Simple method for analyzing the pattern of DNA polymorphism and its application to SNP data of human

In order to analyze the pattern of DNA polymorphism in detail, we have developed a simple method using a new statistic theta(i) which estimates 4Nmu from the number of segregating sites whose allelic nucleotide frequency is i/n among n DNA sequences, where N is the effective population size and mu is the mutation rate per generation per nucleotide site. Under the assumption that mutations are selectively neutral and a population size is constant, the expectation of theta(i) is equal to that of theta, which estimates 4Nmu from the number of segregating sites, so that the distribution of theta(i) is flat. Therefore, the departure of the distribution of theta(i) from the horizontal line, which represents the value of theta, reflects change in population size and natural selection. Results of the coalescent simulation show that the distributions of theta(i) in the populations which experienced expansion and reduction are U-shaped and upside-down U-shaped, respectively. And the distributions of theta(i) in some populations that experienced bottleneck are W-shaped. Furthermore, we have applied this method to the SNP data in the International HapMap Project. Results of data analyses show that the distributions of theta(i) in the CEU (European), CHB and JPT (Asian) populations are different from that in the YRI population (African). From these results of data analyses in nuclear DNA and the pattern of polymorphism in human mitochondrial DNA already known, we infer that the CEU, CHB and JPT populations experienced the bottleneck.     

Population genetic inference using a fixed number of segregating sites: a reassessment

Coalescent theory is commonly used to perform population genetic inference at the nucleotide level. Here, we examine the procedure that fixes the number of segregating sites (henceforth the FS procedure). In this approach a fixed number of segregating sites (S) are placed on a coalescent tree (independently of the total and internode lengths of the tree). Thus, although widely used, the FS procedure does not strictly follow the assumptions of coalescent theory and must be considered an approximation of (i) the standard procedure that uses a fixed population mutation parameter theta, and (ii) procedures that condition on the number of segregating sites. We study the differences in the false positive rate for nine statistics by comparing the FS procedure with the procedures (i) and (ii), using several evolutionary models with single-locus and multilocus data. Our results indicate that for single-locus data the FS procedure is accurate for the equilibrium neutral model, but problems arise under the alternative models studied; furthermore, for multilocus data, the FS procedure becomes inaccurate even for the standard neutral model. Therefore, we recommend a procedure that fixes the theta value (or alternatively, procedures that condition on S and take into account the uncertainty of theta) for analysing evolutionary models with multilocus data. With single-locus data, the FS procedure should not be employed for models other than the standard neutral model.     

The Effect of Change in Population Size on DNA Polymorphism

The expected number of segregating sites and the expectation of the average number of nucleotide differences among DNA sequences randomly sampled from a population, which is not in equilibrium, have been developed. The results obtained indicate that, in the case where the population size has changed drastically, the number of segregating sites is influenced by the size of the current population more strongly than is the average number of nucleotide differences, while the average number of nucleotide differences is affected by the size of the original population more severely than is the number of segregating sites. The results also indicate that the average number of nucleotide differences is affected by a population bottleneck more strongly than is the number of segregating sites.     

Expected coalescence times and segregating sites in a model of glacial cycles

The climatic fluctuations of the Quaternary have influenced the distribution of numerous plant and animal species. Several species suffer population reduction and fragmentation, becoming restricted to refugia during glacial periods and expanding again during interglacials. The reduction in population size may reduce the effective population size, mean coalescence time and genetic variation, whereas an increased subdivision may have the opposite effect. To investigate these two opposing forces, we proposed a model in which a panmictic and a structured phase alternate, corresponding to interglacial and glacial periods. From this model, we derived an expression for the expected coalescence time and number of segregating sites for a pair of genes. We observed that increasing the number of demes or the duration of the structured phases causes an increase in coalescence time and expected levels of genetic variation. We compared numerical results with the ones expected for a panmictic population of constant size, and showed that the mean number of segregating sites can be greater in our model even when population size is much smaller in the structured phases. This points to the importance of population structure in the history of species subject to climatic fluctuations, and helps explain the long gene genealogies observed in several organisms.     

