Heterosis, epistasis and linkage disequilibrium in a wild population of the polymorphic butterfly Danaus chrysippus (L.)

The colour polymorphism of the Danaus chrysippus population at Dar es Salaam, East Africa, is controlled at three major loci, each with two alleles. Two of the loci, one governing ground colour and the other forewing pattern, are closely linked. The third locus, determining hindwing pattern, assorts independently. Thirty-eight broods raised from wild mated pairs, Fl and F2 generations gave 857 offspring of 23 genotypes (out of 27 possible). The forewing length, taken as an index of size, was investigated in relation to the genotype. Heterosis is evident at all three loci. The two linked loci show epistatic interaction of an unexpected kind: double heterozygotes are smaller than heterozygotes at only one locus but larger than double homozygotes. The heterotic effect at the third, unlinked locus is the most pronounced and is additive to that at the other two. Heterosis is more marked in males than females. The possibility that body size has importance in connexion with sexual selection, food resources and mimetic relationships is discussed. Analysis of gene and chromosome frequencies in the wild parents of 61 broods suggests that double heterozygotes for the two linked loci may have heterozygous advantage. Seventy-eight per cent of chromosomes are repulsion phase: thus, there is pronounced linkage disequilibrium which must be maintained by selection as crossing over is almost 296. In particular, the chromosome carrying both dominant alleles in coupling is rare. Consideration of the centres of distribution and present ranges of the alleles at all three loci suggests that three geographical races, aegyptius, dorippus and alcippus, were isolated by forest barriers, during wet periods in the Pleistocene, in south-west, north-east and north-west Africa respectively. They have probably expanded their ranges in the post-glacial period to overlap and interbreed in central and east Africa. Either heterozygous advantage or seasonal (directional) selection or a combination of both is responsible for the persistence of the polymorphism.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Chromosome Interactions in DROSOPHILA MELANOGASTER. II. Total Fitness

In a large experiment, using nearly 200 population cages, we have measured the fitness of Drosophila melanogaster homozygous (1) for the second chromosome, (2) for the third chromosome, and (3) for both chromosomes. Twentyfour second chromosomes and 24 third chromosomes sampled from a natural population were tested. The mean fitness of the homozygous flies is 0.081 ± 0.014 for the second chromosome, 0.080 ± 0.017 for the third chromosome, and 0.079 ± 0.024 for both chromosomes simultaneously. Assuming that fitnesses are multiplicative (the additive fitness model makes no sense in the present case because of the large selection coefficients involved), the expected mean fitness of the homozygotes for both chromosomes is 0.0066; their observed fitness is more than ten times greater. Thus, it appears that synergistic interactions between loci are considerable; and that, consequently, the fitness function substantially departs from linearity. Two models are tentatively suggested for the fitness function: a "threshold" model and a "synergistic" model.—The experiments reported here confirm previous results showing that the concealed genetic load present in natural populations of Drosophila is sufficient to account for the selective maintenance of numerous polymorphisms (of the order of 1000).     

Natural selection on a leaf-shape polymorphism in the ivyleaf morning glory (Ipomoea hederacea)

Leaf shape is one of the most variable plant traits. Previous work has provided much indirect evidence that leaf-shape variation is adaptive and that leaf shape influences thermoregulation, water balance, and resistance to natural enemies. Nevertheless, there is little direct evidence that leaf shape actually affects plant fitness. In this study, we first demonstrate that populations of the ivyleaf morning glory, Ipomoea hederacea, in North and South Carolina are frequently polymorphic at a locus that influences leaf shape. We then employ several field experiments to show that this polymorphism is subject to selection. In two of the experiments, at different sites, heterozygotes enjoyed a fitness advantage over both homozygotes. At a third site, in one year directional selection favored lobed leaves, whereas in a second year the pattern of fitnesses was consistent with similar directional selection or heterozygote superiority. Computer simulations of heterozygote advantage under the high selfing rates of I. hederacea indicate that balancing selection of the magnitude observed can by itself stabilize the polymorphism, although spatially and temporally variable selection may also contribute to its long-term maintenance.     

