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.     

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.     

Fluctuation analysis of mutations to nalidixic acid resistance in Escherichia coli.

Mutations of Escherichia coli from sensitivity to nalidixic acid resistance were studied by fluctuation analysis. The mutant distributions in replicate cultures were not significantly affected either by the age of the carbon-starved preculture used for inocula or by the inoculum size. The data from 23 fluctuation tests (48 cultures each) were pooled. The mean number of mutations per culture was estimated to be 0.71 from the fraction of cultures without mutants or 0.74 and 0.77 by maximum-likelihood estimation based on the two models under consideration. When the pooled data were compared with the theoretical expectations, the fits were unsatisfactory (P < 0.005). The lack of fit was caused mainly by too high a frequency of cultures with between 17 and 32 mutants and too high a frequency of cultures with more than 128 mutants. Possible reasons for the lack of fit and its implications with respect to estimation of mutation rates from fluctuation tests are discussed.     

The rate of mutation of a single gene

The rate of mutation refers to the probability that a unit length of DNA (generally a base pair) mutates with time. Fluctuation analysis or mutant accumulation assays applied to phenotypic changes measure mutation rates of cells. However, only a few phenotypic changes indicative of mutations are known thus limiting the analysis to those rare genes. Direct sequencing overcomes the limitations imposed by phenotypic analysis but is limited by the extensive number of clones or cells that have to be analyzed in fluctuation or mutant accumulation assays. We propose a strategy to determine the rate of mutation of a gene by limited direct sequencing of a few single cells of a defined lineage. To accomplish this, we determined the average number of mutations per position in each DNA length sequenced from the proportion of the non-mutated positions, according to the Poisson process and/or the Taylor series. Measuring the rate of mutation by direct sequencing of genes does not require ascertaining a phenotype and can be applied to any area of the genome in a cell. The approach avoids fluctuation errors.     

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.     

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.     

Detecting population growth, selection and inherited fertility from haplotypic data in humans

The frequency of a rare mutant allele and the level of allelic association between this allele and one or several closely linked markers are frequently measured in genetic epidemiology. Both quantities are related to the time elapsed since the appearance of the mutation in the population and the intrinsic growth rate of the mutation (which may be different from the average population growth rate). Here, we develop a method that uses these two kinds of genetic data to perform a joint estimation of the age of the mutation and the minimum growth rate that is compatible with its present frequency. In absence of demographic data, it provides a useful estimate of population growth rate. When such data are available, contrasts among estimates from several loci allow demographic processes, affecting all loci similarly, to be distinguished from selection, affecting loci differently. Testing these estimates on populations for which data are available for several disorders shows good congruence with demographic data in some cases whereas in others higher growth rates are obtained, which may be the result of selection or hidden demographic processes.     

Mutation carrier testing in Lesch-Nyhan syndrome families: HPRT mutant frequency and mutation analysis with peripheral blood T lymphocytes

Mutations in the X chromosome hypoxanthine-guanine phosphoribosyl transferase (HPRT) gene are responsible for Lesch-Nyhan syndrome and related diseases in humans. Because the gene is on the X chromosome, males are affected and females in the families are at risk of being carriers of the mutation. Because there are so many different mutations that can cause the disease (218 different mutations in 271 families), genetic testing for carrier status of females requires detailed molecular analysis of the familial mutation. This analysis can be complicated by the unavailability of an affected male for study. In addition, when the mutation is a deletion (34 reported instances), molecular analysis in females is difficult because of the two X chromosomes. We have applied a peripheral blood T lymphocyte cloning assay that uses resistance to the purine analogue 6-thioguanine (TG) to measure the frequency of cells in females expressing a mutant HPRT allele to determine mutation carrier status in 123 females in 61 families. In families in which the HPRT mutation was determined and could be easily analyzed in samples from females, we found a mean (+/- SD) mutant frequency of 9.7 (+/- 8.7) x 10(-6) in noncarrier females and 2.9 (+/- 3.0) x 10(-2) in carrier females. The frequency in carrier females is less than the 0.5 expected for nonrandom X inactivation because of in vivo selection against HPRT mutation-expressing T lymphocytes or stem cells during prenatal development. The use of this cloning assay allows determination of the carrier status of females even when the HPRT mutation is not yet known or is difficult to determine in DNA samples from females. This approach provides a rapid assay that yields information on carrier status within 10 days of sample receipt.     

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.     