Gene genealogies in a metapopulation

A simple genealogical process is found for samples from a metapopulation, which is a population that is subdivided into a large number of demes, each of which is subject to extinction and recolonization and receives migrants from other demes. As in the migration-only models studied previously, the genealogy of any sample includes two phases: a brief sample-size adjustment followed by a coalescent process that dominates the history. This result will hold for metapopulations that are composed of a large number of demes. It is robust to the details of population structure, as long as the number of possible source demes of migrants and colonists for each deme is large. Analytic predictions about levels of genetic variation are possible, and results for average numbers of pairwise differences within and between demes are given. Further analysis of the expected number of segregating sites in a sample from a single deme illustrates some previously known differences between migration and extinction/recolonization. The ancestral process is also amenable to computer simulation. Simulation results show that migration and extinction/recolonization have very different effects on the site-frequency distribution in a sample from a single deme. Migration can cause a U-shaped site-frequency distribution, which is qualitatively similar to the pattern reported recently for positive selection. Extinction and recolonization, in contrast, can produce a mode in the site-frequency distribution at intermediate frequencies, even in a sample from a single deme.     

DNA Polymorphism in a Subdivided Population: The Expected Number of Segregating Sites in the Two-Subpopulation Model

Using the two subpopulation model, the expected numbers of segregating sites in a number of DNA sequences randomly sampled from a subdivided population were examined for several types of population subdivisions. It is shown that, in the case where the pattern of migration is symmetrical such as the finite island model, the expected number of segregating sites is independent of the migration rate when two or three DNA sequences are randomly sampled from the same subpopulation, but depends on the migration rate when more than three DNA sequences are sampled. It is also shown that the population subdivision can increase the amount of DNA polymorphism even in a subpopulation in some cases.     

Evidence for Balanced Linkage of X Chromosome Polygenes in a Natural Population of Drosophila

Extensive levels of polygenic variation can be maintained in a population without creating a severe segregational load. One way to account for this is that the alleles are arranged on a chromosome so that different regions balance each other phenotypically. To test whether this occurs in a natural population, we isolated ten Drosophila melanogaster X chromosomes and mapped regions of polygenic activity affecting sternopleural bristle number. The chromosomes fell into a small number of groups based upon the similarity of their distributions of polygenic activity. The results are consistent with a model in which a large proportion of the variation can be attributed to a small number of segregating chromosome regions and in which the chromosomes show internal balance.     

Studying models of balancing selection using phase-type theory

Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyze models with BLS acting on a biallelic locus: an equilibrium model with long-term BLS, a model with long-term BLS and recent changes in population size, and a model of recent BLS. Using phase-type theory, a mathematical tool for analyzing continuous time Markov chains with an absorbing state, we examine how BLS affects polymorphism patterns in linked neutral regions, as summarized by nucleotide diversity, the expected number of segregating sites, the site frequency spectrum, and the level of linkage disequilibrium (LD). Long-term BLS affects polymorphism patterns in a relatively small genomic neighborhood, and such selection targets are easier to detect when the equilibrium frequencies of the selected variants are close to 50%, or when there has been a population size reduction. For a new mutation subject to BLS, its initial increase in frequency in the population causes linked neutral regions to have reduced diversity, an excess of both high and low frequency derived variants, and elevated LD with the selected locus. These patterns are similar to those produced by selective sweeps, but the effects of recent BLS are weaker. Nonetheless, compared to selective sweeps, nonequilibrium polymorphism and LD patterns persist for a much longer period under recent BLS, which may increase the chance of detecting such selection targets. An R package for analyzing these models, among others (e.g., isolation with migration), is available.     

The age of nonsynonymous and synonymous mutations in animal mtDNA and implications for the mildly deleterious theory.

McDonald/Kreitman tests performed on animal mtDNA consistently reveal significant deviations from strict neutrality in the direction of an excess number of polymorphic nonsynonymous sites, which is consistent with purifying selection acting on nonsynonymous sites. We show that under models of recurrent neutral and deleterious mutations, the mean age of segregating neutral mutations is greater than the mean age of segregating selected mutations, even in the absence of recombination. We develop a test of the hypothesis that the mean age of segregating synonymous mutations equals the mean age of segregating nonsynonymous mutations in a sample of DNA sequences. The power of this age-of-mutation test and the power of the McDonald/Kreitman test are explored by computer simulations. We apply the new test to 25 previously published mitochondrial data sets and find weak evidence for selection against nonsynonymous mutations.     

