A potential resolution to the lek paradox through indirect genetic effects

Females often prefer males with elaborate traits, even when they receive no direct benefits from their choice. In such situations, mate discrimination presumably has genetic advantages; selective females will produce offspring of higher genetic quality. Over time, persistent female preferences for elaborate secondary-sexual traits in males should erode genetic variance in these traits, eventually eliminating any benefit to the preferences. Yet, strong female preferences persist in many taxa. This puzzle is called the lek paradox and raises two primary questions: do females obtain genetic benefits for offspring by selecting males with elaborate secondary-sexual characteristics and, if so, how is the genetic variation in these male traits maintained? We suggest that indirect genetic effects may help to resolve the lek paradox. Maternal phenotypes, such as habitat selection behaviours and offspring provisioning, often influence the condition and the expression of secondary-sexual traits in sons. These maternal influences are commonly genetic based (i.e. they are indirect genetic effects). Females choosing mates with elaborate traits may receive ‘good genes’ for daughters in the form of effective maternal characteristics. Recognizing the significance of indirect genetic effects may be important to our understanding of the process and consequences of sexual selection.     

Mate choice for non-additive genetic benefits: a resolution to the lek paradox

In promiscuous mating systems, females often show a consistent preference to mate with one or a few males, presumably to acquire heritable genetic benefits for their offspring. However, strong directional selection should deplete additive genetic variation in fitness and consequently any benefit to expressing the preference by females (referred to as the lek paradox). Here, we provide a novel resolution that examines non-additive genetic benefits, such as overdominance or inbreeding, as a source of genetic variation. Focusing on the inbreeding coefficient f and overdominance effects, we use dynamic models to show that (1) f can be inherited from sire to offspring, (2) populations with females that express a mating preferences for outbred males (low f) maintain higher genetic variation than populations with females that mate randomly, and (3) preference alleles for outbred males can invade populations even when the alleles are associated with a fecundity cost. We show that non-additive genetic variation due to overdominance can be converted to additive genetic variation and becomes “heritable” when the frequencies of alternative homozygous genotypes at fitness loci deviate from equality. Unlike previous models that assume an infinite population size, we now show that genetic drift in finite populations can lead to the necessary deviations in the frequencies of homozygous genotypes. We also show that the “heritability of f,” and hence the benefit to a mating preference for non-additive genetic benefits, is highest in small populations and populations in which a smaller number of loci contribute to fitness via overdominance. Our model contributes to the solution of the lek paradox.     

Genic capture and resolving the lek paradox

The genic capture hypothesis offers a resolution to the question of how genetic variation in male sexually selected traits is maintained in the face of strong female preferences. The hypothesis is that male display traits are costly to produce and hence depend upon overall condition, which itself is dependent upon genes at many loci. Few attempts have been made to test the assumptions and predictions of the genic capture hypothesis rigorously and, in particular, little attention has been paid to determining the genetic basis of condition. Such tests are crucial to our understanding of the maintenance of genetic variation and in the evaluation of recent models that propose a role for sexual selection in the maintenance of sex. Here, we review approaches to testing the link between genetically determined condition and levels of sexual trait expression and consider the probable importance of deleterious mutations.     

The lek paradox is not resolved

In a population at equilibrium in a stable environment under natural selection there can be very little heritability of fitness. In some species, for example, those with leks, females exhibit strong preferences for certain characteristics in their mates, though all they obtain are his genes and these cannot contribute much to the fitness of their offspring. In paternal species, where there are often clear economic benefits in obtaining good mates, mate choice mechanisms do not seem so highly developed. This is the lek paradox. We examine three forces, mutational load, migrational load, and two kinds of linkage disequilibrium, which might generate enough heritability to maintain a mating convention in a nonpaternal species, and we evaluate the strength of the selective forces they generate. Only under rather extreme conditions can these forces be of more than negligible strength.     

