Introduction

Hermaphroditic plant species reproduce with variable rates of selfing, ranging from strict selfing to strict outcrossing (Barrett, 2002). Self-incompatibility (SI) is a reproductive system that prevents self-fertilization. In the case of heteromorphic self-incompatibility, distinct morphologies result in distinct compatibility groups, whereas in the case of homomorphic self-incompatibility, compatible individuals cannot be distinguished by their morphology (de Nettancourt, 1977). Most SI systems depend on physiological mechanisms that prevent pollen germination or pollen tube growth. In sporophytic self-incompatibility (SSI) systems, the compatibility of a pollen grain depends on the diploid genotype of the plant that produced it. In gametophytic self-incompatibility (GSI) systems, the compatibility of a pollen grain depends on its haploid genotype. GSI is more widespread than SSI (Glémin et al., 2001).

Fisher (1941) showed that self-fertilization should have a selective advantage because a selfing genotype will transmit more copies of its genome than a non-selfing genotype (this has been termed the automatic advantage of selfing). However, numerous studies have shown that inbred offspring are less fit than outbred offspring. The relative decrease in the mean fitness of selfed versus outcrossed individuals (inbreeding depression) is generally recognized as the only main factor that counterbalances the selective advantage of selfing (Charlesworth and Charlesworth, 1987).

Consequently, the level of inbreeding depression in populations should have a determining function in the evolution of SI systems. Inbreeding depression decreases as population size becomes smaller due to reduced polymorphism for selection to act on (Bataillon and Kirkpatrick, 2000). Charlesworth and Charlesworth (1979) showed that the number of S-alleles also is important in maintaining SI, because a low number of alleles will limit the number of compatible crosses in the population. A decrease in population size can also cause a reduction in the number of S-alleles (Brennan et al., 2003), and a self-compatible mutant can be positively selected for (Reinartz and Les, 1994). Thus, small populations may be particularly prone to the breakdown of SI due to weak inbreeding depression and low numbers of S-alleles. However, small self-incompatible populations may maintain high levels of inbreeding depression due to a sheltered load of deleterious alleles linked to the S-locus (Glémin et al., 2001). The existence of a sheltered load has been shown experimentally in Solanum carolinense by Stone (2004) and Mena-Alí et al. (2009).

Overall, under a wide range of conditions, SI can evolve to self-compatibility. In effect, the loss of SI systems is very frequent in plant evolution (Igic et al., 2008). However, the reasons for which some species maintain an SI system whereas other species lose it are not completely understood. It has been suggested that asexual reproduction, ‘when an individual produces new individuals that are genetically identical to the ancestor at all loci in the genome, except at those sites that have experienced somatic mutations’ (de Meeûs et al., 2007), has a function in the maintenance or breakdown of SI. Two studies suggest that asexuality could relieve the main selective pressures that favor the breakdown of SI. First, Chen et al. (1997) showed in Australian Droseraceae that all self-incompatible taxa have effective forms of asexual reproduction, whereas the obligatory sexual annual taxon, Drosera glanduligera, is self-compatible. Their interpretation is that self-incompatible forms accumulate recessive lethal polymorphisms, especially in association with biparental inbreeding generated by elevated levels of asexual reproduction. The hypothetical high genetic load constitutes the selective pressure that maintains the outbreeding mechanisms. Second, Vallejo-Marín and O′Brien (2007) hypothesized that asexuality could provide reproductive assurance in cross-fertilizing species subject to pollen limitation. They predicted that cross-fertilizing species subject to pollen limitation would often have some means of asexual reproduction, and they found a strong correlation between SI and asexuality in Solanum (Solanaceae). Conversely, Young et al. (2002) developed a contrasting hypothesis for the case of Rutidosis leiolepis in which they found SSI and high rates of asexual reproduction. They suggested that increased asexual reproduction causes mate limitation by reducing genotype diversity at the S-locus, favoring a breakdown of SI.

