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. 2007 Aug;176(4):2465–2476. doi: 10.1534/genetics.107.076554

Sex-Ratio Evolution in Nuclear-Cytoplasmic Gynodioecy When Restoration Is a Threshold Trait

Maia F Bailey 1,1, Lynda F Delph 1
PMCID: PMC1950646  PMID: 17717197

Abstract

Gynodioecious plant species, which have populations consisting of female and hermaphrodite individuals, usually have complex sex determination involving cytoplasmic male sterility (CMS) alleles interacting with nuclear restorers of fertility. In response to recent evidence, we present a model of sex-ratio evolution in which restoration of male fertility is a threshold trait. We find that females are maintained at low frequencies for all biologically relevant parameter values. Furthermore, this model predicts periodically high female frequencies (>50%) under conditions of lower female seed fecundity advantages (compensation, x = 5%) and pleiotropic fitness effects associated with restorers of fertility (costs of restoration, y = 20%) than in other models. This model explains the maintenance of females in species that have previously experienced invasions of CMS alleles and the evolution of multiple restorers. Sensitivity of the model to small changes in cost and compensation values and to initial conditions may explain why populations of the same species vary widely for sex ratio.

GYNODIOECY, in which populations consist of female and hermaphrodite individuals, is a common breeding system of plants. In species with this breeding system, females have obvious disadvantages as compared to hermaphrodites because they achieve fitness only via female function and are dependent on external pollen arrival to achieve seed set. The question of what maintains females in gynodioecious species has been a topic of study in evolutionary biology since Darwin (1877). Currently, many theoretical models on the evolution of gynodioecy exist. These models make various assumptions about the genetic basis of gynodioecy, such as strict nuclear inheritance of sex (e.g., Ashman 2002; Schultz 2002), cytoplasmic inheritance of sex (e.g., Lloyd 1974), or joint nuclear-cytoplasmic control of sex (e.g., Gouyon et al. 1991; Couvet et al. 1998; Bailey et al. 2003).

Molecular studies of plants with nuclear-cytoplasmic sex determination have shown that mitochondrial genes cause cytoplasmic male sterility (CMS), i.e., female phenotype, while nuclear genes specific to the particular mitochondrial CMS gene can counteract the sterility effect restoring females to hermaphrodites (Chase and Gabay-Laughnan 2004). CMS is very widespread throughout the angiosperms (Kaul 1988) probably because the plant mitochondrial genome has two inverted repeats that allow for intragenomic recombination and the formation of the chimeric genes that cause CMS (Chase and Gabay-Laughnan 2004). Gynodioecy is less frequent than CMS in nature, however, because many species with CMS also have nuclear restorers that block the expression of the CMS trait. These CMS alleles can be uncovered in hybrid crosses among populations or species that are fixed for different CMS types and restorers. In many crop species, special nonrestored CMS lineages have been developed for use in producing hybrid seed. In these crops, the genetics of restoration is often complex and involves many genes (e.g., Dill et al. 1997; Pring et al. 1999; reviewed in Delph et al. 2007).

Natural gynodioecious species that have been studied fall into two types: those with strict nuclear control (e.g., Cucurbita foetidissima, Kohn 1989; Fragaria virginiana, Ashman 1999; Fuchsia excorticata, Godley 1955; Phacelia linearis, Eckhart 1992) and those with joint nuclear-cytoplasmic control (e.g., Beta vulgaris, Boutin-Stadler et al. 1989; Daucus carota, Ronfort et al. 1995; Lobelia siphilitica, Dudle et al. 2001; Plantago coronopus, Koelewijn and Van-Damme 1995; Raphanus sativus, Murayama et al. 2004; Silene vulgaris, McCauley et al. 2000; Thymus vulgaris, Dommée et al. 1978). In both types of gynodioecy, nuclear control of sex determination is generally not simple but involves multiple loci. For example, in gynodioecious species with nuclear sex determination, modifiers are often present (e.g., Eckhart 1992; Ashman 1999), while in nuclear-cytoplasmic gynodioecious species restoration of male fertility can depend on multiple nuclear loci (e.g., Belhassen et al. 1991; Koelewijn and Van-Damme 1995; Charlesworth and Laporte 1998). Researchers have often noted the continuous variation in sex ratio among crosses and have productively analyzed sex-ratio data assuming that restoration is a quantitative trait (e.g.,Taylor et al. 2001; Bailey 2002). A recent study of nuclear-cytoplasmic gynodioecy has shown that quantitative threshold models describe the nuclear portion of sex determination as well as or better than simpler Mendelian models in three well-studied species (Ehlers et al. 2005). Previous models of sex-ratio evolution in gynodioecious species have all assumed simple Mendelian inheritance of nuclear sex-determining genes. These findings suggest the utility of new models given that it is not clear how quantitative inheritance of nuclear factors will change the predictions of existing models.

