A robust estimator of mutation rates

Determination of Mutation Rates with Two Symmetric and Asymmetric Mutation Types

We revisit our earlier paper, with two of the coauthors, in which we proposed an unbiased and consistent estimator μ^n for an unknown mutation rate μ of microorganisms. Previously, we proved that the associated sequence of estimators μ^n converges to μ almost surely pointwise on a nonextinct set Ω0. Here, we show that this sequence converges also in the mean square with respect to conditional probability measure P0·=P·∩Ω0/PΩ0 and that, with respect to P0, the estimator is asymptotically unbiased. We further assume that a microorganism can mutate or turn to a different variant of one of the two types. In particular, it can mean that bacteria under attack by a virus or chemical agent are either perishing or surviving, turning them to stronger variant. We propose estimators for their respective types and show that they are a.s. pointwise and L2-consistent and asymptotically unbiased with respect to measure P0.

New approaches to mutation rate fold change in Luria-Delbrück fluctuation experiments

For nearly eight decades the Luria-Delbrück protocol remains the preferred method for experimentally determining microbial mutation rates. However, earnest development and rigorous applications of statistical methods for mutation rate comparison using fluctuation assay data are a relatively recent phenomenon. While likelihood ratio tests tailored for the fluctuation protocol give investigators appropriate tools, researchers sometimes may prefer to view the comparison of two mutation rates through the lens of the ratio of the two mutation rates. The idea of using the bootstrap technique to construct intervals for mutation rate fold change was proposed nearly a decade ago, but it failed to gain traction partly due to a failure to incorporate likelihood-based estimation. In addition to extending the bootstrap method, this paper proposes two new methods of constructing intervals for mutation rate fold change: a profile likelihood method and a Bayesian method utilizing Monte Carlo Markov chain. All three methods are assessed by large-scale simulations.

Estimating the number of genetic mutations (hits) required for carcinogenesis based on the distribution of somatic mutations

Individual instances of cancer are primarily a result of a combination of a small number of genetic mutations (hits). Knowing the number of such mutations is a prerequisite for identifying specific combinations of carcinogenic mutations and understanding the etiology of cancer. We present a mathematical model for estimating the number of hits based on the distribution of somatic mutations. The model is fundamentally different from previous approaches, which are based on cancer incidence by age. Our somatic mutation based model is likely to be more robust than age-based models since it does not require knowing or accounting for the highly variable mutation rate, which can vary by over three orders of magnitude. In fact, we find that the number of somatic mutations at diagnosis is weakly correlated with age at cancer diagnosis, most likely due to the extreme variability in mutation rates between individuals. Comparing the distribution of somatic mutations predicted by our model to the actual distribution from 6904 tumor samples we estimate the number of hits required for carcinogenesis for 17 cancer types. We find that different cancer types exhibit distinct somatic mutational profiles corresponding to different numbers of hits. Why might different cancer types require different numbers of hits for carcinogenesis? The answer may provide insight into the unique etiology of different cancer types.

Cancer is primarily a result of genetic mutations. Each individual instance of cancer is initiated by a specific combination of a small number of mutations (hits). In trying to identify these combinations of mutations, it is important to know how many hits to look for. However, there are conflicting estimates for the number of hits. We present a fundamentally different model for estimating the number of hits. We found that the number hits ranges from two-eight depending on cancer type. These findings may provide insight into the unique characteristics of different cancer types.

Guidelines for DNA recombination and repair studies: Cellular assays of DNA repair pathways

Understanding the plasticity of genomes has been greatly aided by assays for recombination, repair and mutagenesis. These assays have been developed in microbial systems that provide the advantages of genetic and molecular reporters that can readily be manipulated. Cellular assays comprise genetic, molecular, and cytological reporters. The assays are powerful tools but each comes with its particular advantages and limitations. Here the most commonly used assays are reviewed, discussed, and presented as the guidelines for future studies.

