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Keywords:

  • environmental fluctuations;
  • evolution;
  • evolution of sex;
  • population genetics;
  • theoretical biology

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

Fluctuating selection is often thought to be ineffective in maintaining a genetic polymorphism except when generations overlap, for example when a seed bank causes a storage effect. Here, I demonstrate that fluctuating selection on sex-limited traits automatically includes such a ‘storage effect’ because sex-limited alleles are shielded from selection in the sex where they are not expressed. With analytical calculations and numerical simulations I show that fluctuating selection can maintain a genetic polymorphism in sex-limited traits. Such a protected polymorphism can reduce the cost of sex when female-limited traits are involved. But, this effect will probably be small compared to the two-fold advantage of asexual reproduction unless many polymorphic loci interact or exceptionally strong environmental fluctuations are present. It is argued that genetic polymorphisms maintained by fluctuating selection on sex-limited traits may partly explain the large genetic variance of traits under strong sexual selection.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

Many traits under selection have high heritabilities. One of the most striking examples is the surprisingly large genetic variance that exists for sexually selected traits ( Pomiankowski & Møller, 1995). To explain the maintenance of such genetic polymorphisms, heterozygote advantage, frequency-dependent selection and environmental fluctuations have been proposed. Here, I would like to concentrate on the question of whether environmental fluctuations can lead to the maintenance of genetic variance. In particular, I will examine whether temporal fluctuations in environmental conditions are likely to contribute to the maintenance of genetic variance in sex-limited traits. According to previous theoretical models, temporal fluctuations in environmental conditions are unlikely to maintain a genetic polymorphism ( Hedrick, 1974; Hoekstra, 1975; Hedrick, 1986; Kirzhner et al., 1995 ). In the case of several alleles at a locus under consideration, one allele usually confers the higher geometric mean fitness, and the other alleles at that locus can be expected to decrease in frequency until they disappear. A polymorphism can only be maintained by fluctuating selection when some proportion of the allele with the lower geometric mean fitness is shielded from selection. Such cases are given when the recessive homozygote has lower geometric mean fitness but higher arithmetic mean fitness than the heterozygote ( Haldane & Jayakar, 1963), when the heterozygote has higher geometric mean fitness than both homozygotes ( Gillespie, 1973, 1991), when selection fluctuates both in space and time ( Ewing, 1979), or when some genotypes are shielded from selection in a seed bank ( Chesson & Warner, 1981; Ellner & Hairston, 1994; Sasaki & Ellner, 1995; Ellner & Sasaki, 1996). As sex-limited traits are also partly shielded from selection – they are expressed in one sex only –Sasaki & Ellner (1997) have already mentioned that fluctuating selection should be able to maintain a polymorphism for traits with sex-limited expression.

Many sex-limited traits involve interactions between individuals, and such traits should be especially prone to be influenced by fluctuations in population density. Temporal changes in population density, an almost ubiquitous observation (e.g. Merell & Rodell, 1968; Heath, 1974; Lynch, 1987; Hairstone & Dillon, 1990; Forsman, 1993; Borash et al., 1998 ), will probably cause strong fluctuating selection on sex-limited traits ( Conner, 1989; French & Cade, 1989; Wikelski & Trillmich, 1997). A male with a superior but costly courtship signal might for example be at a disadvantage under low population density because only few females are available and the cost of the signal outweighs the advantage of his attractiveness whereas under higher population density this male might achieve many copulations and the benefit of attractiveness may thus surpass the cost of signal production. Similar arguments can be made for intrasexual competition, mechanisms of female choice or for behaviour used in offspring care. Fluctuating selection and sex limitation of traits are thus both widespread and likely to co-occur and the proposed effect of fluctuating selection on the maintenance of genetic variance should be relevant for a large number of traits.

