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.     

Increased recombination in a multi-locus system during environmental fluctuations

The role of variability of environment as a factor of recombination evolution is studied. It was shown that selection in multilocus system under cyclic or random fluctuations of external conditions results in rapid increase in recombination frequency, even under free recombination between the rec-modificator and the selected system (SS). If so, pair-vise linkage disequilibrium coefficients have unchanged signs and are negative, and the advantage of rec+ alleles is provided by changes in signs of disequilibriums of higher order. The low frequency environmental fluctuations maintain recombination as it is, while high frequency ones induce the substitution of the low level by high. It was established that the recombination promoting mechanism of crossing-over maintenance, due to elimination of non-optimal allelic combinations of SS loci under temporal optimum variations, may be accomplished by smaller genetic load, than under elimination of deleterious mutations.     

On the evolution of recombination and meiosis.

Theories on the evolution of recombination in regard to its ability to increase mean fitness require a consistent source of negative linkage disequilibrium among loci affecting fitness to show an advantage to recombination. Here we present evidence that, at least theoretically, genetic variation for recombination can spread in very large populations under a strictly multiplicative-fitness, deleterious-allele model. The model uses only Mendelian genetics in a multi-locus context to show that a dominant gene for recombination can spread when rare and resist invasion when common. In non-equilibrium populations driven by Muller's ratchet, the gene increases its probability of fixation by increasing the probability of being associated with the best individuals. This occurs at an optimal level of recombination. Its action results in both an immediate and a long-term advantage to recombination amongst the proto-meiotic organisms modelled. The genetic mechanism lends itself naturally to a model for the evolution of meiosis, where modern-day gametes are seen as derivative of ancient unicellular organisms.     

Linkage disequilibrium, selection and recombination at three Loci

Limits to the relationship among linkage disequilibrium, selection and recombination at equilibrium in three-locus, two-allele, deterministic, discrete generation models are determined using linear programming techniques. These results show that the commonly used measures of linkage disequilibrium are not appropriate for a multilocus setting. Additionally, interactions among three loci are important in reducing the strength of selection necessary to maintain a given level of disequilibrium, relative to a two-locus model.     

Sexual recombination under the joint effects of mutation,selection, and random sampling drift

The advantage or disadvantage of sexual reproduction or recombination for the accumulation of mutant genes in a population is studied under the joint effects of recurrent mutations, selection, and random sampling drift. To obtain the rate at which mutant genes are incorporated three different methods are used; numerical integration of Kolmogorov backward equations, simulation of stochastic difference equations, and Monte Carlo experiments. The first two methods are used in a two-locus system to obtain the fixation probability of double mutants and other related quantities under five different selection models. The third one is conducted for a multiple-locus system and the rate of accumulation of mutant genes per locus is studied. Comparison of the results between sexual and asexual populations shows that the effect of recombination depends on initial linkage disequilibrium, mutation rate v, selection intensity s, and population size N_e. The mode of selection is also an important factor and the large effect of recombination is observed when mutant genes are individually deleterious but collectively favorable. Under a given model of selection, the great advantage or disadvantage of recombination is achieved when a large extent of genetic polymorphism is produced not by mutation but by recombination. Extreme values of N_es and N_ev make the effect insignificant. The results of Monte Carlo experiments also reveal the presence of interaction between selection and sampling drift even when the loci segregate independently and selection is multiplicative. Although this interaction is usually small, there are cases in which one locus theory cannot be used freely. In those cases, the effect of recombination is prominent and one locus theory gives an overestimate of the rate.     

Evolution of recombination due to random drift

In finite populations subject to selection, genetic drift generates negative linkage disequilibrium, on average, even if selection acts independently (i.e., multiplicatively) upon all loci. Negative disequilibrium reduces the variance in fitness and hence, by Fisher's (1930) fundamental theorem, slows the rate of increase in mean fitness. Modifiers that increase recombination eliminate the negative disequilibria that impede selection and consequently increase in frequency by "hitchhiking." Thus, stochastic fluctuations in linkage disequilibrium in finite populations favor the evolution of increased rates of recombination, even in the absence of epistatic interactions among loci and even when disequilibrium is initially absent. The method developed within this article allows us to quantify the strength of selection acting on a modifier allele that increases recombination in a finite population. The analysis indicates that stochastically generated linkage disequilibria do select for increased recombination, a result that is confirmed by Monte Carlo simulations. Selection for a modifier that increases recombination is highest when linkage among loci is tight, when beneficial alleles rise from low to high frequency, and when the population size is small.     

