The effects of multilocus balancing selection on neutral variability

We studied the effect of multilocus balancing selection on neutral nucleotide variability at linked sites by simulating a model where diallelic polymorphisms are maintained at an arbitrary number of selected loci by means of symmetric overdominance. Different combinations of alleles define different genetic backgrounds that subdivide the population and strongly affect variability. Several multilocus fitness regimes with different degrees of epistasis and gametic disequilibrium are allowed. Analytical results based on a multilocus extension of the structured coalescent predict that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. Our simulation results show that although variability increases with the number of genetic backgrounds that are maintained in the population, it is reduced by random fluctuations in the frequencies of those backgrounds and does not reach high levels even in very large populations. We also show that previous results on balancing selection in single-locus systems do not extend to the multilocus scenario in a straightforward way. Different patterns of linkage disequilibrium and of the frequency spectrum of neutral mutations are expected under different degrees of epistasis. Interestingly, the power to detect balancing selection using deviations from a neutral distribution of allele frequencies seems to be diminished under the fitness regime that leads to the largest increase of variability over the neutral case. This and other results are discussed in the light of data from the Mhc.     

A quantitative model of DNA fragments generated by ionizing radiation, and possible experimental applications

We derive an equation for observed frequencies of DNA fragments as a function of size. In this derivation, we consider an experimental system where fragments are generated by random, independent double-strand breaks on chromosomes (or other large DNA molecules) and then separated by size on agarose gels. When visualizing these fragments using Southern hybridization techniques (employing a site-specific probe), we predict an intensity distribution that has unusual properties. In particular, peaks in the fragment size distribution depend not only on standard breakage parameters, but also on the location of the hybridization site. Our model is consistent with experimental and theoretical results reported elsewhere, where measurements of peaks are used for the physical mapping of genes. Further, we propose that similar experiments might be suitable for precise measurements of the parameters of double-strand breakage (as an alternative to neutral filter elutions and neutral sucrose gradients) and for testing the assumption of random, independent breakage for different types of radiation.     

Evidence for the non-quasispecies evolution of RNA viruses [corrected

The quasispecies model of RNA virus evolution differs from those formulated in conventional population genetics in that neutral mutations do not lead to genetic drift of the population, and natural selection acts on the mutant distribution as a whole rather than on individual variants. By computer simulation, we show that this model could be inappropriate for many RNA viruses because the neutral sequence space may be too large to allow the formation of a quasispecies distribution. This view is supported by our analysis of gene sequences from vesicular stomatitis virus, which is considered a prototype RNA virus quasispecies. Our results are relevant to the evolution of RNA systems in general.     

Molecular evolution, mutation size and gene pleiotropy: a geometric reexamination

The influence of phenotypic effects of genetic mutations on molecular evolution is not well understood. Neutral and nearly neutral theories of molecular evolution predict a negative relationship between the evolutionary rate of proteins and their functional importance; nevertheless empirical studies seeking relationships between evolutionary rate and the phenotypic role of proteins have not produced conclusive results. In particular, previous studies have not found the expected negative correlation between evolutionary rate and gene pleiotropy. Here, we studied the effect of gene pleiotropy and the phenotypic size of mutations on the evolutionary rate of genes in a geometrical model, in which gene pleiotropy was characterized by n molecular phenotypes that affect organismal fitness. For a nearly neutral process, we found a negative relationship between evolutionary rate and mutation size but pleiotropy did not affect the evolutionary rate. Further, for a selection model, where most of the substitutions were fixed by natural selection in a randomly fluctuating environment, we also found a negative relationship between evolutionary rate and mutation size, but interestingly, gene pleiotropy increased the evolutionary rate as √n. These findings may explain part of the disagreement between empirical data and traditional expectations.     