Contrary to theoretical expectation, sex-limited traits show large genetic variance in the face of strong selection ( Pomiankowski & Møller, 1995). The capture of genetic variance in condition ( Rowe & Houle, 1996) or strong directional selection ( Iwasa & Pomiankowski, 1995) have been proposed as possible mechanisms to explain the maintenance of genetic variance in sexually selected traits. The analytical calculations and simulations described here show that fluctuating selection is likely to contribute to the maintenance of genetic variance in sex-limited traits.

The model

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

Two alleles (A and B) at a single locus were assumed to influence the expression of a sex-limited trait in a haploid-dioecious organism with discrete generations and infinite population size. The two phenotypes produced by these alleles differ in fitness. I assumed that due to environmental fluctuations the fitness of both phenotypes varies temporally and two-environmental conditions (1 and 2) occur in equal frequencies. For the simulations I used cyclical selection with a cycle length of two generations to create a deterministic model. It should be noted that for infinite population size, conditions for polymorphism maintenance are identical for other cycle lengths and for random fluctuations ( Reinhold, 1999). Under condition 1 the fitness of individuals expressing allele A and B were assumed to be t and 1, respectively, and u and 1 under condition 2 (t, u > 0). I defined the strength of fluctuating selection as 1 − √(t/u) for u larger than t and as 1 – √(u/t) for t larger than u.

In haploids, overdominance is not possible so that a protected polymorphism can only be caused by fluctuating selection. To examine whether the results of the haploid model could be extended to the diploid case, I conducted simulations assuming a diploid sexual organism with discrete generations and infinite population size. For these simulations, I assumed multiplicative fitness of the heterozygotes, i.e. relative fitness of the heterozygotes (compared with the fitness of homozygotic individuals for allele B) was assumed to be √t and √u under conditions 1 and 2, whereas relative fitness of homozygotes for allele A was assumed to be t and u.

Using analytical calculations of the described haploid model, I estimated whether there are combinations of selection forces which can lead to a protected polymorphism. According to Reinhold (1999), the mean rate, the frequency of a rare autosomal allele A under fluctuating selection can be expected to increase per generation can be estimated by f(A|B)=√((t/2 + 1/2)(u/2 + 1/2)) when the effect of sex limitation is considered. Whenever f(A|B) is larger than one, allele A can be expected to increase in frequency. If under similar environmental conditions a rare allele B also increases in frequency (that is when f(B|A)=√((1/2t + 1/2)(1/2u + 1/2)) > 1) a protected polymorphism is maintained by fluctuating selection. The combination of possible values for t and u that lead to such a protected polymorphism can thus be calculated.

With the described equations I also estimated the strength of selection working to increase the frequency of a rare allele. These values, estimated for the symmetrical case (t=1/u) and for an asymmetrical case (t=0.8/u) thus give an estimate of the stability of the protected polymorphism. As an additional measure for the stability of the observed polymorphic cases, I introduced a third allele C with fitness effects r and s under conditions 1 and 2 into simulated haploid and polymorphic populations and observed the evolutionary stable allele frequencies.

In order to compare the fitness of asexual and sexual individuals, I compared the average reproductive success of a polymorphic sexual population with the reproductive success of a monomorphic-asexual population. For these simulations I examined the same symmetrical and asymmetrical cases as above. In the simulated cases, males were assumed not to contribute to offspring production and care. A cost of sex of 0.5 was thus assumed and fluctuating selection can only be expected to maintain sexual reproduction if sexual individuals have, on average, two times the reproductive success of asexual individuals.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

Whenever the assumed selection regime was symmetrical, a case in which individuals with allele A have at condition 1 a selective advantage against individuals with allele B that was as large as the advantage of individuals with allele B under condition 2 (i.e. t=1/u), both alleles will remain in the population ( Fig. 1a). Such a protected polymorphism also occurred in cases where the fitness effects of the two alleles were not symmetrical. Without sex-limitation there is only a neutral polymorphism when fitness effects are symmetrical – with sex-limited selection a large area of the possible combinations of fitness effects of alleles A and B lead to a protected polymorphism ( Fig. 1a). Such a protected polymorphism also exists for the diploid case when multiplicative effects of the alleles were assumed ( Fig. 1b).

image

Figure . 1. Description of the selection regimes that lead to a protected polymorphism. The limiting conditions for (a) the haploid case and (b) the diploid case are given as solid curves; within the shaded area between the curves a polymorphism is maintained by fluctuating selection on sex-limited traits. For the diploid case I assumed multiplicative fitness of alleles A and B. For traits expressed in both sexes, a neutral polymorphism can in both cases only exist for the values on the dotted line.

