Microsatellite mutations during the polymerase chain reaction: mean field approximations and their applications

We develop a novel mathematical model for microsatellite mutations during polymerase chain reaction (PCR). Based on the model, we study the first- and second-order moments of the number of repeat units in a randomly chosen molecule after n PCR cycles and their corresponding mean field approximations. We give upper bounds for the approximation errors and show that the approximation errors are small when the mutation rate is low. Based on the theoretical results, we develop a moment estimation method to estimate the mutation rate per-repeat-unit per PCR cycle and the probability of expansion when mutations occur. Simulation studies show that the moment estimation method can accurately recover the true mutation rate and probability of expansion. Finally, the method is applied to experimental data from single-molecule PCR experiments.     

The mutation process of microsatellites during the polymerase chain reaction.

We build a mathematical model for the mutation process of microsatellites during polymerase chain reaction (PCR) using the theory of branching processes. Based on the model, we develop a method to estimate the mutation rate of microsatellites per PCR cycle and the probability of expansion by maximizing a quasi-likelihood of the observed data. We show by simulations that the proposed estimation method can accurately recover the relationship between the mutation rate and number of repeat units. The theoretical basis for the proposed method is also given. We apply the method to experimental data on poly-A and poly-CA repeats.     

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.     

Estimation of the Amount of DNA Polymorphism When the Neutral Mutation Rate Varies among Sites

Knowing the amount of DNA polymorphism is essential to understand the mechanism of maintaining DNA polymorphism in a natural population. The amount of DNA polymorphism can be measured by the average number of nucleotide differences per site (π), the proportion of segregating (polymorphic) site (s) and the minimum number of mutations per site (s*). Since the latter two quantities depend on the sample size, θ is often used as a measure of the amount of DNA polymorphism, where θ = 4Nμ, N is the effective population size and μ is the neutral mutation rate per site per generation. It is known that θ estimated from π, s and s* under the infinite site model can be biased when the mutation rate varies among sites. We have therefore developed new methods for estimating θ under the finite site model. Using computer simulations, it has been shown that the new methods give almost unbiased estimates even when the mutation rate varies among sites substantially. Furthermore, we have also developed new statistics for testing neutrality by modifying Tajima's D statistic. Computer simulations suggest that the new test statistics can be used even when the mutation rate varies among sites.     

A general model of error-prone PCR

In this paper, we generalize a previously-described model of the error-prone polymerase chain reaction (PCR) reaction to conditions of arbitrarily variable amplification efficiency and initial population size. Generalisation of the model to these conditions improves the correspondence to observed and expected behaviours of PCR, and restricts the extent to which the model may explore sequence space for a prescribed set of parameters. Error-prone PCR in realistic reaction conditions is predicted to be less effective at generating grossly divergent sequences than the original model. The estimate of mutation rate per cycle by sampling sequences from an in vitro PCR experiment is correspondingly affected by the choice of model and parameters.     

A Model for Genome Size Evolution

We present a model for genome size evolution that takes into account both local mutations such as small insertions and small deletions, and large chromosomal rearrangements such as duplications and large deletions. We introduce the possibility of undergoing several mutations within one generation. The model, albeit minimalist, reveals a non-trivial spontaneous dynamics of genome size: in the absence of selection, an arbitrary large part of genomes remains beneath a finite size, even for a duplication rate 2.6-fold higher than the rate of large deletions, and even if there is also a systematic bias toward small insertions compared to small deletions. Specifically, we show that the condition of existence of an asymptotic stationary distribution for genome size non-trivially depends on the rates and mean sizes of the different mutation types. We also give upper bounds for the median and other quantiles of the genome size distribution, and argue that these bounds cannot be overcome by selection. Taken together, our results show that the spontaneous dynamics of genome size naturally prevents it from growing infinitely, even in cases where intuition would suggest an infinite growth. Using quantitative numerical examples, we show that, in practice, a shrinkage bias appears very quickly in genomes undergoing mutation accumulation, even though DNA gains and losses appear to be perfectly symmetrical at first sight. We discuss this spontaneous dynamics in the light of the other evolutionary forces proposed in the literature and argue that it provides them a stability-related size limit below which they can act.     

The Amount of DNA Polymorphism Maintained in a Finite Population When the Neutral Mutation Rate Varies among Sites

The expectations of the average number of nucleotide differences per site (π), the proportion of segregating site (s), the minimum number of mutations per site (s*) and some other quantities were derived under the finite site models with and without rate variation among sites, where the finite site models include Jukes and Cantor's model, the equal-input model and Kimura's model. As a model of rate variation, the gamma distribution was used. The results indicate that if distribution parameter α is small, the effect of rate variation on these quantities are substantial, so that the estimates of θ based on the infinite site model are substantially underestimated, where θ = 4Nv, N is the effective population size and v is the mutation rate per site per generation. New methods for estimating θ are also presented, which are based on the finite site models with and without rate variation. Using these methods, underestimation can be corrected.     