The net effect of asexual reproduction on the maintenance or breakdown of SI systems is therefore still unknown. Moreover, it is unclear to what extent the effects of asexual reproduction discussed by the various authors are actually present in partially asexual, self-incompatible populations. Theoretical studies have described the effects of partial asexuality on the inbreeding depression of self-compatible populations (Muirhead and Lande, 1997) and the number of S-alleles (Vallejo-Marín and Uyenoyama, 2008). However, no model has been developed so far to study the joint effect of asexuality, SI and drift on the evolution of diversity and fitness parameters for a hermaphroditic species. Using individual-based simulations, we investigated the effect of (1) partial asexual reproduction and drift on diversity at the S-locus, and (2) combined asexuality, SI and drift on the frequency of deleterious alleles. Thus, our main goal was to characterize the dynamics of two key factors (number of S-alleles and inbreeding depression) for the maintenance of SI in partially asexual, self-incompatible populations. It was not the aim of this work to estimate the probability of invasion of self-compatible alleles or other modifiers of reproduction.

Materials and methods

Genetic model

The model considered in this study was based on the model developed by Glémin et al. (2001). This model consists of a population of N (four population sizes evaluated: 25, 50, 100 and 1000) diploid hermaphroditic individuals with a GSI system. In addition, our model also included asexual reproduction at rate c (probability that an individual is generated by asexual reproduction, seven values evaluated: 0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1). We considered an asexual reproduction event as the production of a new independent individual that is an exact copy of the parental individual (or only different by somatic mutation) (de Meeûs et al., 2007). As in Glémin et al. (2001), each individual genome possessed the S-locus, which regulated SI, and a viability locus, whose state determined individual fitness. Two neutral loci were also included for reference. The four loci were considered to be physically unlinked and were inherited independently through sexual reproduction.

Sexual reproduction events were controlled by the S-locus: crosses between individuals were only possible when the S-allele carried by the pollen grain was different from both S-alleles of the pistil (that is, at least three different S-alleles were necessary in the population for sexual reproduction to occur). To focus on effects specifically attributable to asexuality on the maintenance and breakdown of SI, we assumed unlimited pollen availability and that sexual crosses were always fruitful. Thus, sexual reproduction was only limited when asexuality and genetic drift reduced the number of S-alleles to less than three within a population. In this case, independent of the original rate of asexual reproduction, all individuals reproduced asexually (c=1) until mutation introduced a third S-allele. Then, c was reset to its original value.

Individual fitness f was determined by the viability locus, which had two alleles A and a. Strength of the dominance of allele A over deleterious allele a was regulated by the dominance coefficient h, and strength of the selection against a was regulated by the selection coefficient s. Thus, relative fitness of genotypes were as follows: fAA=1, fAa=1−hs and faa=1−s (0<s<=1 and 0<h<=1/2). Three selection regimes at the viability locus were considered: (1) a neutral case, to be able to study the effects of the interaction between SI and asexuality without the interference of selection, where parameters s and h were set to zero; (2) a case of mildly deleterious and partially recessive mutations, for which values assigned to s and h were, respectively, 0.1 and 0.2; and (3) a case of highly recessive lethal mutations, values assigned to s and h were, respectively, 1 and 0.02.

Mutations were allowed in all loci. Neutral loci mutated at rate μN=10−3, following a k-allele model (k=100). Mutation at the S-locus followed an infinite-allele model (Kimura and Crow, 1964) with rate μS; three values were considered for this rate (μS=10−3, 10−4 and 10−5 as in Glémin et al. (2001)). Mutations at the viability locus occurred at rate μ1=10−3 from A to a and at rate μ2=10−4 from a to A.

Individual-based simulations

Forward-time simulations of a population of N diploid individuals with no overlapping generations were performed. Simulations started with all individuals carrying two unique alleles at the S-locus (2N different S-alleles in the whole population), alleles A or a randomly assigned (with equal probability, 0.5) to the viability locus and any of the k alleles (k=100) randomly assigned (with equal probability, 1/k) to the two neutral loci.