Queuing in space and time reduces the lek paradox on an antelope lek

Lek systems, where females often use centrality to assess male quality, highlight a general paradox in evolutionary biology: how can female preferences for males providing good genes persist when consequential strong directional selection is predicted to deplete additive genetic variance in male quality and thereby obliterate benefits of choosiness? An explanation contributing to the resolution of this lek paradox may be that genetic variance is retained when an indirect mate choice cue, such as centrality on a lek, is an imperfect indicator of male genetic quality. Here I investigate whether the presence of alternative male mating tactics limits the reliability of centrality as a quality indicator in lek-breeding topi antelopes. Whereas males establishing territories directly on the central lek were relatively large, smaller peripheral males regularly shifted their territories centripetally and in this way also occasionally obtained central territories. By such opportunistic queuing, small males could increase their mating success drastically; however, their territorial tenure in the lek centre was relatively short, consistent with moderate competitive ability. These results suggest that male topi antelopes can obtain central lek territories through alternative mating tactics, providing scope for variance in male quality on the central lek. In a separate finding, the mating success of central males was found to increase during territorial tenure, independent of estimated age. The demonstration of queuing in both space and time on a mammalian lek highlights the importance of considering male tactical dynamics over time in order to avoid an inflated appearance of the lek paradox.     

Condition-dependence, genotype-by-environment interactions and the lek paradox

The lek paradox states that maintaining genetic variation necessary for ‘indirect benefit’ models of female choice is difficult, and two interrelated solutions have been proposed. ‘Genic capture’ assumes condition-dependence of sexual traits, while genotype-by-environment interactions (GEIs) offer an additional way to maintain diversity. However, condition-dependence, particularly with GEIs, implies that environmental variation can blur the relationship between male displays and offspring fitness. These issues have been treated separately in the past. Here we combine them in a population genetic model, and show that predictions change not only in magnitude but also in direction when the timing of dispersal between environments relative to the life cycle is changed. GEIs can dramatically improve the evolution of costly female preferences, but also hamper it if much dispersal occurs between the life history stage where condition is determined and mating. This situation also arises if selection or mutation rates are too high. In general, our results highlight that when evaluating any mechanism promoted as a potential resolution of the lek paradox, it is not sufficient to focus on its effects on genetic variation. It also has to be assessed to what extent the proposed mechanism blurs the association between male attractiveness and offspring fitness; the net balance of these two effects can be positive or negative, and often strongly context-dependent.     

Condition-dependence,genotype-by-environment interactions and the lek paradox

The lek paradox states that maintaining genetic variation necessary for 'indirect benefit' models of female choice is difficult, and two interrelated solutions have been proposed. 'Genic capture' assumes condition-dependence of sexual traits, while genotype-by-environment interactions (GEIs) offer an additional way to maintain diversity. However, condition-dependence, particularly with GEIs, implies that environmental variation can blur the relationship between male displays and offspring fitness. These issues have been treated separately in the past. Here we combine them in a population genetic model, and show that predictions change not only in magnitude but also in direction when the timing of dispersal between environments relative to the life cycle is changed. GEIs can dramatically improve the evolution of costly female preferences, but also hamper it if much dispersal occurs between the life history stage where condition is determined and mating. This situation also arises if selection or mutation rates are too high. In general, our results highlight that when evaluating any mechanism promoted as a potential resolution of the lek paradox, it is not sufficient to focus on its effects on genetic variation. It also has to be assessed to what extent the proposed mechanism blurs the association between male attractiveness and offspring fitness; the net balance of these two effects can be positive or negative, and often strongly context-dependent.     

A resolution of the paradox of enrichment

Theoretical studies have shown a paradoxical destabilizing response of predator-prey ecosystems to enrichment, but there is the gap between the intuitive view of nature and this theoretical prediction. We studied a minimal predator-prey system (a two predator-two prey system) in which the paradox of enrichment pattern can vanish; the destabilization with enrichment is reversed, leading to stabilization (a decrease in the amplitude of oscillation of population densities). For resolution of the paradox, two conditions must be met: (1) the same prey species must be preferred as a dietary item by both predator species, creating the potential for high exploitative competition between the predator species, and (2), while both predators are assumed to select their diet in accordance with optimal diet utilization theory, one predator must be a specialist and the other a generalist. In this system, the presence of a less profitable prey species can cause the increase in population oscillation amplitudes associated with increasing enrichment to be suppressed via the optimal diet utilization of the generalist predator. The resulting stabilization is explained by the mitigating effect of the less profitable prey showing better population growth with increasing enrichment on the destabilization underlying the specialist predator and prey relation, thus resolving the paradox of enrichment.     