In this article we describe a new deterministic model of nuclear-cytoplasmic gynodioecy in which sex is jointly determined by two CMS alleles interacting with multiple, diploid nuclear-restorer loci. These restorers act in both a background-sensitive and quantitative threshold manner such that restoration of male function occurs when the number of restorers expressed in the individual's CMS background exceeds a threshold value (Figure 1). These assumptions reflect recent evidence from natural systems and should accurately describe sex-ratio evolution in several well-studied cases of gynodioecy, e.g., P. coronopus, P. lanceolata, S. vulgaris, and T. vulgaris. We find that females are maintained at low frequencies for all biologically relevant parameter values, but that dynamic equilibria and very high periodic female frequencies are also possible.

Figure 1.—

Figure 1.—

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Sex determination is a threshold trait. Individuals with expressed restorer values below 3 are female. The two examples of restorer value distributions are from simulations using a threshold–cost profile and the hybridization scenario. (a) x = 0.3, y = 0; i.e., compensation is set at 0.3 and cost at 0 resulting in 9% females and a mean restorer value of 3.7 in the population. (b) x = 0.3, y = 0.8; i.e., compensation is again 0.3 but cost is 0.8, resulting in 17% females and a mean restorer value of 3.4 in the population.

METHODS

Because tracking multiple genotypes for two separate sets of restorer loci is prohibitive, we have instead used additive-genetic values to track nuclear-restoration ability of individuals and allowed them to make offspring with additive-genetic values distributed around the midparental mean. We assume two CMS types, A and B, and two CMS-specific nuclear-restorer additive-genetic values, a CMS-A restorer value, RA, and a CMS-B restorer value, RB, that are integer values between 0 and 5. Additive-genetic value “genotypes” are given as CMS type, RA value and RB value, e.g., A50 genotype individuals have a CMS-A cytotype, RA = 5 and RB = 0. We only allowed for six levels of restoration for each restorer type because, although restoration often appears additive in nature, we do not expect the number of restorer loci to be very high. Our use of additive-genetic values rather than actual nuclear genotypes means that we assume that all possible nuclear genotype variants are available within populations and cannot fix. In other words, it is impossible for the model to produce a population in which all individuals have the same restorer value. Because of the method of inheritance, even if the only reproducing individuals have the same restorer value, they will produce some offspring with slightly different restorer values. This does not mean that there are appreciable frequencies of all possible restorer values in the population at all times. For example in Figure 1, there are no individuals with restoration value of 0 in either model population at equilibrium. We justify this by pointing out that both standing variation and mutation rates for quantitative traits are generally high (Barton and Keightley 2002) and that pollen transmission among populations can reintroduce any variants lost from individual populations. Sex is determined by the restorer value corresponding to the cytoplasmic background and is a threshold trait such that a corresponding, or “expressed,” restorer value of 3 or more restores male function (Figure 1), e.g., A35, A42, and B05 genotype individuals are all hermaphrodites. All the nuclear loci determining restorer values are assumed to be unlinked such that inheritance of the two restorer values is independent. Although there is some evidence of linked nuclear-restorer alleles in Oryza sativa with the wild abortive type of CMS (Tan et al. 1998), most restorers appear to segregate independently (Chase and Gabay-Laughnan 2004).

