A statistical model to estimate variance in long term-low dose mutation assays: testing of the model in a human lymphoblastoid mutation assay

Long term-low dose mutation assays offer a means to study the genetic effects of environmental mutagens at concentrations relevant to human exposure. These assays involve continuous induction of mutants, serial dilution of cultures and sampling to determine the mutant fraction as a function of time and mutagen concentration. An arithmetic model for the expected variance among identically treated cultures is presented. This model provides means to calculate a predicted variance of the mutant fractions and mutation rates in typical long term-low dose experiments. We have calculated the expected variances of the mutant fraction with this model and compared them to the observed variances among 4 independent experiments in which human lymphoblastoid cells were treated for 5, 10, 15 and 20 days with a non-toxic concentration of the mutagen 4-aminobiphenyl. Mutations at the HPRT locus were measured by determining the 6-thioguanine-resistant mutant fraction. The expected and observed variances of the mutant fractions are in close agreement. This model is adequate to predict the variance of the mutant fraction and should be useful in experimental design and objective evaluation of long term-low dose mutation assays.     

Variance estimation in single-cell mutation assays: comparison to experimental observations in human lymphoblasts at 4 gene loci

The performance of mutation assays with single cells involves a series of separate steps beginning with the induction of mutant cells and ending with the counting of mutant and wild-type colonies. The vatiation among identically treated cultures is here modeled as arising from 3 sources: (1) the number of mutant cells surviving treatment, (2) the number of mutant cells sampled in steps of sampling and growth required in assays involving phenotypic lag, and (3) the number of mutant and nonmutant colonies actually observed.

The arithmetical statements describing the expectation of variance from each step are presented and used to provide means to calculate an expected overall variance for typical experiments.

The model is then tested by comparing its predictions with the observed mutant fractions in human lymphoblastoid cells at the loci coding for 6-thioguanine, ouabain, podophyllotoxin, and 5,6-dichlororibofuranosyl benzimidazole resistances. The model is found to have excellent predictive qualities and should be useful in experimental design of studies involving induction of rare variants in single-cell systems.     

Cloning of a human S-phase cell cycle gene: use of transient expression for screening.

We report here the cloning of a human cell cycle gene capable of complementing a temperature-sensitive (ts) S-phase cell cycle mutation in a Chinese hamster cell line. Cloning was performed as follows. A human genomic library in phage lambda containing 600,000 phages was screened with labeled cDNA synthesized from an mRNA fraction enriched for the specific cell cycle gene message. Plaques containing DNA inserts which hybridized to the cDNA were picked, and their DNAs were assayed for transient complementation in DNA transformation experiments. The transient complementation assay we developed is suitable for most cell cycle genes and indeed for many genes whose products are required for cell proliferation. Of 845 phages screened, 1 contained an insert active in transient complementation of the ts cell cycle mutation. Introduction of this phage into the ts cell cycle mutant also gave rise to stable transformants which grew normally at the restrictive temperature for the ts mutant cells.     

A two-step selection approach for the identification of ligand-binding determinants in cytokine receptors

We have developed a novel cell-based method for the isolation and selection of mutant cytokine receptors with defects in ligand binding and applied it to the human interleukin-4 receptor. The experimental procedure is based upon the functional heterologous expression of receptor mutants in eukaryotic cells followed by a two-step selection procedure. Positive selection for cells that express receptor variants is achieved by means of an agonistic antibody that mediates cell survival through receptor dimerization. An IL-4-coupled toxin is subsequently used to select against cells expressing wild-type receptors. Cells expressing mutant receptors that are unable to bind the cytotoxic ligand survive and can be amplified. The procedure allows the isolation of rare receptor variants from cell pools containing predominantly wild-type cells. This method, which should be equally applicable to similar receptor systems, was used to demonstrate the importance of a critical charged amino acid residue in the human IL-4 receptor alpha-subunit for IL-4-induced receptor activation.     

