Effect of epistasis and linkage on fixation probability in three-locus models: an ancestral recombination-selection graph approach

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination-Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained. The coefficients of the expansion are expressed in terms of the initial state of the population and the epistatic interactions among the selected loci. Under the assumption of weak selection at tightly linked loci, the sign of the leading term, which depends on the signs of epistasis and initial linkage disequilibrium, determines whether an increase in recombination rates increases the chance of ultimate fixation of the new mutant. If mutants are advantageous, this is the case when epistasis is positive or null and the initial linkage disequilibrium is negative, which is an expected state in a finite population under directional selection. Moreover, this is also the case for a neutral mutant modifier coding for higher recombination rates if the same conditions hold at the selected loci. Under the same conditions, deleterious mutants are disfavored for ultimate fixation and neutral modifiers for higher recombination rates still favored. The recombination rates between the modifier locus and the selected loci do not come into play in the leading terms of the approximation for the fixation probability, but they do in higher-order terms.     

Neutral and Non-Neutral Evolution of Drosophila Mitochondrial DNA

To test hypotheses of neutral evolution of mitochondrial DNA (mtDNA), nucleotide sequences were determined for 1515 base pairs of the NADH dehydrogenase subunit 5 (ND5) gene in the mitochondrial DNA of 29 lines of Drosophila melanogaster and 9 lines of its sibling species Drosophila simulans. In contrast to the patterns for nuclear genes, where D. melanogaster generally exhibits much less nucleotide polymorphism, the number of segregating sites was slightly higher in a global sample of nine ND5 sequences in D. melanogaster (s = 8) than in the nine lines of D. simulans (s = 6). When compared to variation at nuclear loci, the mtDNA variation in D. melanogaster does not depart from neutral expectations. The ND5 sequences in D. simulans, however, show fewer than half the number of variable sites expected under neutrality when compared to sequences from the period locus. While this reduction in variation is not significant at the 5% level, HKA tests with published restriction data for mtDNA in D. simulans do show a significant reduction of variation suggesting a selective sweep of variation in the mtDNA in this species. Tests of neutral evolution based on the ratios of synonymous and replacement polymorphism and divergence are generally consistent with neutral expectations, although a significant excess of amino acid polymorphism within both species is localized in one region of the protein. The rate of mtDNA evolution has been faster in D. melanogaster than in D. simulans and the population structure of mtDNA is distinct in these species. The data reveal how different rates of mtDNA evolution between species and different histories of neutral and adaptive evolution within species can compromise historical inferences in population and evolutionary biology.     

Diallelic multilocus model of neutral genes

The evolution of a completely linked diallelic multilocus system of neutral genes in a finite population is studied. A diffusion model incorporating random genetic drift and mutation is used. We neglect the recombination. To begin with, the spectral analysis of the Kolmogorov backward equation for this model is investigated. We apply this to two extreme situations when the number of sites approaches to infinity. One is a DeMoivre-Laplace type approximation and the other is a Poisson type approximation. The former is applied to the study of the simultaneous distribution and evolution of a large number of neutral genes. It is applicable to the distribution of a polygenic character controlled by clustered loci on a chromosome, and we show that it differs from the normal distribution on account of random genetic drift and linkage disequilibrium. The latter is applied to the distribution of the number of segregating sites in DNA nucleotide sequences, and the rate of evolution is obtained.     

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 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.     

Estimating effective population size from samples of sequences: inefficiency of pairwise and segregating sites as compared to phylogenetic estimates.

It is known that under neutral mutation at a known mutation rate a sample of nucleotide sequences, within which there is assumed to be no recombination, allows estimation of the effective size of an isolated population. This paper investigates the case of very long sequences, where each pair of sequences allows a precise estimate of the divergence time of those two gene copies. The average divergence time of all pairs of copies estimates twice the effective population number and an estimate can also be derived from the number of segregating sites. One can alternatively estimate the genealogy of the copies. This paper shows how a maximum likelihood estimate of the effective population number can be derived from such a genealogical tree. The pairwise and the segregating sites estimates are shown to be much less efficient than this maximum likelihood estimate, and this is verified by computer simulation. The result implies that there is much to gain by explicitly taking the tree structure of these genealogies into account.     