Two processes cause variation among parents and offspring for restoration values: variation in the production of gametes and variation among the offspring produced by fusion of gametes with equivalent restoration values. These processes are effects of dominance within nuclear loci and additive effects among loci. Evidence from several species suggests that restorers are dominant even in those species with multiple nuclear-restorer loci (Koelewijn and Van-Damme 1995; Charlesworth and Laporte 1998; Taylor et al. 2001). In our model, individuals with RA = 3 will make gametes in the proportion of one-quarter with RA = 2, one-half with RA = 3, and one-quarter with RA = 4 due to segregation of dominant and recessive alleles. When gametes fuse they primarily produce offspring with the midparental value, but also produce some offspring within ±1 restorer value as a consequence of dominance within loci and additive effects among loci. For example, if the mean RA of two gametes is (2), the offspring will be one-quarter with RA = 1, one-half with RA = 2 and one-quarter with RA = 3. The offspring value will be lower if both parents carry restorers at the same loci and higher if they carry restorers at different loci. If the mean RA of two gametes is (4.5), the offspring will be one-half with RA = 4 and one-half with RA = 5. These assumptions are semiconservative, in that offspring vary from parents much less than is possible in real systems but more consistently than in real systems where alleles can fix and cause offspring to have the same trait values as parents.

Two variables control the dynamics of the model: compensation and cost. Compensation describes the differences in seed fecundity between females and hermaphrodites. We assume that compensation values are equivalent for the two CMS types; therefore, females uniformly make no pollen and have equivalent or higher seed fecundity than hermaphrodites. Compensation is represented by x such that hermaphrodite fecundity equals (1) and female fecundity equals (1 + x). Cost describes variation in fitness among hermaphrodites associated with restorers. In our model, cost is the loss of pollen fitness that is associated with the restorer value that does not correspond to the CMS background of the individual, i.e., the restorers that do not affect sex determination in that individual and are “silent.” We do not consider costs associated with matching, sexually expressed restorers (expressed costs) or pleiotropic effects independent of the CMS background (constitutive costs). In the former case, theoretical work indicates that only recessive expressed costs would support gynodioecy (Bailey et al. 2003); however, we know from naturally occurring restorers in crop plants that pleiotropic effects are usually dominant when present (e.g., Singh and Brown 1991). Moreover, all recorded pleiotropic changes in gene expression associated with nuclear restorers occur when the restorer is present with a mismatched CMS type or occurs in all CMS backgrounds (Delph et al. 2007) suggesting that expressed costs are not a realistic component of gynodioecious systems. Although constitutive costs are both possible and probable, we have not modeled constitutive costs because previous work comparing silent and constitutive costs indicates that they have similar effects on sex-ratio evolution (Bailey et al. 2003).

The assumption that cost acts on pollen fitness is based on data showing that the molecular action of nuclear restorers often occurs in anther tissue (e.g., Abad et al. 1995; Brown 1999; reviewed in Delph et al. 2007). Cost is represented as y such that the hermaphrodites in the population with the highest silent restorer values, e.g., A35 and B53 individuals, have a pollen fitness of (1 – y). The pollen fitness of hermaphrodites with intermediate silent restorer values depends on which cost profile we used (Figure 2). When cost is a threshold trait, CMS-A hermaphrodites with RB = 0, 1, or 2 have pollen fitness of 1; those with RB = 3, 4, or 5 have pollen fitness equal to (1 − y). When cost is an additive trait, individuals vary continuously for pollen fitness as a function of their restorer value. Cost can also be a threshold/additive trait when individuals with silent restorer values above the sex-determination threshold exhibit cost, but that cost is variable, with higher restorer values incurring higher costs (Figure 2).

Figure 2.—

Figure 2.—

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Cost of restoration acts on pollen fitness. Cost is associated only with those loci that do not match the cytoplasmic background and are “silent” in determining sexual phenotype. This effect is modeled in three ways: the threshold, threshold/additive, and additive–cost profiles, which are shown with a solid gray line, dotted black line, and solid black line, respectively.

As the resulting model (appendix) is intractable to directly solving for evolutionary stable strategy (ESS) conditions, we ran computer simulations of an infinite population for 2000 generations for combinations of compensation (x) and cost (y) values across the biologically relevant range of these variables (0 ≤ x ≤ 1, 0 ≤ y < 1). While compensation can exceed 1, when this occurs, the system is vulnerable to the invasion of a nuclear feminizing factor (Gouyon et al. 1991). When cost equals 1, hermaphrodites with silent restorer values of 5 (and others depending on the cost profile) produce no pollen and become another type of female.

