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.     

A stochastic model of mutant growth due to mutation in tumors,based on stem cell considerations

A stochastic model of the evolution of mutant subpopulations from stem cells in a human tumor system is derived. From the model, the growth of mutants (both stem cell mutants and overall mutants) due to mutation of tumor stem cells during growth is explored in detail. This allows one to relate the mutant stem cell and overall tumor mutant cell population sizes. The relation of these average sizes is derived for large tumor size and confirms the result of the model due to MacKillop et al. [4], which is based on three tumor cell subpopulations: stem, transitional, and end cells. Furthermore, the results of the stem cell statistics obtained are the same as those obtained from the filtered Poisson process approach [2].     

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.     

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.     

Measurement of spontaneous mutation rates at the Na+/K+ ATPase locus (ouabain resistance) of human fibroblasts using improved growth conditions

The mutation rate for the Na+/K+ ATPase locus (ouabain resistance, OuaR) in mammalian cells in culture has been reported to be 10-100-fold lower than the mutation rate of other gene loci in culture, such as the hypoxanthine phosphoribosyl transferase (HPRT) locus. Determination of the mutation rate to ouabain resistance is sensitive to culture conditions and the concentration of ouabain used to select mutants. Our improved growth conditions for human cells have permitted absolute cloning efficiencies of 70-90% and population doubling times of 16-17 h with both normal human diploid fibroblasts, KD, and their chemically induced neoplastic derivative, Hut-11A. Ouabain at 10(-7) M was found to be adequate to select for resistant (OuaR) mutants with an absolute recovery efficiency of 54-102%. Under these conditions, the mutation rates to ouabain resistance for human cells were measured and found to be 1-8.5 X 10(-7)/cell/generation for KD cells and 6-13 X 10(-7)/cell/generation for Hut-11A cells. These rates are 5-25 times higher than previously reported for human cells. Improved growth and the use of a lower concentration of ouabain for selection may allow for the increased recovery of OuaR mutants and an improved estimate of the mutation rate at this locus, which is only 2-10-fold less than the mutation rate at the HPRT locus in the same cells.     

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.     

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.     

Mutation-selection balance in mixed mating populations

An approximation to the average number of deleterious mutations per gamete, Q, is derived from a model allowing selection on both zygotes and male gametes. Progeny are produced by either outcrossing or self-fertilization with fixed probabilities. The genetic model is a standard in evolutionary biology: mutations occur at unlinked loci, have equivalent effects, and combine multiplicatively to determine fitness. The approximation developed here treats individual mutation counts with a generalized Poisson model conditioned on the distribution of selfing histories in the population. The approximation is accurate across the range of parameter sets considered and provides both analytical insights and greatly increased computational speed. Model predictions are discussed in relation to several outstanding problems, including the estimation of the genomic deleterious mutation rates (U), the generality of "selective interference" among loci, and the consequences of gametic selection for the joint distribution of inbreeding depression and mating system across species. Finally, conflicting results from previous analytical treatments of mutation-selection balance are resolved to assumptions about the life-cycle and the initial fate of mutations.     

Estimation of spontaneous mutation rates

Spontaneous or randomly occurring mutations play a key role in cancer progression. Estimation of the mutation rate of cancer cells can provide useful information about the disease. To ascertain these mutation rates, we need mathematical models that describe the distribution of mutant cells. In this investigation, we develop a discrete time stochastic model for a mutational birth process. We assume that mutations occur concurrently with mitosis so that when a nonmutant parent cell splits into two progeny, one of these daughter cells could carry a mutation. We propose an estimator for the mutation rate and investigate its statistical properties via theory and simulations. A salient feature of this estimator is the ease with which it can be computed. The methods developed herein are applied to a human colorectal cancer cell line and compared to existing continuous time models.     

Patterns of cell division and the risk of cancer

Epidermal and intestinal tissues divide throughout life to replace lost surface cells. These renewing tissues have long-lived basal stem cell lineages that divide many times, each division producing one stem cell and one transit cell. The transit cell divides a limited number of times, producing cells that move up from the basal layer and eventually slough off from the surface. If mutation rates are the same in stem and transit divisions, we show that minimal cancer risk is obtained by using the fewest possible stem divisions subject to the constraints imposed by the need to renew the tissue. In this case, stem cells are a necessary risk imposed by the constraints of tissue architecture. Cairns suggested that stem cells may have lower mutation rates than transit cells do. We develop a mathematical model to study the consequences of different stem and transit mutation rates. Our model shows that stem cell mutation rates two or three orders of magnitude less than transit mutation rates may favor relatively more stem divisions and fewer transit divisions, perhaps explaining how renewing tissues allocate cell divisions between long stem and short transit lineages.     

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.      

