The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

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.     

The effect of positive assortative mating at one locus on a second linked locus

Summary A model for positive assortative mating based on genotype for one locus is employed to investigate the effect of this mating system on the genotypic structure of a second linked locus as well as on the joint genotypic structure of these two loci. It is shown that the second locus does not attain a precise positive assortative mating structure, but yet it shares a property that is characteristic of positive assortative mating, namely an increase in the frequency of homozygotes over that typically found in panmictic structures. Given any arbitrary genotypic structure for the parental population, the resulting offspring generation possesses a structure at the second locus that does not depend on the recombination frequency, while the joint structure of course does. In case assortative mating as well as linkage are not complete, there exists a unique joint equilibrium state for the two loci, which is characterized by complete stochastic independence between the two loci as well as by Hardy-Weinberg proportions at the second locus. For the second locus alone, Hardy-Weinberg equilibrium is realized if and only if gametic linkage equilibrium and an additionally specified condition are realized.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

A Two-Stage Test for Distinguishing Random,Pseudo-Random and Nonrandom Mating Populations

There is no such an implication that a population in Hardy-Weinberg equilibrium must have undergone random mating. Therefore, it is unequivocal that the usual tests for “Hardy-Weinberg equilibrium” are indeed tests for “random union of gametes” but not for “random mating”. In this paper, utilizing population characteristics expressed in equilibrium state (equilibrium or disequilibrium) and mating behavior (random or nonrandom), a two-stage testing procedure for distinguishing random, pseudo-random and nonrandom mating populations is proposed. At the first stage, a population is tested for Hardy-Weinberg equilibrium. If insignificant result (i.e., in equilibrium) is obtained, then to a second stage the population is further tested for mating behavior. Random mating-pairs data are needed here for analysis instead of random individuals for usual Hardy-Weinberg equilibrium tests. Since distinguishing the three types of mating populations depends on the combined results of two stages, the probability of correct determination of the two-stage tests is discussed by simulation studies.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Comparisons of positive assortative mating and sexual selection models

A series of theoretical models of positive assortative mating and sexual selection are contrasted. It is established that for a dominant trait partial positive assortative mating generally implies some fixation, whereas sexual selection exhibits a unique globally stable polymorphism exhibiting Hardy-Weinberg proportions. The effects of monogamy against polygamy do not qualitatively alter the equilibrium outcomes, although the rate of evolutionary change is generally slowed with monogamy vis-à-vis polygamy. For sexual selection the influence of timing of random mating as against preferential mating causes no change in the equilibrium states, although the rates of convergence can be slowed if sexual selection occurs late in the breeding season. Under assortative mating the timing can alter the equilibrium outcomes. The amount of heterozygosity is always deficient in cases of assortative mating, but always exhibits Hardy-Weinberg ratios under a sexual selection mechanism. This suggests that observations consistent with Hardy-Weinberg equilibrium states cannot preclude ipso facto certain forms of selection forces, including mating patterns and some natural selection structures.     

Variation in mating propensity and fertility in isofemale strains of Drosophila ananassae

Mating propensity and fertility were studied in five laboratory strains of Drosophila ananassae which were established from single females collected from different geographical localities. The results show statistically significant variation among different isofemale lines with respect to mating propensity and fertility. The strains showing greater sexual activity produce more progeny. Thus, there is a positive correlation between mating activity and fertility in D. ananassae. The comparison of mating frequencies of strains and their hybrids reveals the existence of heterosis and reciprocal effects. The data suggest that the males are more subject to intrasexual selection.     

Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

Some multiallele partial assortative mating systems for a polygamous species

The consequences of preferential mating in the presence of partial assortative and sexual selection mechanisms are ascertained for a two-allele one-locus trait involving two phenotype classes C₁ = {all homozygotes} and C₂ = {heterozygotes}. Relevant biological cases may include Burley (1977, Proc. Nat. Acad. Sci. USA 74, 3476–3479), Wilbur et al. (1978, Evolution 32, 264–270), and Singh and Zouros (1978, Evolution 32, 342–353). When the preference rate for the heterozygote exceeds that for homozygotes, it is established that the unique stable state is the central Hardy-Weinberg equilibrium. The rate of approach is faster with sexual selection than for the corresponding model of assortative mating. When the preference rates favor the homozygotes then in this symmetric model of sexual selection two asymmetric Hardy-Weinberg polymorphisms can evolve, and which succeeds depends on initial conditions. The models are also analyzed with natural selection acting on phenotypes superimposed on assortative mating. In this case we can have up to three coexisting stable states involving both fixation alternatives and a central polymorphism. The corresponding model with sexual selection maintains either the central equilibrium as in assortative mating or two asymmetric polymorphic equilibria.     

