Fluctuation analysis: the effect of plating efficiency

This paper supplements an earlier paper which explained how to calculate the probability distribution of the number of mutants that would be observed in a fluctuation test experiment. The formulas in that work give the distributions to be expected under a wide variety of experimental conditions, but the method it uses when only a fraction of the mutants will produce visible colonies are clumsy and inefficient. Here I describe efficient procedures for dealing with that case, provided that the mutation rate per cell division remains constant during the experiment.     

A Bayesian two-level model for fluctuation assay

The fluctuation experiment is an essential tool for measuring microbial mutation rates in the laboratory. When inferring the mutation rate from an experiment, one assumes that the number of mutants in each test tube follows a common distribution. This assumption conceptually restricts the scope of applicability of fluctuation assay. We relax this assumption by proposing a Bayesian two-level model, under which an experiment-wide average mutation rate can be defined. The new model suggests a gamma mixture of the Luria-Delbrück distribution, which coincides with a recently discovered discrete distribution. While the mixture model is of considerable independent interest in fluctuation assay, it also offers a practical Markov chain Monte Carlo method for estimating mutation rates. We illustrate the Bayesian approach with a detailed analysis of an actual fluctuation experiment.     

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.     

New algorithms for Luria-Delbrück fluctuation analysis

Fluctuation analysis is the most widely used approach in estimating microbial mutation rates. Development of methods for point and interval estimation of mutation rates has long been hampered by lack of closed form expressions for the probability mass function of the number of mutants in a parallel culture. This paper uses sequence convolution to derive exact algorithms for computing the score function and observed Fisher information, leading to efficient computation of maximum likelihood estimates and profile likelihood based confidence intervals for the expected number of mutations occurring in a test tube. These algorithms and their implementation in SALVADOR 2.0 facilitate routine use of modern statistical techniques in fluctuation analysis by biologists engaged in mutation research.     

A note on the evaluation of fluctuation experiments

Fluctuation analysis has emerged as a valuable tool for the measurement of mutation rates in single-cell populations. In this paper, we show how to make fuller use of the information supplied by the outcome of a fluctuation experiment. We shall extend Lea and Coulson's theory of the Luria-Delbrück distribution so that it accounts for residual mutation, reduced plating efficiency of mutants, and phenotypic lag, and establish a unifying method for the evaluation of fluctuation experiments in these cases and discuss its limitations. It will be proved that not all factors that might influence the distribution of mutant colonies in a fluctuation experiment can, in effect, be determined simultaneously. Nevertheless, it will be shown that the fluctuation-analytic approach to the measurement of mutation rates may retain its value in comparison with (or may even be superior to) alternative methods. Finally, we give some numerical examples to illustrate our results.     

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.     

Evidence that most radiation-induced HPRT mutants are generated directly by the initial radiation exposure.

Radiation-induced HPRT mutants are generally assumed to arise directly from DNA damage that is misrepaired within a few hours after X-irradiation. However, there is the possibility that mutations result indirectly from radiation-induced genomic instability that may occur several days after the initial radiation exposure. The protocols that commonly employ a 5-7 day expression period to allow for expression of the mutant phenotype prior to replating for selection of mutants would not be able to discriminate between mutants that occurred initially and those that arose during or after the expression period. To address this question, we performed a fluctuation analysis in which synchronous or asynchronous populations of human bladder carcinoma cells were treated with single doses of X-irradiation. For comparison, radiation was delivered during the expression period, either from an initial dose of 1.0 Gy followed by two 1.0 Gy doses separated by 24 h or from disintegrations resulting from I125dU incorporated into DNA. The mutation frequency observed at the time of replating was used to calculate the average number of mutants in the initial irradiated culture by assuming that the mutants were induced directly at the time of irradiation. Then, this average number was used to calculate the fraction of the irradiated cultures that would be predicted by a Poisson distribution to have zero mutants. There was reasonably good agreement between the predicted poisson distribution and the observed distribution for the cultures that received single doses. Moreover, as expected, when cultures were irradiated during the expression period, the fraction of the cultures having zero mutants was significantly less than that predicted by a Poisson distribution. These results indicate that most radiation-induced HPRT mutations are induced directly by the initial DNA damage, and are not the result of radiation-induced instability during the 5-7 day expression period.     