Molecular analysis of hemophilia B in Poland: 12 novel mutations of the factor IX gene

We examined the molecular basis of factor IX deficiency in 53 unrelated Polish patients with hemophilia B. Heteroduplex analysis and direct sequencing of polymerase chain reaction (PCR) products were applied to identify the gene defect. Forty-three different point mutations were detected in the factor IX gene of 47 patients. There were 29 missense mutations, 9 nonsense mutations, 4 splice site mutations and 1 point mutation in the promoter region. Twelve mutations were novel. The results of this study emphasize a very high degree of heterogeneity of hemophilia B.     

An additional allelic variant of the CYP2D6 gene causing impaired metabolism of sparteine

The identification of a novel CYP2D6 allele from a healthy Caucasian poor metabolizer was achieved by using a previously described polymerase chain reaction/single-strand conformation polymorphism strategy. Among the four point mutations that this allele carries, a missense mutation in exon 1 (212 G → A or D6–H) seems to be responsible for the loss of CYP2D6 function. Although the mutation D6-H has a low prevalence in a randomly selected population of healthy Caucasians, its identification should further increase the phenotype prediction rate by genotyping. Received: 14 September 1995 / Revised: 22 November 1995     

Branching is coordinated with mitosis in growing hyphae of Aspergillus nidulans

Filamentous fungi like Aspergillus nidulans can effectively colonize their surroundings by the formation of new branches along the existing hyphae. While growth conditions, chemical perturbations, and mutations affecting branch formation have received great attention during the last decades, the mechanisms that regulates branching is still poorly understood. In this study, a possible relation between cell cycle progression and branching was studied by testing the effect of a nuclei distribution mutation, cell cycle inhibitors, and conditional cell cycle mutations in combination with tip-growth inhibitors and varying substrate concentrations on branch initiation. Formation of branches was blocked after inhibition of nuclear division, which was not caused by a reduced growth rate. In hyphae of a nuclei distribution mutant branching was severely reduced in anucleated hyphae whereas the number of branches per hyphal length was linearly correlated to the concentration of nuclei, in the nucleated hyphae. In wild type cells, branching intensity was increased when the tip extension was reduced, and reduced when growing on poor substrates. In these situations, the hyphal concentration of nuclei was maintained and it is suggested that branching is correlated to cell cycle progression in order to maintain a minimum required cytoplasmic volume per nucleus and to avoid the formation of anucleated hyphae in the absence of nuclear divisions. The presented results further suggest the hyphal diameter as a key point through which the hyphal element regulates its branching intensity in response to the surrounding substrate concentrations.     

Determination of the mutation rate of a retrovirus.

The mutation rate of Rous sarcoma virus (RSV) was measured. Progeny descended from a single virion were collected after one replication cycle, and seven regions of the genome were analyzed for mutations by denaturing-gradient gel electrophoresis. In all, 65,250 nucleotides were screened, yielding nine mutations, and the RSV mutation rate was calculated as 1.4 x 10(-4) mutations per nucleotide per replication cycle. These results indicate that RSV is an extremely mutable virus. We speculate that the mutation rate of a virus may correlate inversely with the effectiveness of vaccination against a given virus and suggest that prevention of retrovirus-mediated disease via vaccination may prove difficult.     

Some Statistical Improvements for Estimating Population Size and Mutation Rate from Segregating Sites in DNA Sequences

In population genetics, under a neutral Wright-Fisher model, the scaling parameter straight theta=4Nmu represents twice the average number of new mutants per generation. The effective population size is N and mu is the mutation rate per sequence per generation. Watterson proposed a consistent estimator of this parameter based on the number of segregating sites in a sample of nucleotide sequences. We study the distribution of the Watterson estimator. Enlarging the size of the sample, we asymptotically set a Central Limit Theorem for the Watterson estimator. This exhibits asymptotic normality with a slow rate of convergence. We then prove the asymptotic efficiency of this estimator. In the second part, we illustrate the slow rate of convergence found in the Central Limit Theorem. To this end, by studying the confidence intervals, we show that the asymptotic Gaussian distribution is not a good approximation for the Watterson estimator.     

Estimating Effective Population Size and Mutation Rate from Sequence Data Using Metropolis-Hastings Sampling

We present a new way to make a maximum likelihood estimate of the parameter 4N(e)μ (effective population size times mutation rate per site, or θ) based on a population sample of molecular sequences. We use a Metropolis-Hastings Markov chain Monte Carlo method to sample genealogies in proportion to the product of their likelihood with respect to the data and their prior probability with respect to a coalescent distribution. A specific value of θ must be chosen to generate the coalescent distribution, but the resulting trees can be used to evaluate the likelihood at other values of θ, generating a likelihood curve. This procedure concentrates sampling on those genealogies that contribute most of the likelihood, allowing estimation of meaningful likelihood curves based on relatively small samples. The method can potentially be extended to cases involving varying population size, recombination, and migration.     

