Male-female interactions in Drosophila melanogaster: model with one-locus and three-alleles

We develop a fertility model of fitness that is general in that it does not assume that the fitnesses of the mating combinations are symmetrical or that they are additive or multilicative (i. e., that they can be inferred from fitnesses of the two genotypes involved in a mating). %he model considers one locus with three alleles. An experimental test with Drosophila rnelanogaster confirms that the fitnesses of the mating types depart from both additivity (or multiplicativity) and symmetry although this last property is of no consequence for the development of analytical models). urnerical simulations yield the same, or very nearly the same, equilibrium freuencies as the analytical model, independently of whether or not Hardy-Weinberg equilibrium Trequencies are assumed at the beginning of each selection cycle.     

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models. The first is a phenotype space in which individuals are characterized by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form a giant connected cluster percolating throughout the phenotype space provided the viability probability is larger than 1/2(n). The second model explicitly describes genotype-to-phenotype and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at the point of phase transition between local and global are most conducive to the formation of the giant cluster. In the third class of models, particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have zero fitness. This setting can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation and is related to K-SAT problems, prominent in computer science. We analyze the conditions for the existence of viable genotypes, their number, as well as the structure and the number of connected clusters of viable genotypes. We show that analysis based on expected values can easily lead to wrong conclusions, especially when fitness correlations are strong. We focus on pairwise incompatibilities between diallelic loci, but we also address multiple alleles, complex incompatibilities, and continuous phenotype spaces. In the case of diallelic loci, the number of clusters is stochastically bounded and each cluster contains a very large sub-cube. Finally, we demonstrate that the discrete NK model shares some signature properties of models with high correlations.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.     

Models of Frequency-Dependent Selection with Mutation from Parental Alleles

Frequency-dependent selection (FDS) remains a common heuristic explanation for the maintenance of genetic variation in natural populations. The pairwise-interaction model (PIM) is a well-studied general model of frequency-dependent selection, which assumes that a genotype’s fitness is a function of within-population intergenotypic interactions. Previous theoretical work indicated that this type of model is able to sustain large numbers of alleles at a single locus when it incorporates recurrent mutation. These studies, however, have ignored the impact of the distribution of fitness effects of new mutations on the dynamics and end results of polymorphism construction. We suggest that a natural way to model mutation would be to assume mutant fitness is related to the fitness of the parental allele, i.e., the existing allele from which the mutant arose. Here we examine the numbers and distributions of fitnesses and alleles produced by construction under the PIM with mutation from parental alleles and the impacts on such measures due to different methods of generating mutant fitnesses. We find that, in comparison with previous results, generating mutants from existing alleles lowers the average number of alleles likely to be observed in a system subject to FDS, but produces polymorphisms that are highly stable and have realistic allele-frequency distributions.     

An n Locus Multiallele Model for Gene Substitution

We discuss the conceptual conflict between a slow series of gene substitutions as the mechanism of evolutionary change, and the apparent need for rapid and coordinated changes at many loci simultaneously in producing complex adaptations. To improve on the limitations of classical theory and accommodate the enormous amount of variability disclosed by electrophoretic studies, we develop a model that can deal with gene substitution at n loci, with numerous alleles at each locus. Fitness is treated somewhat differently from the usual way by allowing it to vary between zero and the number of offspring an individual of a particular species can produce. As maximum fitnesses, we chose five as typical of large mammals, 100 for insects like Drosophila, and 1000 for very prolific species. When our model is applied to the classical problem of determining the number of generations required to change the gene frequency from 0.0001 to 0.9999 (but for 100 loci rather than one), we find that it requires 22,899 generations when maximum fitness is five, 7,984 generations when maximum fitness is 100 and 5,333 generations when it is 1000. This is something of an improvement over the 300,000 generations calculated by Haldane (1957). By allowing the fitnesses in our model to be explicitly frequency dependent, these results are reduced considerably. In addition, allowing varying proportions of the population to inbreed reduces the number of generations required for the classical problem by as much as 50%. We also point out that, given the large amount of observed genetic variation, evolutionary change may not be so much a matter of classical gene substitution as it is of changing from one array of alleles to another. With our model, the array (0.5, 0.15, 0.2, 0.1, 0.05) can be changed to (0.03, 0.1, 0.2, 0.17, 0.5) at 1000 loci in 6,043, 2,108, or 1,408 generations, depending on whether the maximum fitness is five, 100, or 1000. Finally, we note that it is possible to substitute one array for another while continuously favoring heterozygotes.     