At each generation, the number of individuals generated by asexual reproduction x was drawn from a Poisson distribution with mean Nc, and the remaining N−x individuals were generated by sexual reproduction (parents contributing to these two types of offspring were not determined in this step). Genotypes of the N individuals were generated one by one. If a given individual originated by asexual reproduction, a genotype i was randomly drawn from all individuals of the previous generation. That genotype was assigned to the individual with a probability equal to the fitness of genotype i; this will be called the selection step. If the genotype was not successfully assigned, the genotype draft and the selection step was repeated until a genotype passed the selection step. If the individual originated by sexual reproduction, the simulation followed a different procedure. First, an individual from previous generation was randomly drawn to act as a mother and the ovule haploid genotype was randomly generated. Then, a second individual was drawn to act as a father and the pollen haploid genotype was randomly generated. If the S-allele at the pollen grain was not different from both S-alleles of the mother (that is, incompatible pollen), the pollen was discarded. New fathers and pollen grain genotypes were generated until a compatible cross was obtained (that is, pollen availability was unlimited). Then, a diploid genotype i was generated from the haploid genotypes of the ovule and the pollen. The selection step was carried out on genotype i as in the case of asexual reproduction; if the genotype was not assigned the process was repeated by drawing two new parents until a genotype was able to pass the selection step. Once all N genotypes were selected, mutations were applied to each individual. The numbers of mutants at the S-locus and at the two neutral loci were drawn from Poisson distributions with means 2NμS and 2NμN, respectively. At the viability locus, a mutant was formed with a probability μ1 if the allele was A and a probability μ2 if the allele was a. The rate of asexual reproduction was a fixed parameter of our model and did not change with the selection process.

Monitored genetic indices

Simulations were run for 50 000 generations before beginning to monitor the genetic indices, this allowed monitored indices to become stable and avoided the influence of initial conditions in their values (Glémin et al., 2001). From generation 50 000, indices were calculated every 3000 generations until 107 000 generations (that is, 20 times). For the same combination of parameters, 50 replicates of the simulations were performed providing 1000 (indices calculated 20 times in each of the 50 replicates) evaluations of the monitored variables for each set of parameter values. Indices were calculated from all genotypes of the population (that is, they were not estimates, but the true values of the population).

Genetic diversity indices

For each locus we calculated the number of alleles (na), allele frequencies, effective number of alleles ( , where pi was the frequency of allele i), expected heterozygosity (He=1−1/ne), observed heterozygosity (Ho, proportion of heterozygous individuals) and the inbreeding coefficient (FIS=(He−Ho)/He). Theoretical predictions for some of these indices were obtained from the work of several authors for some of the scenarios considered:

(a) Vallejo-Marín and Uyenoyama (2008) studied the effects of partial asexuality in the number of alleles of the S-locus using a diffusion approximation and found that the number of common alleles (nc, which is approximately equivalent to the effective number of S-alleles, ne; Takahata, 1990) is determined by

where M=N(1–c) and θ=2NμS. This equation was solved numerically to obtain the expected effective number of S-alleles in the simulations without selection on the viability locus.

(b) Glémin et al. (2001) obtained the expected FIS for a neutral locus linked to the S-locus as

where fS is the scaling factor of the genealogy of the S-locus and r′ is the net recombination rate between the two loci. For the case of partial asexuality, the number of common alleles was given by Equation (1) and the scaling factor was

(Vallejo-Marín and Uyenoyama, 2008). To correct for the reduction of recombination due to asexuality, we used r′=r(1−c), where r was the recombination rate during sexual reproduction (that is, r=0.5 for unlinked loci). This approach ignored mutation on the neutral locus.

(c) Finally, an expression of the expected FIS of a neutral locus in a self-compatible, partially asexual population was derived following Balloux et al. (2003):

where γ=(1−μN)2 and s is the selfing rate. Selfing rate in Balloux et al. (2003) model is defined as the rate of self-fertilization of individuals; thus a fertilization event between two individuals from the same genet is considered an outcrossing event. Under random mating, s=1/N and for negligible mutation rates, γ=1.

Linkage disequilibrium

The linkage disequilibrium between pairs of loci (S-locus-viability locus and the two neutral loci) was studied with the correlation coefficient between two loci, RGGD, developed by Garnier-Géré and Dillmann (1992), which is particularly appropriate for partially asexual diploids (de Meeûs and Balloux, 2004).

Fitness

Mean fitness of the population () was calculated from the genotype frequencies as , and the average fitness of individuals potentially produced by selfing (Ws), outcrossing (Wo) and asexuality (Wa) were calculated as Wo=1 × pA2+(1−hs) × 2 × pA × pa+(1−s) × pa2 (where pA and pa are the frequencies of alleles A and a) and . Inbreeding depression (δ) and mutation load (L) were calculated as and . In addition, we compared the fitness of individuals produced by asexual reproduction (Wa) to the fitness of individuals produced sexually, using the ratio Wo/Wa. Ws, Wo and Wa were calculated as the fitness of potential offspring if all individuals were produced by selfing, by outcrossing and asexually, respectively. However, these modes of reproduction might not be actually present in the different scenarios considered (for example, in self-incompatible populations, no selfing occurs, but potential Ws and inbreeding depression can be calculated).

