The Advantages of Segregation and the Evolution of Sex

Corresponding author: Sarah P. Otto, 6270 University Blvd., University of British Columbia, Vancouver, BC V6T 1Z4, Canada., otto{at}zoology.ubc.ca (E-mail)

Communicating editor: M. K. UYENOYAMA

	ABSTRACT

TOP ABSTRACT MODEL COMPARISONS TO OTHER MODELS... DISCUSSION APPENDIX LITERATURE CITED

In diploids, sexual reproduction promotes both the segregation of alleles at the same locus and the recombination of alleles at different loci. This article is the first to investigate the possibility that sex might have evolved and been maintained to promote segregation, using a model that incorporates both a general selection regime and modifier alleles that alter an individual's allocation to sexual vs. asexual reproduction. The fate of different modifier alleles was found to depend strongly on the strength of selection at fitness loci and on the presence of inbreeding among individuals undergoing sexual reproduction. When selection is weak and mating occurs randomly among sexually produced gametes, reductions in the occurrence of sex are favored, but the genome-wide strength of selection is extremely small. In contrast, when selection is weak and some inbreeding occurs among gametes, increased allocation to sexual reproduction is expected as long as deleterious mutations are partially recessive and/or beneficial mutations are partially dominant. Under strong selection, the conditions under which increased allocation to sex evolves are reversed. Because deleterious mutations are typically considered to be partially recessive and weakly selected and because most populations exhibit some degree of inbreeding, this model predicts that higher frequencies of sex would evolve and be maintained as a consequence of the effects of segregation. Even with low levels of inbreeding, selection is stronger on a modifier that promotes segregation than on a modifier that promotes recombination, suggesting that the benefits of segregation are more likely than the benefits of recombination to have driven the evolution of sexual reproduction in diploids.

SEXUAL reproduction is widespread among eukaryotes (BELL 1982 ), but why sex evolved and why it is maintained in so many species have remained unresolved questions in evolutionary biology. Paradoxically, sexual reproduction, while common, entails several costs that are avoided by asexuals. Sexual organisms must find and court a mate, must risk disease transmission and predation during mating, and are prone to conflicts between the sexes, including conflicts over parental care (e.g., a partner may contribute few resources to offspring production; MAYNARD SMITH 1978 ) and conflicts over investment in current vs. future reproduction (MOORE and HAIG 1991 ; CHAPMAN et al. 1995 ). A further problem with sexual reproduction is that it breaks up genetic associations that have accumulated over time in response to selection. In a constant environment without mutations or genetic drift, these genetic associations are typically favorable, and theoretical analyses have demonstrated that decreased levels of recombination evolve under such circumstances (FELDMAN 1972 ; ALTENBERG and FELDMAN 1987 ; FELDMAN et al. 1997 ). The resolution to the paradox of sex must, therefore, lie with perturbations—resulting from biotic or abiotic changes in the external environment, mutation, and/or random genetic drift within a population.

Several theoretical studies have examined the evolution and maintenance of genetic mixing in the face of environmental change, mutation, and drift (see reviews by BARTON and CHARLESWORTH 1998 ; OTTO and MICHALAKIS 1998 ; WEST et al. 1999 ; OTTO and LENORMAND 2002 ), but these studies have largely ignored genetic associations within a locus under the assumption that sex evolved to promote recombination among alleles at different loci. Furthermore, those theoretical models investigating the evolution of rates of genetic mixing within a population (so-called "modifier models") have focused almost exclusively on the evolution of recombination rates. Yet sex entails the segregation of alleles at each locus as well as recombination between alleles at different loci. Just as recombination breaks up genetic associations among loci (linkage disequilibria), segregation breaks up genetic associations within a locus (departures from Hardy-Weinberg proportions). Thus, selection could indirectly favor the evolution of sexual reproduction through the effects of sex on one-locus genetic associations. Using a model that allows the allocation to asexual vs. sexual reproduction to depend on a modifier locus, this article investigates when we would expect increased sex to evolve as a consequence of segregation rather than recombination.

Within a randomly mating diploid species, sexual reproduction (meiosis followed by syngamy) breaks down associations between alleles carried on homologous chromosomes at a locus. Indeed, within a large, fully sexual population exhibiting nonoverlapping generations, genetic associations at a locus are completely eliminated and Hardy-Weinberg proportions are attained after one generation of random mating. Within asexual populations or partially sexual populations, however, one-locus genetic associations can persist and accumulate over time. Whenever these genetic associations affect fitness, indirect selection will act on any feature that alters their accumulation, including the level of sexual reproduction. Genetic associations between two alleles (A and a) at a locus (A) are typically measured by the inbreeding coefficient, F, as

(1)

where p_ij and p_k are the frequencies of genotype ij and allele k. As indicated by the first part ofEquation 1, F measures the difference between the observed frequency of a genotype and its expected frequency at Hardy-Weinberg equilibrium. As indicated by the second part ofEquation 1, F also measures whether homozygotes are more frequent (F > 0) or less frequent (F < 0) than expected based on the frequency of heterozygotes within the population. Thus, F can be thought of as a one-locus analog of the gametic-phase linkage disequilibrium (D), which measures whether combinations of alleles at two loci are more or less frequent than expected (see COMPARISONS TO OTHER MODELS AND INTERPRETATION). While F is called an "inbreeding coefficient," processes besides inbreeding can generate departures from F = 0, including selection and drift. While the associations between alleles at a locus generated by selection and drift would not persist in a fully sexual population, they do persist in a population that reproduces asexually as well as sexually.

With one exception (UYENOYAMA and BENGTSSON 1989 , discussed below), all previous models that have investigated the importance of segregation to the evolution of sex have focused on mean fitness comparisons of sexual and asexual populations rather than examining the conditions under which sex evolves within a population. Let us begin by reviewing these mean fitness results, focusing on the three main forms of selection at a locus (heterozygote advantage, purifying selection, and directional selection). First, consider heterozygote advantage. Within an asexual population, the frequency of heterozygotes would rise to fixation, at which point there would be a strong negative one-locus genetic association (F = -1). Within a sexual population, however, the segregation of alleles would break down the genetic association, and the less fit homozygotes would be formed by syngamy each generation. Hence, at equilibrium, a sexual population would suffer a decrease in mean fitness, known as the "segregation load," compared to an asexual population (CROW 1970 ; PECK and WAXMAN 2000 ).