Stable Equilibria at Two Loci in Populations with Large Selfing Rates

The equilibrium structure of two-locus, two-allele models with very large selfing rates is found using perturbation techniques. For free recombination, r = 1/2, the following results hold. If the heterozygotes do not have at least an approximate 30% advantage in fitness relative to homozygotes, a stable equilibrium with all alleles present is possible only if all of the homozygote fitnesses differ at most by approximately the outcrossing rate, t, and all stable polymorphic equilibria have disequilibrium values, D, that are at most on the order of the outcrossing rate. Once the heterozygote fitnesses are above the threshold, there are stable equilibria possible with D near its maximum possible value. The results show that the observed disequilibria in highly selfed plant populations are not likely to result from selection leading to an equilibrium.     

PHYSIOLOGICAL AND BEHAVIORAL ADAPTATION TO VARYING ENVIRONMENTS: A MATHEMATICAL MODEL

We develop a mathematical model to explore the evolution of habitat selection and physiological adaptation in a heterogeneous environment. The model assumes the following conditions: 1) a panmictic population of infinite size; 2) prereproductive individuals mobile enough to move between patches; 3) alleles at one locus code for absence or presence of adaptation to detrimental patches; 4) alleles at a second locus code for absence or presence of behavior(s) that cause avoidance of the detrimental patches; 5) additive effects of alleles controlling physiology and behavior; 6) frequency-independent fitness. Results of the model indicate that nontrivial, polymorphic equilibria do not exist. The pattern of genotypic fitnesses and the initial allelic frequencies can influence whether the population adapts by physiological or behavioral mechanisms, or by both. Linkage between the two loci can alter the outcome of evolution, given specified genotypic fitness values and initial allelic frequencies.     

On the evolution of epistasis II: a generalized Wright-Kimura framework

The evolution of fitness interactions between genes at two major loci is studied where the alleles at a third locus modify the epistatic interaction between the two major loci. The epistasis is defined by a parameter epsilon and a matrix structure that specifies the nature of the interactions. When epsilon=0 the two major loci have additive fitnesses, and when these are symmetric the interaction matrices studied here produce symmetric viabilities of the Wright [1952. The genetics of quantitative variability. In: Reeve, E.C.R., Waddington, C.H. (Eds.), Quantitative Inheritance. Her Majesty's Stationary Office, London]-Kimura [1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278-281] form. Two such interaction matrices are studied, for one of which epistasis as measured by |epsilon| always increases, and for the other it increases when the linkage between the major loci is tight enough and there is initial linkage disequilibrium. Increase of epistasis does not necessarily coincide with increase in equilibrium mean fitness.     

LONG-TERM EXPERIMENTAL EVOLUTION IN ESCHERICHIA COLI. III. VARIATION AMONG REPLICATE POPULATIONS IN CORRELATED RESPONSES TO NOVEL ENVIRONMENTS