How often are polymorphic restriction sites due to a single mutation?

An approximate expression is obtained for the probability that a restriction site, which is polymorphic in a random sample, is a site at which two or more mutations have occurred in the descent to the sample from the most recent common ancestor of the sample. The analysis is based on the assumption that the population from which the sample is obtained is at equilibrium under a selectively neutral Wright-Fisher model. Monte Carlo simulations show that the approximation is quite accurate. For commonly observed levels of genetic variation in humans and in natural populations of Drosophila, it is found that multiple mutations would occur at 5 to 10 percent of polymorphic restriction sites assuming that six-cutter enzymes are used on samples of size 50 to 100. Simulations are also used to investigate the bias and mean square error of four estimators of 4Nu, where N is the population size and u is the neutral mutation rate per nucleotide site. Two of the estimators are biased by approximately 20 percent when levels of variation are similar to those which have been observed in natural populations of Drosophila.     

A method to detect ras point mutations in small subpopulations of cells.

Biochemical assays for ras mutations are capable of detecting a mutant allele only if it is present in at least 5% of cells tested. Further, ras mutation assays which utilize the polymerase chain reaction (PCR) are unable to distinguish a ras mutation in a small population of cells from mutations resulting from Taq DNA polymerase base misincorporation. We used a standard restriction fragment length polymorphism assay of PCR-amplified c-Ki-ras to detect codon 12 mutations in tumor cells and found a cumulative error frequency for Taq DNA polymerase of one codon 12 mutation per 2 X 10(4) molecules of total amplification product. The Taq polymerase-induced mutations were found to be multiple base transitions and represented a constant proportion of the amplification product at each step of the PCR. The ability to detect the in vitro generated mutation was dependent on the number of thermal cycles and the sensitivity of the detection assay. With these considerations in mind, we developed a two-step RFLP assay in which the thermal cycle number was kept low and molecules containing mutations at codon 12 were selectively amplified in the second step. We were able to detect a ras mutation occurring in 1 per 1000 cells (a two log improvement over standard RFLP methods) without detecting mutations resulting from Taq DNA polymerase infidelity.     

Upper limit of the rate and per generation effects of deleterious genomic mutations

Unbiased or upper limit estimates of the rate (U) of genomic mutations to mildly deleterious alleles are crucial in genetic and conservation studies and in human health care. However, only a few estimates of the lower bounds of U are available. We present a fairly robust estimation that yields an upper limit of U and a nearly unbiased estimate of the per generation fitness decline due to new deleterious mutations. We applied the approach to three species of the freshwater microcrustacean Daphnia and revealed that the upper limit of U for egg survivorship is 0.73 (SD = 0.30) in 14 D. pulicaria populations. For the first four clutches, per generation decline in fecundity due to deleterious mutations ranged from 2.2% to 7.8% in 20 D. pulex populations and from 1.1% to 5.1% in 8 D. obtusa populations. These results indicate the mutation pressure is high in natural Daphnia populations. The approach investigated here provides a potential way to quickly and conveniently characterize U and per generation effects of deleterious genomic mutations on fitness or its important components such as fecundity.     

Rate and pattern of mutation at microsatellite loci in maize

Microsatellites are important tools for plant breeding, genetics, and evolution, but few studies have analyzed their mutation pattern in plants. In this study, we estimated the mutation rate for 142 microsatellite loci in maize (Zea mays subsp. mays) in two different experiments of mutation accumulation. The mutation rate per generation was estimated to be 7.7 x 10(-4) for microsatellites with dinucleotide repeat motifs, with a 95% confidence interval from 5.2 x 10(-4) to 1.1 x 10(-3). For microsatellites with repeat motifs of more than 2 bp in length, no mutations were detected; so we could only estimate the upper 95% confidence limit of 5.1 x 10(-5) for the mutation rate. For dinucleotide repeat microsatellites, we also determined that the variance of change in the number of repeats (sigma(m)2) is 3.2. We sequenced 55 of the 73 observed mutations, and all mutations proved to be changes in the number of repeats in the microsatellite or in mononucleotide tracts flanking the microsatellite. There is a higher probability to mutate to an allele of larger size. There is heterogeneity in the mutation rate among dinucleotide microsatellites and a positive correlation between the number of repeats in the progenitor allele and the mutation rate. The microsatellite-based estimate of the effective population size of maize is more than an order of magnitude less than previously reported values based on nucleotide sequence variation.     