Rme1 is necessary for Mi-1-mediated resistance and acts early in the resistance pathway

The tomato gene Mi-1 confers resistance to root-knot nematodes (Meloidogyne spp.), potato aphid, and whitefly. Using genetic screens, we have isolated a mutant, rme1 (resistance to Meloidogyne spp.), compromised in resistance to M. javanica and potato aphid. Here, we show that the rme1 mutant is also compromised in resistance to M. incognita, M. arenaria, and whitefly. In addition, using an Agrobacterium-mediated transient assay in leaves to express constitutive gain-of-function mutant Pto(L205D), we demonstrated that the rme1 mutation is not compromised in Pto-mediated hypersensitive response. Moreover, the mutation in rme1 does not result in increased virulence of pathogenic Pseudomonas syringae or Mi-1-virulent M. incognita. Using a chimeric Mi-1 construct, Mi-DS4, which confers constitutive cell death phenotype and A. rhizogenes root transformation, we showed that the Mi-1-mediated cell death pathway is intact in this mutant. Our results indicate that Rme1 is required for Mi-1-mediated resistance and acts either at the same step in the signal transduction pathway as Mi-1 or upstream of Mi-1.     

An in situ protocol for measuring the expression of chemically-induced mutations in mammalian cells

The generation of expression curves and the evaluation of mutagenic responses of mammalian cells using standard mutagenesis assays can be inaccurate because mutant and wild-type cells are usually mixed during the expression phase. If some mutant progenitors or mutants grow more slowly than the wild-type cells during the expression period, there will be a decrease in the mutant to wild-type ratio with time and the mutant fraction will not accurately represent the number of mutational events that occurred. The mutant fraction may also inaccurately assess the number of mutations if these mutations are expressed over a number of generations during the time before selection. We previously showed that recovery of L5178Y mouse cell mutants is not complete when mutations are allowed to express in suspension because slowly growing mutants and/or mutant progenitors are diluted out during this time (Rudd et al., 1990). In order to more accurately quantitate the mutagenic response of the cells, we developed an in situ procedure which segregates and immobilizes cells during expression. Because of this immobilization, slowly growing mutant progenitors and mutants expressed at different times will have an equal probability of being scored as mutants. Thus, one mutation leads to one mutant colony and the measurement of the mutagenic response of the cells to the chemical accurately reflects the mutational events that occurred. We plated L5178Y tk^+/− mouse cells in semisolid medium immediately after treatment. As the cells grew and formed microcolonies, the selective agent TFT was added as an overlay at specified times, permitting only TFT^r cells to survive. In this procedure, each mutation was captured as an individual colony; consequently, the measured mutation fraction accurately reflected the mutational events that occurred at the selected locus. In addition, the induced mutant colonies arising in the agar are the result of independent mutational events. We previously described the in situ protocol for L5178Y cells and showed that the spontaneous mutation rate measured was 50-fold greater than when the cells expressed the phenotype in suspension (Rudd et al., 1990). From this we concluded that the slow growth phenotype was expressed before TFT resistance. In the present paper, we evaluate the effect of chemical treatment on the mutation fraction as a function of the time to TFT addition. Using the in situ protocol, we generated expression curves for three nucleotide analogs, 5-azacytidine, TFT and AraC. The numbers of TFT^r colonies produced at various times after treatment indicated that chemically-treated cultures had higher mutation fractions than the solvent controls. The maximal differential increase in mutation rate occured between 30 and 60 h for 5-azacytidine and between 20 and 40 h for TFT and AraC. Our results document the feasibility of quantitating induced mutation fractions using the in situ protocol, confirm the mutagenicity of AraC and 5azacytidine and demonstrate the mutagenic activity of TFT at the tk locus. In addition to recovering mutants more accurately than the suspension protocol, the in situ protocol has the advantage of being experimentally less labor and time intensive. Therefore, we believe that this method should be considered for evaluation as an assay to measure the potential mutagenic effects of chemicals in mammalian cells in vitro.     