The Fundamental Theorem of Natural Selection in Ewens'' Sense (Case of Fertility Selection)

We show that the Fundamental Theorem of Natural Selection in Ewens' sense is valid in the case of fertility selection: the additive genetic variance in fertility divided by the mean fertility is exactly equal to the partial change in the mean fertility from the current generation to the next. This partial change is the increase in the mean additive value caused by frequency changes from one generation to the next but keeping unchanged the additive values. The only hypothesis on mating is that it does not affect the allelic frequencies in the sense that these are the same before and after mating in the parental generation, which occurs for a wide range of mating patterns going from random mating to several regular systems of inbreeding and cases of assortative mating. The fertility of couples is determined by the genes at an arbitrary number of loci, and the additive (average) allelic effects are defined by a linear system of equations, which is used to extend Ewens' optimality principle to the case of fertility selection.     

A multi-locus continuous-time selection model

Summary A continuous time selection model is formulated for a diploid monoecious population with multiple alleles at each of an arbitrary number of loci, incorporating differential fertility and mortality as well as arbitrary mating and age structure. The model is simplified in the case of age-independence and for the case of a stable age distribution. The age-independent model is examined in detail for the special case of multiple alleles at each of two loci. This model is analyzed under the assumptions of random mating and additive fertilities, with close attention given to the behavior of the system with respect to Hardy-Weinberg proportions and linkage equilibrium.M. M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Conditions under Which the Mean Fertility Is Maximized When a Population Is at a Stable Equilibrium

It is assumed that there is a population with two alleles at one locus, random mating of adults and selection only involving differential fertilities. By making use of the Kuhn-Tucker theory of optimization under constraints, conditions are derived under which stable equilibrium frequencies x, y and z of the three genotypes are the same as those that maximize the mean fertility of the population. We derive all sets of frequencies of this type for the Hadeler-Liberman symmetric fertility model and all such sets for which at least one genotype is missing for the general model. If the population has frequencies that are initially near those at which there is both a stable equilibrium and maximization of the mean fertility, then the mean fertility (t) at time t is nondecreasing with t as t -> &. It is found that it is possible for the stable equilibrium maximum points (x, y, z) to be one or two points on a ridge on which the mean fertility is maximized or the entire set of points on the ridge. Furthermore, may be smaller on this ridge than at another stable equilibrium point at which is not even locally maximized.     

Nuclear androdioecy and gynodioecy

We formulate two single-locus Mendelian models, one for androdioecy and the other one for gynodioecy, each with 3 parameters: t the male (female) fertility rate of males (females) to hermaphrodites, s the fraction of the progeny derived from selfing; and g the fitness of inbreeders. Each model is expressed as a transformation of a 3 dimensional zygotic algebra, which we interpret as a rational map of the projective plane. We then study the dynamics for the evolution of each reproductive system; and compare our results with similar published models. In this process, we introduce a general concept of fitness and list some of its properties, obtaining a relative measure of population growth, computable as an eigenvalue of a mixed mating transformation for a population in equilibrium. Our results concur with previous models of the evolution of androdioecy and gynodioecy regarding the threshold values above which the sexual polymophism is stable, although the previous models assume constant the fraction of ovules from hermaphrodites that are self pollinated, while we assume constant the fraction of the progeny derived from selfing. A stable androdioecy requires more stringent conditions than a stable gynodioecy if the amount of pollen used for selfing is negligible in comparison with the total amount of pollen produced by hermaphrodites. Otherwise, both models are identical. We show explicitly that the genotype fitnesses depend linearly on their frequencies. Simulations show that any population not at equilibrium always converges to the equilibrium point of higher fitness. However, at intermediate steps, the fitness function occasionally decreases.     

Little effect of delayed mating on fecundity or fertility of female fungus gnats Lycoriella ingenua

The effect of delayed female mating for the mushroom fungus gnat Lycoriella ingenua is investigated. We examine the effect of delaying female mating on the fertility and egg viability of female flies that have a mating delay of 0–5 days after emergence. Male fly age is held constant. Female age does not impact male acceptance and most flies copulate within seconds of pairing. We find that female flies experiencing mating delays of 0–4 days after emergence lay a similar number of eggs onto artificial substrates. Females that experience a mating delay of 5 days lay 54% fewer eggs than those that mate on day 0 (day of emergence). There is no effect of mating delay on the percentage of larvae that emerge. The results of the present study indicate that mating delays have little effect on the fertility or fecundity of the mushroom fungus pest L. ingenua.     

