VARIANCE-INDUCED PEAK SHIFTS

The increase in phenotypic variance that occurs in some populations as a result of bottlenecks and founder events can cause a dramatic increase in the probability of a peak shift from one adaptive state to another. Periods of small population size allow drift in the amount of phenotypic variance. Increases in phenotypic variance, coupled with a constant individual fitness function with multiple peaks, can cause the mean fitness landscape to change from bimodal to unimodal, thereby allowing the population's mean phenotype to change deterministically by selection. As the amount of phenotypic variance is returned to an equilibrium state, the multiple peaks reemerge, but the population has moved from one stable state to another. These variance-induced peak shifts allow punctuational evolution from one peak to another at a rate that can be much higher than that predicted by Wright's shifting-balance process alone.     

On Models of Quantitative Genetic Variability: A Stabilizing Selection-Balance Model

A model of stabilizing selection on a multilocus character is proposed that allows the maintenance of stable allelic polymorphism and linkage disequilibrium. The model is a generalization of Lerner's model of homeostasis in which heterozygotes are less susceptible to environmental variation and hence are superior to homozygotes under phenotypic stabilizing selection. The analysis is carried out for weak selection with a quadratic-deviation model for the stabilizing selection. The stationary state is characterized by unequal allele frequencies, unequal proportions of complementary gametes, and a reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared with Mather's polygenic balance theory, with models that include mutation-selection balance, and others that have been proposed to study the role of linkage disequilibrium in quantitative inheritance.     

Stabilizing selection,mutational bias,and the evolution of sex*

Stabilizing selection around a fixed phenotypic optimum is expected to disfavor sexual reproduction, since asexually reproducing organisms can maintain a higher fitness at equilibrium, while sex disrupts combinations of compensatory mutations. This conclusion rests on the assumption that mutational effects on phenotypic traits are unbiased, that is, mutation does not tend to push phenotypes in any particular direction. In this article, we consider a model of stabilizing selection acting on an arbitrary number of polygenic traits coded by bialellic loci, and show that mutational bias may greatly reduce the mean fitness of asexual populations compared with sexual ones in regimes where mutations have weak to moderate fitness effects. Indeed, mutation and drift tend to push the population mean phenotype away from the optimum, this effect being enhanced by the low effective population size of asexual populations. In a second part, we present results from individual‐based simulations showing that positive rates of sex are favored when mutational bias is present, while the population evolves toward complete asexuality in the absence of bias. We also present analytical (QLE) approximations for the selective forces acting on sex in terms of the effect of sex on the mean and variance in fitness among offspring.     

ASSORTATIVE MATING AND THE ADAPTIVE LANDSCAPE

The ability of a population to shift from one adaptive peak to another was examined for a two-locus model with different degrees of assortative mating, selection, and linkage. As expected, if the proportion of the population that mates assortatively increases, so does its ability to shift to a new peak. Assortative mating affects this process by allowing the mean fitness of a population to increase monotonically as it passes through intermediate gene frequencies on the way to a new, higher, homozygotic peak. Similarly, if the height of the new peak increases or selection against intermediates becomes less severe, the population becomes more likely to shift to a new peak. Close linkage also helps the shift to a new adaptive peak and acts similarly to assortative mating, but it is not necessary for such a shift as was previously thought. When a population shifts to a new peak, the number of generations required is significantly less than that needed to return to the original peak when that happens. The short period of time required may be an explanation for rapid changes in the geological record. Under extremely high degrees of assortative mating, the shift takes longer, presumably because of the difficulty of breaking up less favored allelic combinations.     

The Genetic Covariance between Characters Maintained by Pleiotropic Mutations

A statistical genetic model of a multivariate phenotype is derived to investigate the covariation of pleiotropic mutations with additive effects under the combined action of phenotypic selection, linkage and the mating system. Equilibrium formulas for large, randomly mating populations demonstrate that, when selection on polygenic variation is much smaller than twice the harmonic mean recombination rate between loci with interacting fitnesses, linkage disequilibrium is negligible and pleiotropy is the main cause of genetic correlations between characters. Under these conditions, approximate expressions for the dynamics of the genetic covariances due to pleiotropic mutations are obtained. Patterns of genetic covariance between characters and their evolution are discussed with reference to data on polygenic mutation, chromosomal organization and morphological integration.     

