in the regulatory mechanism for chromosome condensation, was transformed to the temperature-resistant (ts+) phenotype by means of DNA-mediated gene transfer with hamster and mouse DNA. Treatment of mouse DNA with the restriction enzymes EcoRI, HindIII, PstI and SalI, but not with XhoI, almost completely abolished the transforming activity. A fluctuation test, originally devised by Luria and Delbrück, was used to estimate the reversion and transformation frequencies of tsBN2 cultures.     

Fluctuation Analysis: The Probability Distribution of the Number of Mutants under Different Conditions

In the 47 years since fluctuation analysis was introduced by Luria and Delbrück, it has been widely used to calculate mutation rates. Up to now, in spite of the importance of such calculations, the probability distribution of the number of mutants that will appear in a fluctuation experiment has been known only under the restrictive, and possibly unrealistic, assumptions: (1) that the mutation rate is exactly proportional to the growth rate and (2) that all mutants grow at a rate that is a constant multiple of the growth rate of the original cells. In this paper, we approach the distribution of the number of mutants from a new point of view that will enable researchers to calculate the distribution to be expected using assumptions that they believe to be closer to biological reality. The new idea is to classify mutations according to the number of observable mutants that derive from the mutation when the culture is selectively plated. This approach also simplifies the calculations in situations where two, or many, kinds of mutation may occur in a single culture.     

A generalized Luria-Delbrück model

We develop extensions of the Luria-Delbrück model that explicitly consider non-exponential growth of normal cells and a birth-death process with mean exponential or Gompertz growth of mutants. Death of mutant cells can be important in clones arising during cancer progression. The use of a birth-death process for growth of mutant cells, as opposed to a pure birth process as in previous work on the Luria-Delbrück model, leads to a large increase in the extra Poisson variation in the size of the mutant cell populations, which needs to be addressed in statistical analyses. We also discuss connections with previous work on carcinogenesis models.     

The origin of phage virology.

The history of bacteriophage (phage) had its start in 1915, when Twort isolated an unusual filterable and infectious agent from excrete of patients struck by diarrhoea; this discovery was followed by an analogous, and probably independent, finding of d'Hérelle in 1917. For several years phage research made scant progress but great attention was paid to the question of phage nature, which saw the contrast between d'Hérelle and Bordet's views (living against chemical nature, respectively). This situation changed with the independent discovery of lysogeny, in 1925, thanks to Bordet and Bail: this phenomenon was considered of genetical origin, a view that Wollman interpreted by assimilating the properties of phage to those of gene (according to a previous idea of Muller). In the 1930s, Burnet's work opened a new era by demonstrating the occurrence of several species of phages and their antigenic property. In the same period, the physical and chemical characteristics of these viruses were disclosed thanks, in particular, to the work of Schlesinger, who first demonstrated that a virus (phage) was constituted of nucleoproteins. The peculiarity of phage was finally shown after the invention of electron microscope: H. Ruska, in 1940, and Anderson and Luria in the next years, obtained the first images of tailed phages, a finding that strongly helped the investigation on the first steps of the infection process. The decisive impulse to phage virology came from Delbrück, a physicist who entered biology giving it a new arrangement. The so-called "phage group" assembled brilliant minds (Luria, Hershey and Delbrück himself, and later a dozen of other scientists): this group faced three fundamental questions of phage virology, i.e., the mechanisms of attack, multiplication and lysis. In ten years' time, phage virology became an integrant part of molecular biology, also thanks to the discovery of the genetical properties of DNA: in such scientific context, Delbrück, Luria and Hershey's works emerged for the absolute excellence of their results, which led such scientists to Nobel prize. Lysogeny was however neglected by the phage group: this singular property shared by bacteria and phages was instead investigated by Lwoff's group, in Paris, and explained in its fundamental features during the 1950s. The "phage's saga" has gone on being an important division of molecular biology till today, and its history is far from being over.     