Twelve populations of Escherichia coli were founded from a single clone and propagated for 2000 generations in identical glucose-limited environments. During this time, the mean fitnesses of the evolving populations relative to their common ancestor improved greatly, but their fitnesses relative to one another diverged only slightly. Although the populations showed similar fitness increases, they may have done so by different underlying adaptations, or they may have diverged in other respects by random genetic drift. Therefore, we examined the relative fitnesses of independently derived genotypes in two other sugars, maltose and lactose, to determine whether they were homogeneous or heterogeneous in these environments. The genetic variation among the derived lines in fitness on maltose and lactose was more than 100-times greater than their variation in fitness on glucose. Moreover, the glucose-adapted genotypes, on average, showed significant adaptation to lactose, but not to maltose. That pathways for use of maltose and glucose are virtually identical in E. coli, except for their distinct mechanisms of uptake, suggests that the derived genotypes have adapted primarily by improved glucose transport. From consideration of the number of generations of divergence, the mutation rate in E. coli, and the proportion of its genome required for growth on maltose (but not glucose), we hypothesize that pleiotropy involving the selected alleles, rather than random genetic drift of alleles at other loci, was the major cause of the variation among the derived genotypes in fitness on these other sugars.     

Proportions of different habitat types are critical to the fate of a resistance allele

We describe a simple deterministic theoretical framework for analysing the gene frequency evolution of two alternative alleles at a single genetic locus in a habitat comprising two environments in which the genotypes have different relative fitnesses. We illustrate this for adaptation of pest insects, where one allele (resistance to toxins expressed in transgenic crops) is favoured in one environment (transgenic plants) and the other allele (susceptibility to toxins) is favoured in the other environment (‘refuges’ of non-transgenic plants). The evolution of allele frequencies depends on selection pressure because of relative sizes of the environments and relative fitnesses of the genotypes in each environment. We demonstrate that there are critical threshold proportions for habitat division that determine equilibrium allele frequencies. The stability of the system depends on relationships between the relative genotype fitnesses. In some cases, the division of the habitat in exactly the threshold proportions removes selection pressure and maintains polymorphism at all allele frequencies.     

Multiple adaptive substitutions during evolution in novel environments

We consider an asexual population under strong selection-weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. I. Underdominance and stability

It is pointed out that the standard selection models in population genetics all require some form of heterozygote advantage in fitness in order to guarantee the maintenance or stability of genetic polymorphisms. Even more recent results demonstrating the existence of stable two-locus polymorphisms with marginal underdominance at both loci are based on certain epistatically acting heterosis assumptions. This raises the question as to whether heterozygote advantage in fitness is indeed a generally valid principle of maintaining polymorphisms. To avoid ambiguity in definition of heterozygote advantage (overdominance) as it appears in multiallele or multilocus systems, a one-locus-two-allele model is considered. This model allows for sexually asymmetric selection and random mating. It is shown that the model produces globally stable polymorphisms exhibiting underdominance in fitness for a considerable and biologically reasonable range of selection values. Having thus properly refuted the general validity of the common overdominance principle, a modified version is suggested which covers the classical viability selection model and its extension to arbitrary, sexually asymmetric viability and fertility selection. This modified overdominance principle is based on the notion of fractional fitnesses and relates protectedness of biallelic polymorphisms to the extent to which each genotype reproduces its own type. The fact that the model treated displays frequency dependent fitnesses which may change in ranking while approaching equilibrium is discussed in relation to problems of the evolution of overdominance and underdominance.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.     

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.     

Phenogenetic drift and the evolution of genotype-phenotype relationships

Maintenance of a stable two-locus polymorphism is analyzed statistically by fitting a logistic regression with a quadratic function of genotypic fitnesses to the probability for a fitness set to maintain a polymorphism. The regression is fitted using a data set containing information on stable equilibria maintained by 32,00 randomly generated fitness sets with three recombination values (0. 005, 0.05, 0.5). Fitted logistic regressions discriminate with 88 to 90% accuracy between fitness sets maintaining and not maintaining a stable internal equilibrium, which implies the existence of a fitness structure (balance of fitnesses) maintaining a two-locus polymorphism. Aspects of the balance of fitnesses revealed by logistic regressions are discussed. It is demonstrated that logistic regression also discriminates between types of a stable polymorphism: globally stable polymorphism, several simultaneously stable polymorphisms, and stable equilibria in addition to a polymorphic one, which implies that different balances of fitnesses are responsible for the maintenance of different types of polymorphism.     