Alteration of the largest subunit of RNA polymerase II and its effect on chromosome stability in Schizosaccharomyces pombe

A mutation in the RNA polymerase II largest subunit (RpII LS) that is related to abnormal induction of sister chromatid exchange has previously been described the CHO-K1 cell mutant tsTM4. To elucidate the molecular basis of this effect we introduced the mutation into the homologous site in the Schizosaccharomyces pombe rpb1 gene, which encodes RpII LS. Since the tsTM4 mutant exhibited a decrease in the rate of DNA synthesis in cells arrested in S phase at the nonpermissive temperature, we focussed on the study of growth, the cell cycle, and chromosome stability at various temperatures. First, we examined the effects of the mutation on haploid yeast cells. The mutant showed slower growth than the wild type, but cell growth was not arrested at the nonpermissive temperature. When growing cells were shifted to the nonpermissive temperature, an accumulation of cells in G1 and/or G0 was observed. Tetrad analysis suggested that these phenotypes were associated with the mutation. In diploid cells, chromosome instability was detected by loss of intragenic complementation between two alleles of the ade6 gene. An abnormal fraction of cells containing an intermediate DNA content was also observed by FACS analysis. The accumulation of this fraction may reflect the fact that a large number of cells are in S phase or have an abnormal DNA content as a result of chromosome instability. These observations demonstrate that the S. pomberpb1 mutant exhibits a phenotype very similar to that of the CHO-K1 cell mutant tsTM4.     

Bovine papillomavirus mutant temperature sensitive for transformation, replication and transactivation.

The genetic analysis of the papillomaviruses has been hampered by the lack of mutants conditionally defective for viral biological activities. We report here the construction and characterization of a temperature-sensitive papillomavirus mutant. The mutation is predicted to insert the sequence Pro-Arg-Ser-Arg into the N-terminal half of the bovine papillomavirus type 1 (BPV1) ORF E2 protein, the major viral regulatory protein. The cloned mutant viral DNA displays temperature-sensitive defects in the induction of focus formation in mouse C127 cells, in its establishment as an extrachromosomal plasmid and in transactivation of a BPV1 enhancer. Genetic experiments confirm that this pleiotropic phenotype is caused by the insertion mutation in ORF E2 and that the transformation and replication defects of the mutant at 37 degrees C are corrected in trans by wild-type E2 gene activity. Most cell lines stably transformed by the mutant at 32.5 degrees C display a reduced ability to overgrow a monolayer of normal cells following temperature shift to 37 degrees C and the mutant viral DNA after temperature shift is present in decreased copy number and/or in an integrated state. These results provide strong genetic evidence that continued ORF E2 activity is required for maintenance of BPV1-induced transformation and for normal viral DNA replication.     

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.     

Differential ability of a T-antigen transport-defective mutant of simian virus 40 to transform primary and established rodent cells.

The transforming potential and oncogenicity of a simian virus 40 (SV40) mutant affecting T-antigen (T-ag), SV40(cT)-3, was examined in an effort to dissect T-ag functions in transformation. SV40(cT)-3 has a point mutation at nucleotide 4434 that abolishes the transport of T-ag to the nucleus but does not affect its association with the cell surface. Transfection-transformation assays were performed with primary cells and established cell lines of mouse and rat origin. The efficiency of transformation for established cell lines by SV40(cT)-3 was comparable to that of wild-type SV40, indicating that transformation of established cell lines can occur in the absence of detectable amounts of nuclear T-ag. Transformation of primary mouse embryo fibroblasts by SV40(cT)-3 was markedly influenced by culture conditions; the relative transforming frequency was dramatically reduced in assays involving focus formation in low serum concentrations or anchorage-independent growth. Immunofluorescence tests revealed that the transformed mouse embryo fibroblasts partially transport the mutant cT-ag to the cell nucleus. Transformed cell lines induced by SV40(cT)-3 did not differ in growth properties from wild-type transformants. SV40(cT)-3 was completely defective for the transformation of primary baby rat kidney cells, a primary cell type unable to transport the mutant T-ag to the nucleus. The intracellular localization of cellular protein p53 was found to mimic T-ag distribution in all the transformants analyzed. The mutant virus was weakly oncogenic in vivo: the induction of tumors in newborn hamsters by SV40(cT)-3 was reduced in incidence and delayed in appearance in comparison to wild-type SV40. These observations suggest that cellular transformation is regulated by both nuclear and surface-associated forms of SV40 T-ag.     

