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.     

Determining mutation rates in bacterial populations

When properly determined, spontaneous mutation rates are a more accurate and biologically meaningful reflection of underlying mutagenic mechanisms than are mutant frequencies. Because bacteria grow exponentially and mutations arise stochastically, methods to estimate mutation rates depend on theoretical models that describe the distribution of mutant numbers among parallel cultures, as in the original Luria-Delbr]uck fluctuation analysis. An accurate determination of mutation rate depends on understanding the strengths and limitations of these methods, and how to design fluctuation assays to optimize a given method. In this paper we describe a number of methods to estimate mutation rates, give brief accounts of their derivations, and discuss how they behave under various experimental conditions.     

Multiplexing mutation rate assessment: determining pathogenicity of Msh2 variants in Saccharomyces cerevisiae

Despite the fundamental importance of mutation rate as a driving force in evolution and disease risk, common methods to assay mutation rate are time-consuming and tedious. Established methods such as fluctuation tests and mutation accumulation experiments are low-throughput and often require significant optimization to ensure accuracy. We established a new method to determine the mutation rate of many strains simultaneously by tracking mutation events in a chemostat continuous culture device and applying deep sequencing to link mutations to alleles of a DNA-repair gene. We applied this method to assay the mutation rate of hundreds of Saccharomyces cerevisiae strains carrying mutations in the gene encoding Msh2, a DNA repair enzyme in the mismatch repair pathway. Loss-of-function mutations in MSH2 are associated with hereditary nonpolyposis colorectal cancer, an inherited disorder that increases risk for many different cancers. However, the vast majority of MSH2 variants found in human populations have insufficient evidence to be classified as either pathogenic or benign. We first benchmarked our method against Luria–Delbrück fluctuation tests using a collection of published MSH2 missense variants. Our pooled screen successfully identified previously characterized nonfunctional alleles as high mutators. We then created an additional 185 human missense variants in the yeast ortholog, including both characterized and uncharacterized alleles curated from ClinVar and other clinical testing data. In a set of alleles of known pathogenicity, our assay recapitulated ClinVar’s classification; we then estimated pathogenicity for 157 variants classified as uncertain or conflicting reports of significance. This method is capable of studying the mutation rate of many microbial species and can be applied to problems ranging from the generation of high-fidelity polymerases to measuring the frequency of antibiotic resistance emergence.     

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.     

Carbon starvation of Salmonella typhimurium does not cause a general increase of mutation rates.

Mutation rates in bacteria can vary depending on the genetic target studied and the specific growth conditions of the cells. Here, two different methods were used to determine how rates of mutation to antibiotic resistance, auxotrophy, and prototrophy were influenced by carbon starvation on agar plates. The rate of mutation to rifampin resistance was increased by starvation as measured by fluctuation tests, similar to what has been reported previously for Escherichia coli. In contrast, the rates of mutation to various types of auxotrophy were unaffected or decreased as measured by both fluctuation tests and a repeated-streaking procedure. Similarly, the rates of reversion to prototrophy of his and lac nonsense and missense mutations were unaffected by starvation. Thus, mutation rates of different genetic targets can be affected differently by starvation and we conclude that carbon starvation is not generally mutagenic in Salmonella typhimurium.     

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 note on efficient estimation of mutation rates using Luria-Delbrück fluctuation analysis

The maximum likelihood and Luria-Delbrück P0 methods for the estimation of spontaneous mutation rates are compared. The maximum likelihood method is fully efficient, utilizing all available information in a fluctuation experiment, but can be numerically cumbersome. Under certain conditions, there is little loss of efficiency using the P0 method, which is readily implemented numerically. Design considerations should aid investigators in minimizing statistical errors associated with the statistical analysis of fluctuation experiments.     

Model-Based Statistical Procedures for the Analysis of in Vitro Mutagenesis Assays

A number of short-term screening assays for potential chemical carcinogens and mutagens involve the measurement of mutation rates in in vitro cell populations. In this paper, statistical procedures are proposed for the analysis of results from these assays. The procedures, which are based on previously developed stochastic models of cell growth and mutation, include hypothesis tests for the comparison of mutation rates in treated and control cultures, and estimation and hypothesis tests for the parameters of a linear mutagenesis dose-response. Computer simulation is used to validate the methods, and they are illustrated by an analysis of data obtained under the mouse lymphoma mutagenesis protocol.     

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.     

