Implication of nonlinear kinetics on risk estimation in carcinogenesis.

Efforts in estimating carcinogenic risk in humans from long-term exposure to chemical carcinogens have centered on the problem of low-dose extrapolation. For chemicals with metabolites that interact with DNA, it may be more meaningful to relate tumor response to the concentration of the DNA adducts in the target organ rather than to the applied dose. Many data suggest that the relation between tumor response and concentration of DNA adducts in the target organ may be linear. This implies that the nonlinearities of the dose-response curve for tumor induction may be due to the kinetic processes involved in the formation of carcinogen metabolite--DNA adducts. Of particular importance is the possibility that the kinetic processes may show a nonlinear "hockey-stick" like behavior which results from saturation of detoxification or DNA repair processes. The mathematical models typically used for low-dose extrapolation are shown potentially to overestimate risk by several orders of magnitude when nonlinear kinetics are present.

The coalescent process in models with selection and recombination.

The statistical properties of the process describing the genealogical history of a random sample of genes at a selectively neutral locus which is linked to a locus at which natural selection operates are investigated. It is found that the equations describing this process are simple modifications of the equations describing the process assuming that the two loci are completely linked. Thus, the statistical properties of the genealogical process for a random sample at a neutral locus linked to a locus with selection follow from the results obtained for the selected locus. Sequence data from the alcohol dehydrogenase (Adh) region of Drosophila melanogaster are examined and compared to predictions based on the theory. It is found that the spatial distribution of nucleotide differences between Fast and Slow alleles of Adh is very similar to the spatial distribution predicted if balancing selection operates to maintain the allozyme variation at the Adh locus. The spatial distribution of nucleotide differences between different Slow alleles of Adh do not match the predictions of this simple model very well.

Statistical properties of the number of recombination events in the history of a sample of DNA sequences.

Some statistical properties of samples of DNA sequences are studied under an infinite-site neutral model with recombination. The two quantities of interest are R, the number of recombination events in the history of a sample of sequences, and RM, the number of recombination events that can be parsimoniously inferred from a sample of sequences. Formulas are derived for the mean and variance of R. In contrast to R, RM can be determined from the sample. Since no formulas are known for the mean and variance of RM, they are estimated with Monte Carlo simulations. It is found that RM is often much less than R, therefore, the number of recombination events may be greatly under-estimated in a parsimonious reconstruction of the history of a sample. The statistic RM can be used to estimate the product of the recombination rate and the population size or, if the recombination rate is known, to estimate the population size. To illustrate this, DNA sequences from the Adh region of Drosophila melanogaster are used to estimate the effective population size of this species.

On the divergence of alleles in nested subsamples from finite populations.

Within-population variation at the DNA level will rarely be studied by sequencing of loci of randomly chosen individuals. Instead, individuals will usually be chosen for sequencing based on some knowledge of their genotype. Data collected in this way require new sampling theory. Motivated by these observations, we have examined the sampling properties of a finite population model with two mutation processes and with no selection or recombination. One mutation process generates new alleles according to an infinite-alleles model, and the other generates polymorphisms at sites according to an infinite-sites model. A sample of n genes is considered. The stationary distribution of the number of segregating sites in a subsample from one of the allelic classes in the sample conditional on the allelic configuration of the sample is studied. A recursive scheme is developed to compute the moments of this distribution, and it is shown that the distribution is functionally independent of the number of additional alleles in the sample and their respective frequencies in the sample. For the case in which the sample contains only two alleles, the distribution of the number of segregating sites in a subsample containing both alleles conditional on the sample frequencies of the alleles is studied. The results are applied to the analysis of DNA sequences of two alleles found at the Adh locus of Drosophila melanogaster. No significant departure from the neutral model is detected.

Deleterious background selection with recombination.

An analytic expression for the expected nucleotide diversity is obtained for a neutral locus in a region with deleterious mutation and recombination. Our analytic results are used to predict levels of variation for the entire third chromosome of Drosophila melanogaster. The predictions are consistent with the low levels of variation that have been observed at loci near the centromeres of the third chromosome of D. melanogaster. However, the low levels of variation observed near the tips of this chromosome are not predicted using currently available estimates of the deleterious mutation rate and of selection coefficients. If considerably smaller selection coefficients are assumed, the low observed levels of variation at the tips of the third chromosome are consistent with the background selection model.

Transposable Elements in Mendelian Populations. III. Statistical Results.

Methods are proposed for estimating the parameters in the frequency spectrum derived from the model of Langley, Brookfield and Kaplan for the evolution of a transposable element in a finite Mendelian population. Statistical properties of these estimates are studied, and the results are supported with simulation data. The effects on the estimates caused by possible insensitivities of the experimental technique are also discussed. To illustrate the proposed methodology, the data of Montgomery and Langley are analyzed.