Alteration of the largest subunit of RNA polymerase II and its effect on chromosome stability in Schizosaccharomyces pombe

A mutation in the RNA polymerase II largest subunit (RpII LS) that is related to abnormal induction of sister chromatid exchange has previously been described the CHO-K1 cell mutant tsTM4. To elucidate the molecular basis of this effect we introduced the mutation into the homologous site in the Schizosaccharomyces pombe rpb1 gene, which encodes RpII LS. Since the tsTM4 mutant exhibited a decrease in the rate of DNA synthesis in cells arrested in S phase at the nonpermissive temperature, we focussed on the study of growth, the cell cycle, and chromosome stability at various temperatures. First, we examined the effects of the mutation on haploid yeast cells. The mutant showed slower growth than the wild type, but cell growth was not arrested at the nonpermissive temperature. When growing cells were shifted to the nonpermissive temperature, an accumulation of cells in G1 and/or G0 was observed. Tetrad analysis suggested that these phenotypes were associated with the mutation. In diploid cells, chromosome instability was detected by loss of intragenic complementation between two alleles of the ade6 gene. An abnormal fraction of cells containing an intermediate DNA content was also observed by FACS analysis. The accumulation of this fraction may reflect the fact that a large number of cells are in S phase or have an abnormal DNA content as a result of chromosome instability. These observations demonstrate that the S. pomberpb1 mutant exhibits a phenotype very similar to that of the CHO-K1 cell mutant tsTM4. Received: 1 October 1997 / Accepted: 29 December 1997     

Mutations in the gene encoding the adenovirus early region 1B 19,000-molecular-weight tumor antigen cause the degradation of chromosomal DNA

The adenovirus mutant Ad2ts111 has been previously shown to contain a mutation in the early region 2A gene encoding the single-stranded-DNA-binding protein that results in thermolabile replication of virus DNA and a mutation in early region 1 that causes degradation of intracellular DNA. A recombinant virus, Ad2cyt106, has been constructed which contains the Ad2ts111 early region 1 mutation and the wild-type early region 2A gene from adenovirus 5. This virus, like its parent Ad2ts111, has two temperature-independent phenotypes; first, it has the ability to cause an enhanced and unusual cytopathic effect on the host cell (cytocidal [cyt] phenotype) and second, it induces degradation of cell DNA (DNA degradation [deg] phenotype). The mutation responsible for these phenotypes is a single point mutation in the gene encoding the adenovirus early region 1B (E1B) 19,000-molecular-weight (19K) tumor antigen. This mutation causes a change from a serine to an asparagine in the 20th amino acid from the amino terminus of the protein. Three other mutants that affect the E1B 19K protein function have been examined. The mutants Ad2lp5 and Ad5dl337 have both the cytocidal and DNA degradation phenotypes (cyt deg), whereas Ad2lp3 has only the cytocidal phenotype and does not induce degradation of cell DNA (cyt deg+). Thus, the DNA degradation is not caused by the altered cell morphology. Furthermore, the mutant Ad5dl337 does not make any detectable E1B 19K protein product, suggesting that the absence of E1B 19K protein function is responsible for the mutant phenotypes. A fully functional E1B 19K protein is not absolutely required for lytic growth of adenovirus 2 in HeLa cells, and its involvement in transformation of nonpermissive cells to morphological variants is discussed.     

Nonmyristoylated Abl proteins transform a factor-dependent hematopoietic cell line.