Bayesian Procedures for the Estimation of Mutation Rates from Fluctuation Experiments

Bayesian procedures are developed for estimating mutation rates from fluctuation experiments. Three Bayesian point estimators are compared with four traditional ones using the results of 10,000 simulated experiments. The Bayesian estimators were found to be at least as efficient as the best of the previously known estimators. The best Bayesian estimator is one that uses (1/m(2)) as the prior probability density function and a quadratic loss function. The advantage of using these estimators is most pronounced when the number of fluctuation test tubes is small. Bayesian estimation allows the incorporation of prior knowledge about the estimated parameter, in which case the resulting estimators are the most efficient. It enables the straightforward construction of confidence intervals for the estimated parameter. The increase of efficiency with prior information and the narrowing of the confidence intervals with additional experimental results are investigated. The results of the simulations show that any potential inaccuracy of estimation arising from lumping together all cultures with more than n mutants (the jackpots) almost disappears at n = 70 (provided that the number of mutations in a culture is low). These methods are applied to a set of experimental data to illustrate their use.     

Estimation of the mutation rate during error-prone polymerase chain reaction.

Error-prone polymerase chain reaction (PCR) is widely used to introduce point mutations during in vitro evolution experiments. Accurate estimation of the mutation rate during error-prone PCR is important in studying the diversity of error-prone PCR product. Although many methods for estimating the mutation rate during PCR are available, all the existing methods depend on the assumption that the mutation rate is low and mutations occur at different places whenever they occur. The available methods may not be applicable to estimate the mutation rate during error-prone PCR. We develop a mathematical model for error-prone PCR and present methods to estimate the mutation rate during error-prone PCR without assuming low mutation rate. We also develop a computer program to simulate error-prone PCR. Using the program, we compare the newly developed methods with two other methods. We show that when the mutation rate is relatively low (< 10(-3) per base per PCR cycle), the newly developed methods give roughly the same results as previous methods. When the mutation rate is relatively high (> 5 x 10(-3) per base per PCR cycle, the mutation rate for most error-prone PCR experiments), the previous methods underestimate the mutation rate and the newly developed methods approximate the true mutation rate.     

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.     

The influence of premeiotic clusters of mutation on indirect estimations of mutation rate

Based on the hypothesis that rare alleles are in mutation and random loss equilibrium, mutation rate can be indirectly estimated by measuring the number of rare variants and the average existing time of a mutant allele. This method can be applied to estimate the mutation rate in humans. However, this estimation of mutation rate is affected by the presence of premeiotic clusters of mutation. Mutation clusters change both the number of initial mutants and the average existing time of a mutant allele. As a result, the formula indirectly estimating mutation rate should be modified. The influence of premeiotic clusters is more obvious when the population size is small or the average cluster size is big. For example, if the population size is 3,000 and average cluster size is two, instead of one, the mutation rate is increased by about 9.4%.     

Nuclear–mitochondrial epistasis: a gene's eye view of genomic conflict

We use population genetic models to investigate the cooperative and conflicting synergistic fitness effects between genes from the nucleus and the mitochondrion. By varying fitness parameters, we examine the scope for conflict relative to cooperation among genomes and the utility of the “gene's eye view” analytical approach, which is based on the marginal average fitness of specific alleles. Because sexual conflict can maintain polymorphism of mitochondrial haplotypes, we can explore two types of evolutionary conflict (genomic and sexual) with one epistatic model. We find that the nuclear genetic architecture (autosomal, X‐linked, or Z‐linked) and the mating system change the regions of parameter space corresponding to the evolution by sexual and genomic conflict. For all models, regardless of conflict or cooperation, we find that population mean fitness increases monotonically as evolution proceeds. Moreover, we find that the process of gene frequency change with positive, synergistic fitnesses is self‐accelerating, as the success of an allele in one genome or in one sex increases the frequency of the interacting allele upon which its success depends. This results in runaway evolutionary dynamics caused by the positive intergenomic associations generated by selection. An inbreeding mating system tends to further accelerate these runaway dynamics because it maintains favorable host–symbiont or male–female gene combinations. In contrast, where conflict predominates, the success of an allele in one genome or in one sex diminishes the frequency of the corresponding allele in the other, resulting in considerably slower evolutionary dynamics. The rate of change of mean fitness is also much faster with positive, synergistic fitnesses and much slower where conflict is predominant. Consequently, selection rapidly fixes cooperative gene combinations, while leaving behind a slowing evolving residue of conflicting gene combinations at mutation–selection balance. We discuss how an emphasis on marginal fitness averages may obscure the interdependence of allelic fitness across genomes, making the evolutionary trajectories appear independent of one another when they are not.     

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.     

Estimating mutation rate: how to count mutations?

