Maternal investment and male reproductive success in angiosperms: parent-offspring conflict or sexual selection?

It is possible to interpret components of seed development in angiosperms from the perspective of parent-offspring conflict (a special case of kin selection) or sexual selection. Available parent-offspring conflict models predict the evolution of traits determining the outcome of competition among related individuals soliciting maternal resources. In such models, ‘selfishness’ may spread even if it reduces female fecundity and thus population mean fitness may decline. These models are limited, however, because most of them do not simultaneously consider selection among maternal genotypes varying in the tendency to respond to their offspring. Available sexual selection models, in contrast, do consider the joint evolution of polygenic male traits (influencing viability, mating success and fecundity) and female preferences (influencing the mating success of different male phenotypes). These models have shown that male traits may evolve that are non-optimal with respect to viability. Only one recent sexual selection model explicitly incorporates direct fecundity selection upon females; this model concludes that fecundity will be maximized at equilibrium. Hence population mean fitness may decline due to reduced male viability but not due to diminished female fecundity. Available sexual selection models, however, are limited because they do not consider the effects of interactions among relatives. The assumptions and qualitative results of the two types of models are compared and discussed in the context of seed development. Differential allocation of maternal resources among genetically distinct developing seeds may be viewed from the perspective of either. Because the results of the available models of parent-offspring conflict and sexual selection are not wholly consistent and because data confirming the genetic basis of maternal patterns of investment or differential male reproductive success are scant, it is not clear which set of conclusions is most appropriate to apply to plants. To achieve the generality towards which mathematical approaches aspire, new models concerning the evolution of traits influencing resource allocation in plants must incorporate the components of both parent-offspring conflict and sexual selection.     

With selection for fecundity the mean fitness does not necessarily increase

A population with two alleles at one locus is considered. It is assumed that there is random mating of adults and that matings in which a particular pair of genotypes is involved may have a different mean number of offspring, or fecundity, than other types of matings. There is assumed to be no other selection. It is shown that the genotypic frequencies that maximize the mean fecundity of the population are not necessarily the same as the stable equilibrium frequencies. Thus, examples can be found for which the mean fecundity decreases from one generation to the next, and one such example is presented. An example in which there is no stable equilibrium, and the mean fecundity oscillates, is also given.     

Sexual selection when the female directly benefits

Why do females of many species mate with males on the basis of traits apparently detrimental to male survival? The answer may lie in the fact that these male traits are correlated with male condition. We consider the argument that high male condition directly benefits female fecundity and/or viability (e.g. through lower transmission of parasites, improved control of resources, or better paternal care). Using a quantitative genetic model we show how female preferences for male traits that indicate condition can evolve, even if the male traits themselves have deleterious effects on both the male and the female's fecundity. So-called ‘arbitrary preferences’ can spread in this way because male traits subject to sexual selection are often under additional selection to become correlated with condition. At equilibrium the positive effects of male condition on a female's fecundity and the negative effects of the male trait on her fecundity are balanced and the female preference is under stabilizing selection. The male trait will often be correlated with viability, but not with fecundity, even though the preference evolved as a result of differences in male fecundity. The mean fecundity of females is not maximized, and can steadily decline as the male trait and female preference evolve. If the male trait has no direct deleterious effects on female fecundity, as may happen in species with no paternal care, female preferences are under continuous directional selection to increase.     

Investigating the Interactions of Yeast Prions: [SWI+], [PSI+], and [PIN+]

The relationship between quantitative genetics and population genetics has been studied for nearly a century, almost since the existence of these two disciplines. Here we ask to what extent quantitative genetic models in which selection is assumed to operate on a polygenic trait predict adaptive fixations that may lead to footprints in the genome (selective sweeps). We study two-locus models of stabilizing selection (with and without genetic drift) by simulations and analytically. For symmetric viability selection we find that ∼16% of the trajectories may lead to fixation if the initial allele frequencies are sampled from the neutral site-frequency spectrum and the effect sizes are uniformly distributed. However, if the population is preadapted when it undergoes an environmental change (i.e., sits in one of the equilibria of the model), the fixation probability decreases dramatically. In other two-locus models with general viabilities or an optimum shift, the proportion of adaptive fixations may increase to >24%. Similarly, genetic drift leads to a higher probability of fixation. The predictions of alternative quantitative genetics models, initial conditions, and effect-size distributions are also discussed.     