The cellular parameters of leaf development in tobacco: a clonal analysis

The cellular parameters of leaf development in tobacco (Nicotiana tabacum L.) have been characterized using clonal analysis, an approach that provides unequivocal evidence of cell lineage. Our results indicate that the tobacco leaf arises from a group of around 100 cells in the shoot apical meristem. Each of these cells contributes to a unique longitudinal section of the axis and transverse section of the lamina. This pattern of cell lincage indicates that primordial cells contribute more or less equally to the growth of the axis, in contrast to the more traditional view of leaf development in which the leaf is pictured as arising from a group of apical initials. Clones induced prior to the initiation of the lamina demonstrate that the subepidermal layer of the lamina arises from at least six files of cells. Submarginal cells usually divide with their spindles parallel to the margin, and therefore contribute relatively little to the transverse expansion of the lamina. During the expansion of the lamina the orientation and frequency of cell division are highly regulated, as is the duration of meristematic growth. Initially, cell division is polarized so as to produce lineages that are at an oblique angle to the midrib; later cell division is in alternating perpendicular planes. The distribution of clones generated by irradiation at various stages of development indicates that cell division ceases at the tip of the leaf when the leaf is about one tenth its final size, and then ceases in progressively more basal regions of the lamina. Variation in the mutation frequency within the lamina reflects variation in the frequency of mitosis. Prior to the mergence of the leaf the frequency of mutation is maximal near the tip of the leaf and extremely low at its base; after emergence, the frequency of mutation increases at the base of the leaf. In any given region of the lamina the frequency of mutation is highest in interveinal regions, and is relatively low near the margin. Thus, both the orientation and frequency of cell division at the leaf margin indicate that this region plays a minor role in the growth of the lamina.Abbreviation MF mutation frequency     

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.     

Inhibition of cell division in Escherichia coli K-12 by the R-factor R1 and copy mutants of R1.

The effect of the copy number of plasmid R1drd-19 on cell division of Escherichia coli K-12 was studied in populations growing as steady-state cultures at different growth rates, the growth rate being varied by use of different carbon sources. The plasmid copy number was also varied by using copy mutants of the R-factor. The mean cell size was larger in populations carrying an R-factor than in R-factorless populations, an effect that was more pronounced at low growth rates and in populations carrying R-factor copy mutants. The increased cell size was due to formation of elongated cells in a fraction of the population and to an increase in the diameter of all cells. The majority of the cells divided at a normal cell length, but the presence of an R-factor caused some cells to elongate, probably by the uncoupling of chromosome replication and cell division. This can be explained as a competition between the chromosome and plasmid replicons for some replication factor(s), presumably acting on both initiation and elongation of replication. The formation of elongated cells was a reversible process, but occasionally some of the elongated cells reached lengths 20 times that of newborn cells. If cell division did not occur at the normal cell size, the septum was not formed until the cell size was four times that of a newborn cell. When an elongated cell divided, it usually formed a polar septum, thus producing a newborn cell of normal cell length. The ability of plasmid-containing cells to omit one cell division but to retain the capacity of dividing one mass doubling later is compatible with a mechanical model for septum formation and cell division.     

Cell division activity during apical hook development

Growth during plant development is predominantly governed by the combined activities of cell division and cell elongation. The relative contribution of both activities controls the growth of a tissue. A fast change in growth is exhibited at the apical hypocotyl of etiolated seedlings where cells grow at different rates to form a hook-like structure, which is traditionally assumed to result from differential cell elongation. Using new tools we show asymmetric distribution of cell division during early stages of hook development. Cell divisions in the apical hook were predominantly found in subepidermal layers during an early step of hook development, but were absent in mutants exhibiting a hookless phenotype. In addition, during exaggeration of hook curvature, which is mediated by ethylene, a rapid change in the combined activities of cell division and cell elongation was detected. Our results indicate a fast change in cell division activity during apical hook development. We suggest that cell division together with cell elongation contributes to apical hook growth. Our results emphasize the change in the relative contribution of cell division and cell elongation in a fast growing structure like the apical hook.     

Isolation and partial characterization of conditional cell division cycle mutants inChlamydomonas

Summary We have isolated a number of temperature conditional cell division cycle mutants of the unicellular plantChlamydomonas reinhardtii that are defective in single nuclear genes. Cells grow and divide normally at the permissive temperature (21 °C), but arrest in division at the restrictive temperature (33 °C). We have characterized these mutants using DNA probes and immunofluorescence techniques to localize cytoskeletal and microtubule organizing centre proteins. We describe here 3 broad classes of cell cycle mutation which result in cell cycle arrest with: unreplicated DNA (G1 arrest), duplicated DNA (G2 arrest) and multiple nuclei due to defective cytokinesis (cytokinesis arrest). The continuation of nuclear division in mutants blocked in cytokinesis provides support of an earlier hypothesis that stage specific events in theChlamydomonas cell cycle are arranged in separate dependent sequences. The mutants isolated in the present study provide insights into the role of cytoskeletal proteins in the coordination of plant cell division and the means to investigate the molecular mechanisms whereby division by multiple fission is controlled in the unicellular plantChlamydomonas.Abbreviations BB basal bodies - EMS ethylmethane sulphonate - MT microtubule - MTOC Microtubule organizing centre - NBBC nucleus-basal body connector - PAR photosynthetically active radiation     

COMPOUND POISSON APPROXIMATIONS FOR NUMBER OF RARE MUTANTS

Thispaperdiscussesthelimitdistributionofnumberofraremutantsinamutationprocess.Theresultofthepapergeneralizedthatof[1].Meanwhile,theauthoralsodiscusscompoundPoissonapproximationtheoremforakindofrandomsum.     

Regulation of DNA synthesis in bacteria: Analysis of the Bates/Kleckner licensing/initiation-mass model for cell cycle control

Bates and Kleckner have recently proposed that bacterial cell division is a licensing agent for a subsequent initiation of DNA replication. They also propose that initiation mass for DNA replication is not constant. These two proposals do not take into account older data showing that initiation of DNA replication can occur prior to the division event. This critical analysis is derived from measurements of DNA replication during the division cycle in cells growing at different, and more rapid, growth rates. Furthermore, mutants impaired in division can initiate DNA synthesis. The data presented by Bates and Kleckner do not support the proposal that initiation mass is variable, and the proposed pattern of DNA replication during the division cycle of the K12 cells analysed is not consistent with prior data on the pattern of DNA replication during the division cycle.