The number of segregating sites in expanding human populations, with implications for estimates of demographic parameters

The frequency distribution of pairwise differences between sequences of mtDNA has recently been used to estimate the size of human populations before and after a hypothetical episode of rapid population growth and the time at which the population grew. To test the internal consistency of this method, we used three different sets of human mtDNA data and the corresponding demographic parameters estimated from the distribution of pairwise differences to determine by simulation the expected number of segregating sites, S, and its empirical distribution. The results indicate that the observed values of S are significantly lower than expected in two of three cases under the assumption of the infinite-sites model. Further simulations in which mutations were allowed to occur more than once at the same site and in which there was variation in mutation rate among sites show that the expected number of segregating sites can be much lower than under the infinite-site assumption. Nevertheless, the observed value of S is still significantly different from the value expected under the expansion hypothesis in two of three cases.      

Sufficiency of the number of segregating sites in the limit under finite-sites mutation

We show that the number of segregating sites is a sufficient statistic for the scaled mutation parameter (θ) in the limit as the number of sites tends to infinity and there is free recombination between sites. We assume that the mutation parameter at each site tends to zero such than the total mutation parameter (θ) is constant in the limit. Our results show that Watterson’s estimator is the maximum likelihood estimator in this case, but that it estimates a composite parameter which is different for different mutation models. Some of our results hold when recombination is limited, because Watterson’s estimator is an unbiased, method-of-moments estimator regardless of the recombination rate. The quantity it estimates depends on the details of how mutations occur at each site.     

Statistical Properties of a DNA Sample under the Finite-Sites Model

Statistical properties of a DNA sample from a random-mating population of constant size are studied under the finite-sites model. It is assumed that there is no migration and no recombination occurs within the locus. A Markov process model is used for nucleotide substitution, allowing for multiple substitutions at a single site. The evolutionary rates among sites are treated as either constant or variable. The general likelihood calculation using numerical integration involves intensive computation and is feasible for three or four sequences only; it may be used for validating approximate algorithms. Methods are developed to approximate the probability distribution of the number of segregating sites in a random sample of n sequences, with either constant or variable substitution rates across sites. Calculations using parameter estimates obtained for human D-loop mitochondrial DNAs show that among-site rate variation has a major effect on the distribution of the number of segregating sites; the distribution under the finite-sites model with variable rates among sites is quite different from that under the infinite-sites model.     

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.     

Homozygosity in a population of variable size and mutation rate

A formula is obtained for the probability that two genes at a single locus, sampled at random from a population at time t, are of particular types. The model assumed is a diffusion approximation to a neutral Wright-Fisher model in which mutation is not necessarily symmetric and the population size is a function of time. It is shown that for symmetric mutation in a population undergoing a step-function type bottleneck, homozygosity increases with decreasing population size. A formula is given for the distribution of the number of segregating sites occurring in two randomly sampled sequences of completely linked sites, with general mutation at a site and identical mutation structure between sites.We give similar results for a population of fixed size but for which the mutation rate is a function of time, and not necessarily symmetric. We confirm the intuitively clear effect that increasing the mutation rate decreases homozygosity.     