Because model predictions are sensitive to initial genotype frequencies, we simulated two different initial populations, which we refer to as the migration scenario and hybridization scenario, respectively. In the first, we mimic the genotype frequencies that occur when a hermaphrodite migrant from a CMS-B population is introduced into a population of CMS-A hermaphrodites with both ancestral populations having low frequencies of silent restorers (99% A50, 1% B05). In the second set of simulations, we model a hybrid zone between two hermaphrodite populations, both of which carry high frequencies of silent restorers (50% A45, 50% B54). This scenario was suggested by recent evidence from a gynodioecious species that more females are produced in within-population crosses than in among-population crosses even when the populations are known to differ for cytoplasmic variation (Bailey and McCauley 2005). Theory predicts that silent restorers could be found at high frequency in some populations even though there is a cost of silent restorers if the cost is too small to prevent their spread or because of founder effects or drift. We further explored the effects of initial conditions by choosing a set of parameters that yield very different results under the two initial condition scenarios (x = 0.2, y = 0.7, additive cost profile) and re-ran the model using a factorial design with six initial RA values (0 – 5), six RB values (0-5), and six CMS-A frequencies (0.01, 0.10, 0.20, 0.30, 0.40, 0.50). In these runs, RA and RB values were the same in both CMS backgrounds. Because some hermaphrodites must be present for reproduction to occur, we were not able to model populations in which both RA and RB values in the initial population are below 3.

At 2000 generations, simulations had either settled to a point equilibrium or continued to cycle. When the latter occurred, cycles were examined to determine if the maximum and minimum values were stable or changing. If either the maximum or minimum values were not stable, simulations were run for up to 5000 generations to determine if a stable dynamic equilibrium had been achieved. In reporting dynamic equilibria, we have graphed the maximum female frequency value occurring in the stable cycles.

RESULTS

All parameter combinations for all three cost profiles and both initial genotype frequencies predict maintenance of females at above 6% frequency (Figure 3). Depending on the cost profile (threshold, threshold/additive, or additive) and the initial conditions (the migration or hybridization scenarios) assumed, dynamic equilibria are possible and may be present when cost and compensation are very small. Maximum female frequencies in dynamic equilibria cycles exceeded 60% in all cases where dynamic equilibria occurred. The smallest parameter values to give dynamic equilibria are for the migration scenario when cost is a threshold trait. Under these conditions, female frequencies cycle between 9 and 67% when females have 5% higher fecundity than hermaphrodites and hermaphrodites with silent restorer values of 3 or higher have 20% less pollen fitness than other hermaphrodites (i.e., x = 0.05 and y = 0.20).

Figure 3.—

Figure 3.—

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Effects of initial conditions and cost of restoration on female frequencies. Initial genotype frequencies are 99% A50 and 1% B05 in a, c, and e (the migration scenario) while initial frequencies are 50% A45 and 50% B54 in b, d, and f (the hybridization scenario). A threshold–cost profile is modeled in a and b, a threshold/additive–cost profile in c and d, and an additive–cost profile in e and f. When dynamic equilibria occur, the maximum female frequency has been graphed. Dynamic equilibria include all points in which graphed female frequencies are 60% or more.

When both cost and compensation are zero, females are produced as a result of variation between parents and offspring. When compensation is zero, CMS frequencies do not change from the initial conditions as there is no force to increase one type over the other whether or not females are present. When compensation is nonzero, CMS frequencies change when the CMS types have different distributions of expressed restorers because female frequencies and compensation will favor one cytoplasmic type over the other. The speed at which CMS frequencies change, potentially driving cycles of female frequencies, depends on the degree of compensation (Figure 4).

Figure 4.—

Figure 4.—

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The effects of compensation and cost on female-frequency cycles. Thin lines indicate CMS-A female frequencies and thick lines total female frequencies. (a and c) Compensation, 0.10. (b and d) Compensation, 0.50.

Cost primarily determines the transition from point to dynamic equilibria (the sharp increase in female frequency in Figure 3) as well as the mean restoration values at equilibrium (Figures 1 and 4). Although there is some dependence on compensation, dynamic equilibria are most strongly correlated with large costs. Large costs cause the population to evolve toward lower restorer values in general because of selection against hermaphrodites with high silent restorer values. When the distribution of restorer values shifts toward lower values, more females are produced because a higher proportion of the distribution of offspring falls below the threshold for restoration of male function (Figure 1). Cost also affects the speed of female-frequency cycles by determining the size of the selection differential between different classes of hermaphrodites in the population, although this effect on periodicity is small compared to the effect of compensation (Figure 4).