The coalescent process in models with selection.

Statistical properties of the process describing the genealogical history of a random sample of genes are obtained for a class of population genetics models with selection. For models with selection, in contrast to models without selection, the distribution of this process, the coalescent process, depends on the distribution of the frequencies of alleles in the ancestral generations. If the ancestral frequency process can be approximated by a diffusion, then the mean and the variance of the number of segregating sites due to selectively neutral mutations in random samples can be numerically calculated. The calculations are greatly simplified if the frequencies of the alleles are tightly regulated. If the mutation rates between alleles maintained by balancing selection are low, then the number of selectively neutral segregating sites in a random sample of genes is expected to substantially exceed the number predicted under a neutral model.

The "hitchhiking effect" revisited.

The number of selectively neutral polymorphic sites in a random sample of genes can be affected by ancestral selectively favored substitutions at linked loci. The degree to which this happens depends on when in the history of the sample the selected substitutions happen, the strength of selection and the amount of crossing over between the sampled locus and the loci at which the selected substitutions occur. This phenomenon is commonly called hitchhiking. Using the coalescent process for a random sample of genes from a selectively neutral locus that is linked to a locus at which selection is taking place, a stochastic, finite population model is developed that describes the steady state effect of hitchhiking on the distribution of the number of selectively neutral polymorphic sites in a random sample. A prediction of the model is that, in regions of low crossing over, strongly selected substitutions in the history of the sample can substantially reduce the number of polymorphic sites in a random sample of genes from that expected under a neutral model.

The hitchhiking effect on the site frequency spectrum of DNA polymorphisms.

The level of DNA sequence variation is reduced in regions of the Drosophila melanogaster genome where the rate of crossing over per physical distance is also reduced. This observation has been interpreted as support for the simple model of genetic hitchhiking, in which directional selection on rare variants, e.g., newly arising advantageous mutants, sweeps linked neutral alleles to fixation, thus eliminating polymorphisms near the selected site. However, the frequency spectra of segregating sites of several loci from some populations exhibiting reduced levels of nucleotide diversity and reduced numbers of segregating sites did not appear different from what would be expected under a neutral equilibrium model. Specifically, a skew toward an excess of rare sites was not observed in these samples, as measured by Tajima's D. Because this skew was predicted by a simple hitchhiking model, yet it had never been expressed quantitatively and compared directly to DNA polymorphism data, this paper investigates the hitchhiking effect on the site frequency spectrum, as measured by Tajima's D and several other statistics, using a computer simulation model based on the coalescent process and recurrent hitchhiking events. The results presented here demonstrate that under the simple hitchhiking model (1) the expected value of Tajima's D is large and negative (indicating a skew toward rare variants), (2) that Tajima's test has reasonable power to detect a skew in the frequency spectrum for parameters comparable to those from actual data sets, and (3) that the Tajima's Ds observed in several data sets are very unlikely to have been the result of simple hitchhiking. Consequently, the simple hitchhiking model is not a sufficient explanation for the DNA polymorphism at those loci exhibiting a decreased number of segregating sites yet not exhibiting a skew in the frequency spectrum.

On the divergence of members of a transposable element family.

Statistical properties of the amount of divergence of members of a transposable element family are studied. The analysis is based on the model proposed by Langley et al. describing the evolution of a family of selectively neutral transposable elements in a finite haploid population of size 2N. By considering the time back to the most recent common ancestor of two copies, both the probability of identity and the moments of the number of sites that differ between two sampled copies are obtained. Our analytic results are consistent with the numerical results of Ohta for a similar model. The effects of gene conversion are also examined. In agreement with Slatkin, we find that gene conversion has a minimal effect on the probability of identity providing that the rate of deletion is sufficiently large.

Variability of safe dose estimates when using complicated models of the carcinogenic process. A case study: methylene chloride.

Advances in understanding carcinogenesis have led to the development of mathematical models that have biologically interpretable parameters. These models utilize more of the available scientific data than the empirical models routinely employed for quantifying carcinogenic risk. They also require consideration of sources of uncertainty in risk estimates that were previously ignored, such as animal-to-animal variability of physiological and pharmacological constants. A numerical technique is proposed for studying the consequences of incorporating the intrapopulation variability of biologically interpretable parameters into the risk assessment process. To demonstrate the technique, the variability of safe dose estimates for exposure to methylene chloride is considered. The results suggest that intrapopulation variability of the model parameters can increase the variability of safe dose estimates an appreciable amount.

On the divergence of genes in multigene families.