Muller's ratchet and the pattern of variation at a neutral locus

The levels and patterns of variation at a neutral locus are analyzed in a haploid asexual population undergoing accumulation of deleterious mutations due to Muller's ratchet. We find that the movement of Muller's ratchet can be associated with a considerable reduction in genetic diversity below classical neutral expectation. The extent to which variability is reduced is a function of the deleterious mutation rate, the fitness effects of the mutations, and the population size. Approximate analytical expressions for the expected genetic diversity are compared with simulation results under two different models of deleterious mutations: a model where all deleterious mutations have equal effects and a model where there are two classes of deleterious mutations. We also find that Muller's ratchet can produce a considerable distortion in the neutral frequency spectrum toward an excess of rare variants.     

Mutation Accumulation May Only Be a Minor Force in Shaping Life-History Traits,Even when Reproduction Is Sexual

In a previous theoretical study we investigated whether adaptive or non-adaptive processes are more important in the evolution of senescence. We built a model that combined both processes and found that mutation accumulation is important only at those ages where mortality has a negligible impact on fitness. This model, however, was limited to haploid organisms. Here we extend our model by introducing diploidy and sexual reproduction. We assume that only recessive (mutated) homozygotes experience detrimental effects. Our results corroborate our previous conclusions, confirming that life histories are largely determined by adaptive processes. We also found that the equilibrium frequencies of mutated alleles are at higher values than in haploid model, because mutations in heterozygotes are hidden for directional selection. Nevertheless, the equilibrium frequencies of recessive homozygotes that make mutations visible to selection are very similar to the equilibrium frequencies of these alleles in our haploid model. Diploidy and sexual reproduction with recombination slows down approaching selection-mutation balance.     

Compensatory evolution in diploid populations

Compensatory mutations are individually deleterious but harmless in appropriate combinations either at more than two sites within a gene or on separate genes. Considering that dominance effects of selection and heterodimer formation of gene products may affect the rate of compensatory evolution, we investigate compensatory neutral mutation models for diploid populations. Our theoretical analysis on the average time until fixation of compensatory mutations shows that these factors play an important role in reducing the fixation time of compensatory mutations if mutation rates are not low. Compensatory evolution of heterodimers is shown to occur more easily if the deleterious effects of single mutants are recessive.     

Are rare variants responsible for susceptibility to complex diseases?

Little is known about the nature of genetic variation underlying complex diseases in humans. One popular view proposes that mapping efforts should focus on identification of susceptibility mutations that are relatively old and at high frequency. It is generally assumed-at least for modeling purposes-that selection against complex disease mutations is so weak that it can be ignored. In this article, I propose an explicit model for the evolution of complex disease loci, incorporating mutation, random genetic drift, and the possibility of purifying selection against susceptibility mutations. I show that, for the most plausible range of mutation rates, neutral susceptibility alleles are unlikely to be at intermediate frequencies and contribute little to the overall genetic variance for the disease. Instead, it seems likely that the bulk of genetic variance underlying diseases is due to loci where susceptibility mutations are mildly deleterious and where there is a high overall mutation rate to the susceptible class. At such loci, the total frequency of susceptibility mutations may be quite high, but there is likely to be extensive allelic heterogeneity at many of these loci. I discuss some practical implications of these results for gene mapping efforts.     

Maintenance of Genetic Variability under the Pressure of Neutral and Deleterious Mutations in a Finite Population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.     

Interactions of Polycomb and trithorax with cis regulatory regions of Ultrabithorax during the development of Drosophila melanogaster.

The activity of the Ultrabithorax gene is continuously required during imaginal development to maintain the morphogenetic identity of the third thoracic segment of Drosophila. The spatial pattern of Ultrabithorax gene expression depends on certain cis regulatory regions and several trans regulatory genes. Amongst the latter the Polycomb gene is necessary to maintain Ultrabithorax repressed in cells where it was not initially activated and the trithorax gene is required for maintaining the expression of the gene where initially active. We have studied genetic interactions between several Ultrabithorax mutations in coding and cis regulatory regions in combination with Polycomb and trithorax mutations. Our results suggest that Polycomb and trithorax gene products do not interact with Ultrabithorax protein products but interact (directly or indirectly) with specific and discrete cis regulatory regions such as those where anterobithorax and postbithorax but not bithorax mutations map. We discuss possible mechanisms of these interactions.     