A second look at the final number of cells in a fluctuation experiment

In a fluctuation experiment, the number of cells existing in a culture immediately before plating (commonly known as Nt) varies across the parallel cultures. However, most existing mathematical models for fluctuation assay data do not recognize the variation in Nt. Despite repeated attempts in the past to integrate this source of variability in the estimation of microbial mutation rates, several questions of practical importance remain unanswered. The present investigation finds that the variation needs accounting for only when the coefficient of variation for Nt is large, and experimental data suggest that the coefficient of variation is often moderate or small. Moreover, an increase in the inoculum size can reduce the coefficient of variation. Through extensive simulation, several existing methods that accommodate the variation in Nt are compared. It was found that a newly devised likelihood method based on the existing gamma mixture model outperforms other existing methods. The investigation focuses on the estimation of mutation rates using the Lea-Coulson model, under which mutation is selectively neutral; however, the paper also explores the major findings' implications for the comparison of mutation rates using the likelihood ratio test, and for the estimation of mutation rates using the Mandelbrot-Koch model that allows for non-neutral mutations.

Fast maximum likelihood estimation of mutation rates using a birth-death process

Since fluctuation analysis was first introduced by Luria and Delbrück in 1943, it has been widely used to make inference about spontaneous mutation rates in cultured cells. Under certain model assumptions, the probability distribution of the number of mutants that appear in a fluctuation experiment can be derived explicitly, which provides the basis of mutation rate estimation. It has been shown that, among various existing estimators, the maximum likelihood estimator usually demonstrates some desirable properties such as consistency and lower mean squared error. However, its application in real experimental data is often hindered by slow computation of likelihood due to the recursive form of the mutant-count distribution. We propose a fast maximum likelihood estimator of mutation rates, MLE-BD, based on a birth-death process model with non-differential growth assumption. Simulation studies demonstrate that, compared with the conventional maximum likelihood estimator derived from the Luria-Delbrück distribution, MLE-BD achieves substantial improvement on computational speed and is applicable to arbitrarily large number of mutants. In addition, it still retains good accuracy on point estimation.

Combining magnetic sorting of mother cells and fluctuation tests to analyze genome instability during mitotic cell aging in Saccharomyces cerevisiae

Saccharomyces cerevisiae has been an excellent model system for examining mechanisms and consequences of genome instability. Information gained from this yeast model is relevant to many organisms, including humans, since DNA repair and DNA damage response factors are well conserved across diverse species. However, S. cerevisiae has not yet been used to fully address whether the rate of accumulating mutations changes with increasing replicative (mitotic) age due to technical constraints. For instance, measurements of yeast replicative lifespan through micromanipulation involve very small populations of cells, which prohibit detection of rare mutations. Genetic methods to enrich for mother cells in populations by inducing death of daughter cells have been developed, but population sizes are still limited by the frequency with which random mutations that compromise the selection systems occur. The current protocol takes advantage of magnetic sorting of surface-labeled yeast mother cells to obtain large enough populations of aging mother cells to quantify rare mutations through phenotypic selections. Mutation rates, measured through fluctuation tests, and mutation frequencies are first established for young cells and used to predict the frequency of mutations in mother cells of various replicative ages. Mutation frequencies are then determined for sorted mother cells, and the age of the mother cells is determined using flow cytometry by staining with a fluorescent reagent that detects bud scars formed on their cell surfaces during cell division. Comparison of predicted mutation frequencies based on the number of cell divisions to the frequencies experimentally observed for mother cells of a given replicative age can then identify whether there are age-related changes in the rate of accumulating mutations. Variations of this basic protocol provide the means to investigate the influence of alterations in specific gene functions or specific environmental conditions on mutation accumulation to address mechanisms underlying genome instability during replicative aging.

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.

