Fertility and viability at the Sod locus in Drosophila melanogaster: non-additive and asymmetric selection

Experiments were designed to test in Drosophila melanogaster the effect of mating type at the Sod locus on fertility and viability. The experiments show that fertility is neither additive (or multiplicative) nor symmetric, i.e. that the fertility of a mating type cannot be predicted from the average fertility of the two genotypes involved in the mating. There is no significant male x female interaction with respect or progeny viability; but the interaction is significant for productivity, i.e. when fertility and viability are jointly taken into account. There is overdominance with respect to female fertility, but not with respect to male fertility or to viability. There also is alloprocoptic selection with respect to fertility and with respect to productivity, i.e. mating between like homozygotes are less fertile and productive than matings between dissimilar homozygotes. Selection at the Sod locus yields stable polymorphic equilibria, with the frequency of the F allele predicted at P = 0.641 or 0.695, respectively for low and high larval density.     

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.     

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     

Selection for Increased Mutation Rates with Fertility Differences between Matings

Previous studies of mutation modification have considered models in which selection is a result of viability differences that are sex symmetric. The results of a numerical study of a model in which selection is a result of fertility differences between mated pairs demonstrate that the type of selection to which a population is subject can have a significant impact on the evolution of various parameters of the genetic system. When the fertility of matings between individuals with different genotypes exceeds the fertility of at least some of the matings between individuals with the same genotype, selection may favor increased rates of mutation, in contrast to the results from all existing constant viability models with random mating and infinite population size. Increased mutation rates are most frequently favored when forward and back mutation occur at approximately equal rates and when the modifying locus is loosely linked to the selected locus. We present one example in which selection favors increased rates of mutation even though the selection scheme is reducible to one of differential viability between the sexes.     

Low fertility increases descendant socioeconomic position but reduces long-term fitness in a modern post-industrial society

Adaptive accounts of modern low human fertility argue that small family size maximizes the inheritance of socioeconomic resources across generations and may consequently increase long-term fitness. This study explores the long-term impacts of fertility and socioeconomic position (SEP) on multiple dimensions of descendant success in a unique Swedish cohort of 14 000 individuals born during 1915–1929. We show that low fertility and high SEP predict increased descendant socioeconomic success across four generations. Furthermore, these effects are multiplicative, with the greatest benefits of low fertility observed when SEP is high. Low fertility and high SEP do not, however, predict increased descendant reproductive success. Our results are therefore consistent with the idea that modern fertility limitation represents a strategic response to the local costs of rearing socioeconomically competitive offspring, but contradict adaptive models suggesting that it maximizes long-term fitness. This indicates a conflict in modern societies between behaviours promoting socioeconomic versus biological success. This study also makes a methodological contribution, demonstrating that the number of offspring strongly predicts long-term fitness and thereby validating use of fertility data to estimate current selective pressures in modern populations. Finally, our findings highlight that differences in fertility and SEP can have important long-term effects on the persistence of social inequalities across generations.     

The Two-Locus Model with Sex Differences in Recombination

The criteria for stability of the equilibrium with D=0 are obtained for the two locus model with multiplicative or symmetric fitnesses when the recombination values in males and females are different. It is shown that if r is defined to be equal to the average of the recombination values in males and females, then the criteria are exactly the same as in the standard two locus model.-The equilibrium values with D not equal0 are obtained for the symmetric fitness model. At this equilibrium, the absolute value of D is always greater (for the same average recombination value) if the recombination values in males and females differ than if they are equal.     

Fertility selection,genetic selection and evolution

The relationship between fertility selection as measured by the correlation in progeny number between parents and offspring, and selection at individual loci is investigated in humans. Estimates for the magnitude of fertility selection (0.1) and the rate of gene substitution (0.5 gene substitutions per generation per genome) are used in various mathematical models for selection. It is found that the observed magnitude of fertility selection cannot be explained by non‐epistatic directional selection at individual loci. A symmetric quantitative directional selection model is consistent with the observed data. But it is possible that fertility selection does not have a genetic basis.     

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.     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

Fitness and extra-group reproduction in male Verreaux's sifaka: An analysis of reproductive success from 1989-1999

Adult males in social groups often compete with other male group members for access to adult females. In some primate species, males also seek mating opportunities in neighboring social groups. Such extra-group fertilizations (EGFs) provide an additional source of variation in male fitness. This additional component of fitness provided by EGFs must be incorporated into analyses that investigate sources of variation in male lifetime reproductive success. In this study, a model is analyzed in which male fitness over a 10-year sample period is decomposed into additive and multiplicative variance and covariance components. The data come from an ongoing study of a wild population of Verreaux's sifaka (Propithecus verreauxi verreauxi) located at Beza Mahafaly Special Reserve, Southwest Madagascar. Paternity and demographic data for 134 males are used to decompose male fitness into the following three multiplicative components: reproductive lifespan during sample period, fertility, and offspring survival. These multiplicative components are estimated for males reproducing within their resident groups plus (i.e., the additive portion) for males reproducing in neighboring social groups. The analysis shows that variation in fertility makes the largest contribution to variation in total fitness, followed by variation in amount of time spent in sample period (which is a proxy of total reproductive lifespan) and variation in offspring survival. EGFs contribute an important source of variation to male fitness, and numerous factors enhance the opportunities for EGFs in male sifaka. These include female choice, a high degree of home range overlap, and a limited mating season.     

Natural Selection with Nuclear and Cytoplasmic Transmission. I. a Deterministic Model

A deterministic model allowing variation at a nuclear genetic locus in a population segregating two cytoplasmic types is formulated. Additive, multiplicative and symmetric viability matrices are analyzed for existence and stability of equilibria. The protectedness of polymorphisms in both nuclear genes and cytoplasmic types is also investigated in the general model. In no case is a complete polymorphism protected with this deterministic model. Results are discussed in light of the extensive variation in mtDNA that has recently been reported.     