Ascertainment of the effect of differential growth rates of mutants on observed mutant frequencies in X-irradiated mammalian cells An indication for the occurrence of X-ray-induced untargeted mutagenesis

As it is not known to what extent differential growth rates of induced mutants lead to over- and under-representation of mutants in treated populations and thereby affect the determination of mutant frequencies, the mutation induction in X-irradiated L5178Y mouse lymphoma cells was determined via two methods. The first method involves the standard protocol which may suffer from the effect of differential growth rates, while the second method is based upon the fluctuation test in which the differential growth rates can be actually measured. It appeared that the standard protocol led to a mutant frequency that was similar to the mutant frequency determined in the fluctuation test. Therefore, the standard protocol appears to lead to only a minor under-estimation if any. Substantial heterogeneity in growth rates of induced mutants was observed, but the mutants with a selective advantage appear largely to compensate for the mutants that are lost because of selective disadvantage. It was calculated that the chance for isolating the same mutant twice from a treated population had been increased 2.2-fold because of the observed differential growth rates. Therefore, our data indicate that the standard protocol does not lead to serious errors in the determination of mutant frequencies and in the sampling of mutants. The fluctuation tests were also used to determine the spontaneous mutation frequency per cell per generation. The mutation rate appeared more than 10-fold enhanced in X-irradiated cells which may be attributed to the induction of a process of untargeted mutagenesis in mammalian cells.     

Studies on the origin of bacterial viruses. I-IV

I. The Incidence of Phage-Producing Cells in Various B. megatherium Cultures Analyses of small samples containing a few cells each show that lysogenic B. megatherium produces phage particles in groups of from 10 to 1000 depending on the megatherium strain and the culture medium. These groups probably correspond to the number of particles produced by a single cell. The proportion of such phage-producing cells varies from <1 x 10(-10) to about 1 x 10(-2) depending on the megatherium strain and the culture medium. If a culture produces two types of phage, the different types usually appear in separate samples. If mixed samples occur, the number of such samples is about what would be expected for the probability that two separate groups would appear in one sample. This result indicates that the appearance of a distinct phage type is the result of a change in the bacterial cell rather than a change in a phage particle, since in the latter case a mixture of the two types would result. II. The Effect of Ultraviolet Light on the Incidence of Phage-Producing and of Terramycin-Resistant Cells in Various B. megatherium Cultures Low intensity of ultraviolet light increases the proportion of both phage-producing cells and of terramycin-resistant mutants. The increase in phage-producing cells is greater than the increase in terramycin-resistant cells. High intensities of ultraviolet light cause practically all the cells of some B. megatherium strains to produce phage. The number of terramycin-resistant mutants cannot be determined under these conditions. The effect of ultraviolet light varies, depending on the megatherium strain and the culture medium. III. The Effect of Hydrogen Peroxide on the Incidence of Phage-Producing and of Terramycin-Resistant Cells in Various B. megatherium Cultures Low concentrations of hydrogen peroxide increase the number of phage-producing cells and of terramycin-resistant cells, concomitantly, from two to five times. High concentrations of hydrogen peroxide cause almost all the cells of some strains of megatherium to produce phage. IV. Calculation of the Incidence of Phage-Producing Cells The time rate of the appearance of phage particles in normal cultures, or in cultures treated with ultraviolet light or hydrogen peroxide, may be calculated by the same equations which predict the occurrence of terramycin-resistant mutants in B. megatherium cultures. These equations predict that the number of mutants will increase more or less in proportion to the concentration of mutagenic agent, so long as the mutation rate remains very small compared to the growth rate. As the mutation rate approaches the growth rate, there will be a very rapid increase in the proportion of mutants. This explains the striking effect of higher concentrations of mutagenic agents. In order to calculate the results after exposure to strong ultraviolet light or hydrogen peroxide, it is necessary to assume that the change from normal to phage-producing cell occurs without cell division.     