Second, consider purifying selection acting against mutant alleles at a locus. Within a population containing both wild-type (A) and mutant (a) alleles, the relative fitness of a diploid individual can be written as

(2a)

where s is the selection coefficient (0 < s 1) and h is the dominance coefficient (0 h 1). The dominance coefficient, h, measures one-locus fitness interactions on an additive scale. An alternative coefficient that plays a more central role in the evolution of sex measures dominance on a multiplicative scale:

(2b)

On a log scale, measures whether homozygotes are more fit ( > 0) or less fit ( < 0) than expected based on the fitness of heterozygotes. Note that can be thought of as a one-locus analog of epistasis (), which is a measure of fitness interactions between alleles at two loci (see COMPARISONS TO OTHER MODELS AND INTERPRETATION). If mutations recur at frequency µ per gamete per generation and if mating is random among individuals reproducing sexually, the equilibrium frequency of Aa individuals is 2µ/(hs) in both asexual and sexual diploid populations (assuming weak mutation, µ << hs). The mean fitness is then 1 - 2µ. A more exact treatment that keeps track of order µ² terms (CHASNOV 2000 ) indicates, however, that a negative genetic association (F) develops whenever

(3a)

or, equivalently,

(3b)

Thus, when homozygotes are relatively less fit ( < 0), a departure from Hardy-Weinberg develops such that there are fewer homozygotes than expected at equilibrium (F < 0). This one-locus genetic association reduces the genetic variance in fitness, which hinders selection and slightly reduces the mean fitness at equilibrium. Segregation breaks down this detrimental association, causing sexual populations to have a slightly higher mean fitness than asexual populations at equilibrium (CHASNOV 2000 ). Conversely, when homozygotes are relatively more fit ( > 0), homozygotes become more common then expected (F > 0), which slightly increases genetic variance in fitness and the mean fitness at equilibrium. Now, the one-locus genetic associations built up by selection are favorable. Consequently, the mean fitness at equilibrium is higher in asexual populations, which preserve these associations, than in sexual populations. Typically, data on dominance suggest that deleterious mutations are partially recessive (h < ¹/₂; SIMMONS and CROW 1977 ; DENG and LYNCH 1997 ; GARCIA-DORADO et al. 1999 ). Thus, we expect (3) to hold and predict that sexual populations should have a higher equilibrium mean fitness than asexual populations. As long as sex involves random mating, this advantage is negligibly small unless deleterious mutations are very recessive (µ/s h ; CHASNOV 2000 ). When sexual reproduction is accompanied by inbreeding (the union of gametes that are closely related by descent), however, homozygotes become more common than expected at Hardy-Weinberg equilibrium, causing the mean fitness of sexual populations to be substantially higher than that of asexual populations (AGRAWAL and CHASNOV 2001 ).

Third, consider directional selection causing the spread of a favored allele, A, within a population. Although it is nonstandard, I continue to use the fitness regime described by (2) as this makes it easier to recognize parallels between the results with purifying and directional selection. The arguments made in the previous paragraph continue to apply when genetic associations are initially absent but are generated by directional selection. That is, if (3) holds, selection will cause homozygotes to become less common than expected, decreasing the genetic variance and slowing down adaptive evolution. By breaking down the one-locus genetic associations, sexual reproduction speeds up selection and gains a long-term advantage. Conversely, if (3) fails to hold, selection generates excess homozygosity, a genetic association that hastens adaptive evolution. Now, by breaking down the associations, sexual reproduction hinders selection and suffers a long-term disadvantage. This argument assumes that all genotypes are initially present. Imagine instead that A arises as a single mutation in a heterozygous individual. The spread of the favorable allele would then be limited to the fixation of the heterozygote within asexual populations, at which point adaptive evolution would stall until the AA homozygote was generated by a second mutation or mitotic recombination (KIRKPATRICK and JENKINS 1989 ). In contrast, the AA homozygote would be produced immediately by segregation within sexual populations, hastening adaptation and providing sexual populations with a long-term fitness advantage over asexual populations (KIRKPATRICK and JENKINS 1989 ).

The above discussion focuses on the effects of selection on one-locus genetic associations (F) and long-term mean fitness within an asexual population or a sexual population. These results can predict the outcome of competition between sexual and asexual populations, but only if the sexual and asexual populations are ecologically equivalent yet have been reproductively isolated for long enough for genotypes to reach the frequencies expected under each mode of reproduction. To fully understand the evolution of sex, however, we must also ask how the frequency of sex evolves within a population that is capable of both sexual and asexual reproduction, as is common among protists, fungi, algae, plants, and several invertebrate animal groups (BELL 1982 ). This article addresses this question by tracking the frequency of alleles that modify the relative allocation to the two modes of reproduction within a single population. Alleles at such a "modifier" locus (M) could act in any number of ways; for example, they could alter the probability of undergoing mitotic vs. meiotic cell division in unicellular organisms or alter the probability of reproducing via fission, budding, or apomixis in multicellular organisms.

In this article, evolution at the modifier locus is examined with respect to the dynamics at a locus, A, subject to either purifying or directional selection, which exhibit qualitatively similar results. For want of a better term, the A locus is called a "fitness locus." In a companion article, we examine the evolution of sex when the fitness locus is subject to heterozygote advantage (DOLGIN and OTTO 2003 , this issue), where, to our surprise, we also found that a modifier that increases the frequency of sex can spread under reasonable sets of parameters. A similar model was analyzed by UYENOYAMA and BENGTSSON 1989 , although they restricted their attention to lethal deleterious mutations; their results are summarized where parallels exist to this article. As we shall see, evolutionary change at the modifier locus depends strongly on the degree of inbreeding within the population and the degree of dominance and strength of selection at fitness loci. I argue that, for biologically reasonable values of these parameters, selection generally favors the evolution of increased levels of sexual reproduction and that such selection is strong relative to other deterministic forces acting on the evolution of sex.

