Selection-mutation at a diallelic autosomal locus in a dioecious population

The population is assumed to be infinite dioecious with nonoverlapping discrete generations and random mating. It is assumed that the fitnesses and mutation rates are constant, heterozygotes are viable and the mutation rates are less than one-half. It is proved that the allelic frequencies converge to equilibria as the number of generations tends to infinity. The a priori types of phase portraits are determined. The method employed is elementary. The results extend those of [1, 2, 5, 8] to the case of selection-mutation rather than pure selection and those of [7] to the case of an autosomal rather than a sex-linked locus.     

Selection and mutation at a diallelic X-linked locus

Equilibria and convergence of gene frequencies are studied in the case of a diallelic X-linked locus under the influence of selection and mutation. The model used is that of an infinite diploid population with nonoverlapping discrete generations and random mating. It is proved that if the mutation rates and fitnesses are constant and the mutation rates are less than one-third, then global convergence of gene frequencies to equilibria occurs. The phase portraits of the dynamical system describing the change of allelic frequencies from one generation to the next are determined. Convergence of gene frequencies is monotone from a certain generation on if every other generation is skipped. In the case without mutation, our proof of this monotone convergence simplifies G. Palm's original proof [37].     

Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations

The evolutionary dynamics in general two-sex two-phenotype frequency-dependent selection models are studied with respect to underlying multi-allele one-locus genetic systems. Two classes of equilibria come into play: genotypic equilibria, with equilibrium allelic frequencies independent of the phenotype, and phenotypic equilibria, which are characterized by equal mean phenotypic fitnesses. The exact conditions for genotypic equilibria to exist and be stable and for phenotypic equilibria to exist and be evolutionarily attractive are examined. Using adequate definitions of mean fitnesses in general contexts of frequency-dependent selection in dioecious populations, we show that two phenotypes, when they can coexist in the population, tend to balance their fitnesses as far as is allowed by the genetic system as more alleles responsible for phenotype determination are introduced into the population.     

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.     

Effects of stripe rust on the evolution of genetically diverse wheat populations

Summary Eighteen populations, composed of four wheat (Triticum aestivum) varieties that were originally mixed together at equal frequencies, were grown for one-to-three generations at two locations. In addition, pure stands of the four varieties were grown in each year. Populations were either exposed to two stripe rust (Puccinia striiformis) races, protected from stripe rust, or exposed to alternating years of diseased and disease-free conditions. Regression of the logit of a variety's frequency versus generation number was used to calculate the relative fitness of each variety in each population. These analyses suggest that the relative fitnesses of the wheat varieties were affected by disease and geographic location and were constant over time. However, frequency-changes of varieties in the mixtures were negatively correlated with their planting frequencies (0.0001 < P < 0.085 in 14 out of 16 cases), suggesting that fitnesses were frequency-dependent in both the presence and absence of disease. We hypothesize that failure to detect frequency-dependence of fitness in the logit analyses was due to a limited number of generations and a limited range of initial variety frequencies. This is supported by data from longer-term studies in the literature that provide evidence for frequency-dependence of fitness in plant mixtures. Analyses of currently available field data suggest that stable equilibria may be a more likely outcome for mixtures of varieties that are more closely related and/or more uniformly adapted to the environment in which they are grown.Paper No. 9820 of the journal series of the Oregon Agricultural Experiment Station.     

The Maintenance of Single-Locus Polymorphism. IV. Models with Mutation from Existing Alleles

The ability of viability selection to maintain allelic polymorphism is investigated using a constructionist approach. In extensions to the models we have previously proposed, a population is bombarded with a series of mutations whose fitnesses in conjunction with other alleles are functions of the corresponding fitnesses with a particular allele, the parent allele, already in the population. Allele frequencies are iterated simultaneously, thus allowing alleles to be driven to extinction by selection. Such models allow very high levels of polymorphism to evolve: up to 38 alleles in one case. Alleles that are lethal as homozygotes can evolve to surprisingly high frequencies. The joint evolution of allele frequencies and viabilities highlights the necessity to consider more than the current morphology of a population. Comparisons are made with the neutral theory of evolution and it is suggested that failure to reject neutrality using the Ewens-Watterson test cannot be regarded as evidence for the neutral theory.     

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.     

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.     