Pathogen resistance and genetic variation at MHC loci

Abstract.— Balancing selection in the form of heterozygote advantage, frequency-dependent selection, or selection that varies in time and/or space, has been proposed to explain the high variation at major histocompatibility complex (MHC) genes. Here the effect of variation of the presence and absence of pathogens over time on genetic variation at multiallelic loci is examined. In the basic model, resistance to each pathogen is conferred by a given allele, and this allele is assumed to be dominant. Given that s is the selective disadvantage for homozygotes (and heterozygotes) without the resistance allele and the proportion of generations, which a pathogen is present, is e , fitnesses for homozygotes become (1 — s )^(n-1)e and the fitnesses for heterozygotes become (1 — s )^(n-2)e, where n is the number of alleles. In this situation, the conditions for a stable, multiallelic polymorphism are met even though there is no intrinsic heterozygote advantage. The distribution of allele frequencies and consequently heterozygosity are a function of the autocorrelation of the presence of the pathogen in subsequent generations. When there is a positive autocorrelation over generations, the observed heterozygosity is reduced. In addition, the effects of lower levels of selection and dominance and the influence of genetic drift were examined. These effects were compared to the observed heterozygosity for two MHC genes in several South American Indian samples. Overall, resistance conferred by specific alleles to temporally variable pathogens may contribute to the observed polymorphism at MHC genes and other similar host defense loci.     

The evolution of alternative mating strategies in variable environments

Summary We assessed the influence of phenotypic plasticity in age at maturity on the maintenance of alternative mating strategies in male Atlantic salmon,Salmo salar. We calculated the fitness,r, associated with the parr and the anadromous strategies, using age-specific survival data from the field and strategy-specific fertilization data from the laboratory. The fitness of each strategy depended largely on mate competition (numbers of parr per female, i.e. parr frequency) and on age at maturity. Fitness declined with increasing numbers of parr per female with equilibrium frequencies (at which the fitnesses of each strategy are equal) being within the range observed in the wild. Equilibrium parr frequencies declined with decreasing growth rate and increasing age at maturity. Within populations, the existence of multiple age-specific sets of fitness functions suggests that the fitnesses of alternative strategies are best represented as multidimensional surfaces. The points of intersection of these surfaces, whose boundaries encompass natural variation in age at maturity and mate competition, define an evolutionarily stable continuum (ESC) of strategy frequencies along which the fitnesses associated with each strategy are equal. We propose a simple model that incorporates polygenic thresholds of a largely environmentally-controlled trait (age at maturity) to provide a mechanism by which an ESC can be maintained within a population. An indirect test provides support for the prediction that growth-rate thresholds for parr maturation exist and are maintained by stabilizing selection. Evolutionarily stable continua, maintained by negative frequency-dependent selection on threshold traits, provide a theoretical basis for understanding how alternative life histories can evolve in variable environments.     

Genotype-environment interaction for total fitness in <Emphasis Type="Italic">Drosophila</Emphasis>

A fundamental assumption of models for the maintenance of genetic variation by environmental heterogeneity is that selection favours different genotypes in different environments. Here, I use a method for measuring total fitness of chromosomal heterozygotes in Drosophila melanogaster to assess genotype-environment interaction for fitness across two ecologically relevant environments, medium with and without added ethanol. Two-third chromosomes are compared, one from a population selected for ethanol tolerance, and the other from a control population. The results show strong crossing of reaction norms for outbred, total fitness, with the chromosome from the ethanol-adapted population increasing fitness on ethanol-supplemented food, but decreasing fitness on regular food, relative to the chromosome from the control population. Although I did not map the fitness effects below the chromosome level, the method could be adapted for quantitative trait locus mapping, to determine whether a substantial proportion of fitness variation is contributed by loci at which different alleles are favoured in different environments.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

Multilocus Behavior in Random Environments I. Random Levene Models

In this paper the consequences of natural selection acting on several loci simultaneously in a spatially fluctuating environment are described. The fitnesses of the genotypes are assumed to be additive both within and between loci. The environment is assumed to be made up of a very large (effectively infinite) number of patches in which fitnesses are assigned at random. The resulting deterministic model is called a Random Levene Model and its properties are approximate by a system of differential equations. The main equilibrium properites are that (1) the linkage disequilibrium is zero and (2) the correlations in fitness between alleles at different loci are the principle determinants of the dynamic inter-locus interactions. Although there is no epistasis as conventionally defined, the equilibrium state at the two loci are highly interdependent, the governing principle being that two alleles at different loci whose fitness are negatively correlated across environments have a higher overall fitness due to the reduction in their variance in fitness through the negative correlation. When a large number of loci are considered, they naturally fall into correlation groupings which lead to an enhanced likelihood for polymorphism over that predicted by single-locus theory.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. II. Overdominance and instability with random mating