Spontaneous Mutation Rate of Measles Virus: Direct Estimation Based on Mutations Conferring Monoclonal Antibody Resistance

High mutation rates typical of RNA viruses often generate a unique viral population structure consisting of a large number of genetic microvariants. In the case of viral pathogens, this can result in rapid evolution of antiviral resistance or vaccine-escape mutants. We determined a direct estimate of the mutation rate of measles virus, the next likely target for global elimination following poliovirus. In a laboratory tissue culture system, we used the fluctuation test method of estimating mutation rate, which involves screening a large number of independent populations initiated by a small number of viruses each for the presence or absence of a particular single point mutation. The mutation we focused on, which can be screened for phenotypically, confers resistance to a monoclonal antibody (MAb 80-III-B2). The entire H gene of a subset of mutants was sequenced to verify that the resistance phenotype was associated with single point mutations. The epitope conferring MAb resistance was further characterized by Western blot analysis. Based on this approach, measles virus was estimated to have a mutation rate of 9 × 10⁻⁵ per base per replication and a genomic mutation rate of 1.43 per replication. The mutation rates we estimated for measles virus are comparable to recent in vitro estimates for both poliovirus and vesicular stomatitis virus. In the field, however, measles virus shows marked genetic stability. We briefly discuss the evolutionary implications of these results.     

Adaptation of Spirogyra insignis (Chlorophyta) to an extreme natural environment (sulphureous waters) through preselective mutations

Adaptation of Spirogyra insignis (Chlorophyceae) to growth and survival in an extreme natural environment (sulphureous waters from La Hedionda Spa, S. Spain) was analysed by using an experimental model. Photosynthesis and growth of the alga were inhibited when it was cultured in La Hedionda Spa waters (LHW), but after further incubation for several weeks, the culture survived due to the growth of a variant that was resistant to LHW. A Luria-Delbruck fluctuation analysis was carried out to distinguish between resistant filaments arising from rare spontaneous mutations and resistant filaments arising from other mechanisms of adaptation. It was demonstrated that the resistant filaments arose randomly by rare spontaneous mutations before the addition of LHW (preselective mutations). The rate of spontaneous mutation from sensitivity to resistance was 2.7 x 10(-7) mutants per cell division. Since LHW(resistant) mutants have a diminished growth rate, they are maintained in nonsulphureous natural waters as the result of a balance between new resistants arising from spontaneous mutation and resistants eliminated by natural selection. Thus, recurrence of rare spontaneous preselective mutations ensures the survival of the alga in sulphureous waters.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

A novel model for studying Baculovirus infection process

In this paper, a new Ansatz for modelling the Baculovirus infection cycle is presented. The base of this model is the cell cycle distribution at the time of infection. It is possible to calculate the growth of the culture and the initiation of virus processing by considering cell cycle distribution. By taking into account the length of the viral genome and the polymerase activity, it is possible to calculate the virus production rate, which underlies a logistic growth. In the present work, a new hypothesis explaining the accelerated death rates of infected cells has been introduced. This assumption provides the possibilities of performing calculation without any fixed time intervals. The simulation was tested by comparing experimental data with the model prediction. Therefore, cell cycle distributions over the culture time and the growth behaviour of infected and non-infected insect cells were measured. A model, Baculovirus coding for GFP was employed for the present investigation, as it allows tracking the infection and determining the effectiveness of the infection, which is highly dependent on the cell density at the time of infection (TOI). Furthermore, the new model is is taken to simulate data gained from literature about virus release and adsorption. The new assumptions make the model more independent to fit into different cultivation systems.     