The effect of selection on esterase allozymes in a barley population

Changes in gene and genotypic frequencies at four esterase loci were monitored over 25 generations in Composite Cross V, an experimental population of barley, to obtain experimental evidence concerning the balance of forces responsible for: (1) the marked differences in allelic frequencies among barleys from different ecogeographical regions of the world; and (2) the extensive allelic variation found within local populations of barley. Analyses of the highly significant changes in allelic frequencies which occurred in CCV showed they were due to directional selection favoring particular alleles and not to mutation, migration or genetic drift. The results show that intense balancing selection, featuring consistent excesses of heterozygotes, also occurred in CCV. It is concluded that among the factors of neo-Darwinian evolution, natural selection plays the predominant role in determining the observed patterns of allelic variation in the barley species as a whole.     

A general multilocus two-allele model for epistatic selection

Observations show that evolutionary processes often relate to multilocus epistatic selection. Here we develop further the approach suggested earlier by Zhivotovsky and Gavrilets to admit arbitrary multilocus epistasis. The obtained dynamic equations for allelic frequencies and gametic disequilibria are represented in a simple form. If selection is weak, this result extends Wright’s evolutionary equation to the case of cis-trans effects and sex differences in both recombination rates and genotypic fitnesses. Additionally to Wright’s equations for allelic frequencies, we derive equations for the gametic disequilibrium terms. We also give a general expression for the gametic disequilibria in a quasi-linkage state.     

Effects of genetic background on response to selection in experimental populations of Arabidopsis thaliana

The extent to which genetic background can influence allelic fitness is poorly understood, despite having important evolutionary consequences. Using experimental populations of Arabidopsis thaliana and map-based population genetic data, we examined a multigeneration response to selection in populations with differentiated genetic backgrounds. Replicated experimental populations of A. thaliana with genetic backgrounds derived from ecotypes Landsberg and Niederzenz were subjected to strong viability and fertility selection by growing individuals from each population at high density for three generations in a growth chamber. Patterns of genome-wide selection were evaluated by examining deviations from expected frequencies of mapped molecular markers. Estimates of selection coefficients for individual genomic regions ranged from near 0 to 0.685. Genomic regions demonstrating the strongest response to selection most often were selected similarly in both genetic backgrounds. The selection response of several weakly selected regions, however, appeared to be sensitive to genetic background, but only one region showed evidence of positive selection in one background and negative selection in another. These results are most consistent with models of adaptive evolution in which allelic fitnesses are not strongly influenced by genetic background and only infrequently change in sign due to variation at other loci.     

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.     

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.     

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.     

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.     

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.     

A general model to explore complex dominance patterns in plant sporophytic self-incompatibility systems

We developed a general model of sporophytic self-incompatibility under negative frequency-dependent selection allowing complex patterns of dominance among alleles. We used this model deterministically to investigate the effects on equilibrium allelic frequencies of the number of dominance classes, the number of alleles per dominance class, the asymmetry in dominance expression between pollen and pistil, and whether selection acts on male fitness only or both on male and on female fitnesses. We show that the so-called "recessive effect" occurs under a wide variety of situations. We found emerging properties of finite population models with several alleles per dominance class such as that higher numbers of alleles are maintained in more dominant classes and that the number of dominance classes can evolve. We also investigated the occurrence of homozygous genotypes and found that substantial proportions of those can occur for the most recessive alleles. We used the model for two species with complex dominance patterns to test whether allelic frequencies in natural populations are in agreement with the distribution predicted by our model. We suggest that the model can be used to test explicitly for additional, allele-specific, selective forces.     

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.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

Estimation of effective generation interval in Drosophila population cages

An approximation method for calculating the effective generation interval in populations with overlapping generations was presented and illustrated with Drosophila melanogaster data where a sustained heterosis (or pseudo-overdominance) was observed for sex-linked genes balanced by an inversion. An equilibrium frequeney was established in replicated 2-bottle population eages which was different for males and females. The fitnesses of the genotypes were thus estimable and the expected gene frequencies with diserete generations were calculated. The effective generation interval was estimated by fitting the observed frequencies to those expected by minimizing the lack of fit chisquare. The best fit resulted if the interval was allowed to increase as the population approached a stable age distribution. The best estimate indicated that there was an average interval of 14.75 days per generation in the first 5 generations and 16.25 days thereafter. This estimate is particular to 2-bottle eages; as the number of bottles increases the expected generation interval is also expected to inerease.Journal Paper Number 79-5-211 of the University of Kentucky Agricultural Experiment Station, and Journal Paper Number 7942 of the Purdue University Agricultural Experiment Station.