PEAK SHIFTS PRODUCED BY CORRELATED RESPONSE TO SELECTION

Traits may evolve both as a consequence of direct selection and also as a correlated response to selection on other traits. While correlated response may be important for both the production of evolutionary novelty and in the build-up of complex characters, its potential role in peak shifts has been neglected empirically and theoretically. We use a quantitative genetic model to investigate the conditions under which a character, Y, which has two alternative optima, can be dragged from one optimum to the other as a correlated response to selection on a second character, X. High genetic correlations between the two characters make the transition, or peak shift, easier, as does weak selection tending to restore Y to the optimum from which it is being dragged. When selection on Y is very weak, the conditions for a peak shift depend only on the location of the new optimum for X and are independent of the strength of selection moving it there. Thus, if the “adaptive valley” for Y is very shallow, little reduction in mean fitness is needed to produce a shift. If the selection acts strongly to keep Y at its current optimum, very intense directional selection on X, associated with a dramatic drop in mean fitness, is required for a peak shift. When strong selection is required, the conditions for peak shifts driven by correlated response might occur rarely, but still with sufficient frequency on a geological timescale to be evolutionarily important.     

Dynamics of Genetic Variability in Two-Locus Models of Stabilizing Selection

We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let V(g) be the genic variance of the trait and C(L) be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (V(g), C(L)) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.     

A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait

The equilibrium structure of an additive, diallelic multilocus model of a quantitative trait under frequency- and density-dependent selection is derived. The trait is under stabilizing selection and mediates intraspecific competition as induced, for instance, by differential resource utilization. It is assumed that stabilizing selection is weak, but the strength of competition may be arbitrary relative to it. Density dependence is caused by population regulation, which may be of a very general kind. The number and effects of loci are arbitrary, and stabilizing selection is not necessarily symmetric with respect to the range of phenotypic values. All previously studied models of intraspecific competition for a continuum of resources known to the author reduce to a special case of the present model if overall selection is weak. Therefore, in this case our results are applicable as approximations to all these models. Our central result is the (nearly) complete characterization of the equilibrium and stability structure in terms of all parameters. It is derived under the sole assumption that selection is weak enough relative to recombination to ignore linkage disequilibrium. In particular, necessary and sufficient conditions on the strength of competition relative to stabilizing selection are found that ensure the maintenance of multilocus polymorphism and the occurrence of disruptive selection. In this case, explicit formulas for the number of polymorphic loci at equilibrium, the allele frequencies, the genetic variance, and the strength of disruptive selection are obtained. For two loci, the effects of linkage are investigated analytically; for several loci, they are studied numerically.     

GENOTYPE-ENVIRONMENT INTERACTION AND THE EVOLUTION OF PHENOTYPIC PLASTICITY

Studies of spatial variation in the environment have primarily focused on how genetic variation can be maintained. Many one-locus genetic models have addressed this issue, but, for several reasons, these models are not directly applicable to quantitative (polygenic) traits. One reason is that for continuously varying characters, the evolution of the mean phenotype expressed in different environments (the norm of reaction) is also of interest. Our quantitative genetic models describe the evolution of phenotypic response to the environment, also known as phenotypic plasticity (Gause, 1947), and illustrate how the norm of reaction (Schmalhausen, 1949) can be shaped by selection. These models utilize the statistical relationship which exists between genotype-environment interaction and genetic correlation to describe evolution of the mean phenotype under soft and hard selection in coarse-grained environments. Just as genetic correlations among characters within a single environment can constrain the response to simultaneous selection, so can a genetic correlation between states of a character which are expressed in two environments. Unless the genetic correlation across environments is ± 1, polygenic variation is exhausted, or there is a cost to plasticity, panmictic populations under a bivariate fitness function will eventually attain the optimum mean phenotype for a given character in each environment. However, very high positive or negative correlations can substantially slow the rate of evolution and may produce temporary maladaptation in one environment before the optimum joint phenotype is finally attained. Evolutionary trajectories under hard and soft selection can differ: in hard selection, the environments with the highest initial mean fitness contribute most individuals to the mating pool. In both hard and soft selection, evolution toward the optimum in a rare environment is much slower than it is in a common one. A subdivided population model reveals that migration restriction can facilitate local adaptation. However, unless there is no migration or one of the special cases discussed for panmictic populations holds, no geographical variation in the norm of reaction will be maintained at equilibrium. Implications of these results for the interpretation of spatial patterns of phenotypic variation in natural populations are discussed.     

The Evolution of Multilocus Systems under Weak Selection

The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << c(min), where s and c(min) denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t(1) ~ (ln s)/ln(1 - c(min)) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s -> 0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s(2)). Then after a time t(2) ~ 2t(1), the linkage disequilibria are nearly constant, their rate of change being O(s(2)). Furthermore, with an error of O(s(2)), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s(3)), then the single-generation change in the mean fitness is ΔW = W(-1)V(g) + O(s(3)), where V(g) designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s(2)) evolves at the rate ΔZ = W(-1)C(g) + O(s(2)), where C(g) represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t(2), the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.     

