Adaptive Topography in Fertility-Viability Selection Models: The Haplodiploid Case

A linear combination of partial changes of mean fitnesses from one generation to the next one is shown to be approximately equal to the additive genetic variance in fitness after enough generations and away from equilibrium in random mating haplodiploid populations under arbitrary weak frequency-dependent selection on sex-differentiated viability of individuals and sex-differentiated fertility of matings controlled at a single multiallelic locus. The result can be applied to X-linked locus models in diploid populations. The result is used to deduce approximate adaptive topographies far frequency-independent selection models in the cases of nonsex-differentiated fertilities and multiplicative sex-differentiated fertilities and for kin selection models in family-structured populations under the assumptions of single insemination and multiple insemination of females. Multiple insemination creates frequency-dependent selection regimes.     

The Evolution of One- and Two-Locus Systems. II

Weak selection in a monoecious population is studied in two multiallelic panmictic models. In the first, a single locus is considered with continuous time and age-independent fertilities and mortalities. If the fertilities of the various matings and the genotypic mortalities may be expressed with an error at most of the second order in s (i.e., O(s ²)), where s is the intensity of selection, as sums of terms corresponding to the different genotypes and alleles, respectively, then after several generations the deviations from Hardy-Weinberg proportions are of O(s²). In the second model, two loci are treated with discrete nonoverlapping generations. It is shown that if the epistatic parameters are of O(s²), then after several generations the linkage disequilibria are reduced to O(s²). Assuming only weak selection, it is proved that in both models, after several generations, the total change in mean fitness is generally positive. It is likely that the exclusion of the initial period is usually unnecessary in natural populations. Exceptions are discussed.     

Wild O chromosomes of Drosophila subobscura from different geographic regions have different effects on viability

By means of the marker strain Va/Ba wild chromosomes O of Drosophila subobscura were extracted from eight natural populations situated on a north-south gradient from Sweden or Scotland to Tunesia. Lethal frequencies and viability effects of the wild chromosomes O were studied in homozygous and random heterozygous combinations. In accordance with results from other Drosophila species random heterozygotes were always more viable than homozygotes. The viability-determining polygene system proved, however, dominant to some degree. Geographic differences became apparent especially with respect to three different characteristics: (1) The lethal frequencies for chromosomes O from central populations are higher than for those from northern and southern marginal populations; (2) Mean viabilities of non-lethal homozygotes and random heterozygotes are lower for central than for marginal populations; (3) The increase of viability through heterozygosity is more pronounced in the northern populations than in the others. The differences are thought to be mainly due to differences in the adaptation strategy of marginal and central populations. The viability fitness components seem of more importance for the marginal populations while fertility components may be of greater weight under central conditions. The geographic variability of the viability polygene system is finally compared with that of other genetic traits in D. subobscura.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

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.     

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.     

Heterosis for viability,fecundity, and male fertility in Drosophila melanogaster: comparison of mutational and standing variation.

If genetic variation for fitness traits in natural populations ("standing" variation) is maintained by recurrent mutation, then quantitative-genetic properties of standing variation should resemble those of newly arisen mutations. One well-known property of standing variation for fitness traits is inbreeding depression, with its converse of heterosis or hybrid vigor. We measured heterosis for three fitness traits, pre-adult viability, female fecundity, and male fertility, among a set of inbred Drosophilia melanogaster lines recently derived from the wild, and also among a set of lines that had been allowed to accumulate spontaneous mutations for over 200 generations. The inbred lines but not the mutation-accumulation (MA) lines showed heterosis for pre-adult viability. Both sets of lines showed heterosis for female fecundity, but heterosis for male fertility was weak or absent. Crosses among a subset of the MA lines showed that they were strongly differentiated for male fertility, with the differences inherited in autosomal fashion; the absence of heterosis for male fertility among the MA lines was therefore not caused by an absence of mutations affecting this trait. Crosses among the inbred lines also gave some, albeit equivocal, evidence for male fertility variation. The contrast between the results for female fecundity and those for male fertility suggests that mutations affecting different fitness traits may differ in their average dominance properties, and that such differences may be reflected in properties of standing variation. The strong differentiation among the MA lines in male fertility further suggests that mutations affecting this trait occur at a high rate.     