Different cost profiles and initial condition scenarios differ in both the placement and shape of the transition between point equilibria and dynamic equilibria. The threshold-cost profile predicts dynamic equilibria over the largest proportion of our parameter space. Both the threshold- and additive-cost profiles allow for periodic female frequencies up to 91% when cost and compensation are large while the threshold/additive profile predicts maximum female frequencies around 83% when cost and compensation are large. The hybridization scenario differs from the migration scenario in that it predicts dynamic equilibria only when cost and compensation are near 1 for the threshold and threshold/additive cost profiles and not at all for the additive cost profile. We explored these effects of initial conditions further using the additive cost profile with compensation at 20% and cost at 70%. For these parameter values, predicted maximum female frequencies were either 83% with a dynamic equilibrium or a stable 12% (Figure 5). Point equilibria occurred when initial restorer values were both high (≥3) and when both CMS types were common in the initial population (CMS-A ≥ 0.20).

Figure 5.—

Figure 5.—

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Dynamic or point equilibria results depend on initial restorer values and CMS frequencies. Model predictions of maximum female frequencies under standard parameter values (x = 0.20, y = 0.70, additive cost profile) but with varying initial conditions. (a) CMS-A = 0.01, CMS-B = 0.99. (b) CMS-A = 0.10, CMS-B = 0.90. (c) CMS-A = 0.20, CMS-B = 0.80. (d) CMS-A = 0.30, CMS-B = 0.70. (e) CMS-A = 0.40, CMS-B = 0.60. (f) CMS-A = 0.50, CMS-B = 0.50.

DISCUSSION

Our results from this model are roughly similar to previous model results in which restoration is controlled by one nuclear locus per CMS type (Gouyon et al. 1991; Bailey et al. 2003) verifying the utility of the current model. However, the present results differ from previous models in two important ways. First, we found that predicted female frequencies are nonzero for all parameter values. Second, dynamic equilibria are possible under smaller costs and compensation values (as little as x = 0.05, y = 0.20) than have been found in other models (e.g., x = 0.6, y = 0.37 in Bailey et al. 2003). Furthermore, we found that dramatically different predictions of equilibrium sex ratios can be made depending on initial population genotype frequencies. These novel findings may explain many common sex-ratio patterns in gynodioecious species.

In our model of sex-ratio evolution in nuclear-cytoplasmic gynodioecious species, we find that quantitative restoration in populations with multiple CMS types always allows for the maintenance of females. When cost is small or zero, females are produced as an effect of gametic segregation and recombination in crosses between hermaphrodites rather than as a result of direct selection to increase CMS frequencies or reduce restorer frequencies. This is similar to the maintenance of females when there is overdominance at a single Mendelian locus for sex determination (Ross and Gregorius 1985; Bailey et al. 2003).

Very high female frequencies can be explained by this model even when cost and compensation are small; however, these values depend on how cost is distributed among hermaphrodite genotypes (the cost profile) and the initial genotype frequencies (the migration or hybridization scenarios). Dynamic equilibria occur when selection differentials among individuals in the model population are large enough to drive restorer value cycles in which RA and RB distributions alternate in temporary shifts below the threshold value. When RA and RB distributions differ, one CMS type will be disproportionately female and rapidly increase in frequency as a result of compensation. Next, selection to increase the mean value of expressed restorers for the common CMS type causes a decline in female frequency and concomitant increase in the number of restored hermaphrodites. Selection among hermaphrodites then reduces the mean value of silent restorers because individuals with high silent restorer values have low pollen fitness. Once the mean of the distribution of silent restorer values falls below that of the currently expressed restorer, individuals with the minority CMS type are more likely to be female and produce more seeds than individuals with the majority CMS type due to compensation, which begins a new cycle. The occurrence of dynamic equilibria depends on starting conditions, compensation and cost; however, cost appears to be more important than compensation in causing cycles to occur.

Initial conditions cause the model to result in point or dynamic equilibria depending on whether there are sufficiently high fitness differentials among individuals in the initial generations of the model. This is illustrated in the effect of the hybridization scenario on model outcomes when the additive-cost profile is used. Under these simulation conditions, hermaphrodites present in the early generations of the model all have high silent restorer values and have small pollen-fitness differences. Because these fitness differentials are small, restorer frequencies change slowly and stop changing before individuals with low restorer values, i.e., high fitness hermaphrodites and females, are produced at high frequencies. Because individuals with extreme fitness values are rare, selection differentials between individuals are small and the change in gene frequencies between generations is small. When extremely fit or extremely unfit genotypes are not present in populations, female-frequency cycles cannot start because of a lack of evolutionary momentum.