	MODEL

TOP ABSTRACT MODEL COMPARISONS TO OTHER MODELS... DISCUSSION APPENDIX LITERATURE CITED

Consider two loci, a modifier locus M and a fitness locus A, within a diploid population with nonoverlapping generations. To track changes in allele frequencies at these loci, we begin by censusing at the juvenile stage, before selection, and then proceed through selection, mutation, and reproduction. Let x_ij equal the frequency of juveniles that carry haplotypes i and j (where i and j equal 1 for haplotype MA, 2 for Ma, 3 for mA, and 4 for ma). I assume that all loci are autosomal, that selection does not depend on the sex of the parent, and that there is no selection at the haploid or gametic stage. Consequently, I assume that x_ij = x_ji and keep track of x_ij for j i, only. Thus, for example, the frequency of MM AA individuals is x₁₁ but the frequency of MM Aa individuals is 2x₁₂. At this point, selection occurs according toEquation 2aEquation 2b. Let _ij equal the frequency after selection of adults carrying haplotypes i and j. Thus,

etc., where is the mean fitness within the population. At this stage, mutations from allele A to a occur at rate µ, regardless of the mode of reproduction. Mutations from allele a to A may also occur, but they are ignored because, at mutation-selection balance, allele a is so rare that the frequency of revertants to A is vanishingly small. In the case of a favorable allele spreading through a population, mutations assert a very small influence on the dynamics and are ignored. Let _ij equal the genotype frequencies after selection and mutation, where

etc.

At this point, reproduction occurs. The probability that an individual reproduces sexually depends on its genotype at the modifier locus, M:

If an individual of genotype ij does not reproduce sexually, which occurs with probability 1 - , then it contributes directly to the frequency of juveniles of genotype ij in the next generation (x^'_ij). If the individual reproduces sexually, meiosis occurs with recombination between the M and A loci at rate r. In many organisms with both sexual and asexual reproduction, including most sexual protists, fungi, algae, and nonseed plants, sex involves an alternation of generations between haploid and diploid phases (BELL 1982 ). I thus assume that meiosis generates haploid gametophytes, among whom the frequency of haplotype i is given by y_i. The frequency of MA haploids, for example, would be

(4)

where is the average allocation of the diploid population to sexual reproduction,

(5)

The haploid phase is assumed to be limited in scope, and selection in this phase is ignored. Genetically identical gametes are then produced by each haploid. The probability that any two gametes unite to form a zygote depends on the mating system. In this model, gametes undergo random union with probability 1 - f or inbreed with probability f. Random union produces diploids of genotype ij with probability y_iy_j. Inbreeding occurs among the gametes of a haploid gametophyte, resulting in the production of genotype ii from haploids of genotype i with probability y_i. I refer to this form of inbreeding as gametophytic selfing (this corresponds to "intragametophytic selfing" in the terminology of KLEKOWSKI 1969 ). Gametophytic selfing is only one mechanism by which inbreeding can occur. Inbreeding also occurs when there is sporophytic selfing (where the gametes of a diploid adult are mixed at random), mating among kin, and/or spatial population structure. It is important to keep in mind that the rate of inbreeding (f) is a measure of who mates with whom, whereas the inbreeding coefficient (F) measures a departure from Hardy-Weinberg proportions regardless of the cause of this departure. While all forms of inbreeding generate an excess of homozygosity (positive F), gametophytic selfing does this to the greatest degree (100% of offspring are homozygous). Nevertheless, it is expected that qualitatively similar results to the current model would be observed with other mechanisms of inbreeding.

The overall contribution to the juveniles of the next generation through sexual reproduction is then weighted by the population's average allocation to sex, . Thus, the frequency of MM AA juveniles in the next generation equals

(6a)

and the frequency of MM Aa juveniles (including both 12 and 21 genotypes) in the next generation equals

(6b)

Recursion equations (6c)–(6j) for the remaining diploid juveniles were derived similarly (available upon request). Table 1 summarizes the notation. For the case of s = 1, these recursions are identical to those developed by UYENOYAMA and BENGTSSON 1989 when inbreeding is absent (f = 0) but differ when inbreeding is present, because they assume sporophytic selfing rather than gametophytic selfing.

Throughout the analyses, the recursions (6) are used to determine when a modifier that increases the frequency of sex would spread within a population. To begin, I analyze the case where the population is at a mutation-selection balance at the A locus with the M allele fixed at the modifier locus. I then determine the conditions under which a new modifier allele, m, can spread if it alters the level of sex within the population. The results differ substantially depending on whether or not there is inbreeding (i.e., f = 0 or f 0), so these cases are discussed in turn. Next, I turn to the case of directional selection where a beneficial allele, A, is increasing in frequency within a population, assuming that mutation is a negligible force. Finally, connections are drawn between the results of this model of segregation and models of recombination and ploidy evolution. Mathematica 3.0 (WOLFRAM 1991 ) packages that were used to derive the results and to perform numerical analyses are available upon request.

Mutation-selection balance:
The equilibrium: When allele M is fixed, the recursions (6) reach a mutation-selection balance at which the AA genotype predominates as long as selection is stronger than mutation. Throughout, I assume that mutation is a weak force, that inbreeding, when present, is large relative to the mutation rate (f >> µ), and that hs and fs are not both small relative to µ. At equilibrium, genotypic frequencies remain constant (), and I denote the equilibrium frequencies by _ij. To find _ij, assume that mutation is rare, allowing us to expand and solve _ij in terms of µ. With M fixed, only three genotypes are present, and their frequencies at equilibrium are, to order µ²,

(7)

Here and throughout this article, c_i denotes a function, defined in Table 2, that is positive (or zero) under the stated assumptions. These functions do not necessarily have any biological meaning and are used solely to simplify the presentation of the equations. Note that when inbreeding is absent (f = 0), c₁/c₂ is 1/h, and ₁₂ equals the familiar µ/(hs) plus terms of order µ².

At this mutation-selection balance, one-locus genetic associations are generated by both selection and inbreeding. When mating is random (i.e., no selfing; f = 0), the one-locus genetic association calculated at the equilibrium described byEquation 7 equals

(8)

where O(µ²) denotes terms of the order µ² or smaller. Thus, as found by CHASNOV 2000 , the sign of determines the sign of (seeEquation 2aEquation 2b andEquation 3aEquation 3b). When is negative, the genotypes with the more extreme fitness (AA and aa) have a lower mean fitness on a log scale than the intermediate genotypes (Aa) and consequently become underrepresented within the population (_f=0 becomes negative), with the reverse holding when is positive.

