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.     

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.     

Mate choice for nonadditive genetic benefits and the maintenance of genetic diversity in song sparrows

The lek paradox asserts that strong directional selection via female choice should deplete additive genetic variation in fitness and consequently any benefit to females expressing the preference. Recently, we have provided a novel resolution to the paradox by showing that nonadditive genetic effects such as overdominance can be inherited from parent to offspring, and populations with females that express a mating preference for outbred males maintain higher genetic variation than populations with females that mate randomly. Here, we test our dynamic model using empirical data previously published from a small island population of song sparrows (Melospiza melodia). The model assumes that fitness and male trait expression display overdominance effects. The results demonstrate that female choice for outbred males mediated by directional selection on song repertoire size provides a heritable benefit to offspring through reduced inbreeding depression. Within the population, we estimate the heritability of the inbreeding coefficient to be 0.18 ± 0.08 (SD). Furthermore, we show that mate choice for outbred males increases fitness‐related genetic variation in the population by 12% and thereby reduces inbreeding depression by 1% per generation in typical years and upwards of 15% in severe years. Thus, mate choice may help to stave off population extinction in this and other small populations.     

Genic capture and the genetic basis of sexually selected traits in the zebra finch

Abstract The lek paradox, in which female choice erodes genetic variation in male sexually selected traits, is a fundamental issue in sexual selection. If females gain only genetic benefits from preferentially having their ova fertilized by males with particular traits, what maintains variation in these traits? Under strong directional selection mediated through mate choice, the alleles for beneficial male traits are expected to go to fixation and exhibit little variation. A theoretical solution to the lek paradox is the genic capture hypothesis which states that: costly male traits subject to female choice are condition dependent, that male condition is dependent on genes at many loci and exhibits additive genetic variance, and that positive genetic correlations exist between sexually selected traits and condition. Using a captive population of the zebra finch Taeniopygia guttata, we tested two key predictions from this model: (1) that genetic variance exists in beak color which is a sexually selected trait, but also in condition and immune function, and (2) that positive genetic correlations exist between condition and beak color, and between beak color, condition, and immune function. Genetic parameters were estimated from a large breeding experiment involving 81 sires, 972 offspring, a pedigree of 1526 individuals, using the animal model. We employed the following index of body condition: residuals from a log‐log plot of body mass on tarsus length following a standardized and extended period of exercise, in which residual mass is known to reflect fat and protein reserves. Our results were broadly consistent with the genic capture hypothesis because we found (1) additive genetic variation in beak color and immune function and condition, and (2) positive genetic correlations between condition and beak color, and between condition, beak color, and several assays of immune responsiveness. However, both of these results need qualification. In the first case we identified an important general problem in estimating the coefficient of additive genetic variance (CV_A) in body condition. In the second case, although most of the genetic correlations were positive as predicted, only some were statistically significant, possibly due to our relatively small sample sizes, because genetic correlations typically have large standard errors and therefore require very large samples to be statistically significant. The statistically significant, positive genetic correlations included those between beak color and immune function (response to tetanus), and between immune function (response to tetanus) and condition, both of which indicate that females gain good genes from mating with males in good condition and/or with a redder beak color. We discuss the implications of our results for devising more rigorous but pragmatic tests of the genic capture hypothesis.     

On the resolution of the lek paradox

Directional female mate choice is expected to deplete additive genetic variation in male traits. This should preclude such trait-based choice from resulting in genetic benefits to offspring, and yet genetic benefits are the explanation for the choice. This evolutionary conundrum is known as the lek paradox. Newly proposed resolutions to this paradox aim to unravel mechanisms that contribute to the persistence of genetic variance in traits under directional female mate choice.     

Mate choice evolution, dominance effects, and the maintenance of genetic variation

Female mate choice influences the maintenance of genetic variation by altering the mating success of males with different genotypes. The evolution of preferences themselves, on the other hand, depends on genetic variation present in the population. Few models have tracked this feedback between a choice gene and its effects on genetic variation, in particular when genes that determine offspring viability and attractiveness have dominance effects. Here we build a population genetic model that allows comparing the evolution of various choice rules in a single framework. We first consider preferences for good genes and show that focused preferences for homozygotes evolve more easily than broad preferences, which allow heterozygous males high mating success too. This occurs despite better maintenance of genetic diversity in the latter scenario, and we discuss why empirical findings of superior mating success of heterozygous males consequently do not immediately lead to a better understanding of the lek paradox. Our results thus suggest that the mechanisms that help maintain genetic diversity also have a flipside of making female choice an inaccurate means of producing the desired kind of offspring. We then consider preferences for heterozygosity per se, and show that these evolve only under very special conditions. Choice for compatible genotypes can evolve but its selective advantage diminishes quickly due to frequency-dependent selection. Finally, we show that our model reproduces earlier results on selfing, when the female choice strategy produces assortative mating. Overall, our model indicates that various forms of heterozygote-favouring (or variable) female choice pose a problem for the theory of sexual ornamentation based on indirect benefits, rather than a solution.     