When cycles do occur in our model, they appear robust against the loss of females and cytoplasmic genetic variation. Minimum female frequencies are above 6% for all point equilibria and above 8% for all parameter values giving dynamic equilibria. Frequencies of CMS types vary more widely than female frequencies during cycles (between 1 and 99% at high cost and compensation values); however, this may not indicate a higher likelihood of loss of cytoplasmic variation as recent models (Wade and McCauley 2005) and evidence (McCauley et al. 2005) of paternal leakage in gynodioecious species indicates that CMS variation, once acquired by a population, may be buffered from loss by rare paternal transmission of cytoplasmic genes.

Because we used additive-genetic values rather than nuclear genotypes, we are unable to directly evaluate the effect of quantitative restoration on nuclear genetic variation. In general, drift could cause the loss of either restorer or maintainer alleles, limiting the number of nuclear loci segregating in any one population for sex determination. However, when costs of restoration are small, females and maintainer alleles are at low frequencies such that maintainer alleles will be more likely than restorer alleles to be lost as a consequence of drift. Over time, this combination of selection and drift could cause the loss of females. At this point, further drift and selection among hermaphrodites carrying different CMS types will either cause the loss of CMS types that express costs associated with the fixed restorer alleles or the expression of some level of cost in all hermaphrodites in the population. In natural populations the loss of diversity among sex-determining genes due to drift could lead to a return to all-hermaphrodite populations (hermaphrodity) with latent CMS gene(s) or could contribute to an Allee effect via losses in pollen fitness if costs of restoration become fixed.

Our model suggests that if multiple CMS-restorer systems evolve and if nuclear restoration involves many loci, females and genetic variation will be maintained in populations that contain at least two CMS types, thereby explaining the persistence of gynodioecy. Our model simulates the outcome of gene flow between two all-hermaphrodite populations in which sex determination is controlled by a single fixed CMS allele and multiple nuclear-restorer alleles. We need to ask, then, how common are all-hermaphrodite populations with CMS and multiple nuclear restorers?

As noted in the introduction, many crop genomes have been shown to harbor both CMS and multiple nuclear-restorer alleles. This underlying complexity of sex determination is perhaps surprising as the natural breeding system of these crops is hermaphrodity. This complexity probably arises during the process of evolving back to hermaphrodity from gynodioecy after a CMS invasion. As soon as a CMS gene is present in a population, there is strong selection for any nuclear mutation that can counteract the phenotypic effects of CMS. It makes intuitive sense that the first restorers to evolve and begin to spread may be costly and that they would be replaced or modified by less costly restorers over time resulting in a population with one CMS type, multiple restorers, and only hermaphrodite individuals.

Evidence from non-crop species also supports our assumption of quantitative restoration. Studies of the genetics of restoration in natural gynodioecious populations have repeatedly rejected simple Mendelian models (e.g., Belhassen et al. 1991; Charlesworth and Laporte 1998; Dudle et al. 2001). A recent article by Ehlers et al. (2005) confirmed that quantitative models of sex determination predict sex ratios as well if not better than multilocus Mendelian models in three well-studied gynodioecious species. Therefore, evidence from both crop and wild species confirms that our model's starting conditions probably accurately describe a stage in the evolution of many gynodioecious species.

Our model predicts that once seed migration occurs between two populations fixed for different CMS alleles and their corresponding multiple nuclear restorers, the combination of CMS polymorphism and inheritance patterns of quantitative traits will produce females at a rate determined by the genetic variance due to dominance relationships within restorer loci (dominance variance) and additive effects among loci (epistatic variance). When compensation is nonzero, female frequencies increase. Very high female frequencies (above 50%) are also possible when compensation is nonzero and costs are sufficiently high to cause dynamic equilibria. Therefore, if we need to appeal to dynamic equilibria to explain high female frequencies in a particular gynodioecious species, we need to measure both compensation and cost for comparison to model predictions. Measurements of compensation can be difficult because female fecundity advantages may be obscured by pollen limitation (Ashman 2000) and/or seed-set plasticity among hermaphrodites (Delph 2003). For most cases, researchers may choose to merely demonstrate that compensation is nonzero and instead focus on measuring cost as our model predictions are more sensitive to this parameter.