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

Two formulations of Luria and Delbrück's mutation model have been in common use since the 1940s. While mathematicians focused their attention on the formulation of Lea and Coulson that assumes asynchronous cell growth, biologists found more appealing the formulation of Haldane that assumes synchronous cell growth. This article attempts to solve several outstanding issues for the latter formulation. First, it provides an exact, closed-form expression for the mutant distribution by correcting a minor error in the literature. Second, it presents a novel algorithm for computing the mutant distribution, which leads to novel methods for computing point and interval estimates of mutation rates based on the maximum likelihood principle. Third, it critically examines existing methods based on the mean number of mutants. Finally, it compares the two formulations to underline their strengths and shortcomings.     

Gene amplification in Chinese hamster DNA repair deficient mutants

In order to study the possible relationship between gene amplification and DNA repair we analyzed the amplification of the CAD gene in four mutants hypersensitive to UV light (CHO43RO, CHO7PV, UV5 and UV61) isolated in vitro from Chinese hamster cell lines (CHO-K1 and AA8). These mutants are characterized by different defects in the nucleotide excision repair mechanism and represent complementation groups 1, 9, 2, and 6 respectively. To evaluate the amplification ability of each cell line we measured the rate of appearance of PALA resistant clones with the Luria and Delbrück fluctuation test. Resistance to PALA is mainly due to amplification of the CAD gene. In the mutants CHO43RO, UV5 and CHO7PV we reproducibly found an amplification rate lower than in the parental cell lines (2–5 times), while in UV61 the amplification rate was about 4 times higher. This result indicates that each mutant is characterized by a specific amplification ability and that the unefficient removal of UV induced DNA damage can be associated with either a higher or a lower amplification rate. However, the analysis of randomly isolated CHO-K1 clones with normal UV sensitivity has shown variability in their amplification ability, making it difficult to relate the specific amplification ability of the mutants to the DNA repair defect and suggesting clonal heterogeneity of the parental population.     

Cell migration in development and disease

A recent meeting at the Max Delbrück Center in Berlin, Germany provided a forum to discuss the molecular mechanisms of cell migration in a broad range of contexts including chemotaxis, development, immunity, and cancer.     

Improved inference of mutation rates: I. An integral representation for the Luria-Delbrück distribution

The estimation of mutation rates is ordinarily performed using results based on the Luria-Delbrück distribution. There are certain difficulties associated with the use of this distribution in practice, some of which we address in this paper (others in the companion paper, Oprea and Kepler, Theor. Popul. Biol., 2001). The distribution is difficult to compute exactly, especially for large values of the random variable. To overcome this problem, we derive an integral representation of the Luria-Delbrück distribution that can be computed easily for large culture sizes. In addition, we introduce the usual assumption of very small probability of having a large proportion of mutants only after the generating function has been computed. Thus, we obtain information on the moments for the more general case. We examine the asymptotic behavior of this system. We find a scaling or "standardization" technique that reduces the family of distributions parameterized by three parameters (mutation rate, initial cell number, and final cell number) to a single distribution with no parameters, valid so long as the product of the mutation rate and the final culture is sufficiently large. We provide a pair of techniques for computing confidence intervals for the mutation rate. In the second paper of this series, we use the distribution derived here to find approximate distributions for the case where the cell cycle time is not well-described as an exponential random variable as is implicitly assumed by Luria-Delbrück distribution.     

Update on estimation of mutation rates using data from fluctuation experiments

This note discusses a minor mathematical error and a problematic mathematical assumption in Luria and Delbrück'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.     

The mutagenic action of nitroimidazoles. IV. A comparison of the mutagenic action of several nitroimidazoles and some imidazoles.

The mutagenic action of 51 imidazoles was investigated. The fluctuation test of Luria and Delbrück was used, with Klebsiella pneumoniae as test organism. 8 compounds, including 5 with a weak mutagenic action in the fluctuation test, were also investigated by the Ames test in which Salmonella typhimurium TA100 was used. Of the 51 imidazoles examined, 33 were nitroimidazoles. 31 of the latter appeared to be mutagenic, whereas out of the 18 other imidazoles without a nitro group only 2 were mutagenic. Several of the substances tested for mutagenicity showed an antimicrobial activity. No direct relationship between antimicrobial action, growth inhibition and mutagenicity was established. With methyl-nitroimidazoles a relationship was found between the chemical structure and mutagenic action. However, when the nitroimidazoles had a more complex chemical structure, a relationship between this structure and mutagenicity could not be established.     