Validation of machine learning approach for direct mutation rate estimation

Mutations are the primary source of all genetic variation. Knowledge about their rates is critical for any evolutionary genetic analyses, but for a long time, that knowledge has remained elusive and indirectly inferred. In recent years, parent–offspring comparisons have yielded the first direct mutation rate estimates. The analyses are, however, challenging due to high rate of false positives and no consensus regarding standardized filtering of candidate de novo mutations. Here, we validate the application of a machine learning approach for such a task and estimate the mutation rate for the guppy (Poecilia reticulata), a model species in eco-evolutionary studies. We sequenced 4 parents and 20 offspring, followed by screening their genomes for de novo mutations. The initial large number of candidate de novo mutations was hard-filtered to remove false-positive results. These results were compared with mutation rate estimated with a supervised machine learning approach. Both approaches were followed by molecular validation of all candidate de novo mutations and yielded similar results. The ML method uniquely identified three mutations, but overall required more hands-on curation and had higher rates of false positives and false negatives. Both methods concordantly showed no difference in mutation rates between families. Estimated here the guppy mutation rate is among the lowest directly estimated mutation rates in vertebrates; however, previous research has also found low estimated rates in other teleost fishes. We discuss potential explanations for such a pattern, as well as future utility and limitations of machine learning approaches.     

Crow-Kimura模型中位点突变率对误差阈的随机效应

为了补充Eigen模型和Crow-Kimura模型的随机效应研究,Crow-Kimura模型中的位点突变率被处理成高斯分布随机变量,从而研究误差阈值的特征以及误差阈值的扩展与随机突变率涨落强度之间的关系. 准物种浓度和群体序参数分析表明,在位点突变率涨落较大时,误差阈值不再是相变点,而是平滑的转变区域. 定量分析表明,随机Crow-Kimura模型中转变区宽度与涨落强度之间的关系是非线性的. 将Crow-Kimura模型与Eigen模型的随机特征进行比较发现,在两个模型中适应值随机化使得转变区域的宽度和随机变量涨落强度之间的关系是线性的,而位点突变率随机化中两者的关系是非线性的（指数）. 对于随机化的Crow-Kimura模型,适应值随机化与位点突变率随机化引起的误差阈扩展效应相当. 对于随机Eigen模型,误差阈的扩展效应则主要是由位点突变率的随机化引起的. 之后,本文概述了Eigen模型和Crow Kimura模型中适应值和位点突变率随机化对误差阈值随机效应的影响,并讨论了上述结果对抗病毒策略、癌症治疗和动植物育种的重要意义.     

A Deterministic Approximation to the Number of Mutants in Growing Clones

The use of an approximation to the median of the Poisson distribution to represent each occurrence of mutations in a growing clone permits the prediction of the number of mutants per clone without the limitations imposed by more heuristic expressions. Its application to the evaluation of mutation rates yields results comparable to those obtained by fluctuation analysis.     

QUANTITATIVE GENETICS OF BRYOZOAN PHENOTYPIC EVOLUTION. II. ANALYSIS OF SELECTION AND RANDOM CHANGE IN FOSSIL SPECIES USING RECONSTRUCTED GENETIC PARAMETERS

The roles of natural selection and random genetic change in the punctuated phenotypic evolution of eight Miocene-Pliocene tropical American species of the cheilostome bryozoan Metrarabdotos are analyzed by quantitative genetic methods. Trait heritabilities and genetic covariances reconstructed by partitioning within- and among-colony phenotypic variance are similar to those previously obtained for living species of the cheilostome Stylopoma using breeding data. The hypothesis that differences in skeletal morphology between species of Metrarabdotos are entirely due to mutation and genetic drift cannot be rejected for reasonable rates of mutation maintained for periods brief enough to account for the geologically abrupt appearances of these species in the fossil record. Except for one pair of species, separated by the largest morphologic distance, directional selection acting alone would require unrealistically high rates of selective mortality to be maintained for these periods. Thus, directional selection is not strongly implicated in the divergence of Metrarabdotos species. Within species, rates of net phenotypic change are slow enough to require stabilizing selection, but mask large, relatively rapid fluctuations, all of which, however, can be attributed to chance departures from the mean phenotype by mutation and genetic drift, rather than to tracking environmental fluctuation by directional selection. The results are consistent with genetic models involving shifts between multiple adaptive peaks on which phenotypes remain more or less static through long-term stabilizing selection. Regardless of the degree to which directional selection may be involved in peak shifts, phenotypic differentiation is thus related to processes different than the pervasive stabilizing selection acting within species.     