Mutation rate is an essential parameter in genetic research. Counting the number of mutant individuals provides information for a direct estimate of mutation rate. However, mutant individuals in the same family can share the same mutations due to premeiotic mutation events, so that the number of mutant individuals can be significantly larger than the number of mutation events observed. Since mutation rate is more closely related to the number of mutation events, whether one should count only independent mutation events or the number of mutants remains controversial. We show in this article that counting mutant individuals is a correct approach for estimating mutation rate, while counting only mutation events will result in underestimation. We also derived the variance of the mutation-rate estimate, which allows us to examine a number of important issues about the design of such experiments. The general strategy of such an experiment should be to sample as many families as possible and not to sample much more offspring per family than the reciprocal of the pairwise correlation coefficient within each family. To obtain a reasonably accurate estimate of mutation rate, the number of sampled families needs to be in the same or higher order of magnitude as the reciprocal of the mutation rate.     

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.     

Effects of the shape of distribution of mutant effect in nearly neutral mutation models

Models of the theory of nearly neutral mutation incorporate a continuous distribution of mutation effects in contrast to the theory of purely neutral mutation which allows no mutations with intermediate effects. Previous studies of one such model, namely the house-of-cards mutation model, assumed normal distribution of mutation effect. Here I study the house-of-cards mutation model in random-mating finite populations using the weak-mutation approximation, paying attention to the effects of the distribution of mutant effects. The average selection coefficient, substitution rate and average heterozygosity in the equilibrium and transient states were studied mainly by computer simulation. The main findings are: (i) Very rapid decrease of the substitution rate and very slow approach to equilibrium as selection becomes stronger are characteristics of assuming normal distribution of mutant effect. If the right tail of the mutation distribution decays more rapidly than that of the normal distribution, the decrease of substitution rate becomes slower and equilibrium is achieved more quickly. (ii) The dispersion index becomes smaller or larger than 1 depending on the time and the intensity of selection, (iii) LetN be the population size. When selection is strong the ratio of 4N times the substitution rate to the average heterozygosity, which is expected to be 1 under neutrality, is larger than 1 in earlier generations but becomes less than 1 in later generations. These findings show the importance of the distribution of mutant effect and time in determination of the behaviour of various statistics frequently used in the study of molecular evolution.     

Joint effects of self-fertilization and population structure on mutation load, inbreeding depression and heterosis

Both the spatial distribution of organisms and their mode of reproduction have important effects on the change in allele frequencies within populations. In this article, we study the combined effect of population structure and the rate of partial selfing of organisms on the efficiency of selection against recurrent deleterious mutations. Assuming an island model of population structure and weak selection, we express the mutation load, the within- and between-deme inbreeding depression, and heterosis as functions of the frequency of deleterious mutants in the metapopulation; we then use a diffusion model to calculate an expression for the equilibrium probability distribution of this frequency of deleterious mutants. This allows us to derive approximations for the average mutant frequency, mutation load, inbreeding depression, and heterosis, the simplest ones being Equations 35-39 in the text. We find that population structure can help to purge recessive deleterious mutations and reduce the load for some parameter values (in particular when the dominance coefficient of these mutations is <0.2-0.3), but that this effect is reversed when the selfing rate is above a given value. Conversely, within-deme inbreeding depression always decreases, while heterosis always increases, with the degree of population subdivision, for all selfing rates.     

Comparing analysis methods for mutation-accumulation data: a simulation study

We simulated single-generation data for a fitness trait in mutation-accumulation (MA) experiments, and we compared three methods of analysis. Bateman-Mukai (BM) and maximum likelihood (ML) need information on both the MA lines and control lines, while minimum distance (MD) can be applied with or without the control. Both MD and ML assume gamma-distributed mutational effects. ML estimates of the rate of deleterious mutation had larger mean square error (MSE) than MD or BM had due to large outliers. MD estimates obtained by ignoring the mean decline observed from comparison to a control are often better than those obtained using that information. When effects are simulated using the gamma distribution, reducing the precision with which the trait is assayed increases the probability of obtaining no ML or MD estimates but causes no appreciable increase of the MSE. When the residual errors for the means of the simulated lines are sampled from the empirical distribution in a MA experiment, instead of from a normal one, the MSEs of BM, ML, and MD are practically unaffected. When the simulated gamma distribution accounts for a high rate of mild deleterious mutation, BM detects only approximately 30% of the true deleterious mutation rate, while MD or ML detects substantially larger fractions. To test the robustness of the methods, we also added a high rate of common contaminant mutations with constant mild deleterious effect to a low rate of mutations with gamma-distributed deleterious effects and moderate average. In that case, BM detects roughly the same fraction as before, regardless of the precision of the assay, while ML fails to provide estimates. However, MD estimates are obtained by ignoring the control information, detecting approximately 70% of the total mutation rate when the mean of the lines is assayed with good precision, but only 15% for low-precision assays. Contaminant mutations with only tiny deleterious effects could not be detected with acceptable accuracy by any of the above methods.