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When one of the two alleles was assumed to be initially rare, it increased in frequency for values of t and u that lead to a protected polymorphism. For the symmetrical case, the selective advantage of the rare allele is larger than 5% when the strength of fluctuating selection exceeds 0.5 ( Fig. 2). Whenever the strength of fluctuating selection is larger than 0.5, the effect of fluctuating selection should thus overcome the effect of drift except in populations with very low size or when the two alleles are very similar in their fitness effects (t ≈ u, Fig. 2). If a rare allele has lower arithmetic mean fitness than the other allele, stronger fluctuations in selection will be necessary to achieve similarly strong selection in favour of the rare allele ( Fig. 2).

image

Figure . 2. Influence of the strength of fluctuating selection on the advantage (per generation) of the rare allele in the symmetrical case (i.e. t=1/u, thick line) and a nonsymmetrical case (t=0.8/u, thin line). The dotted line indicates neutrality, the frequency of the allele under consideration increased for values above the line and decreased for values below the line.

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To examine the stability of the observed polymorphism, I also observed the fate of a third allele C introduced into a population polymorphic for the alleles A and B. Here, all possible combinations of 0.2 ≤ r, s, t, u ≤ 2 (with increments of 0.2) were examined provided they led to a stable polymorphism between alleles A and B before the introduction of allele C. After the introduction of allele C, one of the three alleles was always lost when the fitness values of the three alleles differed. Even when allele C was assumed to have similar-average fitness than the other two alleles (that is when t*u and r*s were assumed to be similar or equal to 1) one of the three alleles decreased in frequency and became extinct: When allele A had an advantage against allele B under condition 1 (i.e. t > 1) allele C was able to invade when r/s > t/u or when r/s < 1; in the first case allele A and in the second case allele B disappeared. In all cases where similar average fitness values were assumed, the allele with the intermediate fitness effects of fluctuating selection thus went extinct.

The observed genetic polymorphism maintained by fluctuating selection leads to an increase in mean population fitness. I compared this advantage with the fitness of asexual individuals where sex limitation cannot cause a protected polymorphism. With moderate environmental fluctuations the observed protected polymorphism contributes to reduce the cost of sex ( Fig. 3). But this advantage of sexual reproduction can only overcome the cost of sex under extreme fluctuations of selection in time.

image

Figure . 3. Average reproductive rate of sexual individuals relative to asexual individuals with fluctuating selection of different strength. For the simulation it was assumed that males do not contribute to offspring production and care so that the cost of sex can be given as a factor of two. The two lines represent the symmetrical case (i.e. t=1/u, thick line) and a nonsymmetrical case (t=0.8/u, thin line).

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

It has often been argued that temporal fluctuations in selection cannot explain the existence of a genetic polymorphism ( Hedrick, 1974; Hoekstra, 1975; Hedrick, 1986; Kirzhner et al., 1995 ). When two alleles influenced by fluctuating selection differ in their fitness effects, the allele with the higher geometric mean fitness is expected to increase to fixation. Here, I argue that the outcome of selection differs, when sex-limited traits are assumed to be influenced by temporal fluctuations. The presented model shows that a protected polymorphism can evolve when sex-limited traits are under fluctuating selection. This effect of sex limitation on the stability of a polymorphism is caused by a storage effect that automatically occurs when traits are expressed in only one sex. In the other sex, these alleles are shielded from selection, because they are not expressed and thus can serve a role analogous to a seed bank ( Ellner & Hairston, 1994; Sasaki & Ellner, 1995; Ellner & Sasaki, 1996). A similar but weaker effect occurs, when a trait is expressed in both sexes and when selection differs in strength between the sexes. Denoting whet(x, i) and whom(x, i) the fitness in generation i and sex x of the rare heterozygote and the common homozygote, a polymorphism is stabilized by fluctuating selection when the geometrical mean of [whet(♀, i)/whom(♀, i) + whet(♂, i)/whom(♂, i)]/2 over generations is larger than one for both types of homozygotes.