A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

Multinomial-Sampling Models for Random Genetic Drift

Three different derivations of models with multinomial sampling of genotypes in a finite population are presented. The three derivations correspond to the operation of random drift through population regulation, conditioning on the total number of progeny, and culling, respectively. Generations are discrete and nonoverlapping; the diploid population mates at random. Each derivation applies to a single multiallelic locus in a monoecious or dioecious population; in the latter case, the locus may be autosomal or X-linked. Mutation and viability selection are arbitrary; there are no fertility differences. In a monoecious population, the model yields the Wright-Fisher model (i.e., multinomial sampling of genes) if and only if the viabilities are multiplicative. In a dioecious population, the analogous reduction does not occur even for pure random drift. Thus, multinomial sampling of genotypes generally does not lead to multinomial sampling of genes. Although the Wright-Fisher model probably lacks a sound biological basis and may be inaccurate for small populations, it is usually (perhaps always) a good approximation for genotypic multinomial sampling in large populations.     

Selection with gene-cytoplasm interactions. I. Maintenance of cytoplasm polymorphisms

General conditions for the protectedness of gene-cytoplasm polymorphisms are considered for a biallelic model with two cytoplasm types and under the assumption that nuclear polymorphisms cannot be maintained in the presence of only one cytoplasm type. Analytical results involving male fertilities, female fertilities, viabilities and selfing rates are obtained, and numerical results show spiral and cyclic behavior of population trajectories. It is shown that a maternally inherited cytoplasmic polymorphism cannot be maintained in the absence of a nuclear polymorphism, and that a gene-cytoplasm polymorphism can only be maintained if the population shows sexual asymmetry, i.e. , if the ratio of male to female fertility varies among genotypes. Thus, the classical viability selection model does not allow gene-cytoplasm polymorphisms.     

Approaches to the Hardy-Weinberg manifold

Let fertilities and death rates be additive, let fertilities be positive, and let mating be random in the Nagylaki-Crow continuous model of selection at a multiallelic locus in a monoecious population. Then polymorphisms are in Hardy-Weinberg proportions. If some fertilities vanish, there is an example of a diallelic polymorphism that is not in Hardy-Weinberg proportions. If the fertilities are larger, in one sense or another, than the difference between any two death rates, then convergence to the Hardy-Weinberg manifold is shown. If, in addition, the Malthusian parameters are constant, and only a finite number of equilibria exist, then global convergence to equilibria is proved.     

A Numerical Simulation of the One-Locus, Multiple-Allele Fertility Model

Numerical simulations were performed to determine the equilibrium behavior of the one-locus fertility model in which fitness is considered as a property of a pair of mating diploids. A series of patterns of "fertility matrices" were considered for a single locus with two to six alleles. From these simulations, 19 different statistics were collected that characterize, at equilibrium, the heterozygosity, the mean fitness and the fate of populations begun at the allele-frequency centroid. For more than one-half of the trajectories produced by random fertility matrices, there was a decrease in the mean fitness at some time on the way to equilibrium. The mean number of alleles maintained at equilibrium increased only slightly with matrix dimension. Despite the potential for fertility models to display multiple stable equilibria, random fertility models maintain fewer distinct stable points than do random one-locus viability models. Pleiotropic models were also considered with fertility and viability selection operating sequentially within each generation. Most of the equilibrium statistics (with the exception of mean fertility) for the pleiotropic model were intermediate between the corresponding random viability and fertility models.     