Background selection and population differentiation

A general analytical formula is derived, which predicts the effects of background selection on population differentiation at a neutral locus as a result of its linkage with selected loci of deleterious mutations. The theory is based on the assumptions of random mating, multiplicative fitness, and weak selection in hermaphrodite plants in the island model of population structure. The analytical results show that Fst at the neutral locus increases as a result of the effects of background selection, regardless of the dependence or independence among linked background selective loci. The increment in Fst is closely related to the magnitude of linkage disequilibria between the neutral locus and selected loci, and can be estimated by the ratio of Fst with background selection to Fst without background selection minus one. The steady-state linkage disequilibrium between a neutral locus and a selected locus in subpopulations, primarily attained by gene flow, decreases with the recombination rate, and can be enhanced when there are dependence among linked selected loci. Monte Carlo computer simulations with two- and three-locus models show that the analytical formulae perform well under general conditions. Application of the present theory may aid in analyzing the genome-wide mapping of the effect of background selection in terms of Fst.     

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.     

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.     

Meiosis and the evolution of recombination at low mutation rates

The classical understanding of recombination is that in large asexual populations with multiplicative fitness, linkage disequilibrium is negligible, and thus there is no selective agent driving an allele for recombination. This has led researchers to recognize the importance of synergistic epistatic selection in generating negative linkage disequilibrium that thereby renders an advantage to recombination. Yet data on such selection is equivocal, and various works have shown that synergistic epistasis per se, when left unquantified in its magnitude or operation, is not sufficient to drive the evolution of recombination. Here we show that neither it, nor any mechanism generating negative linkage disequilibrium among fitness-related loci, is necessary. We demonstrate that a neutral gene for recombination can increase in frequency in a large population under a low mutation rate and strict multiplicative fitness. We work in a parameter range where individuals have, on average, less than one mutation each, yet recombination can still evolve. We demonstrate this in two ways: first, by examining the consequences of recombination correlated with misrepaired DNA damage and, second, by increasing the probability of recombination with declining fitness. Interestingly, the allele spreads without repairing even a single DNA mutation.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

The Evolutionary Advantage of Recombination. II. Individual Selection for Recombination

Based on the Fisher-Muller theory of the evolution of recombination, an argument can be constructed predicting that a recessive allele favoring recombination will be favored, if there are either favorable or deleterious mutants occurring at other loci. In this case there is no clear distinction between individual and group selection. Computer simulation of populations segregating for recessive or dominant recombination alleles showed selection favoring recombination, except in the case of a dominant recombination allele with deleterious background mutants. The relationship of this work to parallel investigations by Williams and by Strobeck, Maynard Smith, and Charlesworth is explored. All seem to rely on the same phenomenon. There seems no reason to assume that the evolution of recombination must have occurred by group selection.     

Expected Linkage Disequilibrium for a Neutral Locus Linked to a Chromosomal Arrangement

The expected value of the squared linkage disequilibrium is derived for a neutral locus associated with a chromosomal arrangement that is maintained in the population by strong balancing selection. For a given value of recombination, the expected squared linkage disequilibrium is shown to decrease as the intensity of selection maintaining the arrangement increases. The transient behavior of the expected square linkage disequilibrium is also derived. This theory applies to loci that are closely linked to inversions in Drosophila species and to loci closely linked to the differential segments of the translocation complexes in ring-forming species of Oenothera. In both cases the strong linkage disequilibria that have been observed in natural populations can be explained by random drift.     

Gene Flow and Selection in a Two-Locus System

A model of gene flow and selection in two linked loci is analyzed. The problems considered are the effects of linkage on the clines in frequencies at the two loci and the role of gene flow in producing linkage disequilibrium between the loci. Also, the possible significance of linkage as a mechanism for permitting a population of "track" spatial changes in the environment is considered. The results are that when the recombination fraction between the loci is of the same order of magnitude as the selection coefficients or smaller, then linkage is important in determining the gene frequencies and a substantial amount of linkage disequilibrium is present in the cline. Depending on the spatial pattern of selection on the two loci, linkage can either decrease or increase a population's response to local selection.     