Spontaneous and Ethyl Methanesulfonate-Induced Mutations Controlling Viability in DROSOPHILA MELANOGASTER. II. Homozygous Effect of Polygenic Mutations

Polygenic mutations affecting viability were accumulated on the second chromosome of Drosophila melanogaster by treating flies with EMS in successive generations. The treated chromosomes were later made homozygous and tested for their effects on viability by comparison of the frequency of such homozygotes with that of other genotypes in the same culture. The treated wild-type chromosomes were kept heterozygous in Pm/+ males by mating individual males in successive generations to Cy/Pm females. The number of generations of accumulation was 1 to 30 generations, depending on the concentration of EMS. A similar experiment for spontaneous polygenic mutations was also conducted by accumulating mutations for 40 generations. The lower limit of the spontaneous mutation rate of viability polygenes is estimated to be 0.06 per second chromosome per generation, which is about 12 times as high as the spontaneous recessive lethal mutation rate, 0.005. EMS-induced polygenic mutations increase linearly with the number of treated generations and with the concentration of EMS. The minimum mutation rate of viability polygenes is about 0.017 per 10(-4)m, which is only slightly larger than the lethal rate of 0.013 per 10(-4) m. The maximum estimate of the viability reduction of a single mutant is about 6 to 10 percent of the normal viability. The data are consistent with a constant average effect per mutant at all concentrations, but this is about three times as high as that for spontaneous mutants. It is obvious that one can obtain only a lower limit for the mutation rate, since some mutants may have effects so near to zero that they cannot be detected. The possibility of measuring something other than the lower limit is discussed. The ratio of the load due to detrimental mutants to that caused by lethals, the D/L ratio, is about 0.2 to 0.3 for EMS-induced mutants, as compared to about 0.5 for spontaneous mutants. This is to be expected if EMS treatment produces a large fraction of small deletions and other chromosome rearrangements which are more likely to be lethal.     

Secondary mutations M36I and A71V in the human immunodeficiency virus type 1 protease can provide an advantage for the emergence of the primary mutation D30N

Development of resistance mutations in enzymatic targets of human immunodeficiency virus 1 (HIV-1) hampers the ability to provide adequate therapy. Of special interest is the effect mutations outside the active site of HIV-1 protease have on inhibitor binding and virus viability. We engineered protease mutants containing the active site mutation D30N alone and with the nonactive site polymorphisms M36I and/or A71V. We determined the K(i) values for the inhibitors nelfinavir, ritonavir, indinavir, KNI272, and AG1776 as well as the catalytic efficiency of the mutants. Single and double mutation combinations exhibited a decrease in catalytic efficiency, while the triple mutant displayed catalytic efficiency greater than that of the wild type. Variants containing M36I or A71V alone did not display a significant change in binding affinities to the inhibitors tested. The variant containing mutation D30N displayed a 2-6-fold increase in K(i) for all inhibitors tested, with nelfinavir showing the greatest increase. The double mutants containing a combination of mutations D30N, M36I, and A71V displayed -0.5-fold to +6-fold changes in the K(i) of all inhibitors tested, with ritonavir and nelfinavir most affected. Only the triple mutant showed a significant increase (>10-fold) in K(i) for inhibitor nelfinavir, ritonavir, or AG-1776 displaying 22-, 19-, or 15-fold increases, respectively. Our study shows that the M36I and A71V mutations provide a greater level of inhibitor cross-resistance combined with active site mutation D30N. M36I and A71V, when present as natural polymorphisms, could aid the virus in developing active site mutations to escape inhibitor binding while maintaining catalytic efficiency.     

Estimate of the mutation rate per nucleotide in humans

Many previous estimates of the mutation rate in humans have relied on screens of visible mutants. We investigated the rate and pattern of mutations at the nucleotide level by comparing pseudogenes in humans and chimpanzees to (i) provide an estimate of the average mutation rate per nucleotide, (ii) assess heterogeneity of mutation rate at different sites and for different types of mutations, (iii) test the hypothesis that the X chromosome has a lower mutation rate than autosomes, and (iv) estimate the deleterious mutation rate. Eighteen processed pseudogenes were sequenced, including 12 on autosomes and 6 on the X chromosome. The average mutation rate was estimated to be approximately 2.5 x 10(-8) mutations per nucleotide site or 175 mutations per diploid genome per generation. Rates of mutation for both transitions and transversions at CpG dinucleotides are one order of magnitude higher than mutation rates at other sites. Single nucleotide substitutions are 10 times more frequent than length mutations. Comparison of rates of evolution for X-linked and autosomal pseudogenes suggests that the male mutation rate is 4 times the female mutation rate, but provides no evidence for a reduction in mutation rate that is specific to the X chromosome. Using conservative calculations of the proportion of the genome subject to purifying selection, we estimate that the genomic deleterious mutation rate (U) is at least 3. This high rate is difficult to reconcile with multiplicative fitness effects of individual mutations and suggests that synergistic epistasis among harmful mutations may be common.