Measuring selection in human populations using the growth rate per generation

Estimates of the speed of evolution between generations depend on the association between individual traits and a measure of fitness. The two most frequently used measures of fitness are the net reproduction rate and the 1-year growth factor implied by the fertility and mortality rates. Results based on the two lead to very different results. The reason is that the 1-year growth factor is not a measure of change between generations. Therefore, studies of changes between generations should use the amount of growth over the length of a generation. This is especially important for studies of human populations because of the long length of generation. In addition, estimates based on a single year''s growth are overly sensitive to data on individuals who fail to reproduce. The effects of using a generational measure are demonstrated using data from Kenya and Ukraine. These results demonstrate that using a 1-year growth rate to measure fitness leads to estimates that understate the rate at which evolution changes the characteristics of a human population.     

Selection on viability of individuals heterozygous for the temperature-sensitive lethal mutation l(2)M167(DTS) in experimental populations of Drosophila melanogaster

In experiments on introduction of mutation l(2)M167(DTS) in Drosophila melanogaster populations, larval and pupal viability and developmental rate are limiting factors determining the intensity of selection on the l(2)M167(DTS) mutation. Notwithstanding the rapid elimination of the mutation from the population, positive selection for viability was shown, which increased fitness of the mutation carriers in generations. The fitness component viability was estimated in individuals l(2)M167(DTS)/+; relative to that of wild-type individuals, it varied from 0.1 to 1. Factors affecting this trait in overcrowded populations were found.     

Genetics of acheiropodia (the handless and footless families of Brazil). VII. Population dynamics.

Since carriers of the acheiropodia gene cannot be distinguished from noncarriers, parents and normal sibs of affected individuals have been used to estimate the fitness of heterozygotes. No significant difference in biologic fitness (viability and fertility) between normal sibs and the general population could be detected. A comparison between acheiropods and their normal sibs showed the following: (1) a nonsignificant difference in stillbirth rate; (2) a higher mortality rate of acheiropods in the first 5 years of life; (3) a relative viability not larger than .7; (4) a relative fertility no greater than .14, due to "cosmetic effects"; and (5) a fitness of .10 or lower. The total number of acheiropodia genes in Brazil has been calculated as 25,000 in the 1970s. The data are compatible with an extremely low mutation rate and a very stable locus. It is suggested that all Brazilian acheiropods can be traced to a single mutation. A conservative estimate of the number of acheiropods to appear in the future in Brazil is 14,000 with an extinction time of no less than 2,300 generations or almost 70,000 years. A variety of other parameters have been calculated.     

Viabilities of Third Chromosomes of DROSOPHILA PSEUDOOBSCURA Differing in Relative Competitive Fitness

Arrowhead (AR) third chromosome arrangements of Drosophila pseudoobscura, whose competitive fitnesses had been determined in population cages, were tested for their genetic loads in homozygous, heterozygous (homokaryotypic), and heterokaryotypic (AR/CH) combinations. The results showed that their competitive population cage performances were correlated to their viabilities as homozygotes but were not correlated to their viabilities as heterozygotes or as heterokaryotypes. However, the results do not fit in too simply with the mutational model of population structure, since the improvement of homozygous viability with increased competitive fitness was not accompanied by a significant degree of dominance as measured by the regression of viabilities of heterozygotes on homozygotes. Only the AR chromosomes derived from the population with poorest competitive fitness showed marked partial dominance (h=.35). The viabilities of heterokaryotypes were markedly uniform for all chromosomes tested and produced significantly greater numbers of flies per culture than the homokaryotypes. In general, the results show that the ranking of relative competitive fitnesses of these chromosomes is not a simple extrapolation of their viabilities, although marked changes in the populations tested have occurred. It is proposed that the differences in competitive fitness, homozygous viability, and degree of dominance observed among these chromosomes, arise from differences in genetic variability which enable different linkage relationships to be established for genes affecting these attributes.     