No Evidence for Heritability of Male Mating Latency or Copulation Duration across Social Environments in Drosophila melanogaster

A key assumption underpinning major models of sexual selection is the expectation that male sexual attractiveness is heritable. Surprisingly, however, empirical tests of this assumption are relatively scarce. Here we use a paternal full-sib/half-sib breeding design to examine genetic and environmental variation in male mating latency (a proxy for sexual attractiveness) and copulation duration in a natural population of Drosophila melanogaster. As our experimental design also involved the manipulation of the social environment within each full-sibling family, we were able to further test for the presence of genotype-by-environment interactions (GEIs) in these traits, which have the potential to compromise mate choice for genetic benefits. Our experimental manipulation of the social environment revealed plastic expression of both traits; males exposed to a rival male during the sensitive period of adult sexual maturation exhibited shorter mating latencies and longer copulation durations than those who matured in isolation. However, we found no evidence for GEIs, and no significant additive genetic variation underlying these traits in either environment. These results undermine the notion that the evolution of female choice rests on covariance between female preference and male displays, an expectation that underpins indirect benefit models such as the good genes and sexy sons hypotheses. However, our results may also indicate depletion of genetic variance in these traits in the natural population studied, thus supporting the expectation that traits closely aligned with reproductive fitness can exhibit low levels of additive genetic variance.     

Genetic quality and sexual selection: an integrated framework for good genes and compatible genes

Why are females so choosy when it comes to mating? This question has puzzled and marveled evolutionary and behavioral ecologists for decades. In mating systems in which males provide direct benefits to the female or her offspring, such as food or shelter, the answer seems straightforward — females should prefer to mate with males that are able to provide more resources. The answer is less clear in other mating systems in which males provide no resources (other than sperm) to females. Theoretical models that account for the evolution of mate choice in such nonresource-based mating systems require that females obtain a genetic benefit through increased offspring fitness from their choice. Empirical studies of nonresource-based mating systems that are characterized by strong female choice for males with elaborate sexual traits (like the large tail of peacocks) suggest that additive genetic benefits can explain only a small percentage of the variation in fitness. Other research on genetic benefits has examined nonadditive effects as another source of genetic variation in fitness and a potential benefit to female mate choice. In this paper, we review the sexual selection literature on genetic quality to address five objectives. First, we attempt to provide an integrated framework for discussing genetic quality. We propose that the term ‘good gene’ be used exclusively to refer to additive genetic variation in fitness, ‘compatible gene’ be used to refer to nonadditive genetic variation in fitness, and ‘genetic quality’ be defined as the sum of the two effects. Second, we review empirical approaches used to calculate the effect size of genetic quality and discuss these approaches in the context of measuring benefits from good genes, compatible genes and both types of genes. Third, we discuss biological mechanisms for acquiring and promoting offspring genetic quality and categorize these into three stages during breeding: (i) precopulatory (mate choice); (ii) postcopulatory, prefertilization (sperm utilization); and (iii) postcopulatory, postfertilization (differential investment). Fourth, we present a verbal model of the effect of good genes sexual selection and compatible genes sexual selection on population genetic variation in fitness, and discuss the potential trade-offs that might exist between mate choice for good genes and mate choice for compatible genes. Fifth, we discuss some future directions for research on genetic quality and sexual selection.     

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.     

The lek mating system of the worm pipefish (Nerophis lumbriciformis): a molecular maternity analysis and test of the phenotype‐linked fertility hypothesis

The origin and maintenance of mating preferences continues to be an important and controversial topic in sexual selection research. Leks and lek‐like mating systems, where individuals gather in particular spots for the sole purpose of mate choice, are particularly puzzling, because the strong directional selection imposed by mate choice should erode genetic variation among competing individuals and negate any benefit for the choosing sex. Here, we take advantage of the lek‐like mating system of the worm pipefish (Nerophis lumbriciformis) to test the phenotype‐linked fertility hypothesis for the maintenance of mating preferences. We use microsatellite markers to perform a parentage analysis, along with a mark–recapture study, to confirm that the worm pipefish has an unusual mating system that strongly resembles a female lek, where females display and males visit the lek to choose mates. Our results show that the most highly ornamented females occupy positions near the centre of the breeding area, and males mating with these females receive fuller broods with larger eggs compared to males mating with less‐ornamented females. We also conduct a laboratory experiment to show that female ornaments are condition‐dependent and honestly signal reproductive potential. Overall, these results are consistent with the predictions of a sex‐independent version of the phenotype‐linked fertility hypothesis, as male preference for female ornaments correlates with fertility benefits.     