Influence of Gene Interaction on Complex Trait Variation with Multilocus Models

Although research effort is being expended into determining the importance of epistasis and epistatic variance for complex traits, there is considerable controversy about their importance. Here we undertake an analysis for quantitative traits utilizing a range of multilocus quantitative genetic models and gene frequency distributions, focusing on the potential magnitude of the epistatic variance. All the epistatic terms involving a particular locus appear in its average effect, with the number of two-locus interaction terms increasing in proportion to the square of the number of loci and that of third order as the cube and so on. Hence multilocus epistasis makes substantial contributions to the additive variance and does not, per se, lead to large increases in the nonadditive part of the genotypic variance. Even though this proportion can be high where epistasis is antagonistic to direct effects, it reduces with multiple loci. As the magnitude of the epistatic variance depends critically on the heterozygosity, for models where frequencies are widely dispersed, such as for selectively neutral mutations, contributions of epistatic variance are always small. Epistasis may be important in understanding the genetic architecture, for example, of function or human disease, but that does not imply that loci exhibiting it will contribute much genetic variance. Overall we conclude that theoretical predictions and experimental observations of low amounts of epistatic variance in outbred populations are concordant. It is not a likely source of missing heritability, for example, or major influence on predictions of rates of evolution.     

Testing the neutral fixation of hetero-oligomerism in the archaeal chaperonin CCT

The evolutionary transition from homo-oligomerism to hetero-oligomerism in multimeric proteins and its contribution to function innovation and organism complexity remain to be investigated. Here, we undertake the challenge of contributing to this theoretical ground by investigating the hetero-oligomerism in the molecular chaperonin cytosolic chaperonin containing tailless complex polypeptide 1 (CCT) from archaea. CCT is amenable to this study because, in contrast to eukaryotic CCTs where sub-functionalization after gene duplication has been taken to completion, archaeal CCTs present no evidence for subunit functional specialization. Our analyses yield additional information to previous reports on archaeal CCT paralogy by identifying new duplication events. Analyses of selective constraints show that amino acid sites from 1 subunit have fixed slightly deleterious mutations at inter-subunit interfaces after gene duplication. These mutations have been followed by compensatory mutations in nearby regions of the same subunit and in the interface contact regions of its paralogous subunit. The strong selective constraints in these regions after speciation support the evolutionary entrapment of CCTs as hetero-oligomers. In addition, our results unveil different evolutionary dynamics depending on the degree of CCT hetero-oligomerism. Archaeal CCT protein complexes comprising 3 distinct classes of subunits present 2 evolutionary processes. First, slightly deleterious and compensatory mutations were fixed neutrally at inter-subunit regions. Second, sub-functionalization may have occurred at substrate-binding and adenosine triphosphate-binding regions after the 2nd gene duplication event took place. CCTs with 2 distinct types of subunits did not present evidence of sub-functionalization. Our results provide the 1st in silico evidence for the neutral fixation of hetero-oligomerism in archaeal CCTs and provide information on the evolution of hetero-oligomerism toward sub-functionalization in archaeal CCTs.     

Ancestral inference on gene trees under selection

The extent to which natural selection shapes diversity within populations is a key question for population genetics. Thus, there is considerable interest in quantifying the strength of selection. A full likelihood approach for inference about selection at a single site within an otherwise neutral fully linked sequence of sites is described here. A coalescent model of evolution is used to model the ancestry of a sample of DNA sequences which have the selected site segregating. The mutation model, for the selected and neutral sites, is the infinitely many-sites model where there is no back or parallel mutation at sites. A unique perfect phylogeny, a gene tree, can be constructed from the configuration of mutations on the sample sequences under this model of mutation. The approach is general and can be used for any bi-allelic selection scheme. Selection is incorporated through modelling the frequency of the selected and neutral allelic classes stochastically back in time, then using a subdivided population model considering the population frequencies through time as variable population sizes. An importance sampling algorithm is then used to explore over coalescent tree space consistent with the data. The method is applied to a simulated data set and the gene tree presented in Verrelli et al. (2002).     