Is there any evidence that traits with sex differential selection are often polymorphic? There are some well-known cases of polymorphic traits that are sex limited. Among them is the polymorphism between satellite and calling males in frogs and crickets, body size polymorphism in males, in crustaceans and fish, sneaker and guarding males in fish and beetles and colour polymorphism in birds (see Andersson, 1994 for references). But, there are also polymorphic traits that are expressed in both sexes. And, any polymorphism found in sex-limited traits or other traits can also be caused by frequency-dependent selection. When rare types have a mating advantage or are preyed upon less frequently, selection will also lead to a polymorphism when fluctuating selection is not involved. One possibility to test the proposed hypothesis would be to examine the influence of fluctuating selection on genetic variance in sex limited and in other traits. The suggested effect of fluctuating selection on sex-limited traits should result in a stronger correlation between the strength of fluctuating selection and the genetic variance in sex-limited traits than in traits expressed in both sexes. Another way to test the proposed hypothesis would be to examine whether alleles with sex-limited expression show a larger heterozygosity than other traits. Presently, I am not aware of any existing data that would allow to examine whether fluctuating selection actually contributes to the maintenance of genetic variance in sex-limited traits.

The presented analysis shows that fluctuating selection can lead to a protected polymorphism when only two alleles with specific fitness effects are assumed to co-occur. But how will selection work on such a polymorphism when new alleles are introduced by mutation? Can the protected polymorphism be invaded by these new alleles? To answer this question, a third allele was introduced into a population polymorphic for two alleles. When the new allele was assumed to code for a phenotype with similar geometric mean fitness, invasion mainly depended on the strength of fluctuating selection on the new allele. In these cases, the new allele was only able to invade if it was more strongly influenced by fluctuating selection, and as a result of the invasion of the new allele one of the resident alleles went extinct. But, a protected polymorphism for all three alleles can also result when more than two conditions are assumed. Fluctuating selection on sex-limited traits thus preferentially maintains alleles with more extreme fluctuations in fitness and fluctuating selection can therefore lead to disruptive selection.

Fluctuating selection on sex-limited traits can only lead to a protected polymorphism when there are two sexes. In an asexual population there are no males and a storage effect thus cannot exist. Consequently, fluctuating selection will decrease any existing polymorphism. In a sexual population under fluctuating selection, a polymorphic allele will increase the mean reproductive success in the population and will decrease the cost of sex. This positive effect on the average reproductive success in the population is usually small. Only under very strong fluctuating selection can the cost of sex be offset by the increased mean fitness in the population that is due to the maintained polymophism. As there may be several independent sex-limited traits, one should note that the total effect of all polymorphic alleles together might be much larger than the effect of a single allele.

In conclusion, fluctuating selection of sex-limited traits can lead to the existence of a protected polymorphism. This effect may contribute to explain the maintenance of large genetic variance of traits under strong sexual selection. And, such a polymorphism will reduce the cost of sex when female-limited traits are involved. As alleles with more extreme effects of fluctuating selection are more likely to invade, fluctuating selection on sex-limited alleles may in addition cause disruptive selection.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography

I am grateful to Theo Bakker, Leif Engqvist, Joachim Kurtz, Bernhard Misof, Akira Sasaki and two anonymous reviewers for discussion and their helpful comments on previous drafts of the manuscript.

Bibliography

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Bibliography
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