Many gynodioecious species are self-fertile allowing hermaphrodites, but not females, to produce offspring by self-fertilization. Self-fertilization causes two related phenomena: increased homozygosity and lower relative fitness of the resulting offspring as compared to offspring resulting from outcrosses if inbreeding depression exists. Because our model uses additive genetic values and considers only these values in calculating fitness, we cannot incorporate these effects of self-fertilization. In general, self-fertilization would probably lead to enriching the population for extreme additive genetic values and thereby increase the frequency of females. However, this adjustment of the model would not capture the possible synergies between inbreeding depression and sex. Several researchers have shown that self-fertilized hermaphrodites produce higher frequencies of females than outcrossed hermaphrodites (Emery and McCauley 2002; Bailey and McCauley 2005; Glaettli and Goudet 2006). Emery and McCauley (2002) argue that females are more common among offspring produced by self-fertilization because restorer alleles tend to be dominant; thus, self-fertilization increases female frequencies via the increased production of homozygous recessive genotypes. Therefore, David McCauley has suggested (personal communication to M. Bailey) that average female fitness could decline relative to hermaphrodite fitness in self-compatible populations because the majority of females would be the result of self-fertilization and suffer from inbreeding depression. An individual-based simulation or analytic model that could link fitness of individuals in a given generation to the degree of inbreeding in the previous generation would be needed to theoretically explore this hypothesis.

For most nuclear-cytoplasmic gynodioecious species, we predict that populations at equilibrium without CMS variation will be all-hermaphrodite unless compensation is very high (x > 1), allowing for nuclear-controlled gynodioecy (Lewis 1941). Populations with multiple CMS types may also be all-hermaphrodite at equilbrium if restoration is simple and costs are small or if silent restorers are at high frequencies. Females can be maintained at equilibrium if multiple CMS types are present and either (1) restoration is controlled by few loci and restorer alleles are associated with costs (y ≥ 0.37) (Gouyon et al. 1991; Bailey et al. 2003), or (2) restoration is complex. When restoration is complex and compensation is slight but nonzero, we show here that high female frequencies (above 50%) can be maintained under equilibrium conditions if hermaphrodites that have many restorer alleles have at least 20% pollen loss as compared to hermaphrodites that have fewer restorer alleles (y ≥ 0.20).

Our model prediction that females will be produced at a low rate as an effect of quantitative inheritance of restoration may explain the presence of low frequencies of females in many gynodioecious species (e.g., 0 – 13% in D. carota, Ronfort et al. 1995; 0 – 12% in Hirschfeldia incana, Horovitz and Beiles 1980; 0 – 11% in Iris douglasiana, Uno 1982; 0 – 15% in S. littorea, Guitián and Medrano 2000). The sensitivity of our model predictions to small changes in cost and compensation values and the effect of initial conditions may explain why dramatically different female frequencies are often observed among populations of gynodioecious species. Consistent variation among populations for cost or compensation values could result from ecological effects on reproduction, such as pollen limitation. Founder effects, drift, and isolation may limit selection differentials among individuals by eliminating genotypes with extreme fitness values, thus preventing the evolution of restorer-value cycles.

TABLE A1.

The variables used in this model

Variable Definition
aAi The frequency of CMS-A individuals with RA value i in the population
aBi The frequency of CMS-B individuals with RA value i in the population
bAj The frequency of CMS-A individuals with RB value j in the population
bBj The frequency of CMS-B individuals with RB value j in the population
cA The frequency of CMS-A in the population
cB The frequency of CMS-B in the population
fA The frequency of CMS-A females in the population
fB The frequency of CMS-B females in the population
hA The frequency of CMS-A hermaphrodites in the population
hB The frequency of CMS-B hermaphrodites in the population
aAi, ovules The frequency of ovules made by CMS-A individuals with RA value i
aBi, ovules The frequency of ovules made by CMS-B individuals with RA value i
bAj, ovules The frequency of ovules made by CMS-A individuals with RB value j
bBj, ovules The frequency of ovules made by CMS-B individuals with RB value j
aAi, pollen The frequency of pollen grains made by CMS-A individuals with RA value i
aBi, pollen The frequency of pollen grains made by CMS-B individuals with RA value i
bAj, pollen The frequency of pollen grains made by CMS-A individuals with RB value j
bBj, pollen The frequency of pollen grains made by CMS-B individuals with RB value j
ai,ovules The frequency of ovules with RA value i among all ovules
bj,ovule The frequency of ovules with RB value j among all ovules
ai,pollen The frequency of pollen grains with RA value i among all pollen
bj,pollen The frequency of pollen grains with RB value j among all pollen
ai′ The frequency of individuals with RA value i in the next generation
bj′ The frequency of individuals with RB value j in the next generation
cA′ The frequency of individuals with CMS-A in the next generation
cB′ The frequency of individuals with CMS-B in the next generation
x compensation The excess seed fecundity of females compared to hermaphrodites
y cost A loss of pollen fitness associated with silent restorer values above the threshold value
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Acknowledgments