N-terminal myristoylation can promote the association of proteins with the plasma membrane, a property that is required for oncogenic variants of Src and Abl to transform fibroblastic cell types. The P210bcr/abl protein of chronic myelogenous leukemia cells is not myristoylated and does not stably transform NIH 3T3 fibroblasts; however, it will transform lymphoid and myeloid cell types in vitro and in vivo, suggesting that myristoylation is not required for Abl variants to transform hematopoietic cells. To test this hypothesis, we introduced point mutations that disrupt myristoylation into two activated Abl proteins, v-Abl and a deletion mutant of c-Abl (delta XB), and examined their ability to transform an interleukin-3-dependent lymphoblastoid cell line, Ba/F3. Neither of the nonmyristoylated Abl proteins transformed NIH 3T3 fibroblasts, but like P210bcr/abl, both were capable of transforming the Ba/F3 cells to factor independence and tumorigenicity. Nonmyristoylated Abl variants did not associate with the plasma membrane in the transformed Ba/F3 cells. These results demonstrate that Abl proteins can transform hematopoietic cells in the absence of membrane association and suggest that distinct functions of Abl are required for transformation of fibroblast and hematopoietic cell types.     

RAG-2-deficient mice lack mature lymphocytes owing to inability to initiate V(D)J rearrangement.

We have generated mice that carry a germline mutation in which a large portion of the RAG-2 coding region is deleted. Homozygous mutants are viable but fail to produce mature B or T lymphocytes. Very immature lymphoid cells were present in primary lymphoid organs of mutant animals as defined by surface marker analyses and Abelson murine leukemia virus (A-MuLV) transformation assays. However, these cells did not rearrange their immunoglobulin or T cell receptor loci. Lack of V(D)J recombination activity in mutant pre-B cell lines could be restored by introduction of a functional RAG-2 expression vector. Therefore, loss of RAG-2 function in vivo results in total inability to initiate V(D)J rearrangement, leading to a novel severe combined immune deficient (SCID) phenotype. Because the SCID phenotype was the only obvious abnormality detected in RAG-2 mutant mice, RAG-2 function and V(D)J recombinase activity, per se, are not required for development of cells other than lymphocytes.     

Mutation induction in a mouse lymphoma cell mutant sensitive to 4-nitroquinoline 1-oxide and ultraviolet radiation

The mutant mouse lymphoma cell Q31, which is sensitive to 4-nitroquinoline 1-oxide and ultraviolet radiation (UV), was compared with the parental L5178Y cell for the effect of caffeine and mutation induction after UV irradiation. Caffeine potentiated the lethal effect of UV in both cell strains to a similar extent, indicating that the defective process in Q31 cells was caffeine-insensitive. UV-induced mutation to 6-thioguanine resistance was determined in L5178Y and Q31 cells. The maximal yield of mutants was obtained 7 days post-irradiation in L5178Y cells and 14 days in Q31 cells for higher UV doses. It appears that a much longer time is required for the mutant cells than for the parental cells for full expression of the resistance phenotype even at equitoxic UV doses. A substantially higher frequency in induced mutations was observed in Q31 cells than in L5178Y cells at a given dose of UV. A plot of induced mutation frequency as a function of logarithm of surviving fraction again indicates hypermutability of Q31 cells as compared with the parental strain. In contrast, X-rays induced a similar frequency of mutations to 6-thioguanine resistance in L5178Y and Q31 cells.     

An optimal weighted aggregated association test for identification of rare variants involved in common diseases

The advent of next generation sequencing technologies allows one to discover nearly all rare variants in a genomic region of interest. This technological development increases the need for an effective statistical method for testing the aggregated effect of rare variants in a gene on disease susceptibility. The idea behind this approach is that if a certain gene is involved in a disease, many rare variants within the gene will disrupt the function of the gene and are associated with the disease. In this article, we present the rare variant weighted aggregate statistic (RWAS), a method that groups rare variants and computes a weighted sum of differences between case and control mutation counts. We show that our method outperforms the groupwise association test of Madsen and Browning in the disease-risk model that assumes that each variant makes an equally small contribution to disease risk. In addition, we can incorporate prior information into our method of which variants are likely causal. By using simulated data and real mutation screening data of the susceptibility gene for ataxia telangiectasia, we demonstrate that prior information has a substantial influence on the statistical power of association studies. Our method is publicly available at http://genetics.cs.ucla.edu/rarevariants.     

Mitochondria harbouring mutant mtDNA--a cuckoo in the nest?