Fixation probabilities

To interpret the results, we calculated the probabilities of four events: fixation of alleles A and a in the viability locus, ‘fixation’ of viability locus heterozygote (Aa) and fixation of a genotype in the S-locus (reduction of number of S-alleles to two, this was only monitored in self-incompatible populations). These fixation probabilities were estimated on 1000 evaluations of the monitored variables as the proportion of times in which pa=0, pa=1 and Ho=1 for the viability locus, or n=2 for the S-locus.

Results

Effect of asexuality and drift on linkage disequilibrium between loci

Probably the most obvious effect of partial asexual reproduction was the reduction of recombination. Asexuality generated identical genotypes in offspring, so allele associations were transmitted in the same way that they would be transmitted if they were physically linked throughout the whole genome. In fully asexual populations, maximum linkage disequilibrium was expected (equivalent to the linkage disequilibrium between fully linked loci) and in partially asexual populations, recombination was reduced proportionally to the rate of asexual reproduction. Figure 1 shows how nonrandom associations of alleles increased with asexual reproduction rates. Linkage disequilibrium levels also depended on population size because drift produced a departure from the expected values of frequencies of allele combinations (Figure 1). Strong drift might have even caused some combinations of alleles to be absent in the population.

Figure 1
figure 1

Linkage disequilibrium (RGDD index, Garnier-Géré and Dillmann, 1992) at increasing rates of asexual reproduction. Box plots represent median (black line), first and third quartiles (box) and 5 and 95% percentiles (whiskers) of 1000 observations from simulations performed at seven rates of asexual reproduction (c=0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1) and four population sizes (N=25, 50, 100 and 1000).(a) RGDD index between S-locus and viability locus (s=0.1, h=0.2); (b) RGDD index between two neutral loci.

Full size image

In small populations, some generations with complete allelic association started to appear at asexual reproduction rates of 0.8 and were predominant at rates of 0.99 and higher. In larger populations, complete allelic association was only found in the fully asexual scenario. These associations of alleles were observed between loci subject to selection and between neutral loci, which showed that linkage disequilibrium was not a product of selective processes (Figures 1a and b).

Physical linkage between the S-locus and loci with deleterious alleles can decrease the number of S-alleles (Uyenoyama, 2003). It can also dramatically increase inbreeding depression in small populations (Glémin et al., 2001). Therefore, for the present model, it was important to discern whether the linkage disequilibrium generated by asexual reproduction is sufficient for the selective forces of the S-locus to interact with the viability locus. The results are presented below taking this perspective into account.

Effect of asexuality and drift on diversity parameters: number of S-alleles and heterozygote excess

The effective number of S-alleles decreased with asexuality (Figure 2) as shown analytically by Vallejo-Marín and Uyenoyama (2008). Balancing selection on the S-locus only occurred during sexual reproduction events. Because these events were scarce at high rates of asexual reproduction, the influence of balancing selection (which promotes high allelic diversity) on the population diminished. This can temporarily stop sexual reproduction by reducing the number of S-alleles to two (that is, fixation of an S-locus genotype, Table 1).

Figure 2
figure 2

Effective number of S-alleles at increasing rates of asexual reproduction. Box plots represent median (black line), first and third quartiles (gray box) and 5 and 95% percentiles (whiskers) of 1000 observations from the simulations performed at six rates of asexuality (c=0, 0.5, 0.8, 0.9, 0.99 and 1), population size N=25 (a) and N=100 (b), mutation rate μS=10−5, s=0 and h=0 for the viability locus. Gray line represents the theoretical expected number of common S-alleles obtained numerically from Equation (1) (Equation 3.7 in Vallejo-Marín and Uyenoyama (2008)).

Full size image
Table 1 Genotype fixation probability at the S-locus under two different selection regimes at the viability locus
Full size table

No significant differences in the number of S-alleles were found among the three different selection regimes for the viability locus (frequency distributions of the number of S-alleles among different selection regimes were undistinguishable in a Kolmogorov–Smirnov test, P-value>0.98). It must be noted that small populations (with high linkage disequilibrium) were fixed for the wild-type allele A most of the time (Table 2b) and this may explain the weak influence of deleterious alleles on the S-locus.