A slow selection analysis of two locus, two allele traits.

A deterministic, continuous time model for the dynamics of two locus, two allele Mendelian traits in a large randomly mating diploid population is derived. The model allows for frequency and time dependent birth and death rates. It is analyzed under the assumption that the selective forces acting in the population are small. Slow selection approximations to the system's solution are then constructed. Two particular cases are studied. First, when linkage between loci is tight, the population is shown to rapidly approach Hardy-Weinberg proportions, which then may vary on a (slow) time scale determined by differential fitness. In the case of constant birth and death rates, a measure of the population's fitness is shown to increase on the slow time scale after an initial rapid adjustment. The second case considered is for loose linkage; a population near linkage equilibrium is studied. It is shown that the epistatic parameters cancel and that the results agree with the tight linkage case to leading order. The linkage disequlibrium is described in both cases.     

THE DISTRIBUTION OF PHENOTYPIC VARIANCE WITH INBREEDING

Fifty-two inbred populations of Drosophila melanogaster, each founded from a single pair, and a large number of control, outbred flies were measured for fitness and a set of six traits. A survey of the literature on the effects of inbreeding and population bottlenecks demonstrates that the commonly observed pattern of an apparent variance among characters and among species in changes of phenotypic variance may in fact be largely the result of sampling error, given the pattern of change that we demonstrate within a species for the same character. In our study, population bottlenecks on average decrease the amount of phenotypic variance for a suite of wing characteristics and size, but there is large and significant variation among lines in the amount of phenotypic variance. As a result, several lines increased in variance in spite of the average decrease. Interestingly, the changes in phenotypic variance for fitness are in sharp contrast to those seen for phenotypic variance for morphological traits. The amount of phenotypic variance for fitness varies highly significantly among lines but, on average, is increased by bottlenecks. The changes in phenotypic variance as a result of population bottlenecks are large enough to significantly affect the probability of peak shifts by the variance-induced peak shift model.     

The effect of epistasis on the structure of hybrid zones

Within hybrid zones that are maintained by a balance between selection and dispersal, linkage disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium causes selection on each locus to act on all other loci, thereby steepening clines, and generating a barrier to gene flow. Diffusion models predict simple relations between the strength of linkage disequilibrium and the dispersal rate, sigma, and between the barrier to gene flow, B, and the reduction in mean fitness, W. The aim of this paper is to test the accuracy of these predictions by comparison with an exact deterministic model of unlinked loci (r = 0.5). Disruptive selection acts on the proportion of alleles from the parental populations (p,q): W = exp[-S(4pq)beta], such that the least fit genotype has fitness e-s. Where beta < 1, fitness is reduced for a wide range of intermediate genotypes; where beta > 1, fitness is only reduced for those genotypes close to p = 0.5. Even with strong epistasis, linkage disequilibria are close to sigma 2p'ip'j/rij, where p'i, p'j are the gradients in allele frequency at loci i, j. The barrier to gene flow, which is reflected in the steepening of neutral clines, is given by [formula: see text] where r, the harmonic mean recombination rate between the neural and selected loci, is here 0.5. This is a close approximation for weak selection, but underestimates B for strong selection. The barrier is stronger for small beta, because hybrid fitness is then reduced over a wider range of p. The widths of the selected clines are harder to predict: though simple approximations are accurate for beta = 1, they become inaccurate for extreme beta because, then, fitness changes sharply with p. Estimates of gene number, made from neutral clines on the assumption that selection acts against heterozygotes, are accurate for weak selection when beta = 1; however, for strong selection, gene number is overestimated. For beta > 1, gene number is systematically overestimated and, conversely, when beta < 1, it is underestimated.     

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.     

Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). 1 demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.Supported by PHS Grant ROI GM 32130     

RESPONSE TO SELECTION IN PARTIALLY SELF-FERTILIZING POPULATIONS. II. SELECTION ON MULTIPLE TRAITS

The structured linear model (SLM) is generalized to treat selection on multiple, correlated characters. Four different causes of phenotypic correlations are distinguished by the SLM: environmental covariance, identity disequilibrium, pleiotropy, and linkage disequilibrium. Each is characterized by distinct variables because they have different implications for character evolution. Correlations due to identity disequilibrium and linkage disequilibrium depend on both the mating system and the selection regime. As a consequence, they will evolve rapidly under selection. Correlations due to pleiotropy or environmental factors will evolve more slowly and are characterized by parameters that can be estimated from comparisons among relatives. These parameters include several novel “inbreeding covariance components” that emerge from the interaction of inbreeding and genetic dominance. Although data are limited, current estimates suggest that the expression of these components may substantially alter the pattern of multitrait evolution in self-fertilizing populations.     