ESTIMATING PHENOTYPIC SELECTION IN AGE‐STRUCTURED POPULATIONS BY REMOVING TRANSIENT FLUCTUATIONS

An extension of the selection differential in the Robertson–Price equation for the mean phenotype in an age‐structured population is provided. Temporal changes in the mean phenotype caused by transient fluctuations in the age‐distribution and variation in mean phenotype among age classes, which can mistakenly be interpreted as selection, will disappear if reproductive value weighting is applied. Changes in any weighted mean phenotype in an age‐structured population may be decomposed into between‐ and within‐age class components. Using reproductive value weighting the between‐age class component becomes pure noise, generated by previous genetic drift or fluctuating selection. This component, which we call transient quasi‐selection, can therefore be omitted when estimating age‐specific selection on fecundity or viability within age classes. The final response can be computed at the time of selection, but can not be observed until lifetime reproduction is realized unless the heritability is one. The generality of these results is illustrated further by our derivation of the selection differential for the continuous time age‐structured model with general age‐dependent weights. A simple simulation example as well as estimation of selection components in a house sparrow population illustrates the applicability of the theory to analyze selection on the mean phenotype in fluctuating age‐structured populations.     

Population variation in sexual selection and its effect on size allometry in two dung fly species with contrasting sexual size dimorphism

Body size is one of the most important quantitative traits under evolutionary scrutiny. Sexual size dimorphism (SSD) in a given species is expected to result if opposing selection forces equilibrate differently in both sexes. We document variation in the intensity of sexual and fecundity selection, male and female body size, and thus SSD among 31 and 27 populations of the two dung fly species, Scathophaga stercoraria and Sepsis cynipsea, across Switzerland. Whereas in S. cynipsea females are larger, the SSD is reversed in S. stercoraria. We comprehensively evaluated Fairbairn and Preziosi's (1994) general, three-tiered scenario, hypothesizing that sexual selection for large male size is the major driving force of SSD allometry within these two species. Sexual selection intensity on male size in the yellow dung fly, S. stercoraria, was overall positive, greater, and more variable among populations than fecundity selection on females. Also, sexual selection intensity in a given population correlated positively with mean male body size of that population for both the field-caught fathers and their laboratory-reared sons, indicating a response to selection. In S. cvnipsea, sexual selection intensity on males was lower overall and significantly positive, about equal in magnitude, but more variable than fecundity selection on females. However, there was no correlation between the intensity of sexual selection and mean male body size among populations. In both species, the laboratory-reared offspring indicate genetic differentiation among populations in body size. Despite fulfillment of all key prerequisites, at least in S. stercoraria, we did not find hypoallometry for SSD (Rensch's rule, i.e., greater evolutionary divergence in male size than female size) for the field-caught parents or the laboratory-reared offspring: Female size was isometric to male size in both species. We conclude that S. cynipsea does not fit some major requirements of Fairbairn and Preziosi's (1994) scenario, whereas for S. stercoraria we found partial support for it. Failure to support Rensch's rule within the latter species may be due to phylogenetic or other constraints, power limitations, erroneous estimates of sexual selection, insufficient genetic isolation of populations, or sex differences in viability selection against large size.     

Population trajectories for single locus additive fecundity selection and related selection models

A selection model which comprises models of additive fecundities as well as models of viability, fecundity, or differential mating selection acting only in one sex, is investigated for an autosomal gene locus in a population reproducing in nonoverlapping generations. The recurrence equations and basic properties of the genotypic population trajectories and equilibrium points are formulated for the multiallelic case. For the diallelic case, the trajectory development is discussed in more detail, and it is proven that every population trajectory converges to a Hardy-Weinberg equilibrium point.     

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.     