A stochastic model for estimation of mutation rates in multiple-replication proliferation processes

In this paper we propose a stochastic model based on the branching process for estimation and comparison of the mutation rates in proliferation processes of cells or microbes. We assume in this model that cells or microbes (the elements of a population) are reproduced by generations and thus the model is more suitably applicable to situations in which the new elements in a population are produced by older elements from the previous generation rather than by newly created elements from the same current generation. Cells and bacteria proliferate by binary replication, whereas the RNA viruses proliferate by multiple replication. The model is in terms of multiple replications, which includes the special case of binary replication. We propose statistical procedures for estimation and comparison of the mutation rates from data of multiple cultures with divergent culture sizes. The mutation rate is defined as the probability of mutation per replication per genome and thus can be assumed constant in the entire proliferation process. We derive the number of cultures for planning experiments to achieve desired accuracy for estimation or desired statistical power for comparing the mutation rates of two strains of microbes. We establish the efficiency of the proposed method by demonstrating how the estimation of mutation rates would be affected when the culture sizes were assumed similar but actually diverge.      

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.     

A comprehensive model of mutations affecting fitness and inferences for Arabidopsis thaliana

As the ultimate source of genetic variation, spontaneous mutation is essential to evolutionary change. Theoretical studies over several decades have revealed the dependence of evolutionary consequences of mutation on specific mutational properties, including genomic mutation rates, U, and the effects of newly arising mutations on individual fitness, s. The recent resurgence of empirical effort to infer these properties for diverse organisms has not achieved consensus. Estimates, which have been obtained by methods that assume mutations are unidirectional in their effects on fitness, are imprecise. Both because a general approach must allow for occurrence of fitness-enhancing mutations, even if these are rare, and because recent evidence demands it, we present a new method for inferring mutational parameters. For the distribution of mutational effects, we retain Keightley's assumption of the gamma distribution, to take advantage of the flexibility of its shape. Because the conventional gamma is one sided, restricting it to unidirectional effects, we include an additional parameter, rho, as an amount it is displaced from zero. Estimation is accomplished by Markov chain Monte Carlo maximum likelihood. Through a limited set of simulations, we verify the accuracy of this approach. We apply it to analyze data on two reproductive fitness components from a 17-generation mutation-accumulation study of a Columbia accession of Arabidopsis thaliana in which 40 lines sampled in three generations were assayed simultaneously. For these traits, U approximately/= 0.1-0.2, with distributions of mutational effects broadly spanning zero, such that roughly half the mutations reduce reproductive fitness. One evolutionary consequence of these results is lower extinction risks of small populations of A. thaliana than expected from the process of mutational meltdown. A comprehensive view of the evolutionary consequences of mutation will depend on quantitatively accounting for fitness-enhancing, as well as fitness-reducing, mutations.     

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.     

用~(125)I标记肽图估计近年来流行的甲型流感病毒血凝素的变异率

将一种灵敏而简便的~(125)Ⅰ标记肽图分析方法用于估计近年来流行的5株甲1型和S株甲3型流感病毒血凝素(HA)的变异率。经计算这两种亚型病毒HA的年平均氨基酸变异率分别为0.47％和0.44％。但在1986年分离株与1985年分离株之间则有较大的变化,对HA的变异率与流行病学资料的关系进行了讨论。     

PERSPECTIVE: SPONTANEOUS DELETERIOUS MUTATION

Mildly deleterious mutation has been invoked as a leading explanation for a diverse array of observations in evolutionary genetics and molecular evolution and is thought to be a significant risk of extinction for small populations. However, much of the empirical evidence for the deleterious-mutation process derives from studies of Drosophila melanogaster, some of which have been called into question. We review a broad array of data that collectively support the hypothesis that deleterious mutations arise in flies at rate of about one per individual per generation, with the average mutation decreasing fitness by about only 2% in the heterozygous state. Empirical evidence from microbes, plants, and several other animal species provide further support for the idea that most mutations have only mildly deleterious effects on fitness, and several other species appear to have genomic mutation rates that are of the order of magnitude observed in Drosophila. However, there is mounting evidence that some organisms have genomic deleterious mutation rates that are substantially lower than one per individual per generation. These lower rates may be at least partially reconciled with the Drosophila data by taking into consideration the number of germline cell divisions per generation. To fully resolve the existing controversy over the properties of spontaneous mutations, a number of issues need to be clarified. These include the form of the distribution of mutational effects and the extent to which this is modified by the environmental and genetic background and the contribution of basic biological features such as generation length and genome size to interspecific differences in the genomic mutation rate. Once such information is available, it should be possible to make a refined statement about the long-term impact of mutation on the genetic integrity of human populations subject to relaxed selection resulting from modern medical procedures.