Statistical Method for Testing the Neutral Mutation Hypothesis by DNA Polymorphism

The relationship between the two estimates of genetic variation at the DNA level, namely the number of segregating sites and the average number of nucleotide differences estimated from pairwise comparison, is investigated. It is found that the correlation between these two estimates is large when the sample size is small, and decreases slowly as the sample size increases. Using the relationship obtained, a statistical method for testing the neutral mutation hypothesis is developed. This method needs only the data of DNA polymorphism, namely the genetic variation within population at the DNA level. A simple method of computer simulation, that was used in order to obtain the distribution of a new statistic developed, is also presented. Applying this statistical method to the five regions of DNA sequences in Drosophila melanogaster, it is found that large insertion/deletion (greater than 100 bp) is deleterious. It is suggested that the natural selection against large insertion/deletion is so weak that a large amount of variation is maintained in a population.     

Historical Selection, Amino Acid Polymorphism and Lineage-Specific Divergence at the G6pd Locus in Drosophila Melanogaster and D. Simulans

The nucleotide diversity across 1705 bp of the G6pd gene is studied in 50 Drosophila melanogaster and 12 D. simulans lines. Our earlier report contrasted intraspecific polymorphism and interspecific differences at silent and replacement sites in these species. This report expands the number of European and African lines and examines the pattern of polymorphism with respect to the common A/B allozymes. In D. melanogaster the silent nucleotide diversity varies 2.8-fold across localities. The B allele sequences are two- to fourfold more variable than the derived A allele, and differences between allozymes are twice as among B alleles. There is strong linkage disequilibrium across the G6pd region. In both species the level of silent polymorphism increases from the 5' to 3' ends, while there is no comparable pattern in level of silent site divergence or fixation. The neutral model is not rejected in either species. Using D. yakuba as an outgroup, the D. melanogaster lineage shows a twofold greater rate of silent fixation, but less than half the rate of amino acid replacement. Lineage-specific differences in mutation fixation are inconsistent with neutral expectations and suggest the interaction of species-specific population size differences with both weakly advantageous and deleterious selection.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

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.     

Linkage disequilibrium among multiple neutral alleles produced by mutation in finite population.

An analysis is undertaken for a finite random mating population of the linkage disequilibrium between two loci, at both of which all alleles are neutral, all mutant alleles differ from existing ones and several may be segregating at any time. Formulae are derived for the expected total squared disequilibrium, measured as the sum of squares of disequilibria between all pairs of alleles. The ratio of this quantity to the expected value of the product of the heterozygosities at the two loci is similar to that obtained previously by Ohta and Kimura for two nucleotide sites at each of which not more than two mutant types can segregate at any time.     

Polymorphism and Balancing Selection at Major Histocompatibility Complex Loci

Amino acid replacements in the peptide-binding region (PBR) of the functional major histocompatibility complex (Mhc) genes appear to be driven by balancing selection. Of the various types of balancing selection, we have examined a model equivalent to overdominance that confers heterozygote advantage. As discussed by A. Robertson, overdominance selection tends to maintain alleles that have more or less the same degree of heterozygote advantage. Because of this symmetry, the model makes various testable predictions about the genealogical relationships among different alleles and provides ways of analyzing DNA sequences of Mhc alleles. In this paper, we analyze DNA sequences of 85 alleles at the HLA-A, -B, -C, -DRB1 and -DQB1 loci with respect to the number of alleles and extent of nucleotide differences at the PBR, as well as at the synonymous (presumably neutral) sites. Theory suggests that the number of alleles that differ at the sites targeted by selection (presumably the nonsynonymous sites in the PBR) should be equal to the mean number of nucleotide substitutions among pairs of alleles. We also demonstrate that the nucleotide substitution rate at the targeted sites relative to that of neutral sites may be much larger than 1. The predictions of the presented model are in surprisingly good agreement with the actual data and thus provide means for inferring certain population parameters. For overdominance selection in a finite population at equilibrium, the product of selection intensity (s) against homozygotes and the effective population size (N) is estimated to be 350-3000, being largest at the B locus and smallest at the C locus. We argue that N is of the order of 10(5) and s is several percent at most, if the mutation rate per site per generation is 10(-8).     