Genotype x environment interaction for male attractiveness in an acoustic moth: evidence for plasticity and canalization

The lek paradox arises when choosy females deplete the genetic variance for male display traits from a population, yet substantial additive genetic variation (V_A) in male traits persists. Thus, the lek paradox can be more generally stated as one of the most fundamental evolutionary questions: What maintains genetic variation in natural populations? One solution to this problem may be found in the condition‐dependent nature of many sexually selected traits. Genotype × environment (G × E) interactions can maintain V_A under conditions of environmental heterogeneity provided certain restrictions are met, although antagonistic pleiotropy has also been proposed as a mechanism. Here, we provide evidence for G × E interactions and against the role of antagonistic pleiotropy in the maintenance of V_A for sexually selected traits. Using inbred lines of the lesser waxmoth Achroia grisella, we measured V_A for song attractiveness, condition and development rate under different competitive environments and found that genotypes differed in their plasticity. We argue that variation persists in natural populations because G × E interactions prevent any one variant from producing the optimal phenotype across all environments.     

Predators, reproductive parasites, and the persistence of poor males on leks

Lekking males are thought to face strong directional selectionon secondary sexual traits. How variation in male traits canpersist under these conditions remains problematic (the lekparadox). Here, we present several game-theoretic models thatshow that avoidance of costly and mobile predators, sneakers,or brood parasites (enemies) leads to variation in female choice.This can result in maintenance of variation in male quality."Enemies" will congregate around higher quality males. Femalesmust then trade-off the benefits of mating with high-qualitymales against the increased risk of enemies. At equilibrium,the models predict a positive correlation between the qualityof a male and the proportions of both enemies and females visitinghim. In the first model, we use this framework to predict thelowest quality male on the lek that will receive any matings.In the second model, we examine the influence of this female-enemygame on the maintenance of variation in male quality. Low-qualitymales are likely to persist when enemies are costly to femalesor occur at high density, and when there is some spatial structureon the lek, so that neighboring males are typically of similarquality. If enemies are more costly to males than to females,high-quality males may benefit from receiving fewer female visits.In the third model, we consider the special case when enemiesare male reproductive parasites. These models illustrate theimportance of considering the simultaneous decisions of multipleplayers in mate choice games.     

Heterozygosity-based assortative mating in blue tits (Cyanistes caeruleus): implications for the evolution of mate choice

The general hypothesis of mate choice based on non-additive genetic traits suggests that individuals would gain important benefits by choosing genetically dissimilar mates (compatible mate hypothesis) and/or more heterozygous mates (heterozygous mate hypothesis). In this study, we test these hypotheses in a socially monogamous bird, the blue tit (Cyanistes caeruleus). We found no evidence for a relatedness-based mating pattern, but heterozygosity was positively correlated between social mates, suggesting that blue tits may base their mating preferences on partner''s heterozygosity. We found evidence that the observed heterozygosity-based assortative mating could be maintained by both direct and indirect benefits. Heterozygosity reflected individual quality in both sexes: egg production and quality increased with female heterozygosity while more heterozygous males showed higher feeding rates during the brood-rearing period. Further, estimated offspring heterozygosity correlated with both paternal and maternal heterozygosity, suggesting that mating with heterozygous individuals can increase offspring genetic quality. Finally, plumage crown coloration was associated with male heterozygosity, and this could explain unanimous mate preferences for highly heterozygous and more ornamented individuals. Overall, this study suggests that non-additive genetic traits may play an important role in the evolution of mating preferences and offers empirical support to the resolution of the lek paradox from the perspective of the heterozygous mate hypothesis.     

Genetic variance of sexually selected traits in waxmoths: maintenance by genotype x environment interaction