A possible resolution of the gating paradox

We introduce a Markov model for the gating of membrane channels. The model features a possible solution to the so-called gating current paradox, namely that the bell-shaped curve that describes the voltage dependence of the kinetics is broader than expected from, and shifted relative to, the sigmoidal curve that describes the voltage dependence of the activation. The model also predicts some temperature dependence of this shift, but presence of the latter has not been tested experimentally so far.     

Male killing can select for male mate choice: a novel solution to the paradox of the lek

In lekking species, intense directional selection is applied to aspects of the male genotype by female choice. Under conventional quantitative genetics theory, the expectation is that this will lead to a rapid loss in additive genetic variance for the trait in question. However, despite female choice, male variation is maintained and hence it pays females to continue choosing. This has been termed the ''paradox of the lek''. Here we present a theoretical analysis of a putative sex-role-reversed lek in the butterfly Acraea encedon. Sex-role reversal appears to have come about because of infection with a male-killing Wolbachia. The bacterium is highly prevalent in some populations, such that there is a dearth of males. Receptive females form dense aggregations, and it has been suggested that males preferentially select females uninfected with the bacterium. As with more conventional systems, this presents a theoretical problem exactly analogous to the lek paradox, namely what maintains female variation and hence why do males continue to choose? We model the evolution of a male choice gene that allows discrimination between infected and uninfected females, and show that the stable maintenance of both female variation and male choice is likely, so long as males make mistakes when discriminating between females. Furthermore, our model allows the maintenance, in a panmictic population, of a male killer that is perfectly transmitted. This is the first model to allow this result, and may explain the long-term persistence of a male killer in Hypolimnas bolina.     

A resolution of the mutation load paradox in humans

Current information on the rate of mutation and the fraction of sites in the genome that are subject to selection suggests that each human has received, on average, at least two new harmful mutations from its parents. These mutations were subsequently removed by natural selection through reduced survival or fertility. It has been argued that the mutation load, the proportional reduction in population mean fitness relative to the fitness of an idealized mutation-free individual, allows a theoretical prediction of the proportion of individuals in the population that fail to reproduce as a consequence of these harmful mutations. Application of this theory to humans implies that at least 88% of individuals should fail to reproduce and that each female would need to have more than 16 offspring to maintain population size. This prediction is clearly at odds with the low reproductive excess of human populations. Here, we derive expressions for the fraction of individuals that fail to reproduce as a consequence of recurrent deleterious mutation () for a model in which selection occurs via differences in relative fitness, such as would occur through competition between individuals. We show that is much smaller than the value predicted by comparing fitness to that of a mutation-free genotype. Under the relative fitness model, we show that depends jointly on U and the selective effects of new deleterious mutations and that a species could tolerate 10's or even 100's of new deleterious mutations per genome each generation.     

Loop fold structure of proteins: resolution of Levinthas paradox

According to Levinthal a protein chain of ordinary size would require enormous time to sort its conformational states before the final fold is reached. Experimentally observed time of folding suggests an estimate of the chain length for which the time would be sufficient. This estimate by order of magnitude fits to experimentally observed universal closed loop elements of globular proteins - 25-30 residues.     

To lek or not to lek: mating strategies of male fallow deer

We studied the mating system of fallow deer (Dama dama) for6 years in central Italy. Males in this population could defendterritories that were either single, clumped in leks, or satelliteto leks. The most highly successful males in our study werein leks. When we considered all males, there were no significantdifferences in average copulatory success according to territorytype because many lek males did not achieve any copulations,which were seen in only a few lek territories. The variancein copulatory success, however, was much greater for leks thanelsewhere. Single territories were occupied for shorter timesduring the rut than lek territories. Fighting among males wasmore frequent in the lek, even when we excluded highly successfullek males from the analysis. Chases of nonterritorial malesand harem size were correlated with the number of copulationsachieved by individual males, but did not vary according toterritory type. Copulatory success of some individuals increasedwith age, but there were no age differences among males holdingdifferent types of territories. Satellite males switched tolek territoriality in the course of one rut, but switches fromsingle territory to lek territory were rare. We suggest thatmales in single territories are inferior competitors that selecta low-risk, lowbenefit strategy, whereas those in lek territorieswhere no copulations were seen may be attempting to establishthemselves on the lek to increase their copulatory success infuture years.     