Selfing and other forms of inbreeding (f > 0) generate a positive one-locus genetic association,

(9)

For weak selection, the inbreeding coefficient given by (9) approaches f, as expected for a neutral locus under gametophytic selfing. Note that the one-locus genetic association will typically be orders of magnitude larger when generated by nonrandom mating (9) than when generated by selection alone (8).

Stability analysis without inbreeding: To determine whether a modifier allele that alters an organism's reproductive allocation () to sexual vs. asexual reproduction will invade or disappear when introduced at low frequency within a population, I performed a local stability analysis on the recursions (6) in the vicinity of the equilibrium (7) (for a primer on stability analysis, see Appendices in BULMER 1994 or ROUGHGARDEN 1979 ). The fate of a rare modifier allele (m) depends on the eigenvalues () of the local stability matrix of (6). If all eigenvalues are less than one in magnitude, the m allele declines in frequency over time. Conversely, if at least one eigenvalue is greater than one, allele m will spread within the population. Without inbreeding (f = 0), invasion of the modifier allele at a geometric rate is predicted to occur only when the following eigenvalue is greater than one:

(10)

In contrast to c_i, the d_i denote functions (also defined in Table 2) that are known to change sign depending on the parameter values. If the new modifier allele, m, increases the frequency of sex (₂ > ₁), it will invade if _f=0 is greater than one, which requires that < 0 and d₀ > 0. In other words, there must be an intermediate level of dominance for sex to be favored:

(11)

The parameter range in which sex is favored shrinks as selection becomes weaker, with both the left- and right-hand side of (11) approaching ¹/₂ as s goes to zero. Sex is favored over the broadest range of parameters when deleterious mutations are lethal (s = 1), in which case modifiers that increase allocation to sexual reproduction spread for all dominance coefficients with tight linkage (r = 0) and for h > ₂/(2 + ₂) with loose linkage (r = ¹/₂). For lethal deleterious mutations (s = 1),Equation 11 is equivalent to Equation A2.2a in UYENOYAMA and BENGTSSON 1989 . Although (11) appears not to depend on the current level of sex (₁), the inequalities are easiest to satisfy when the modifier causes Mm heterozygotes to engage in a low frequency of sex (small ₂), which requires that the initial population be primarily asexual for the modifier to increase the frequency of sex (₂ > ₁). For dominance coefficients outside of the range given by (11), selection favors a decreased level of sex. These conditions are illustrated in Fig 1. Considering the case of weak selection and partial recessivity of deleterious mutations as the most biologically relevant, these results indicate that modifiers that increase the frequency of sex would be selected against when inbreeding is absent. The strength of this selection is, however, extremely weak (O(µ²)).

View larger version (28K): In this window In a new window Download PPT slide   Figure 1. Conditions under which a rare modifier that changes the frequency of sex without inbreeding (f = 0) spreads within a population based onEquation 10. Along the topmost curve, there are multiplicative fitness interactions within a locus ( = 0). For 1 = 0.5, modifiers that increase the frequency of sex spread only in the shaded region, i.e., when is negative but weak. This region expands in less sexual populations (1 = 0.1; dashed curve) and contracts in more sexual populations (1 = 0.9; thin solid curve). Other parameters are 2 = 1 + 0.01, r = 1/2.

Stability analysis with inbreeding: Inbreeding dramatically alters the conditions under which sex is favored by uncoupling the sign of the genetic associations (F) from the form of selection (compareEquation 8 andEquation 9). A second local stability analysis was performed to determine when evolution favors an increase in the frequency of sexual reproduction given that sex involves selfing or inbreeding (f > 0). The analysis indicates that a new modifier allele spreads at a geometric rate only when the following eigenvalue is greater than one:

(12)

Invasion thus requires that d₁ be positive.

Fig 2 Fig 3 Fig 4 illustrate the conditions under which a modifier that increases the frequency of sex is able to spread. The evolution of sexual reproduction is favored in two regions. In region 1, selection is weak and deleterious mutations are recessive (bottom left-hand sides in Fig 2 Fig 3 Fig 4), and in region 2, selection is strong and deleterious mutations are dominant (top right-hand sides in Fig 2 Fig 3 Fig 4).

View larger version (35K): In this window In a new window Download PPT slide   Figure 2. Conditions under which a modifier that changes the frequency of sex spreads within an inbreeding population. In A, the modifier is recessive (2 = 1; 3 = 1 + 0.01); in B, the modifier is additive (2 = 1 + 0.005; 3 = 1 + 0.01); in C, the modifier is dominant (2 = 1 + 0.01; 3 = 2). When sexual and asexual reproduction are equally frequent (1 = 0.5), increased sex is favored within the shaded areas. Typically, there are two regions in which a modifier that increases the frequency of sex spreads: (1) Deleterious mutations are weakly selected and partially recessive and (2) deleterious mutations are strongly selected and partially dominant. The boundary between the two regions () is indicated on the x-axis for 1 = 0.5. These regions shift to the left in less sexual populations (1 = 0.1; dashed curves) and shift to the right in more sexual populations (1 = 0.9; thin solid curves). Increasing the modifier's level of dominance contracts the first region and expands the second region, to the point that the first region entirely disappears when the modifier is completely dominant (C). Other parameters are f = 0.05, r = 0.5.

View larger version (34K): In this window In a new window Download PPT slide   Figure 3. The effect of the inbreeding rate on the conditions under which a modifier of sex spreads. The two regions in which sex is favored are shaded for intermediate inbreeding levels (f = 0.1), with their boundary occurring at . These regions shift to the right in less inbred populations (f = 0.0001; dashed curves) and shift to the left in more inbred populations (f = 0.5; thin solid curves). The regions in which sex is favored depend only weakly on f, unless inbreeding is common. While the curves are drawn usingEquation 12, which assumes that f >> µ, nearly identical curves are generated by exact numerical calculations of the eigenvalues with µ = 10-6. Other parameters are 1 = 0.5, 2 = 1 + 0.005, 3 = 1 + 0.01, and r = 0.5.