Consequences of genetic linkage for the maintenance of sexually antagonistic polymorphism in hermaphrodites

When selection differs between males and females, pleiotropic effects among genes expressed by both sexes can result in sexually antagonistic selection (SA), where beneficial alleles for one sex are deleterious for the other. For hermaphrodites, alleles with opposing fitness effects through each sex function represent analogous genetic constraints on fitness. Recent theory based on single‐locus models predicts that the maintenance of SA genetic variation should be greatly reduced in partially selfing populations. However, selfing also reduces the effective rate of recombination, which should facilitate selection on linked allelic combinations and expand opportunities for balancing selection in a multilocus context. Here, I develop a two‐locus model of SA selection for simultaneous hermaphrodites, and explore the joint influence of linkage, self‐fertilization, and dominance on the maintainance of SA polymorphism. I find that the effective reduction in recombination caused by selfing significantly expands the parameter space where SA polymorphism can be maintained relative to single‐locus models. In particular, linkage facilitates the invasion of male‐beneficial alleles, partially compensating for the “female‐bias” in the net direction of selection created by selfing. I discuss the implications of accounting for linkage among SA loci for the maintenance of SA genetic variation and mixed mating systems in hermaphrodites.     

Selection intensity against deleterious mutations in RNA secondary structures and rate of compensatory nucleotide substitutions

A two-locus model of reversible mutations with compensatory fitness interactions is presented; single mutations are assumed to be deleterious but neutral in appropriate combinations. The expectation of the time of compensatory nucleotide substitutions is calculated analytically for the case of tight linkage between sites. It is shown that selection increases the substitution time dramatically when selection intensity Ns > 1, where N is the diploid population size and s the selection coefficient. Computer simulations demonstrate that recombination increases the substitution time, but the effect of recombination is small when selection is weak. The amount of linkage disequilibrium generated in the process of compensatory substitution is also investigated. It is shown that significant linkage disequilibrium is expected to be rare in natural populations. The model is applied to the mRNA secondary structure of the bicoid 3' untranslated region of Drosophila. It is concluded that average selection intensity Ns against single deleterious mutations is not likely to be much larger than 1.     

Nonuniform linkage disequilibrium within a 1,500-kb region of the human immunoglobulin heavy-chain complex.

We have characterized 10 VH polymorphic loci of the VH2, VH3, VH4, and VH5 families. Eight of 10 VH polymorphisms were found to be insertion/deletion polymorphisms, probably the result of nonhomologous recombination over the course of evolution of the current human VH repertoire. The 10 VH polymorphic loci were analyzed in 10 three-generation and 10 two-generation Canadian caucasoid families. Linkage disequilibrium (allelic association) was measured between pairs of VH polymorphic loci, and 12 significant associations were found. The degree of linkage disequilibrium measured between IGH polymorphic loci was then compared with the physical distance separating the loci. The physical distance between IGH polymorphic loci does not entirely determine the degree of linkage disequilibrium between polymorphic loci. Two regions, one in the VH region (between VH3f-2 and VH5-2 and one in the CH region (between C delta and C gamma 3), were found to have linkage disequilibrium values approximately 1/3,000 of that observed in other portions of the IGH region. The previous identification of recombinants in the C delta-to C gamma 3 region indicates that these areas of low linkage disequilibrium are consistent with the presence of recombination hot spots. The observed high amount of recombination in the subtelomeric portion of chromosome 14 therefore appears to be the result of specific hot spots for recombination, rather than a general increase in recombination in this region.     

Recombination disequilibrium in ideal and natural populations

Following Hardy-Weinberg disequilibrium (HWD) occurring at a single locus and linkage disequilibrium (LD) between two loci in generations, we here proposed the third genetic disequilibrium in a population: recombination disequilibrium (RD). RD is a measurement of crossover interference among multiple loci in a random mating population. In natural populations besides recombination interference, RD may also be due to selection, mutation, gene conversion, drift and/or migration. Therefore, similarly to LD, RD will also reflect the history of natural selection and mutation. In breeding populations, RD purely results from recombination interference and hence can be used to build or evaluate and correct a linkage map. Practical examples from F₂, testcross and human populations indeed demonstrate that RD is useful for measuring recombination interference between two short intervals and evaluating linkage maps. As with LD, RD will be important for studying genetic mapping, association of haplotypes with disease, plant breading and population history.