Multiplicative Auditory Spatial Receptive Fields Created by a Hierarchy of Population Codes

A multiplicative combination of tuning to interaural time difference (ITD) and interaural level difference (ILD) contributes to the generation of spatially selective auditory neurons in the owl''s midbrain. Previous analyses of multiplicative responses in the owl have not taken into consideration the frequency-dependence of ITD and ILD cues that occur under natural listening conditions. Here, we present a model for the responses of ITD- and ILD-sensitive neurons in the barn owl''s inferior colliculus which satisfies constraints raised by experimental data on frequency convergence, multiplicative interaction of ITD and ILD, and response properties of afferent neurons. We propose that multiplication between ITD- and ILD-dependent signals occurs only within frequency channels and that frequency integration occurs using a linear-threshold mechanism. The model reproduces the experimentally observed nonlinear responses to ITD and ILD in the inferior colliculus, with greater accuracy than previous models. We show that linear-threshold frequency integration allows the system to represent multiple sound sources with natural sound localization cues, whereas multiplicative frequency integration does not. Nonlinear responses in the owl''s inferior colliculus can thus be generated using a combination of cellular and network mechanisms, showing that multiple elements of previous theories can be combined in a single system.     

Sperm competition games: a comparison of loaded raffle models and their biological implications

Our main aim is to compare the additive model, due to Mesterton-Gibbons, and the multiplicative model, due to Parker, of sperm allocation under sperm competition, when other influences are treated in the same way. We first review these (and other) models and their foundations, leading to a generalization of the multiplicative model. Sperm is assumed to cost energy, and this constraint is incorporated differently in the two models. These give the same results in the random-roles situation when the males occupy roles (of first and second to mate) randomly: the number of sperm ejaculated in the favoured role is greater than that in the disfavoured role by an amount that depends on the effect of sperm limitation (i.e. the probability that there is insufficient sperm to ensure full fertility). If the latter is negligible, or the fertilization raffle fair, this difference is zero, as Parker found originally. In the constant roles situation (where males of a particular type always occupy the same role) the predictions differ: the additive model has the same predictions as in the random roles case, but the multiplicative model predicts that males of the type occupying the favoured role ejaculate less than males of the type occupying the disfavoured role, in accord with Parker's original conclusion. The fitnesses of the two types of male can be calculated in the multiplicative model: the fitness of the favoured male is usually higher, even if he has to expend more energy in "finding" a female, e.g. through fighting, etc. These conclusions relate to inter-male behaviour (i.e. of different male types), as distinct from intra-male behaviour (i.e. of a given male when in different roles). We analyse situations in which one male type has some probability of acting in its less usual role: calculations with varying amounts of sperm limitation are presented. It is found that the presence of a male of a different type has an effect on intra-male ejaculate behaviour, which also depends critically on the role usually occupied. We conclude that the multiplicative model is the more accurate model and provides more information. Some experimental data on sperm numbers are used to find the effects of sperm limitation. For species which conform to the loaded raffle model, sperm limitation typically has small or negligible effects: in this case, we argue that empiricists should look for equal ejaculates in the two roles when studying random role situations; when roles are occupied non-randomly average sperm expenditure should be greater by male types typically occupying the disfavoured role, but within a male type, expenditure should be greater in the role it typically occupies.     

Detection of deleterious genotypes in multigenerational studies. III. Estimation of selection components in highly selfing populations

New paradigms in genetics have increased the chance of finding genes that appear redundant but in fact may have been preserved due to a small level of positive selection potential acting during each generation. Monitoring changes in genotypic frequencies within and between generations allows the dissection of the fertility, viability and meiotic drive selection components acting on such genes in natural and experimental populations. Here, a formal maximum likelihood procedure is developed to identify and estimate these selection components in highly selfing populations by fitting the time-dependent solutions for genotypic frequencies to observed multigenerational counts. With adult census alone, we can not simultaneously estimate all three selection components considered. In such cases, we instead consider a hierarchy of 11 models with either fewer selection components, complete dominance, or multiplicative meiotic drive with a single parameter. We identify the best-fitting of these models by applying likelihood ratio tests to nested models and Akaike's Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to non-nested models. With seed census, fertility and viability selection are not distinguishable and thus can only be estimated jointly. A combination of joint seed and adult census data allows us to estimate all three selection components simultaneously. Simulated data validate the estimation procedure and provide some practical guidelines for experimental design. An application to Arabidopsis data establishes that viability selection is the major selective force acting on the ACT2 actin gene in laboratory-grown Arabidopsis populations.     

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

Log-normal variation belts for growth curves

Prediction (confidence) or tolerance belts compound the uncertainty of sample estimates with the estimated extent of individual variation. The latter is therefore better described by variation belts, in which sample estimates are simply substituted for population parameters. Variation belts can provide valuable graphical indications concerning the goodness of fit of postulated error models. While multiplicative least-squares (MLS) methods appear appropriate in principle for biological growth, they are unsatisfactory in practice when logarithmically transformed data are heteroscedastic. Heteroscedastic multiplicative error models can be fitted by iteratively reweighted multiplicative least squares (IRMLS), but unacceptable negative or infinite residual variance estimates and unreasonably wide variation belts are occasionally obtained. These difficulties can be prevented by constrained iteratively reweighted multiplicative least squares (CIRMLS). Examples are presented concerning the metabolic allometry of white rats, the somatic growth of male elephant seals, and the growth of an experimental population of Paramecium caudatum.     

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.