Statistical properties of the amount of divergence of genes in multigene families are studied. The model considered is an infinite-site neutral model with unbiased intrachromosomal conversion, unbiased interchromosomal conversion, and recombination. By considering the time back to the most recent common ancestor of two genes, both the probability of identity and the moments of S, the number of sites that differ between two sampled genes, are obtained. We find that if recombination rates are large or conversion is always interchromosomal, then the expectation of S is 4N mu n where N is the population size, mu is the rate of mutation per generation per gene and n is the number of genes in the gene family, as the conversion rates approach zero, the moments of divergence do not approach the moments of divergence with conversion rates equal to zero, and it is possible for a decrease in the rate of intrachromosomal conversion to result in a higher probability of identity, but a greater mean divergence of the two genes.

Power calculations for a general class of tests of linkage and association that use nuclear families with affected and unaffected sibs.

Family-based tests of association are now often used when trying to fine-map a disease susceptibility locus. Recently, several tests of linkage and association have been proposed that use nuclear families with multiple affected and unaffected sibs rather than just case-parent triads. In this paper we propose a test that generalizes these previous tests. Formulae are derived to calculate the power of the test for a randomly mating population. These power calculations are used to determine conditions under which it is advantageous to include unaffected sibs in the analysis.

Removing the sampling restrictions from family-based tests of association for a quantitative-trait locus.

One strategy for localization of a quantitative-trait locus (QTL) is to test whether the distribution of a quantitative trait depends on the number of copies of a specific genetic-marker allele that an individual possesses. This approach tests for association between alleles at the marker and the QTL, and it assumes that association is a consequence of the marker being physically close to the QTL. However, problems can occur when data are not from a homogeneous population, since associations can arise irrespective of a genetic marker being in physical proximity to the QTL-that is, no information is gained regarding localization. Methods to address this problem have recently been proposed. These proposed methods use family data for indirect stratification of a population, thereby removing the effect of associations that are due to unknown population substructure. They are, however, restricted in terms of the number of children per family that can be used in the analysis. Here we introduce tests that can be used on family data with parent and child genotypes, with child genotypes only, or with a combination of these types of families, without size restrictions. Furthermore, equations that allow one to determine the sample size needed to achieve desired power are derived. By means of simulation, we demonstrate that the existing tests have an elevated false-positive rate when the size restrictions are not followed and that a good deal of information is lost as a result of adherence to the size restrictions. Finally, we introduce permutation procedures that are recommended for small samples but that can also be used for extensions of the tests to multiallelic markers and to the simultaneous use of more than one marker.

A Monte Carlo procedure for two-stage tests with correlated data.

One strategy for mapping disease loci using marker-disease associations is to test for association with case-control samples and follow up a positive result with a family-based test. Using a family-based test in the second stage can help provide protection against false-positive results that can result from use of inappropriate controls and provides assurance that association identified in the first stage is occurring between linked loci. It is crucial for this two-stage strategy that the first stage be as powerful as possible to detect association since only positive results are tested in the second stage. In certain situations, the power of the first-stage test can be increased by combining the case-control and family data. However, this introduces correlation between the first- and second-stage tests, and treating them as independent tests causes a bias. Here we propose a Monte Carlo method that accounts for the correlation and provides the correct significance level for the second-stage test. We also discuss the use of a two-stage procedure when doing a genome scan for the data presented in the Genetic Analysis Workshop 9 study.

The coalescent process and background selection.

Some statistical properties of gene trees are described for a model with background deleterious mutations. It is argued that the history of a small sample of genes under this model is a continuous time Markov chain that quickly reaches stationarity. This observation leads to simple expressions for the expected nucleotide diversity and suggests that the frequency spectrum in small samples should be approximately the same as under a strict neutral model. The results concerning expected nucleotide diversity are compared with observed variation on the third chromosome of Drosophila melanogaster.

Are moment bounds on the recombination fraction between a marker and a disease locus too good to be true? Allelic association mapping revisited for simple genetic diseases in the Finnish population.

In the past several years, allelic association has helped map a number of rare genetic diseases in the human genome. A commonly used upper bound on the recombination fraction between the disease gene and an associated marker is known to be biased downward, so there is the possibility that an investigator could be misled. This upper bound is based on a moment equation that can be derived within the context of a Poisson branching process, so its performance can be compared with a recently proposed likelihood bound. We show that the confidence level of the moment upper bound is much lower than expected, while the confidence level of the likelihood bound is in line with expectation. The effects of mutation at either the marker or disease locus on the upper bounds are also investigated. Results indicate that mutation is not an important force for typical mutation rates, unless the recombination fraction between the marker and disease locus is very small or the disease allele is very rare in the general population. Finally, the impact of sample size on the likelihood bound is investigated. The results are illustrated with data on 10 simple genetic diseased in the Finnish population.