Manipulating lek size and composition using decoys: an experimental investigation of lek evolution models

Four theoretical models have been proposed to account for the origin and maintenance of leks: hotspot, female preference, hotshot, and black hole models. Each has been validated in particular cases, and most are not mutually exclusive; therefore, it has been difficult to contrast and separate them, empirically and experimentally. By using decoys to mimic natural leks in the little bustard, artificial leks attracted wild birds. Then, by manipulating artificial lek size and structure (sex ratio, male phenotype), the study of responses of wild males and females allowed us to test specific predictions derived from the four classical models of lek evolution. The hotspot model was not supported because female decoys did not attract wild males. Conversely, hotshot males do exist in this species (attracting both wild females and males), as does a female preference for a particular lek size (four males). Finally, males aggressive toward decoys attracted fewer females, consistent with one of the mechanisms by which the black hole model may work. Therefore, three models of lek evolution were partly or fully supported by our experimental results: hotshot, female preference, and black hole models. We suggest that these models actually fit within each other, ensuring the evolution, functioning, and long-term maintenance of leks.     

Male rank and optimal lek size

Widemo and Owens presented a model that calculates the expectedcopulation rates of males on leks of a range of sizes. Theyclaim that a negative relationship between lek size and malemating skew will result in low-ranking males having greateroptimal lek sizes than higher ranking rivals. Widemo and Owensoffered no proof of their claim, and their model assumes thatthe rank of a male does not change as lek size increases, whereasin reality, rank may change as more males arrive. We presenta general model that allows rank to change as lek size increases.We show that the crucial determinant of whether optimal leksize increases with male rank is whether relative competitive differencesincrease with lek size. Contrary to the claim of Widemo andOwens, the relationship between skew and lek size has no directbearing on the optimal levels of aggregation of males of differentrank. We show that a negative relationship between skew andlek size can exist even when high-ranking males have the greatestoptimal lek sizes.     

Effect of substrate roughness on D spacing supports theoretical resolution of vapor pressure paradox.

The lamellar D spacing has been measured for oriented stacks of lecithin bilayers prepared on a variety of solid substrates and hydrated from the vapor. We find that, when the bilayers are in the L(alpha) phase near 100% relative humidity, the D spacing is consistently larger when the substrate is rougher than when it is smooth. The differences become smaller as the relative humidity is decreased to 80% and negligible differences are seen in the L(beta') phase. Our interpretation is that rough substrates frustrate the bilayer stack energetically, thereby increasing the fluctuations, the fluctuational repulsive forces, and the water spacing compared with stacks on smooth surfaces. This interpretation is consistent with and provides experimental support for a recently proposed theoretical resolution of the vapor pressure paradox.     

Male signalling and lek attractiveness in the Mediterranean fruit fly

I tested the null hypothesis that females of the Mediterranean fruit fly, Ceratitis capitata, were equally likely to visit leks composed of males with high versus low mating success. In laboratory trials, the observed distribution of matings among males differed significantly from that expected by chance, owing primarily to the higher than expected numbers of individuals with low (mated 0-1 days over 6 consecutive observation days) or high (mated 4 or more days) mating scores. I termed these two groups as 'low' and 'high' maters, respectively. In the field, greater numbers of female sightings were made at artificial leks of high maters than low maters. This result apparently reflected a greater calling propensity among high maters. Slopes of female sightings versus calling level did not differ significantly between leks of low and high maters, suggesting that the observed relationship between calling activity and female sightings was independent of male mating status. Following the same protocol, I conducted a second experiment to examine whether males used the signals of conspecific males to locate lek sites and, if so, whether signal attractiveness varied with male mating ability. Attraction of males to calling conspecifics was far weaker than that observed for females, and over five different trials a total of only seven male sightings were made at any of the established leks. Copyright 2000 The Association for the Study of Animal Behaviour.