View larger version (36K): In this window In a new window Download PPT slide   Figure 4. The effect of recombination on the conditions under which a modifier of sex spreads within an inbreeding population. The two regions in which sex is favored are shaded for an intermediate recombination rate between the modifier and selected loci (r = 0.1). Region 1 contracts and region 2 expands when the loci are more tightly linked (r = 0.01; dashed curves), and the converse is observed for looser linkage (r = 0.5; thin solid curves). Other parameters are 1 = 0.5, 2 = 1 + 0.005, 3 = 1 + 0.01, and f = 0.05.

In examining the figures, I noted that, for each combination of parameters, there was a value of s, at which, simultaneously, the curve delimiting region 1 crossed the h = 0 axis and the curve delimiting region 2 crossed the h = 1 axis. At this exact point, called , sex was never favored, regardless of the dominance coefficient (e.g., occurs at s = 0.31 in Fig 4). Because the leading eigenvalue equals one along these curves, I determined the value of by setting h to either 0 or 1 (the result was the same) in (12) and solving _f>0 = 1 for s, obtaining

(13)

, which marks the boundary between regions 1 and 2, varies as a function of the frequency of sex (_i, explored in Fig 2) and the level of inbreeding (f, explored in Fig 3), but it is constant as a function of the rate of recombination between the modifier and fitness loci (r, explored in Fig 4). Note that the cutoff lies between zero and one for "directional modifiers," a term that I use to denote a modifier allele, m, that increases the frequency of sex (_{1 2 3}) or decreases it (_{1 2 3}).

Equation 13 allows us to determine how the cutoff between regions in which sex is favored varies with changing parameter values. For the following, I assume that the modifier is directional and define ₂ = ₁ + h_M and ₃ = ₁ + , where measures the homozygous effect of the modifier allele on the frequency of sex and h_M (0 h_M 1) measures the dominance of the modifier. From (13), it can be shown that

1. d/df 0. Higher rates of selfing/inbreeding decrease the cutoff, making it less likely that weakly selected, partially recessive mutations favor sex (see Fig 3).

2. d/d 0. Stronger modifiers increase the cutoff, making it more likely that weakly selected, partially recessive mutations favor sex.

3. d/dh_M < 0. More dominant modifiers decrease the cutoff, making it less likely that weakly selected, partially recessive mutations favor sex. In the special case of a fully dominant modifier, the cutoff goes to zero (Fig 2C).

4. d/d₁ 0. Higher rates of sex within the initial population increase the cutoff, making it more likely that weakly selected, partially recessive mutations favor sex (see Fig 2).

5. d/dr = 0. The cutoff does not depend on the recombination rate. Although the cutoff does not change, it is possible to show that the region in which sex is favored to the left of the cutoff expands in area for increasing recombination, while the region to the right of the cutoff decreases in area for increasing recombination (as seen in Fig 4). Thus, looser linkage makes it more likely that weakly selected, partially recessive mutations favor sex.

Next, let us consider the stability criterion for three cases of special interest. First, when selection is weak, the governing eigenvalue becomes

(14)

Thus, for weak selection, a modifier allele that increases allocation to sexual reproduction spreads whenever deleterious mutations are partially recessive (0 h < ¹/₂), as long as the rare modifier is not fully dominant. Second, when sexual reproduction involves high levels of selfing (f near 1), the governing eigenvalue becomes

(15)

Thus, a rare modifier allele that causes more sex (₃ - ₁ > 0) is always able to invade if inbreeding is high enough among the individuals that reproduce sexually (barring h = 0). Third, if the modifier introduces a small amount of sex ( << 1) into a fully asexual population, the governing eigenvalue becomes

(16)

Thus, a weak modifier allele is always able to invade an asexual population (barring h = 0). Strong modifiers that cause a substantial amount of sex within an otherwise asexual population spread, however, only if dominance (h) is sufficiently high.

Although the above analysis assumes gametophytic selfing, UYENOYAMA and BENGTSSON 1989 obtained similar results assuming sporophytic selfing and lethal mutations (s = 1). In the absence of selection, sporophytic selfing at rate b generates an inbreeding coefficient of F = b/(2 - b) (UYENOYAMA and BENGTSSON 1989 ), while gametophytic selfing at rate f results in F = f. Thus, to compare the two forms of selfing, I set f = b/(2 - b). BothEquation 12 and their results indicate that, when s = 1, sex is favored as long as h is greater than a threshold value (see right-hand edge of Fig 2 Fig 3 Fig 4). This threshold value differs quantitatively but not qualitatively between the two analyses.

In contrast to the case where inbreeding was absent, these results indicate that modifiers that increase the frequency of sex are positively selected when inbreeding is present for the most biologically relevant case of weak selection and partial recessivity of deleterious mutations, as long as a rare modifier is not fully dominant.

Simulation check: Deterministic simulations of the recursions were run using Mathematica 3.0 (WOLFRAM 1991 ) to confirm that the above stability analyses correctly identified the conditions under which sex is favored. The parameters chosen were identical to those in Fig 1 and Fig 2, with h and s set to every combination of {0.01, 0.1, 0.2, 0.3, ... , 0.8, 0.9, 1.0} and with µ = 10^-6. The frequencies of AA, Aa, and aa genotypes were set to the mutation-selection balance described by (7). The frequencies of MM, Mm, and mm genotypes were set to p²_M(1 - f) + fp_M, 2p_Mp_m(1 - f), and p²_m(1 - f) + fp_m, respectively, with the frequency of allele m (p_m = 1 - p_M) set to 0.001 and the selfing rate (f) set to 0 (for Fig 1) or 0.05 (for Fig 2). Linkage disequilibrium between the M and A loci was initially set to zero. The simulations were run for 10,000 generations with f = 0 and for 1000 generations with f = 0.05, and the total change in the modifier frequency was scored. For every parameter combination examined except four cases where no appreciable change in modifier frequency occurred (all near the curves with F = 0), the modifier rose in frequency when predicted by the regions delimited in Fig 1 and Fig 2.