The fixed-size Luria-Delbruck model with a nonzero death rate

What is the expected number of mutants in a stochastically growing colony once it reaches a given size, N? This is a variant of the famous Luria-Delbruck model which studies the distribution of mutants after a given time-lapse. Instead of fixing the time-lapse, we assume that the colony size is a measurable quantity, which is the case in many in-vivo oncological and other applications. We study the mean number of mutants for an arbitrary cell death rate, and give partial results for the variance. For a restricted set of parameters we provide analytical results; we also design a very efficient computational method to calculate the mean, which works for most of the parameter values, and any colony size, no matter how large. We find that a cellular population with a higher death rate will contain a larger number of mutants than a population of equal size with a smaller death rate. Also, a very large population will contain a larger percentage of mutants; that is, irreversible mutations act like a force of selection, even though here the mutants are assumed to have no selective advantage. Finally, we investigate the applicability of the traditional, 'fixed-time' approach and find that it approximates the 'fixed-size' problem whenever stochastic effects are negligible.     

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.     

Measuring bidirectional mutation

The estimation of mutation rates is usually based on a model in which mutations are rare independent Poisson events. Back-mutation of mutants, an even rarer event, is ignored. In the hypermutating B cells of the immune system, mutation between phenotypes exhibiting, vs. not exhibiting, surface immunoglobulin is common in both directions. We develop three strategies for the estimation of mutation rates under circumstances such as these, where mutation rates in both directions are estimated simultaneously. Our model for the growth of a cell culture departs from the classical assumption of cell division as a memoryless (Poisson) event; we model cell division as giving rise to sequential generations of cells. On this basis, a Monte-Carlo simulation is developed. We develop also a numerical approach to calculating the probability distribution for the proportion of mutants in each culture as a function of forward- and backward-mutation rates. Although both approaches are too computationally intensive for routine laboratory use, they provide the insight necessary to develop and evaluate a third, 'hand-calculator' approach to extracting mutation rate estimators from experiments of this type.     

Poisson-like Fluctuation Patterns of Revertants of Leucine Auxotrophy (Leu-500) in Salmonella Typhimurium Caused by Delay in Mutant Cell Division

Leu(+) mutants from Salmonella typhimurium leu-500 strain MA412 arise at high frequencies and mutant colonies appear over a broad range of time on selective plates. This observation suggested that these Leu(+) mutants might be induced or ``directed.' If such a mechanism was responsible, mutants should originate on selective plates rather than in the preceding culture in nonselective conditions and should give rise to Poisson-like fluctuation curves upon plating of sister cultures on selective medium. Poisson-like distribution profiles were indeed observed for Leu(+) mutants of S. typhimurium MA412. However, an explanation for the observed Poisson-like fluctuation patterns without a need for selection-induced mutations was found. Microscopical analysis and cell mass/viable count measurements showed that the size of Leu(+) mutant cells was often much larger than those of nonmutants. This size difference was a stable characteristic of a large proportion of Leu(+) mutants, was observed both in stationary and growing culture and did not measurably affect the division rates of the cells in nutrient broth. As the transition from normal-sized nonmutant to oversized mutant cells during the nonselective culture phase of the fluctuation experiment may have been accompanied by a period with no or few completed cell division cycles, the number of mutant offspring may have been smaller than that of sibling nonmutants. Such underrepresentation of mutants in the final culture is expected to give rise to Poisson-like fluctuation patterns without invoking ``directed' mutations.     

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.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Distribution of mutants in aggregates of cultured cells

The analysis of the distribution of mutants in an exponentially growing culture of cells that are aggregated into clumps of homogeneous size is described, given the mutation rate and a random process by which clumps divide to produce progeny. The mean and standard deviation of the proportion of clumps with a given number of mutant cells at a particular time are calculated. Since the standard deviation tends to be much smaller than the mean, the following conclusions can be drawn. Aggregation lowers the number of mutant-containing clumps in cultures grown to a standard number of cells, but raises the number of mutant-containing clumps in cultures grown to a standard number of clumps. In the absence of mutation, or at low mutation rates, clumps tend to become pure types (normal or mutant). The probability of finding pure, nonmutant-containing clumps, however, is approximately the initial fraction of nonmutant cells (given realistic forward and back mutation rates). Also, in terms of the given process, it is possible to compute the probability that all the cells in an aggregate descend from a single, common parent cell within a given number of generations, and thus to calculate the probability that all the cells in a clone grown from an aggregate descend from a single cell within a known number of generations.