Frequency of chromosome polymorphism for pericentric inversions and B-chromosomes in Spanish populations of Rattus rattus frugivorus

Inversion frequencies in chromosomes 16 and 18 and B-chromosome frequency have been studied in three populations of Rattus rattus frugivorus.In two of these, Cuenca and San Pedro del Pinatar, the frequencies of homozygous and heterozygous individuals do not differ significantly from the Hardy-Weinberg equilibrium for both chromosome pairs. By contrast, in the Vega de Granada population there are fewer heterozygous and more homozygous individuals than expected on the basis of the Hardy-Weinberg distribution, although the frequency distributions of karyotypes in these three populations are not significantly different.In relation to the B chromosome, the Cuenca and San Pedro populations have frequencies of B-carrying animals of 0.25 and 0.22 respectively, the Vega de Granada population of 0.80.     

On Nonrandom Mating Systems for Attaining Hardy-Weinberg Equilibrium

LI (1988) showed that random mating is a sufficient, not a necessary condition for the Hardy-Weinberg principle. A nonrandom mating population that behaves like a random mating population is thus called a ‘pseudo-random mating population’ by him. The pseudo-random mating system studied by him has been focused on those populations in which the parental generation is in Hardy-Weinberg proportions. In other words, the mating type frequency deviations from random mating for each parental genotype add up to zero. In this article these restrictions are relaxed and new pseudo-random mating systems that immediately yield Hardy-Weinberg offspring are also obtained. This is possible because reciprocal crosses have identical segregation probabilities for an autosomal locus, and the manipulation of the combined frequency of reciprocal crosses does not change the gene frequency of the population. A comparison of these new patterns with that of Li is given in the Discussion.     

CYTONUCLEAR THEORY FOR HAPLODIPLOID SPECIES AND X-LINKED GENES. II. STEPPING-STONE MODELS OF GENE FLOW AND APPLICATION TO A FIRE ANT HYBRID ZONE

We develop cytonuclear, hybrid zone models for haplodiploid species or X-linked genes in diploid species using a stepping-stone framework of migration, in which migration rates vary with both direction and sex. The equilibrium clines for the allele frequencies, cytonuclear disequilibria, and frequencies of pure parental types are examined for species with diagnostic markers, under four important migration schemes: uniform migration of both sexes in both directions, greater migration of both sexes from one direction, greater migration of females, and greater migration of males. Of the three cytonuclear variables examined, the allele frequency clines are the most informative in differentiating among the various migration patterns. The cytonuclear disequilibria and the frequency of the pure parental types tend to be useful only in revealing directional asymmetries in migration. The extent of hybrid zone subdivision has quantitative but not qualitative effects on the distribution of cytonuclear variables, in that the allele frequency clines become more gradual, the cytonuclear disequilibria decrease in magnitude, and the frequencies of pure parentals decline with increasing subpopulation number. Also, the only major difference between the X-linked and haplodiploid frameworks is that a higher frequency of pure parentals is found when considering haplodiploids, in which male production does not require mating. The final important theoretical result is that censusing after migration yields greater disequilibria and parental frequencies than censusing after mating. We analyzed cytonuclear data from two transects from a naturally occurring hybrid zone between two haplodiploid fire ant species, Solenopsis invicta and S. richteri, using our stepping-stone framework. The frequency of S. invicta mtDNA exceeds the frequency of the S. invicta nuclear markers through much of this hybrid zone, indicating that sex differences in migration or selection may be occurring. Maximum-likelihood estimates for the migration rates are very high, due to an unexpectedly large number of pure parental types in the hybrid zone, and differ substantially between the two transects. Overall, our model does not provide a good fit, in part because the S. invicta–S. richteri hybrid zone has not yet reached equilibrium.     

Continuous selective models

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.     

MATING PATTERN AND FITNESS-COMPONENT ANALYSIS ASSOCIATED WITH INVERSION POLYMORPHISM IN A NATURAL POPULATION OF DROSOPHILA BUZZATII

Direct studies of mating success or mating pattern associated with Mendelian factors rarely have been carried out in nature. From the samples taken for the standard analyses of selection components, it is not usually possible to obtain the mating table, and only directional selection for male mating success can be detected. Both processes, mating pattern and differential mating probability, together with other fitness components, have been investigated for the inversion polymorphism of a natural population of the cactophilic species Drosophila buzzatii. Two independent samples of adult flies were collected: nonmating or single individuals (base population) and mating pairs (mating population). All individuals were karyotyped for the second and fourth chromosomes. A sequence of models with increasing simplicity was fitted to the data to test null hypotheses of no selection and random union of gametes and karyotypes. The main results were (1) no deviations from random mating were found; (2) differential mating probability was nonsignificant in both sexes; (3) inversion and karyotypic frequencies did not differ between sexes; and (4) karyotypic frequencies did not depart from Hardy-Weinberg expectations. These results are discussed in light of complementary evidence showing the need for interpreting with caution no-effect hypotheses such as the ones tested here. The use of complementary selective tests in these studies is suggested.