Comparison of DNA and protein polymorphisms between humans and chimpanzees

To examine the nucleotide diversity at silent (synonymous + intron + untranslated) and non-silent (nonsynonymous) sites in chimpanzees and humans, genes at six nuclear loci from two chimpanzees were sequenced. The average silent diversity was 0.19%, which was significantly higher than that in humans (0.05%). This observation suggests a significantly larger effective population size and a higher extent of neutral polymorphism in chimpanzees than in humans. On the other hand, the non-silent nucleotide diversity is similar in both species, resulting in a larger fraction of neutral mutations at non-silent sites in humans than in chimpanzees. Other types of polymorphism data were collected from the literature or databases to examine whether or not they are consistent with the nuclear DNA sequence polymorphism observed here. The nucleotide diversity at both silent and non-silent sites in mitochondrial (mt) DNA genes was compatible with that of the nuclear genes. Microsatellite loci showed a similar high extent of heterozygosity in both species, perhaps due to the combined effect of a high mutation rate and a recent population expansion in humans. At protein loci, humans are more heterozygous than chimpanzees, and the estimated fraction of neutral alleles in humans (0.84) is much larger than that in chimpanzees (0.26). These data show that the neutral fraction in non-silent changes is relatively large in the human population. This difference may be due to a relaxation of the functional constraint against proteins in the human lineage. To evaluate this possibility, it will be necessary to examine nucleotide sequences in relation to the physiological or biochemical properties of proteins.     

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.     

A novel analytical method for evolutionary graph theory problems

Evolutionary graph theory studies the evolutionary dynamics of populations structured on graphs. A central problem is determining the probability that a small number of mutants overtake a population. Currently, Monte Carlo simulations are used for estimating such fixation probabilities on general directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic framework for computing fixation probabilities for strongly connected, directed, weighted evolutionary graphs under neutral drift. We show how this framework can also be used to calculate the expected number of mutants at a given time step (even if we relax the assumption that the graph is strongly connected), how it can extend to other related models (e.g. voter model), how our framework can provide non-trivial bounds for fixation probability in the case of an advantageous mutant, and how it can be used to find a non-trivial lower bound on the mean time to fixation. We provide various experimental results determining fixation probabilities and expected number of mutants on different graphs. Among these, we show that our method consistently outperforms Monte Carlo simulations in speed by several orders of magnitude. Finally we show how our approach can provide insight into synaptic competition in neurology.     

A multilocus sequence survey in Arabidopsis thaliana reveals a genome-wide departure from a neutral model of DNA sequence polymorphism

The simultaneous analysis of multiple genomic loci is a powerful approach to studying the effects of population history and natural selection on patterns of genetic variation of a species. By surveying nucleotide sequence polymorphism at 334 randomly distributed genomic regions in 12 accessions of Arabidopsis thaliana, we examined whether a standard neutral model of nucleotide sequence polymorphism is consistent with observed data. The average nucleotide diversity was 0.0071 for total sites and 0.0083 for silent sites. Although levels of diversity are variable among loci, no correlation with local recombination rate was observed, but polymorphism levels were correlated for physically linked loci (<250 kb). We found that observed distributions of Tajima's D- and D/D(min)- and of Fu and Li's D-, D*- and F-, F*-statistics differed significantly from the expected distributions under a standard neutral model due to an excess of rare polymorphisms and high variances. Observed and expected distributions of Fay and Wu's H were not different, suggesting that demographic processes and not selection at multiple loci are responsible for the deviation from a neutral model. Maximum-likelihood comparisons of alternative demographic models like logistic population growth, glacial refugia, or past bottlenecks did not produce parameter estimates that were more consistent with observed patterns. However, exclusion of highly polymorphic "outlier loci" resulted in a fit to the logistic growth model. Various tests of neutrality revealed a set of candidate loci that may evolve under selection.     