Part I of the present series demonstrates that globally stable polymorphic equilibria may show underdominance in Darwinian fitness. Hence, overdominance in fitness can no longer be conceived of as a necessary condition for the stability of a polymorphism. In the present paper, the question is posed as to whether overdominance is at least sufficient for this stability. A population of randomly mating individuals is considered, where selection operates uniquely through differential fecundities of particular mating types and may generate either a heterozygote excess or deficit relative to Hardy-Weinberg proportions. It turns out that both unstable central overdominance and stable central underdominance are possible and that their occurrence is strongly related to an excess or a deficiency of heterozygotes in the vicinity of the regions of instability or stability. As one consequence, the above suggested sufficiency of heterozygote superiority is not valid, even in random mating populations. Based on the results of both papers of this series, which demonstrate the inadequacy of over- and underdominance as indicators of stability or instability, a modified overdominance principle is discussed. This principle states that a biallelic polymorphism is maintained if the heterozygote is superior in its degree of "heterogamous self-replication" to the degrees of "autogamous self-replication" of the corresponding homozygotes. It is derived with the help of fractional fitnesses, and it is pointed out that certain ratios of these may be more useful for finding evolutionary constants which govern the maintenance of genetic polymorphisms than are ratios of total fitnesses.     

Linkage Modification with Mixed Random Mating and Selfing: A Numerical Study

Although recombination cannot increase under conditions of random mating or complete selfing in regimes of constant selection, with mixed random mating and selfing, selection for increased recombination can occur. For some fitness regimes there may be selection for reduced recombination with both low and high degrees of selfing but selection for increased recombination with moderate degrees of selfing. With some fitness regimes there is a historical effect: depending on which equilibrium a population starts from, there may be selection for either increased or decreased recombination. In other cases the direction of selection may be determined by the present state of individuals within the population. If recombination is already fairly limited, there may be selection for further reduction. If recombination is already fairly frequent, there may be selection for increased recombination. For certain symmetric viability systems there may be an intermediate value of the recombination fraction between 0 and 0.5 toward which the population will evolve. Although it is not yet possible to classify precisely those fitness matrices that can exhibit selection for increased recombination, it does appear that selection for increased recombination can occur only if at least two of the double homozygotes are less fit than would be expected on the basis of a comparison of the fitnesses of the single and double heterozygotes on an additive scale.     

Biology of a duplicate gene system with glucose 6-phosphate dehydrogenase activity in Drosophila melanogaster: Genetic analysis and differences in fitness components and reaction to environmental parameters among Zw genotypes

There are two structural forms of glucose 6-phosphate dehydrogenase activity in Drosophila melanogaster. Whether one or the other or both show in vitro (and probably in vivo) activity depends on the genotype of a sex-linked locus (Zw). In this article, the relative fitnesses of heterozygotes (with both electromorphs active) and homozygotes (with activity demonstrable for only one or the other electromorph) for the Zw locus are described. It is shown that the relative fitness of heterozygotes increases with increase in population density, or degree of crowding and trophic stress, and that the mean development times of Zw heterozygotes are lower than those of the Zw homozygotes. In addition, and perhaps accounting for the fitness and viability excess of the heterozygotes, one set of evidence strongly suggests that they are better buffered against trophic stress than the homozygotes.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

Polymorphism in random environments

A two-allele diploid model is described in which the fitnesses of the three genotypes are stationary stochastic processes. It is shown that a stable polymorphism will occur if the geometric mean fitness of the heterozygote exceeds that of both homozygotes. It is possible for the mean fitness of the population to be lower in polymorphic than in the associated monomorphic populations.     

Polymorphism at two loci through selection for linear metric deviation.