Genealogies and weak purifying selection

The assumption that selection alters the genealogical tree of a sample of alleles from a population relative to the neutral expectation underlies several "tests of neutrality." Two recent papers have studied the effect of purifying selection; their suggestive but incomplete results indicate that, in the single site case, the shape of a gene genealogy for a locus may differ only from the neutral expectation. We verify this finding for weak selection using the "ancestral selection graph." We consider a wider range of models, including both a four-allele single-site model and an infinite-sites model. Our results confirm the previous claim for the symmetric-mutation single site model. We emphasize, however, that a neutral-seeming genealogy is consistent with detectable effects of selection on the distribution of allele frequences within the sample. With selection operating, the information about a sample cannot be reduced to the genealogy. As a result, a distinction needs to be made between the selected sites themselves, for which the genealogy offers insufficient information, and linked neutral variation. This distinction seems to have been overlooked in previous papers, yet it has significant implications for the interpretation of data on DNA sequence variation. In particular, it predicts that under purifying selection, the frequency spectrum of neutral mutations will not reflect the skew toward rare polymorphisms at replacement sites even if there is no recombination between them. We caution, however, that the effect of weak selection on the genealogy is specific to the model; a (more realistic) model of multiple linked sites could lead to a more distorted genealogy than is observed for a single site.     

Use of restriction fragment length polymorphisms for genetic counseling: Population genetic considerations

Two-locus population genetic models are analyzed to evaluate the utility of restriction fragment length polymorphisms for purposes of genetic counseling. It is shown that the linkage disequilibrium between a neutral marker and a tightly linked overdominant mutant will increase rapidly as the mutant moves to its polymorphic equilibrium. The linkage disequilibrium decays for deleterious recessive mutants. Two measures involving the linkage disequilibrium are investigated to determine how much information the transmission of the neutral marker provides about the transmission of the selected gene. In certain kinds of matings, where the parental two-locus genotypes and linkage phases are known, it is possible to determine whether or not a progeny is homozygous for the selected gene on the basis of the fetal genotype at the marker locus. A quantity of primary interest is the fraction of matings between individuals heterozygous for the selected gene in which exact diagnosis can be made in this way. The expected proportion of such matings, taken over all two-locus matings involving heterozygotes at the selected locus, is calculated as a function of the gene frequencies at the two loci and the linkage disequilibrium between them. This expected value is maximized when the linkage disequilibrium is at its maximum in absolute value. Fewer than half of all matings are informative if the linkage disequilibrium is small in magnitude or if the gene frequencies at the two loci are quite different. Consideration is also given to various conditional measures of association that may be useful when the parental two-locus genotypes are unknown. The results suggest that the utility of tightly linked neutral marker genes in predicting the transmission of a selected gene is generally less when selection acts against a recessive gene than for overdominant selection.     

A strategy to discover genes that carry multi-allelic or mono-allelic risk for common diseases: a cohort allelic sums test (CAST)

A method is described to discover if a gene carries one or more allelic mutations that confer risk for any specified common disease. The method does not depend upon genetic linkage of risk-conferring mutations to high frequency genetic markers such as single nucleotide polymorphisms. Instead, the sums of allelic mutation frequencies in case and control cohorts are determined and a statistical test is applied to discover if the difference in these sums is greater than would be expected by chance. A statistical model is presented that defines the ability of such tests to detect significant gene-disease relationships as a function of case and control cohort sizes and key confounding variables: zygosity and genicity, environmental risk factors, errors in diagnosis, limits to mutant detection, linkage of neutral and risk-conferring mutations, ethnic diversity in the general population and the expectation that among all exonic mutants in the human genome greater than 90% will be neutral with regard to any effect on disease risk. Means to test the null hypothesis for, and determine the statistical power of, each test are provided. For this "cohort allelic sums test" or "CAST", the statistical model and test are provided as an Excel program, CASTAT(c) at . Based on genetics, technology and statistics, a strategy of enumerating the mutant alleles carried in the exons and splice sites of the estimated approximately 25,000 human genes in case cohort samples of 10,000 persons for each of 100 common diseases is proposed and evaluated: A wide range of possible conditions of multi-allelic or mono-allelic and monogenic, multigenic or polygenic (including epistatic) risk are found to be detectable using the statistical criteria of 1 or 10 "false positive" gene associations approximately 25,000 gene-disease pair-wise trials and a statistical power of >0.8. Using estimates of the distribution of both neutral and gene-inactivating nondeleterious mutations in humans and the sensitivity of the test to multigenic or multicausal risk, it is estimated that about 80% of nullizygous, heterozygous and functionally dominant gene-common disease associations may be discovered. Limitations include relative insensitivity of CAST to about 60% of possible associations given homozygous (wild type) risk and, more rarely, other stochastic limits when the frequency of mutations in the case cohort approaches that of the control cohort and biases such as absence of genetic risk masked by risk derived from a shared cultural environment.     