Evolutionary stable strategy: We turn now to the long-term evolution of the system and ask whether there is a level of sex at which the population will remain and be stable to invasion by any new allele that arises and modifies the frequency of sex. This level of sex represents an evolutionary stable strategy (ESS; MAYNARD SMITH 1982 ). Because the strength of selection acting on the modifier is negligibly weak in the absence of inbreeding, we focus only on the case with inbreeding (analysis without inbreeding is available upon request).

When inbreeding is present, we must determine whether a value of ₁ exists (^*₁) that cannot be invaded by any modifier allele causing the frequency of sex to change to ₂ in heterozygotes and ₃ in homozygotes from the eigenvalue (12). Let us begin with the border solutions . From (16), an asexual population can be invaded by modifiers that introduce sex at sufficiently low rates. Thus, a fully asexual population is never an ESS. A population that is fully sexual is stable to invasion by any weak modifier allele when inbreeding is common, specifically, when

(17)

as illustrated in Fig 5. Numerical analyses suggest that, if the above condition holds and weak modifiers are unable to invade, then strong modifiers (₂, ₃ << 1) are also unable to invade.

View larger version (24K): In this window In a new window Download PPT slide   Figure 5. The ESS level of sex with inbreeding. For a given level of inbreeding (f), complete sexuality is an ESS when selection is sufficiently strong (to the right of the contours;Equation 17). For example, with f = 1, complete sexuality is an ESS over the entire parameter range, while, when f = 0.1, it is an ESS only in the last region at the top right of the graph. To the left of the contours there is no ESS, as there are always some combinations of 2 and 3 that allow a modifier to invade. A shows the case with free recombination between the modifier and fitness locus (r = 1/2); B shows the case with complete linkage (r = 0).

Interestingly, full sexuality is the only ESS with allele M fixed of this model with inbreeding. Any intermediate value of ^*₁ can always be invaded by some weak modifier if we allow all possible levels of dominance for the modifier. This can be shown by noting that an intermediate ESS must satisfy both

but these describe two different equations in one unknown (^*₁) that cannot be satisfied simultaneously. That this might be true can be gleaned from Fig 2. Consider the case where f = 0.05, ₁ = 0.9, r = ¹/₂, s = 0.4, and h = 0.097. This case falls on the solid line in Fig 2B, indicating that a weak additive modifier that increases or decreases sex cannot invade. Fig 2A indicates, however, that a weak recessive modifier that increases the frequency of sex could invade, and Fig 2C indicates that a weak dominant modifier that decreases the frequency of sex could invade. Given that there is no reason to believe that modifier alleles that alter the allocation to sexual and asexual reproduction would exhibit a particular dominance level, we conclude that there is no possible ESS in partially inbreeding populations with both sexual and asexual reproduction. Instead, we predict that the level of sexuality should fluctuate up and down over evolutionary time, depending on the exact sequence of modifier alleles that appear within the population. Nevertheless, the long-term average level of sexuality will depend on the selection parameters, and we can infer from the local stability analyses (see Fig 2 Fig 3 Fig 4) that sexual reproduction will be more common over evolutionary time when dominance (h) and selection coefficients (s) are both low or both high.

Genome-wide strength of selection: Here I estimate the genome-wide strength of selection acting on a modifier of sex assuming free recombination between all loci. Consider L fitness loci scattered throughout the genome, with no linkage disequilibrium between them, as might be expected if the fitness effects of each locus are independent and multiply together (MAYNARD SMITH 1968 ; ESHEL and FELDMAN 1970 ). The strength of indirect selection acting on a modifier allele through its effects on segregation at any one locus may be defined as - 1, where is the leading eigenvalue. It can be shown that, if the modifier allele is rare and selection is weak, measures the asymptotic rate at which a modifier changes in frequency:

Under the above assumptions, each fitness locus has only a small and independent effect on the frequency of the modifier, so we may sum the over the number of fitness loci (L) to get the genome-wide indirect effect of selection on a modifier of sex ().

In the absence of inbreeding, the genome-wide indirect selection on a rare modifier is

(18a)

(fromEquation 10). For each locus, the values of h, s, and µ will differ. Thus, this sum depends on the joint distribution of these parameters, which is unknown. Assuming, for the sake of argument, that there is little variance in each parameter and that selection is weak, the total strength of indirect selection on the modifier becomes

(18b)

where U is the mutation rate per diploid genome per generation and a bar over a parameter denotes its average value. This genome-wide force selects against sex but is exceedingly weak (proportional to the per-locus mutation rate) unless mutations are very nearly recessive, such that the denominator in (18a) is on the order of .

Much stronger selection on the modifier is observed when inbreeding is present. For weak selection (s << 1; fromEquation 14), the genome-wide selective force on a rare modifier is approximately

(19a)

As long as the modifier is not completely dominant and as long as deleterious mutations are partially recessive (0 < ), weak selection against deleterious mutations favors the evolution of sex with a force that is proportional to the genome-wide mutation rate times the effect of the modifier (₃ - ₂) times the inbreeding coefficient. The above calculations fail, however, to take into account the wide variation in dominance and selection coefficients among mutations. To make accurate predictions regarding the effects of segregation on the evolution of sex requires us to integrate over the joint distribution of h and s. Although this distribution is unknown, data from Drosophila suggest that a small percentage of deleterious mutations (5%) are lethal, and these tend to be more highly recessive (h 0.02–0.03) than mildly deleterious mutations (SIMMONS and CROW 1977 ; CHARLESWORTH and CHARLESWORTH 1999 ). Thus, it is worth asking whether the impact of relatively rare, lethal mutations outweighs the impact of mildly deleterious mutations on the evolution of sex. The genome-wide selective force on a modifier arising from such lethals can be simplified by assuming a weak additive modifier inEquation 12, yielding

(19b)

Except in populations with very little sex (₁ small), lethal mutations that are highly recessive tend to select against sex, but given that lethals account for only a fraction of deleterious mutations, (19b) tends to represent a smaller selective force than (19a) does. For example, if ₁ = 0.5, f = 0.05, h = 0.1 for weakly deleterious mutations, h = 0.02 for lethal deleterious mutations, and U_lethal is 5% of the total deleterious mutation rate, the strength of selection acting on a modifier arising from lethal mutations is only 10% of that arising from weakly deleterious mutations. Thus the combined force of many mild deleterious mutations and few lethal mutations still tends to favor the evolution of sex.