When traits experience directional selection, such as that imposed by sexual selection, their genetic variance is expected to diminish. Nonetheless, theory and findings from sexual selection predict and demonstrate that male traits favored by female choice retain substantial amounts of additive genetic variance. We explored this dilemma through an ecological genetic approach and focused on the potential contributions of genotype x environment interaction (GEI) to maintenance of additive genetic variance for male signal characters in the lesser waxmoth, Achroia grisella (Lepidoptera: Pyralidae). We artificially selected genetic variants for two male signal characters, signal rate (SR) and peak amplitude (PA), that influence female attraction and then examined the phenotypic plasticity of these variants (high- and low-SR and high- and low-PA lines) under a range of environmental conditions expected in natural populations. Our split-family breeding experiments indicated that two signal characters, SR and PA, and several developmental characters in both high- and low-SR and high- and low-PA lines displayed considerable phenotypic plasticity among the environments tested. Moreover, strong GEIs leading to crossover between high- and low-SR lines were found for SR and developmental period. Therefore, neither high- nor low-SR genetic variants would achieve maximum attractiveness and fitness in every environment, and those variants producing unattractive signals with low SRs under normal conditions may remain in populations provided that gene flow across environments or generation overlap are sufficiently high. We speculate that the phenotypic plasticity for SR and developmental period is adaptive in A. grisella populations experiencing a range of temperature and density conditions. Females mating with attractive (high-SR) males may be assured of obtaining good genes because these males sire offspring that develop more rapidly and a crossover for developmental period may parallel that for SR. Such parallel crossovers may be expected wherever good-genes sexual selection mechanisms operate.     

Female guppies agree to differ: phenotypic and genetic variation in mate-choice behavior and the consequences for sexual selection

Variation among females in mate choice may influence evolution by sexual selection. The genetic basis of this variation is of interest because the elaboration of mating preferences requires additive genetic variation in these traits. Here we measure the repeatability and heritability of two components of female choosiness (responsiveness and discrimination) and of female preference functions for the multiple ornaments borne by male guppies (Poecilia reticulata). We show that there is significant repeatable variation in both components of choosiness and in some preference functions but not in others. There appear to be several male ornaments that females find uniformly attractive and others for which females differ in preference. One consequence is that there is no universally attractive male phenotype. Only responsiveness shows significant additive genetic variation. Variation in responsiveness appears to mask variation in discrimination and some preference functions and may be the most biologically relevant source of phenotypic and genetic variation in mate-choice behavior. To test the potential evolutionary importance of the phenotypic variation in mate choice that we report, we estimated the opportunity for and the intensity of sexual selection under models of mate choice that excluded and that incorporated individual female variation. We then compared these estimates with estimates based on measured mating success. Incorporating individual variation in mate choice generally did not predict the outcome of sexual selection any better than models that ignored such variation.     

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.     

The capture of heritable variation for genetic quality through social competition

In theory, females of many species choose mates based on traits that are indicators of male genetic quality. A fundamental question in evolutionary biology is why genetic variation for such indicator traits persists despite strong persistent selection imposed by female preference, which is known as the lek paradox. One potential solution to the lek paradox suggests that the traits that are targets of mate choice should evolve condition-dependent expression and that condition should have a large genetic variance. Condition is expected to exhibit high genetic variance because it is affected by a large number of physiological processes and hence, condition-dependent traits should 'capture' variation contributed by a large number of loci. We suggest that a potentially important cause of variation in condition is competition for limited resources. Here, we discuss a pair of models to analyze the evolutionary genetics of traits affected by success in social competition for resources. We show that competition can contribute to genetic variation of 'competition-dependent' traits that have fundamentally different evolutionary properties than other sources of variation. Competition dependence can make traits honest indicators of genetic quality by revealing the relative competitive ability of males, can provide a component of heritable variation that does not contribute to trait evolution, and can help maintain heritable variation under directional selection. Here we provide a general introduction to the concept of competition dependence and briefly introduce two models to demonstrate the potential evolutionary consequences of competition-dependent trait expression.     

EVOLUTION OF MATE CHOICE FOR GENOME-WIDE HETEROZYGOSITY

The extent to which indirect genetic benefits can drive the evolution of directional mating preferences for more ornamented mates, and the mechanisms that maintain such preferences without depleting genetic variance, remain key questions in evolutionary ecology. We used an individual-based genetic model to examine whether a directional preference for mates with higher genome-wide heterozygosity ( H ), and consequently greater ornamentation, could evolve and be maintained in the absence of direct fitness benefits of mate choice. We specifically considered finite populations of varying size and spatial genetic structure, in which parent–offspring resemblance in heterozygosity could provide an indirect benefit of mate choice. A directional preference for heterozygous mates evolved under broad conditions, even given a substantial direct cost of mate choice, low mutation rate, and stochastic variation in the link between individual heterozygosity and ornamentation. Furthermore, genetic variance was retained under directional sexual selection. Preference evolution was strongest in smaller populations, but weaker in populations with greater internal genetic structure in which restricted dispersal increased local inbreeding among offspring of neighboring females that all preferentially mated with the same male. These results suggest that directional preferences for heterozygous or outbred mates could evolve and be maintained in finite populations in the absence of direct fitness benefits, suggesting a novel resolution to the lek paradox.