Response of the phiX174 am3, cs70 transgene to acute and chronic ENU exposure: implications for protocol design

Studies of other transgenic assays have shown that time after treatment is a very important variable in the analysis of mutation frequencies but that eventually a plateau frequency is reached, indicating that the mutations are neutral. This neutrality is very important for the design of both experiments and testing protocols. Here we show that the phiX174 am3, cs70 transgene gives qualitatively similar results to the other transgenes studied after exposure of the mice to N-ethyl-N-nitrosourea. In the small intestine, the mutant frequency induced by an acute dose did not change significantly from 10 to 70 days post-treatment, indicating that the mutations induced are, indeed, neutral. Likewise, the mutant frequency increased linearly with duration of exposure to ENU at a constant rate. Mutant frequencies obtained were 10 times higher from the chronic exposure than produced by a nearly lethal acute dose. As in previous comparisons of a transgene and the endogenous Dlb-1 locus in the small intestine, the chronic exposure was much more effective at increasing the sensitivity of the transgene than of the endogenous gene. The Dlb-1 locus shows more complex kinetics in this strain, as in others, with mutations initially accumulating at a slower rate, indicating a differential repair of genetic damage.     

The effects of weak selection on neutral diversity at linked sites

The effects of selection on variability at linked sites have an important influence on levels and patterns of within-population variation across the genome. Most theoretical models of these effects have assumed that selection is sufficiently strong that allele frequency changes at the loci concerned are largely deterministic. These models have led to the conclusion that directional selection for selectively favorable mutations, or against recurrent deleterious mutations, reduces nucleotide site diversity at linked neutral sites. Recent work has shown, however, that fixations of weakly selected mutations, accompanied by significant stochastic changes in allele frequencies, can sometimes cause higher diversity at linked sites when compared with the effects of fixations of neutral mutations. This study extends this work by deriving approximate expressions for the mean conditional times to fixation and loss of mutations subject to selection, and analyzing the conditions under which selection increases rather than reduces these times. Simulations are used to examine the relations between diversity at a neutral site and the fixation and loss times of mutations at a linked site that is subject to selection. It is shown that the long-term level of neutral diversity can be increased over the purely neutral value by recurrent fixations and losses of linked, weakly selected dominant or partially dominant favorable mutations, or linked recessive or partially recessive deleterious mutations. The results are used to examine the conditions under which associative overdominance, as opposed to background selection, is likely to operate.     

Lines of descent in the diffusion approximation of neutral Wright-Fisher models

This paper studies lines of descent in the diffusion approximation of neutral Wright-Fisher models where the mutation rate away from each gene per generation is the same. Here a line of descent begins with a single gene and has branches at each generation where genes are reproduced from a parent in the line. New mutations are not included in a line of descent but are considered to begin a new line. The joint distribution of the number of lines of descent surviving in a population from time 0 to time t and the frequencies in these lines is derived. Expected times between loss of lines of descent are found. The distribution of the number of lines of descent in a sample from the population is derived. This leads to the distribution of the number of types in a sample from a nonstationary infinite alleles population.