The above discussion assumes that sex entails no direct fitness costs (e.g., costs associated with searching for and courting mates, producing males, etc.). We can incorporate such fitness costs, , by multiplying the terms inEquation 6aEquation 6b representing sexual reproduction by (1 - ) and renormalizing. To simplify the analysis, I repeated the local stability analysis assuming a weak modifier and weak selection against deleterious mutations. Modifier alleles now change in frequency in response to two forces: the direct costs of sex (measured by ) and the indirect effects of altering segregation patterns at selected loci (measured by per locus and per genome). Only if the net effect is positive ( + > 0) will a modifier allele spread. To determine , the local stability analysis was performed by fixing the A allele at the selected locus (i.e., by setting µ = 0) and by defining - 1, yielding

(20)

which is negative for a modifier that increases the frequency of sex.Equation 20 equals the difference between the cost of sex paid by the old and new modifier alleles and is small whenever the modifier only slightly alters the frequency of sex. Next, mutations were reincorporated into the model, and a stability analysis was performed near the mutation-selection equilibrium. The indirect effect of the modifier per locus was defined as - 1 - , which was summed across loci to get the net indirect effect of the modifier, , again ignoring variation in the parameters. In the absence of inbreeding (f = 0), is only on the order of the per-locus mutation rate and hence is negligible relative to the costs of sex. With inbreeding, however, the indirect selective force on a modifier is

(21)

which differs slightly from (19a) because of the assumption of a weak modifier (reflected in the ₁ term) and because the cost of sex reduces the efficacy of sexual reproduction in breaking down genetic associations [reflected in the (1 + )/(1 - ) term]. Overall, the effects of a modifier on segregation will overwhelm the costs of sex only if the sum of (20) and (21) is positive. For a modifier that increases the frequency of sex, this requires that mutations be partially recessive (0 h < ¹/₂) and

(22)

Condition (22) indicates that a modifier that causes a slight increase in the frequency of sex can invade despite a twofold cost of sex ( = ¹/₂) as long as the genome-wide deleterious mutation rate (U) is high enough and/or the frequency of sex (₁) is initially low enough. As an example, when f = 0.05 and h = 0.1, the current allocation to sexual reproduction must be < 54% if U = 1 or 7% if U = 0.1 for sex to evolve. These calculations indicate that the advantages of segregation can be strong enough within inbreeding populations to select for costly sex, especially when sex is currently rare. Counterintuitively, the cost of sex does not always make condition (22) harder to satisfy. This is because the cost of sex reduces the effective level of genetic mixing within a population, causing the initial population to be similar to a more asexual population in which the advantages of sex are greater. The above assumes, however, that the modifier is weak, so that the modifier alleles differ very little in the cost of sex imposed upon them; numerical examples suggest that the costs of sex are less likely to be counterbalanced by the benefits of segregation for modifier alleles that cause large increases in the frequency of sex. The above also assumes that selection at the fitness loci (s) is weak. With stronger selection, both the intrinsic costs of sex and the effects of sex on segregation can select against modifiers that increase the frequency of sex (see Fig 2 Fig 3 Fig 4). Nevertheless, this analysis indicates that the inclusion of substantial costs of sex is not fatal to the hypothesis that the consequences of segregation might have shaped the evolution and maintenance of sex.

Directional selection:
Quasi-linkage equilibrium (QLE): Segregation could also provide an advantage to sex when populations are adapting to new environments. Insight into the dynamics of nonequilibrium populations can be gained using a method introduced by KIMURA 1965 , known as a quasi-linkage equilibrium (QLE) analysis (see BARTON and TURELLI 1991 ). The critical assumption made in a QLE analysis is that genetic associations reach an approximate balance between the forces that generate associations (e.g., selection, drift, and inbreeding) and those that break them down (e.g., sex and recombination) as long as the forces breaking down associations are sufficiently strong. Under this condition, genetic associations rapidly reach quasi-equilibrium at every point along the allele frequency trajectory. To solve for the QLE level of association, one sets the change in each association to zero and solves for its quasi-equilibrium value as a function of the current allele frequencies. Throughout the following I use the central-moment association measures as defined in BARTON and TURELLI 1991 and KIRKPATRICK et al. 2002 . These describe the gametic-phase linkage disequilibrium (D = y₁y₄ - y₂y₃), the linkage disequilibrium between alleles on homologous chromosomes, the departure from Hardy-Weinberg at locus A (the numerator of F inEquation 1), the departure from Hardy-Weinberg at locus M, the association between a modifier allele and the departure from Hardy-Weinberg at the viability locus, the association between a viability allele and the departure from Hardy-Weinberg at the modifier locus, and the association between the departure from Hardy-Weinberg at the modifier locus and the departure from Hardy-Weinberg at the viability locus. These seven association measures, along with the frequencies of the A and m alleles (p_A and p_m, respectively), provide nine independent equations that completely describe the dynamics and can be used to replace the genotypic frequencies, x_ij, inEquation 6a HREF="#FD6b">Equation 6b.

QLE without inbreeding: To find the QLE without inbreeding (f = 0), it was assumed that selection is weak [s = O()] and that the modifier is weak [₂ = ₁ + O() and ₃ = ₁ + O()], where is some small term ( << 1). The QLE values for each association measure were solved, keeping terms up to O(²), and then used to determine the change in frequency of the modifier allele (). The resulting equation for the per-generation change in the modifier is

(23)

where

(The derivation ofEquation 23 assumes that h is not very near ¹/₂. For nearly additive beneficial alleles, seeEquation 31.)Equation 23 indicates that, under weak selection, modifiers that increase the frequency of sex are always selected against. To leading order in s,Equation 23 is identical to the per-generation change in the modifier expected at mutation-selection balance (1 - _f=0 fromEquation 10) under the combined set of assumptions: The modifier is weak, m is rare, and p_A is near the equilibrium described byEquation 7. Thus, under directional selection as well as at a mutation-selection balance, weak selection on locus A generates indirect selection against a modifier allele that increases allocation to sexual reproduction as long as inbreeding is absent.