A model of phenotypic stabilising selection in which the fitness of an individual depends solely on its phenotype, and not directly on its genetic constitution, is explored algebraically for a system of two linked loci of unequal effect. It is found that selection for metric deviation gives rise to polymorphic gametefrequency equilibria for a variety of fitness regimes. Stability of non-trivial equilibria occurs for a wide range of parameter sets. Stability is facilitated by close linkage and inequality between gene effects. It is suggested that, in general genetic variation may be maintained under stabilising selection when the fitness of double heterozygotes exceeds that of the phenotypically intermediate homozygotes.     

Mutation-Selection Balance with Stochastic Selection

Diffusion theory has been used to analyze a model of mutation-selection balance in which the selection process is assumed to be stochastic in time. The limiting outcome of the mutation-stochastic selection process is determined qualitatively by the geometric mean fitnesses of the genotypes, and the conditions for fixation or polymorphism are similar to those that determine the outcome of the mutation-selection process when selection is constant. However, in the case of a completely recessive allele, detailed numerical study of the polymorphism associated with stochastic selection has shown that the average allele frequency maintained is greater than the equilibrium frequency expected when selection is constant, even when the geometric mean fitness of the recessive homozygotes is identical in the stochastic and deterministic models. Thus, allele frequencies in natural populations that are too high to be plausibly explained by a balance between mutation and constant selection can be accounted for if selection is stochastic.     

Should Individual Fitness Increase with Heterozygosity?

Natural selection influences not only gamete frequencies in populations but also the multilocus fitness structures associated with segregating gametes. In particular, only certain patterns of multilocus fitnesses are consistent with the maintenance of stable multilocus polymorphisms. This paper offers support for the proposition that, at stable, viability-maintained, multilocus polymorphisms, the fitness of a genotype tends to increase with the number of heterozygous loci it contains. Average fitness always increases with heterozygosity at stable product equilibria (i.e., those without linkage disequilibrium) maintained by either additive or multiplicative fitness schemes. Simulations suggest that it "generally" increases for arbitrary fitness schemes. The empirical literature correlating allozyme heterozygosity with fitness-correlated traits is discussed in the light of these and other theoretical results.     

Fitness of allozyme variants in Drosophila pseudoobscura III. Possible factors contributing to the maintenance of polymorphisms in nature

Experiments have been performed to investigate the mechanisms maintaining enzyme polymorphisms in natural populations. We have measured effects on fitness of genotypic variants at three loci, Est-5, Odh, and Mdh-2, in D. pseudoobscura. Significant differences exist among the genotypes in the rate of development from egg-to-adult; there is also indication of differences in larval survival. In a population segregating for allelic variants at all three loci, there is indication that segregation distortion at meiosis or some form of gametic selection might be involved. The relative fitnesses of alternative genotypes are reversed when either different fitness components are considered, or the genotypic frequencies are changed, or the larval density is increased. These fitness reversals may contribute to the maintenance of the polymorphisms, and may account for cyclical oscillations of allozyme frequencies observed in natural populations.Research supported by U.S. Public Health Service Research Fellowship (1F05 TWO 1991-01) to D.M. and by contract AT(04-3)34 with the U.S. Atomic Energy Commission. Adress reprint requests from Europe to D.M.; from elsewhere to F.J.A.     

The Effect of Level of Heterozygosity on the Performance of Hybrids between Isogenic Lines of Barley

Sixteen "isogenic" lines of Atlas 46 barley differing in one to four short chromosome segments, and 16 heterozygotes obtained by crossing these lines to male-sterile Atlas, were used to study the effect of level of heterozygosity on performance. In field tests conducted in four environments (two planting dates in two years) significant differences were found among the homozygous isogenic lines for the traits seed yield, kernel weight, tiller number, plant height, and heading time; thus each of the marked chromosome segments carries genes which, when homozygous, affect these quantitative characters. It was also found that heterozygotes produced more and heavier kernels and were taller and earlier than homozygotes but there was no clear indication that the degree of heterosis increased as the number of heterozygous segments increased from one to five. Degree of heterosis was, however, strongly affected by the environment, by allelic state at each segment (especially the segment marked by the two-row, six-row spike locus), and also by genotype for other marked segments. These results indicate that heterosis in barley has a more complex structure than can be adequately represented by simple models, such as the multiplicative model in which fitnesses are the product of fitnesses at individual loci, or threshold models in which optimum fitness is approached asymptotically as the number of heterozygous loci increases.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.