Accumulation of neutral mutations in growing cell colonies with competition

Neutral mutations play an important role in many biological processes including cancer initiation and progression, the generation of drug resistance in bacterial and viral diseases as well as cancers, and the development of organs in multicellular organisms. In this paper we study how neutral mutants are accumulated in nonlinearly growing colonies of cells subject to growth constraints such as crowding or lack of resources. We investigate different types of growth control which range from "division-controlled" to "death-controlled" growth (and various mixtures of both). In division-controlled growth, the burden of handling overcrowding lies with the process of cell-divisions, the divisions slow down as the carrying capacity is approached. In death-controlled growth, it is death rate that increases to slow down expansion. We show that division-controlled growth minimizes the number of accumulated mutations, and death-controlled growth corresponds to the maximum number of mutants. We check that these results hold in both deterministic and stochastic settings. We further develop a general (deterministic) theory of neutral mutations and achieve an analytical understanding of the mutant accumulation in colonies of a given size in the absence of back-mutations. The long-term dynamics of mutants in the presence of back-mutations is also addressed. In particular, with equal forward- and back-mutation rates, if division-controlled and a death-controlled types are competing for space and nutrients, cells obeying division-controlled growth will dominate the population.

On an unbiased and consistent estimator for mutation rates

Spontaneous mutations are stochastic events. The mutation rate, defined as mutations per genome per replication, is generally very low, and it is widely accepted that spontaneous mutations occur at defined, but different, rates in bacteriophage and in bacterial, insect, and mammalian cells. The calculation of mutation rates has proved to be a significant problem. Mutation rates can be calculated by following mutant accumulation during growth or from the distribution of mutants obtained in parallel cultures. As Luria and Delbrück described in 1943, the number of mutants in parallel populations of bacterial cells varies widely depending on when a spontaneous mutation occurs during growth of the culture. Since 1943, many mathematical refinements to estimating rates, called estimators, have been described to facilitate determination of the mutation rate from the distribution or frequency of mutants detected following growth of parallel cultures. We present a rigorous mathematical solution to the mutation rate problem using an unbiased and consistent estimator. Using this estimator we demonstrate experimentally that mutation rates can be easily calculated by determining mutant accumulation, that is, from the number of mutants measured in two successive generations. Moreover, to verify the consistency of our estimator we conduct a series of simulation trials that show a surprisingly rapid convergence to the targeted mutation rate (reached between 25th and 30th generations).

Estimating mutation rates in low-replication experiments

Since mutation rate is a key biological parameter, its proper estimation has received great attention for decades. However, instead of the mutation rate, many authors opt for reporting the average mutant frequency, a less meaningful quantity. This is because the standard methods to estimate the mutation rate, derived from the Luria and Delbrück's fluctuation analysis, ideally require high-replication experiments to be applied; a requirement often unattainable due to constraints of time, budget or sample availability. But the main problem with mutant frequency, apart from being less informative, is its poor reproducibility; an especially marked defect when the chosen average is the arithmetic mean. Several authors tried to avoid this by employing other averages (such as the median or the geometric mean) or discarding outliers, though as far as we know nobody has evaluated which method performs best under low-replication settings. Here we use computer simulations to compare the performance of different methods used in low-replication experiments (≤4 cultures). Besides the customary averages of mutant frequency, we also tested two well-known fluctuation methods. Contrary to common practice, our results support that fluctuation methods should be applied in such circumstances, as they perform as well as or better than any average of mutant frequency. In particular, experimentalists will benefit from using MSS maximum likelihood in low-replication experiments because it: (i) provides more reproducible results, (ii) allows for direct estimation of mutation rate and (iii) allows for the application of conventional statistics.

Bootstrap estimation of confidence intervals on mutation rate ratios

The fluctuation test is a useful tool for estimating the mutation rate of cells. However, statistical methods for comparing mutation rate estimates between different strains or conditions have not yet been fully developed. Methods exist for placing confidence intervals on estimates of the number of mutational events in cultures for a given strain and set of conditions, but placing confidence intervals on mutation rate is complicated by differences in the final number of cells in culture between parallel cultures. Additionally, confidence intervals on individual mutation rate estimates are not always the most useful statistical tool when comparing two or more different strains or conditions. We present a bootstrap method for estimating confidence intervals on the quotient of two mutation rates determined from two fluctuation test experiments using experimental and control strains or conditions. We use Monte Carlo simulations to validate this method over a wide range of mutation rates and for empirically measured variation in the estimates of final number of cells in culture. Furthermore, we provide the computational tools to implement the bootstrap method described here on experimental fluctuation test data and to evaluate this method for experimental parameters other than those considered herein.