A Monte Carlo simulation model for studying evolution in age-structured populations

This model provides for any number of genotypes defined by age-specific survival and fecundity rates in a population with completely overlapping generations and growing under the control of density-governing functions affecting survival or fecundity. It is tested in situations involving two alleles at one locus. Nonselection populations at Hardy–Weinberg equilibrium obey the ecogenetic law; i.e., each genotype follows Lotka's law regarding rate of increase and stable age distribution as if it were an independent true-breeding population. Populations experiencing age- and density-independent selection approximate this situation, and the changes in gene frequency are predicted by relative fitnesses bases on λ, the finite rate of increase of the genotypes. Polymorphic gene equilibria occurring at steady-state population sizes are determined by fitnesses based on R, the net reproductive rate. In examples involving differences in generation time produced by age-dependent differences in fecundity, the allele associated with longer generation time may be favored or opposed by selection, depending on whether the density-governing factor controlling population size affects survival or fecundity. If such genotypes have similar R's, a genetic equilibrium may be established if the population is governed by a density function acting upon fecundity. Received: August 23, 1999 / Accepted: July 13, 2000     

Evolution of late-life fecundity in Drosophila melanogaster

Late-life fecundity has been shown to plateau at late ages in Drosophila analogously to late-life mortality rates. In this study, we test an evolutionary theory of late life based on the declining force of natural selection that can explain the occurrence of these late-life plateaus in Drosophila. We also examine the viability of eggs laid by late-age females and test a population genetic mechanism that may be involved in the evolution of late-life fecundity: antagonistic pleiotropy. Together these experiments demonstrate that (i) fecundity plateaus at late ages, (ii) plateaus evolve according to the age at which the force of natural selection acting on fecundity reaches zero, (iii) eggs laid by females in late life are viable and (iv) antagonistic pleiotropy is involved in the evolution of late-life fecundity. This study further supports the evolutionary theory of late life based on the age-specific force of natural selection.     

Selection responses of means and inbreeding depression for female fecundity in Drosophila melanogaster suggest contributions from intermediate-frequency alleles to quantitative trait variation

The extent to which quantitative trait variability is caused by rare alleles maintained by mutation, versus intermediate-frequency alleles maintained by balancing selection, is an unsolved problem of evolutionary genetics. We describe the results of an experiment to examine the effects of selection on the mean and extent of inbreeding depression for early female fecundity in Drosophila melanogaster. Theory predicts that rare, partially recessive deleterious alleles should cause a much larger change in the effect of inbreeding than in the mean of the outbred population, with the change in inbreeding effect having an opposite sign to the change in mean. The present experiment fails to support this prediction, suggesting that intermediate-frequency alleles contribute substantially to genetic variation in early fecundity.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Evolution and stability of the G-matrix on a landscape with a moving optimum

In quantitative genetics, the genetic architecture of traits, described in terms of variances and covariances, plays a major role in determining the trajectory of evolutionary change. Hence, the genetic variance-covariance matrix (G-matrix) is a critical component of modern quantitative genetics theory. Considerable debate has surrounded the issue of G-matrix constancy because unstable G-matrices provide major difficulties for evolutionary inference. Empirical studies and analytical theory have not resolved the debate. Here we present the results of stochastic models of G-matrix evolution in a population responding to an adaptive landscape with an optimum that moves at a constant rate. This study builds on the previous results of stochastic simulations of G-matrix stability under stabilizing selection arising from a stationary optimum. The addition of a moving optimum leads to several important new insights. First, evolution along genetic lines of least resistance increases stability of the orientation of the G-matrix relative to stabilizing selection alone. Evolution across genetic lines of least resistance decreases G-matrix stability. Second, evolution in response to a continuously changing optimum can produce persistent maladaptation for a correlated trait, even if its optimum does not change. Third, the retrospective analysis of selection performs very well when the mean G-matrix (G) is known with certainty, indicating that covariance between G and the directional selection gradient beta is usually small enough in magnitude that it introduces only a small bias in estimates of the net selection gradient. Our results also show, however, that the contemporary G-matrix only serves as a rough guide to G. The most promising approach for the estimation of G is probably through comparative phylogenetic analysis. Overall, our results show that directional selection actually can increase stability of the G-matrix and that retrospective analysis of selection is inherently feasible. One major remaining challenge is to gain a sufficient understanding of the G-matrix to allow the confident estimation of G.     