The authors thank Curt Lively for his comments on the model and manuscript. This work was supported by a National Science Foundation grant to L.F.D. (DEB-0210971).

APPENDIX

To calculate the change in genotype frequencies from one generation to the next, we need to calculate the different gamete types that are produced and in what proportions and then the types of zygotes that these gametes can produce under random mating. In the following discussion we develop the recursion equations used for our model under the assumption of a threshold-cost profile (Figure 2). The equations are the same for the other cost profiles if the pollen-fitness values from Figure 2 are substituted in the equations calculating the distribution of genotypes among pollen grains.

Gamete production:

In calculating the types of gametes made, we need to consider sexual phenotype, compensation, and cost effects. To mimic the effect of segregation of dominant and recessive alleles during gametogenesis, we allow individuals to make a portion of gametes with ±1 restorer value. For example, an individual with RA value of 1 makes Inline graphicRA = 0, Inline graphicRA = 1, and Inline graphicRA = 2 gametes.

Both female and hermaphrodite individuals make ovules, but compensation affects only females. There are 20 total ovule values, but we show only the calculations for the RA values. The calculations for RB values are similar.

For CMS-A individuals, only females produce ovules with RA values of 0 and 1 (see Table A1 for variable definitions):

graphic file with name M4.gif

Both female and hermaphrodite CMS-A individuals make ovules with RA values of 2 and 3:

graphic file with name M5.gif

CMS-A hermaphrodites are the only CMS-A individuals that make ovules with RA values of 4 and 5:

graphic file with name M6.gif

Among CMS-B individuals, both females and hermaphrodites produce ovules with all six RA values. For example:

graphic file with name M7.gif

We can calculate the rest of the RA values produced by CMS-B individuals by substituting the correct parental frequencies in the first term.

Only hermaphrodites produce pollen, but cost only affects some hermaphrodites according to the value of the silent restorer. Again, we show the calculations for the RA values but not the RB value calculations as they are similar. Among CMS-A individuals, RA values affect sex but do not affect cost. No CMS-A individuals produce pollen with RA values of 0 or 1:

graphic file with name M8.gif

This models a restoration system that is sporophytic in that CMS-A individuals with a RA value of 3 produce some pollen grains with a RA value of 2. For the rest of the RA values, pollen production by CMS-A individuals depends only on the relative frequencies of the various hermaphrodite genotypes:

graphic file with name M9.gif

When we consider RA values among pollen grains produced by CMS-B individuals, we must remember that cost will affect the production from individuals with RA values of 3 and higher:

graphic file with name M10.gif

We can then calculate standardized RA and RB values among gametes by adding together production from CMS-A and CMS-B individuals and dividing by the total gamete production of the population. This last step adjusts values so that total frequencies will sum to one after compensation and cost affects.

graphic file with name M11.gif

Zygotic frequencies from gamete frequencies:

The frequencies of restorer values in the next generation can now be calculated from the ovule and pollen frequencies. Cytotype frequencies can be calculated using CMS and CMS-specific sex frequencies.

Remember that restorer values of zygotes can vary from the midparental mean due to additive effects among restorer loci. This effect allows for many different ovule and pollen combinations to result in a particular offspring restorer value. For example, the total frequency of all genotypes with an RA value of 3 in the next generation is

graphic file with name M12.gif

Note that if the average of the RA values for the two gametes is 2.5, 3, or 3.5, one-half of their offspring have a RA value of 3. If the average value of the two gametes is 2 or 4, only one-quarter of the offspring have a RA value of 3.

Cytotype frequencies can be calculated from the ovule production of CMS-A individuals divided by total ovule production:

graphic file with name M13.gif

Once we have calculated the new values of ai′, bi′, and cA′, we can produce a new generation of our simulation.

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