How strong is this force? As the A allele rises in frequency from p_A,0 at time 0 to p_A,T at time T, the cumulative change in the modifier allele would be

(24)

Under the assumptions of weak selection and frequent sexual reproduction, we may approximate the per-generation change in p_A by the differential equation

(25)

where

Transforming the independent variable inEquation 24 from time (t) to allele frequency (p_A), usingEquation 25 and integrating, we get

(26a)

If the beneficial allele rises from a low initial frequency (p_A,0 near 0) to a high final frequency (p_A,T near 1), the above simplifies to

(26b)

The factor in braces is nearly quadratic in shape and falls from ¹/₂ at h = 0 to 0 at h = ¹/₂ and then rises back to ¹/₂ at h = 1. Thus, the total strength of selection on the modifier arising per selective sweep is < -s/(2²₁) when inbreeding is absent. Consequently, the total amount of selection acting on the modifier locus M amounts to less than one generation's worth of selection on locus A, unless sex is rare. Note, however, that the QLE approximation will break down if sex is so rare that selection builds genetic associations faster than sex breaks them down.

QLE with inbreeding: A similar QLE analysis was conducted assuming that sexual reproduction occurs with inbreeding. The following equations for the change in the frequency of the modifier must be added to the above QLE results without inbreeding. Unless inbreeding levels are low [O() or smaller], however, inbreeding causes a greater change in the modifier and so the previous terms may be neglected. Per generation, the modifier allele changes in frequency by

(27)

where

For this QLE approximation to be valid, sex must be frequent relative to the rate of inbreeding (₁ >> f); otherwise the genetic associations are slow to reach steady state.Equation 27 indicates that a directional modifier allele that increases the frequency of sex will spread whenever h < ¹/₂ under weak selection. Again, to leading order in s,Equation 27 is identical to the per generation change in the modifier expected at mutation-selection balance (1 - _f>0 fromEquation 12) under the combined set of assumptions. There is, however, a key biological difference: The requirement that h be <¹/₂ implies that deleterious mutations must be partially recessive but beneficial mutations must be partially dominant for sex to be favored.

Equation 27 may be integrated over a selective sweep usingEquation 25 to rewrite time in terms of the allele frequency, p_A, where now g(p_A) = (1 - f)((1 - h)(1 - p_A) + hp_A) + f. The total change in the modifier per generation is then

(28)

For a directional modifier that increases the frequency of sex,Equation 28 is positive at h = 0, declines with increasing h, reaches 0 at h = ¹/₂, and becomes negative for h > ¹/₂. In the case where the beneficial allele rises from a low initial frequency (p_A,0 near 0) to a high final frequency (p_A,T near 1), (28) becomes approximately

(29)

where the approximation works best near h = ¹/₂ and underestimates the change in the modifier for h near zero or one.Equation 28 andEquation 29 indicate that each selective sweep within a genome causes a modifier allele that promotes sexual reproduction to rise in frequency, as long as beneficial alleles are weakly selected and partially dominant. Now the total amount of selection acting on the modifier depends not on s but on f and will be substantial when inbreeding rates are high.

Simulation check: The above QLE predictions were compared to deterministic simulations, which were performed as described for the case of a mutation-selection balance with the following exceptions. The A allele started at a frequency of 0.001, and the simulations were run until A reached a frequency of 0.999. Mutations were ignored. The initial frequencies of AA, Aa, and aa genotypes were set to p²_A(1 - f) + fp_A, 2p_Ap_a(1 - f), and p²_a(1 - f) + fp_a, respectively. Finally, weak selection (s = 0.01) and high initial frequencies of sex (₁ = 0.5 or 0.9) were assumed, as required for the QLE analysis to be valid. Fig 6 illustrates that the QLE analysis accurately predicts the total change in the modifier observed in simulations. Further simulations (available upon request) demonstrate, however, that the QLE predictions can be off by as much as a factor of five if either s or ₁ is set to 0.1.

View larger version (22K): In this window In a new window Download PPT slide   Figure 6. The total change in frequency of a modifier allele that increases allocation to sexual reproduction over the course of a selective sweep. The plots scale the total change as pm,total/(pMpm), which represents the selection gradient acting on the modifier, assuming a weak additive modifier (3 = 1 + 2). A is without inbreeding (f = 0; usingEquation 26a); B is with inbreeding (f = 0.05; using the sum ofEquation 26a andEquation 28). The long dashed curves are the analytical results and the circles are the simulation results when sexual and asexual reproduction are equally frequent (1 = 0.5). The solid curves are the analytical results, and the squares are the simulation results when sex is initially common (1 = 0.9). Other parameters are 2 = 1 + 0.005, 3 = 1 + 0.01, r = 0.5, pA,0 = 0.001, pA,T = 0.999, pM,0 = 0.999, and s = 0.01. As selection becomes stronger, the analytical results become less accurate.

As noted afterEquation 23 andEquation 27, the QLE results with directional selection are equivalent to those obtained at a mutation-selection balance when selection is assumed to be weak. Simulations were performed to explore whether the results remain similar under stronger selection. Without inbreeding, simulations indicate that the answer is "yes" (Fig 7A). With inbreeding, however, the conditions that favor sex at mutation-selection balance and under directional selection differ, especially as selection becomes stronger (Fig 7B). The discrepancy diminishes when the simulations are allowed to run for longer while the a allele is rare, as is the case at mutation-selection balance. Nevertheless, the qualitative result remains that sex, with inbreeding, is favored when dominance levels are low and selection is weak or when dominance levels are high and selection is strong.

View larger version (45K): In this window In a new window Download PPT slide   Figure 7. Conditions under which a modifier allele that promotes sexual reproduction increased in frequency over the course of a selective sweep. Such conditions are denoted by a "+"; a "-" indicates that the modifier decreased in frequency over the course of the simulations. The simulations of a selective sweep are compared to the conditions under which sex is favored at a mutation-selection balance (shaded regions), on the basis ofEquation 11 when inbreeding is absent (f = 0; A) and on the basis ofEquation 12 when inbreeding is present (f = 0.05; B). Other parameters are 1 = 0.5, 2 = 1 + 0.005, 3 = 1 + 0.01, r = 0.5, pA,0 = 0.001, pA,T = 0.999, pM,0 = 0.999, FA,0 = f, and FM,0 = f.