Genic Variation in Male Haploids under Deterministic Selection

Genic variation in male haploids and male diploids was compared assuming constant fitnesses (derived from computer-generated random numbers) and infinite population size. Several models were studied, differing by the fitness correlation between the sexes (r_s) and genotypes (r_g), and by the intensity of selection as measured by the coefficient of variation (CV) of the fitness distribution. Genic variation was quantified using the proportion of polymorphic loci, P, the gene diversity at polymorphic loci, H_p, and the gene diversity over all loci, H_a. The two genetic systems were compared via variation ratios: variation in male haploidy/variation in male diploidy.—P and H_a were markedly lower for male-haploids than for male diploids, the variation ratios declining with increasing r_s, r_g and CV, but the two genetic systems were similar for H_p. Except for male diploids with r_s = 1, the two sexes had different equilibrium gene frequencies but the sample sizes required to detect such differences reliably were larger than usually possible in surveys of natural populations.—Data from natural populations fit the above trends qualitatively, but the variation ratios are much lower than those from our analyses, except that for H_p, which is higher when Drosophila is excluded. Also, the frequency distribution of most common alleles from electrophoretic data has a deficiency of intermediate frequencies compared to that from the computer-generated sets of fitnesses, possibly reflecting either the influence of stochastic processes shifting frequencies away from equilibrium or the involvement of alleles under selection-mutation balance.——While electrophoretic data suggest that Drosophila has unusually high levels of genic variation, unusually low levels of genic variation in male haploids compared with male diploids are not strongly indicated. However, if further data confirm male haploids as having low levels of genic variation, likely explanations are that the bulk of electrophoretically detected variation involves fixed-fitness balancing selection, selection-mutation balance involving slightly deleterious recessive alleles, new favorable male haploid alleles moving more rapidly to fixation than under male diploidy and thus carrying linked loci to fixation faster, or some combination of these possible factors.     

INBREEDING DEPRESSION,GENETIC LOAD,AND THE EVOLUTION OF OUTCROSSING RATES IN A MULTILOCUS SYSTEM WITH NO LINKAGE

We studied deterministic models of multilocus systems subject to mutation–selection balance with all loci unlinked, and with multiplicative interactions of the loci affecting fitness, in partially self-fertilizing populations. The aim was to examine the fitnesses of the zygotes produced by outcrossing and by selling, and the magnitude of inbreeding depression, in populations with different levels of inbreeding. The fates of modifiers of the outcrossing rate were also examined. With biologically plausible parameter values, inbreeding depression can be very large in moderately selfing populations, particularly when the mutant alleles are fairly recessive and selection is weak. A modifier allele reducing the selfing rate can be favored under these circumstances. In more inbred populations, inbreeding depression is lower, and selection favors alleles that increase the selfing rate. When inbreeding depression is caused by mutant alleles with strong selective disadvantage, modifiers causing large increases in selfing can often be favored even when the inbreeding depression exceeds one-half, though in these circumstances modifiers increasing selfing by smaller amounts are usually eliminated. Weaker selection appears to be more favorable to the maintenance of outcrossing.     

Frequency-dependent selection and the maintenance of genetic variation: exploring the parameter space of the multiallelic pairwise interaction model

When individuals' fitnesses depend on the genetic composition of the population in which they are found, selection is then frequency dependent. Frequency-dependent selection (FDS) is often invoked as a heuristic explanation for the maintenance of large numbers of alleles at a locus. The pairwise interaction model is a general model of FDS via intraspecific competition at the genotypic level. Here we use a parameter-space approach to investigate the full potential for the maintenance of multiallelic equilibria under the pairwise interaction model. We find that FDS maintains full polymorphism more often than classic constant-selection models and produces more skewed equilibrium allele frequencies. Fitness sets with some degree of rare advantage maintained full polymorphism most often, but a wide variety of nonobvious fitness patterns were also found to have positive potential for polymorphism. An example is put forth suggesting possible explanations for multiallelic polymorphisms maintained despite positive FDS on individual alleles.