Selection in diplo-haplonts

A deterministic selection model is considered for diplohaplontic populations which reproduce in non-overlapping generations and form zygotes by random fusion of gametes. The model allows for vegetative propagation as well as for viability and fertility selection in both the haploid and diploid phases. Fertility selection in the haploid phases is permitted to differ according to sex, and all selection coefficients are assumed to be constant over the generations. The main result obtained is that this model, including all of the above selective forces, is formally and analytically equivalent to and thus shares all of the properties of the “classical” viability selection model. This result is essentially due to the fact that when haplogenotypic frequencies in the sporophytes are considered in consecutive generations, all of the different selection forces can be expressed with the help of a single selection coefficient for each diplogenotype. The benefits from this simplification are demonstrated with the help of two examples for a single, multiallelic locus. Restricting selection to the haplophases only, it is shown that fertility selection acting in opposing directions in the sexes can lead to stable multiallelic polymorphisms without assuming spatial heterogeneity of the environment. However, it is unlikely that the conditions for this will be realized in actual populations for more than two alleles. The second example is concerned with the problem of maintaining several sexual types (males, females and bisexuals) in a population, with special emphasis on the conditions for the establishment of dioecious or monoecious systems of sexuality or a mixture of these.     

Selection by Fertility in DROSOPHILA PSEUDOOBSCURA

Fertility, the component of selection due to female fecundity and male mating success, differed significantly among the ST/ST, ST/AR, and AR/AR karyotypes in experimental populations and varied with karyotypic frequency. In relation to ST/AR, ST/ST females and males had higher fertilities at low frequency; AR/AR males and females were at a significant fertility disadvantage at intermediate frequency, while at low and at high frequencies their fertilities matched or exceeded that of the heterokaryotype. These fertility differences were comparable in size to viability differences previously reported for D. pseudoobscura karyotypes. Differential fertility seems likely to be an important element, perhaps just as important as differential viability, in the balancing selection that maintains the chromosomal polymorphism in this species.     

Natural selection on age-specific fertilities in human females: comparison of individual-level fitness measures.

Lifetime reproductive success and timing of reproduction are key components of life-history evolution. To understand the evolution of reproductive schedules, it is important to use a measure of fitness that is sensitive both to reproductive quantity and reproductive timing. There is a contradiction between the theory, which mainly focuses on the rate measures of fitness (r and lambda), and empirical studies, which mainly use lifetime reproductive success (LRS), or some of its correlates, as a fitness measure. We measured phenotypic selection on age-specific fertilities in three pre-modern human populations using individually estimated finite rate of increase, er (lambda). We found that lambda and lifetime reproductive success ranked individuals differently according to their fitness: for example, a female giving birth to four children at a young age may actually have a higher fitness than a female giving birth to six children at a greater age. Increase in fertility at the young age classes (15-19 years) was favoured by selection, but the intensity of selection on fertility was higher in the older age classes (20-30 years), where the variance in fertility was highest. Hence, variation in fertility in the older age classes (20-30) was actually responsible for most of the observed variation in fitness among the individuals. Additionally, more than 90% of variation in fitness (lambda) was attributable to individual differences in LRS, whereas only about 5% of all variation in fitness was due to differences in the reproductive schedule. The rate-sensitive fitness measure did not significantly challenge the importance of total fertility as a component of fitness in humans. However, the rate-sensitive measure clearly allowed more accurate estimation of individual fitness, which may be important for answering some more specific questions.     

A theoretical and numerical assessment of genetic variability

The equilibrium behavior of one-locus viability selection models is studied numerically. The selection schemes include randomly chosen viabilities, viabilities chosen to measure a hypothetical distance between the alleles making up the genotype and viabilities that obey various allelic dominance relations. From 3 to 8 alleles are considered. Among the key conclusions are (1) equilibria that are most polymorphic do not usually have the highest mean fitness, (2) the more structure there is in the choice of the viability model, the greater is the level of polymorphism at equilibrium, and (3) for the numbers of alleles chosen here, the equilibrium reached by iteration from the centroid of the allele frequency simplex is the best predictor of the equilibrium attainable from randomly chosen starting vectors. Preliminary evidence shows that this is not the case for 16 alleles.