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.     

Proliferation model dependence in fluctuation analysis: the neutral case

We discuss the evaluation of Luria-Delbrück fluctuation experiments under Bellman-Harris models of cell proliferation. It is shown that under certain very natural assumptions concerning the life-time distributions and the offspring distributions of mutant and non-mutant cells, the suitably normed and centered number of mutants contained in a large culture of bacteria (or the like) converges to a certain stable random variable with index 1. The result obtains under the assumption that the mutation under consideration is “neutral” in the sense that on average and in the long run, mutant cells produce the same number of offspring as non-mutant cells.     

GENETIC ADAPTATION AND ACCLIMATION OF PHYTOPLANKTON ALONG A STRESS GRADIENT IN THE EXTREME WATERS OF THE AGRIO RIVER–CAVIAHUE LAKE (ARGENTINA)1

We tested if different adaptation strategies were linked to a stress gradient in phytoplankton cells. For this purpose, we studied the adaptation and acclimation of Dictyosphaerium chlorelloides (Naumann) Komárek et Perman (Chlorophyta) and Microcystis aeruginosa (Kütz.) Kütz. (Cyanobacteria) to different water samples (from extremely acid, metal‐rich water to moderate stressful conditions) of the Agrio River–Caviahue Lake system (Neuquén, Argentina). Both experimental strains were isolated from pristine, slightly alkaline waters. To distinguish between physiological acclimation and genetic adaptation (an adaptive evolution event), a modified Luria‐Delbrück fluctuation analysis was carried out with both species by using as selective agent sample waters from different points along the stress gradient. M. aeruginosa did not acclimate to any of the waters tested from different points along the stress gradient nor did D. chlorelloides to the two most acidic and metal‐rich waters. However, D. chlorelloides proliferated by rapid genetic adaptation, as the consequence of a single mutation (5.4 × 10^?7 resistant mutants per cell per division) at one locus, in less extreme water and also by acclimation in the least extreme water. It is hypothesized that the stress gradient resulted in different strategies of adaptation in phytoplankton cells from nonextreme waters. Thus, very extreme conditions were lethal for both organisms, but as stressful conditions decreased, adaptation of D. chlorelloides cells was possible by the selection of resistant mutants, and in less extreme conditions, by acclimation.     

Physics and the origins of molecular biology

Bohr, Delbrück and Schrödinger were physicists who had important influences on biology in the second half of the twentieth century. They thought that future studies of the gene might reveal new principles or paradoxes, analogous to the wave/particle paradox of light propagation, or even new physical laws. This stimulated several physicists to enter the field of biology. Delbrück founded the bacteriophage group which provided one of the roots of molecular biology. Another was X-ray crystallography which led to the discovery of DNA structure. The strength and success of molecular biology came from the many interactions between geneticists, physicists, chemists and biochemists. It was also characterized by a powerful combination of theoretical and experimental approaches.     

Improved inference of mutation rates: II. Generalization of the Luria-Delbrück distribution for realistic cell-cycle time distributions

In the first paper of this series (Kepler and Oprea, Theor. Popul. Biol. 2001) we found a continuum approximation of the Luria-Delbrück distribution in terms of a scaled variable related to the proportion of mutants in the culture. Here we show that the Luria-Delbrück distribution is inaccurate when realistic division processes are being considered due to the non-Markovian character of the cell cycle. We derive the expectation of the proportion of mutants in the culture for arbitrary cell-cycle time distributions. We then introduce a two-parameter generalization of the continuum Luria-Delbrück distribution for two of the more commonly used cell-cycle time distributions: gamma and shifted exponential. We obtain the generalized distribution by defining a map from the actual parameters to "effective" parameters. The effective mutation rate is obtained analytically, while the effective population size is obtained by fitting simulation data. Our simulations show that the second parameter depend mostly on the coefficient of variation of the cell-cycle time distribution.