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.     

Large number of replacement polymorphisms in rapidly evolving genes of Drosophila. Implications for genome-wide surveys of DNA polymorphism

We present a survey of nucleotide polymorphism of three novel, rapidly evolving genes in populations of Drosophila melanogaster and D. simulans. Levels of silent polymorphism are comparable to other loci, but the number of replacement polymorphisms is higher than that in most other genes surveyed in D. melanogaster and D. simulans. Tests of neutrality fail to reject neutral evolution with one exception. This concerns a gene located in a region of high recombination rate in D. simulans and in a region of low recombination rate in D. melanogaster, due to an inversion. In the latter case it shows a very low number of polymorphisms, presumably due to selective sweeps in the region. Patterns of nucleotide polymorphism suggest that most substitutions are neutral or nearly neutral and that weak (positive and purifying) selection plays a significant role in the evolution of these genes. At all three loci, purifying selection of slightly deleterious replacement mutations appears to be more efficient in D. simulans than in D. melanogaster, presumably due to different effective population sizes. Our analysis suggests that current knowledge about genome-wide patterns of nucleotide polymorphism is far from complete with respect to the types and range of nucleotide substitutions and that further analysis of differences between local populations will be required to understand the forces more completely. We note that rapidly diverging and nearly neutrally evolving genes cannot be expected only in the genome of Drosophila, but are likely to occur in large numbers also in other organisms and that their function and evolution are little understood so far.     

Analysis of a genetic hitchhiking model, and its application to DNA polymorphism data from Drosophila melanogaster

Begun and Aquadro have demonstrated that levels of nucleotide variation correlate with recombination rate among 20 gene regions from across the genome of Drosophila melanogaster. It has been suggested that this correlation results from genetic hitchhiking associated with the fixation of strongly selected mutants. The hitchhiking process can be described as a series of two-step events. The first step consists of a strongly selected substitution wiping out linked variation in a population; this is followed by a recovery period in which polymorphism can build up via neutral mutations and random genetic drift. Genetic hitchhiking has previously been modeled as a steady-state process driven by recurring selected substitutions. We show here that the characteristic parameter of this steady-state model is alpha v, the product of selection intensity (alpha = 2Ns) and the frequency of beneficial mutations v (where N is population size and s is the selective advantage of the favored allele). We also demonstrate that the steady-state model describes the hitchhiking process adequately, unless the recombination rate is very low. To estimate alpha v, we use the data of DNA sequence variation from 17 D. melanogaster loci from regions of intermediate to high recombination rates. We find that alpha v is likely to be > 1.3 x 10(-8). Additional data are needed to estimate this parameter more precisely. The estimation of alpha v is important, as this parameter determines the shape of the frequency distribution of strongly selected substitutions.      

Fixations of the HIV-1 env gene refute neutralism: New evidence for pan-selective evolution

We examined 103 nucleotide sequences of the HIV-1 env gene, sampled from 35 countries and tested: I) the random (neutral) distribution of the number of nucleotide changes; II) the proportion of bases at molecular equilibrium; III) the neutral expected homogeneity of the distribution of new fixated bases; IV) the hypothesis of the neighbor influence on the mutation rates in a site. The expected random number of fixations per site was estimated by Bose-Einstein statistics, and the expected frequencies of bases by matrices of mutation-fixation rates. The homogeneity of new fixations was analyzed using χ2 and trinomial tests for homogeneity. Fixations of the central base in trinucleotides were used to test the neighbor influence on base substitutions. Neither the number of fixations nor the frequencies of bases fitted the expected neutral distribution. There was a highly significant heterogeneity in the distribution of new fixations, and several sites showed more transversions than transitions, showing that each nucleotide site has its own pattern of change. These three independent results make the neutral theory, the nearly neutral and the neighbor influence hypotheses untenable and indicate that evolution of env is rather highly selective.