Mutations of the mitochondrial DNA (mtDNA) are associated with a number of human diseases. To become relevant in terms of pathology, a mutation must generally affect at least 50-70% of mtDNA molecules in a tissue. One way to reach this level is by inheritance. Mitotic segregation of mtDNA in the female germline can result in large increases in the percentage of mutant mtDNA between generations. A different explanation is required if a particular mtDNA mutation accumulates over time in somatic cells. We discuss the possibility that mutant mtDNA, by causing deficient oxidative phosphorylation, may become preferentially replicated and may thus thrive in the cell like a cuckoo in the nest. However, despite preferential replication, a de novo mtDNA mutation will be confined to that particular cell or a small clone of daughter cells. Significant accumulation can only occur if the cell harbouring the mutant mtDNA undergoes malignant transformation and therefore starts proliferating continuously. This type of amplification of mutant mtDNA has recently been demonstrated in certain bone marrow disorders (myelodysplastic syndromes) and in colon cancer cell lines. Finally, in postmitotic tissues, an inherited mutation which is present in virtually all cells of the tissue, may accumulate through replicative advantage. This may contribute to the development of degenerative diseases.     

Low level of cellular protein phosphorylation by nontransforming overproduced p60c-src.

We have previously found that Rous sarcoma virus variants in which the viral src (v-src) gene is replaced by the cellular src (c-src) gene have no transforming activity. In this study, we analyzed the basis for the inability of the p60c-src overproduced by these variants to transform cells. Phosphorylations of tyrosine residues in total cell protein or in cellular 34K protein are known to be markedly enhanced upon infection with wild-type Rous sarcoma virus. We found that these tyrosine phosphorylations were only slightly increased in the c-src-containing virus-infected cells, whereas both levels were significantly increased by infection with wild-type Rous sarcoma virus, or transforming mutant viruses which are derived from c-src-containing viruses by spontaneous mutation. Phosphorylation at tyrosine 416 of p60 itself was also extremely low in overproduced p60c-src and high in p60s of transforming mutant viruses. In immunoprecipitates with monoclonal antibody, the overproduced p60c-src had much lower casein tyrosine kinase activity than did p60v-src. We previously showed that p60 myristylation and plasma membrane localization may be required for cell transformation. p60c-src was similar to transforming p60s in these properties. These results strongly suggest that the low level of tyrosine phosphorylation by overproduced p60c-src accounts for its inability to transform cells.     

Mutant adenovirus type 9 E4 ORF1 genes define three protein regions required for transformation of CREF cells.

Human adenovirus type 9 (Ad9) elicits exclusively estrogen-dependent mammary tumors in rats, and an essential oncogenic determinant for this virus is Ad9 E4 open reading frame 1 (9ORF1), which encodes a 125-residue cytoplasmic protein with cellular growth-transforming activity in vitro. In this study, we engineered 48 different mutant 9ORF1 genes in an attempt to identify regions of this viral protein essential for transformation of the established rat embryo fibroblast cell line CREF. In initial assays with CREF cells, 17 of the 48 mutant 9ORF1 genes proved to be severely defective for generating transformed foci but only 7 of these defective genes expressed detectable amounts of protein. To further examine the defects of the seven mutant proteins, we selected individual cell pools of stable CREF transformants for the wild-type and mutant 9ORF1 genes. Compared to cell pools expressing the wild-type 9ORF1 protein, most cell pools expressing mutant proteins displayed decreased growth in soft agar, and all generated significantly smaller tumors in syngeneic animals. The altered amino acid residues of the seven mutant 9ORF1 polypeptides clustered within three separate regions referred to as region I (residues 34 to 41), region II (residues 89 to 91), and C-terminal region III (residues 122 to 125). By using indirect immunofluorescence, we also assessed whether the mutant proteins localized properly to the cytoplasm of cells. The region I and region II mutants displayed approximately wild-type subcellular localizations, whereas most region III mutants aberrantly accumulated within the nucleus of cells. In summary, we have identified three 9ORF1 protein regions necessary for cellular transformation and have demonstrated that C-terminal region III sequences significantly influence the proper localization of the 9ORF1 polypeptide in cells.     

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.