Environmental heterogeneity generates fluctuating selection on a secondary sexual trait

In any population in which resources are limiting, the allocation of resources toward increased reproductive success may generate costs to survival [1-8]. The relationship between a sexually selected trait and fitness will therefore represent a balance between its relative associations with fecundity versus viability [3, 6, 7]. Because the risk of mortality in a population is likely to be heavily determined by ecological conditions, survival costs may vary as a function of the prevailing environment [7]. As a result, for populations experiencing heterogeneous ecological conditions, there may not be a single optimal level of allocation toward reproduction versus survival [9]. Here, we show that early viability and fecundity selection act in opposing directions on a secondary sexual trait and that their relative magnitude depends upon ecological conditions, generating fluctuating selection. In a wild population of Soay sheep (Ovis aries), phenotypic and genetic associations between male horn growth and lifetime reproductive success were positive under good environmental conditions (because of increased breeding success) and negative under poor environmental conditions (because of reduced survival). In an unpredictable environment, high allocation to early horn growth is a gamble that will only pay off if ensuing conditions are favorable. Such fluctuating selection may play an important role in preventing the erosion of genetic variance in secondary sexual traits.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Mathematical models in genetics

In this study, we present some of the basic ideas of population genetics. The founders of population genetics are R.A. Fisher, S. Wright, and J. B.S. Haldane. They, not only developed almost all the basic theory associated with genetics, but they also initiated multiple experiments in support of their theories. One of the first significant insights, which are a result of the Hardy–Weinberg law, is Mendelian inheritance preserves genetic variation on which the natural selection acts. We will limit to simple models formulated in terms of differential equations. Some of those differential equations are nonlinear and thus emphasize issues such as the stability of the fixed points and time scales on which those equations operate. First, we consider the classic case when selection acts on diploid locus at which wу can get arbitrary number of alleles. Then, we consider summaries that include recombination and selection at multiple loci. Also, we discuss the evolution of quantitative traits. In this case, the theory is formulated in respect of directly measurable quantities. Special cases of this theory have been successfully used for many decades in plants and animals breeding.     

Natural Selection and Y-Linked Polymorphism

Several population genetic models allowing natural selection to act on Y-linked polymorphism are examined. The first incorporates pleiotropic effects of a Y-linked locus, such that viability, segregation distortion, fecundity and sexual selection can all be determined by one locus. It is shown that no set of selection parameters can maintain a stable Y-linked polymorphism. Interaction with the X chromosome is allowed in the next model, with viabilities determined by both X- and Y-linked factors. This model allows four fixation equilibria, two equilibria with X polymorphism and a unique point with both X- and Y-linked polymorphism. Stability analysis shows that the complete polymorphism is never stable. When X- and Y-linked loci influence meiotic drive, it is possible to have all fixation equilibria simultaneously unstable, and yet there is no stable interior equilibrium. Only when viability and meiotic drive are jointly affected by both X- and Y-linked genes can a Y-linked polymorphism be maintained. Unusual dynamics, including stable limit cycles, are generated by this model. Numerical simulations show that only a very small portion of the parameter space admits Y polymorphism, a result that is relevant to the interpretation of levels of Y-DNA sequence variation in natural populations.     

Evolution of the segregation ratio: modification of gene conversion and meiotic drive

We compare the evolutionary pressures that direct the modification of gene conversion and meiotic drive at loci subject to purifying and overdominant viability selection. Gene conversion differs from meiotic drive in that modifers do not affect their own segregation ratios, even when linked to the viability locus. Segregation distortion generates gametic level disequilibria between alleles at the viability locus and modifiers of gene conversion and meiotic drive: enhancers of segregation distortion become positively associated with driven alleles. Suppression of gene conversion evolves if the driven allele is marginally disadvantageous (overdominant viability selection), and higher rates evolve if the driven alleles are relatively advantageous (purifying viability selection). Gametic disequilibria permit enhancers of meiotic drive that are linked to the driven locus to promote their own segregation. We attribute the failure of genetic modifiers of gene conversion and meiotic drive to maximinize mean fitness to the generation of such associations.     

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.     

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