The Effect of Deleterious Mutations on Neutral Molecular Variation

Selection against deleterious alleles maintained by mutation may cause a reduction in the amount of genetic variability at linked neutral sites. This is because a new neutral variant can only remain in a large population for a long period of time if it is maintained in gametes that are free of deleterious alleles, and hence are not destined for rapid elimination from the population by selection. Approximate formulas are derived for the reduction below classical neutral values resulting from such background selection against deleterious mutations, for the mean times to fixation and loss of new mutations, nucleotide site diversity, and number of segregating sites. These formulas apply to random-mating populations with no genetic recombination, and to populations reproducing exclusively asexually or by self-fertilization. For a given selection regime and mating system, the reduction is an exponential function of the total mutation rate to deleterious mutations for the section of the genome involved. Simulations show that the effect decreases rapidly with increasing recombination frequency or rate of outcrossing. The mean time to loss of new neutral mutations and the total number of segregating neutral sites are less sensitive to background selection than the other statistics, unless the population size is of the order of a hundred thousand or more. The stationary distribution of allele frequencies at the neutral sites is correspondingly skewed in favor of rare alleles, compared with the classical neutral result. Observed reductions in molecular variation in low recombination genomic regions of sufficiently large size, for instance in the centromere-proximal regions of Drosophila autosomes or in highly selfing plant populations, may be partly due to background selection against deleterious mutations.     

The evolution of isochores: evidence from SNP frequency distributions

The large-scale systematic variation in nucleotide composition along mammalian and avian genomes has been a focus of the debate between neutralist and selectionist views of molecular evolution. Here we test whether the compositional variation is due to mutation bias using two new tests, which do not assume compositional equilibrium. In the first test we assume a standard population genetics model, but in the second we make no assumptions about the underlying population genetics. We apply the tests to single-nucleotide polymorphism data from noncoding regions of the human genome. Both models of neutral mutation bias fit the frequency distributions of SNPs segregating in low- and medium-GC-content regions of the genome adequately, although both suggest compositional nonequilibrium. However, neither model fits the frequency distribution of SNPs from the high-GC-content regions. In contrast, a simple population genetics model that incorporates selection or biased gene conversion cannot be rejected. The results suggest that mutation biases are not solely responsible for the compositional biases found in noncoding regions.     

Efficient algorithms for counting and reporting segregating sites in genomic sequences.

The number of segregating sites provides an indicator of the degree of DNA sequence variation that is present in a sample, and has been of great interest to the biological, pharmaceutical and medical professions. In this paper, we first provide linear- and expected-sublinear-time algorithms for finding all the segregating sites of a given set of DNA sequences. We also describe a data structure for tracking segregating sites in a set of sequences, such that every time the set is updated with the insertion of a new sequence or removal of an existing one, the segregating sites are updated accordingly without the need to re-scan the entire set of sequences.     

Second-order moments of segregating sites under variable population size

The identification of genomic regions that have been exposed to positive selection is a major challenge in population genetics. Since selective sweeps are expected to occur during environmental changes or when populations are colonizing a new habitat, statistical tests constructed on the assumption of constant population size are biased by the co-occurrence of population size changes and selection. To delimit this problem and gain better insights into demographic factors, theoretical results regarding the second-order moments of segregating sites, such as the variance of segregating sites, have been derived. Driven by emerging genomewide surveys, which allow the estimation of demographic parameters, a generalized version of Tajima's D has been derived that takes into account a previously estimated demographic scenario to test single loci for traces of selection against the null hypothesis of neutral evolution under variable population size.     

The Effects of Background and Interference Selection on Patterns of Genetic Variation in Subdivided Populations

It is well known that most new mutations that affect fitness exert deleterious effects and that natural populations are often composed of subpopulations (demes) connected by gene flow. To gain a better understanding of the joint effects of purifying selection and population structure, we focus on a scenario where an ancestral population splits into multiple demes and study neutral diversity patterns in regions linked to selected sites. In the background selection regime of strong selection, we first derive analytic equations for pairwise coalescent times and F_ST as a function of time after the ancestral population splits into two demes and then construct a flexible coalescent simulator that can generate samples under complex models such as those involving multiple demes or nonconservative migration. We have carried out extensive forward simulations to show that the new methods can accurately predict diversity patterns both in the nonequilibrium phase following the split of the ancestral population and in the equilibrium between mutation, migration, drift, and selection. In the interference selection regime of many tightly linked selected sites, forward simulations provide evidence that neutral diversity patterns obtained from both the nonequilibrium and equilibrium phases may be virtually indistinguishable for models that have identical variance in fitness, but are nonetheless different with respect to the number of selected sites and the strength of purifying selection. This equivalence in neutral diversity patterns suggests that data collected from subdivided populations may have limited power for differentiating among the selective pressures to which closely linked selected sites are subject.