EVOLUTION OF GENERALISTS AND SPECIALISTS IN SPATIALLY HETEROGENEOUS ENVIRONMENTS

Quantitative genetic models are used to investigate the evolution of generalists and specialists in a coarse-grained environment with two habitat types when there are costs attached to being a generalist. The outcomes for soft and hard selection models are qualitatively different. Under soft selection (e.g., for juvenile or male-reproductive traits) the population evolves towards the single peak in the adaptive landscape. At equilibrium, the population mean phenotype is a compromise between the reaction that would be optimal in both habitats and the reaction with the lowest cost. Furthermore, the equilibrium is closer to the optimal phenotype in the most frequent habitat, or the habitat in which selection on the focal trait is stronger. A specialist genotype always has a lower fitness than a generalist, even when the costs are high. In contrast, under hard selection (e.g., for adult or female-reproductive traits) the adaptive landscape can have one, two, or three peaks; a peak represents a population specialized to one habitat, equally adapted to both habitats, or an intermediate. One peak is always found when the reaction with the lowest cost is not much different from the optimal reaction, and this situation is similar to the soft selection case. However, multiple peaks are present when the costs become higher, and the course of evolution is then determined by initial conditions, and the region of attraction of each peak. This implies that the evolution of specialization and phenotypic plasticity may not only depend on selection regimes within habitats, but also on contingent, historical events (migration, mutation). Furthermore, the evolutionary dynamics in changing environments can be widely different for populations under hard and soft selection. Approaches to measure costs in natural and experimental populations are discussed.     

Pleiotropic Stabilizing Selection Limits the Number of Polymorphic Loci to at Most the Number of Characters

We demonstrate that, in a model incorporating weak Gaussian stabilizing selection on n additively determined characters, at most n loci are polymorphic at a stable equilibrium. The number of characters is defined to be the number of independent components in the Gaussian selection scheme. We also assume linkage equilibrium, and that either the number of loci is large enough that the phenotypic distribution in the population can be approximated as multivariate Gaussian or that selection is weak enough that the mean fitness of the population can be approximated using only the mean and the variance of the characters in the population. Our results appear to rule out antagonistic pleiotropy without epistasis as a major force in maintaining additive genetic variation in a uniform environment. However, they are consistent with the maintenance of variability by genotype-environment interaction if a trait in different environments corresponds to different characters and the number of different environments exceeds the number of polymorphic loci that affect the trait.     

Enzyme Polymorphism and Cyclic Parthenogenesis in DAPHNIA MAGNA. II. Heterosis following Sexual Reproduction

Cyclical parthenogenesis exaggerates the force of selection relative to recombination and will therefore enhance interlocus effects. Observations of electrophoretic variation in a natural population of Daphnia magna Straus (Crustacea: Cladocera) are interpreted in this light. Sexual reproduction led to Hardy-Weinberg equilibrium, but heterozygote excesses rapidly developed at each of three observed loci during subsequent parthenogenesis. Homozygote fecundity was often lower than that of heterozygotes; this may have been the cause of some of the observed frequency changes. The superior fitness of the enzyme heterozygotes does not imply that selection was necessarily acting on the enzyme loci themselves, since apparent heterosis is the expected result of linkage disequilibrium.     

Population dynamics of the additive polygenic system under truncation selection]

Common features of the equations describing dynamics of the additive polygenic system under truncation selection are summarized. A combination of parameters playing the role of the effective selective pressure on the ith polygenic locus was revealed. The product of mean relative fitnesses of the individual polygenic loci, [formula: see text], was shown to play the role of relative mean fitness of the polygenic population. This value depends on the measurable parameters of the character distribution in the population: [formula: see text]. It was shown that under the constant population number during truncation selection, the characteristic of the best genotype increases, [formula: see text]; which is also a product of the frequencies of preferable genotypes at individual polygenic loci. This value plays the role of the proportion of the number of the best ("champion") genotype in the population. In fact, this is the champion genotype polygene consensus pattern frequency, which a priori indicates the possibility of the champion pattern fixation. The analogue of Haldane's dilemma for the polygenic system which restrict the number of polygenes simultaneously subjected to adaptive evolution [formula: see text] was obtained for the case of constant effective population number (Ne = const).