Table 2 Fixation probabilities at the viability locus in a self-incompatible population under two different selection regimes
Full size table

Another general effect of partial asexual reproduction in diploids was the reduction of allele segregation. This can increase heterozygosity by the independent accumulation of mutations on the two alleles of an asexual lineage (Pamilo, 1987; Birky, 1996). To quantify this effect, we measured the inbreeding coefficient FIS, which compares observed and expected heterozygosities.

Theoretical equilibrium values for FIS at a neutral locus (unlinked to the S-locus) were very similar in self-compatible and self-incompatible populations (Figure 3), and, for large population sizes, the simulated populations followed the same pattern (Figure 3b). However, for small population sizes, drift generated strong linkage disequilibrium between the S-locus and other loci. Thus, balancing selection on the S-locus had a hitchhiking effect on other loci, increasing their heterozygosity. This explained the contrasting FIS values on simulated self-compatible and self-incompatible populations (Figure 3a). Analytical predictions do not reflect this dramatic difference because they do not account for linkage disequilibrium. FIS decreased with asexuality and this decrease was enhanced by SI in small populations, particularly noticeable at asexual reproduction rates higher than 0.8. This decrease in FIS observed in neutral loci also occurred at the viability locus (both for the mildly deleterious and lethal recessive cases), which implied an increase in the frequency of the deleterious allele (Figure 4).

Figure 3
figure 3

Inbreeding coefficient FIS at a neutral locus at increasing rates of asexual reproduction. Box plots represent median (black line), first and third quartiles (white and gray boxes, for self-compatible and self-incompatible populations, respectively) and 5 and 95% percentiles (whiskers) of 1000 observations from the simulations performed at six rates of asexuality (c=0, 0.5, 0.8, 0.9, 0.99 and 1), mutation rates μS=10−5 and μN=10−3, and population size (a) N=25 and (b) N=100. Theoretical equilibrium values for FIS at a neutral locus in a self-compatible, random-mating population (dotted black line, from Equation (3) neglecting mutation) and in a self-incompatible population (continuous gray line, from Equation (2)) are also represented.

Full size image
Figure 4
figure 4

Frequency of deleterious alleles at increasing rates of asexual reproduction under two selection regimes. Box plots represent median (black line), first and third quartiles (white and gray boxes, for self-compatible and self-incompatible populations, respectively) and 5 and 95% percentiles (whiskers) of 1000 observations from the simulations performed at seven rates of asexuality (c=0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1), mutation rates μS=10−5 and μN=10−3. Population size and viability locus coefficients: (a) N=25, s=0.1, h=0.2; (b) N=25, s=1, h=0.02; (c) N=100, s=0.1, h=0.2; and (d) N=100, s=1, h=0.02.

Full size image

Effect of asexuality and drift on fitness

For both self-compatible and self-incompatible populations, mutation load globally increased with asexuality and population size, but a greater variance among generations and populations was observed for very small populations (Figure 5). As mentioned above, the frequency of the deleterious allele increased with asexuality in small self-incompatible populations, due to a hitchhiking effect of the S-locus over the viability locus. In the SI system, when the population size was small enough, the mutation load increased with asexual reproduction rates; this load resulted from the increase of the deleterious allele frequency (Figures 5a and c). Under similar conditions in the self-compatible system, mutation loads were lower than the mutation load observed in the self-incompatible system (Figures 5b and d). In contrast, inbreeding depression had similar values in self-incompatible and self-compatible populations (Figure 6), showing some increases with asexual reproduction rate for lethal recessive mutations that was more apparent in large populations.

Figure 5
figure 5

Mutation load at increasing rates of asexual reproduction under two different selection regimes according to reproductive system. Box plots represent median (black line), first and third quartiles (box) and 5 and 95% percentiles (whiskers) of 1000 observations from simulations performed at seven rates of asexual reproduction (c=0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1) and four population sizes (N=25, 50, 100 and 1000). (a) Self-incompatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (b) self-compatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (c) self-incompatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02) and (d) self-compatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02).

Full size image
Figure 6
figure 6

Inbreeding depression at increasing rates of asexual reproduction and under two different selection regimes according to reproductive system. Box plots represent median (black line), first and third quartiles (box) and 5 and 95% percentiles (whiskers) of 1000 observations from simulations performed at seven rates of asexual reproduction (c=0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1) and four population sizes (N=25, 50, 100 and 1000). (a) Self-incompatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (b) self-compatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (c) self-incompatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02) and (d) self-compatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02).

Full size image

The average fitness was higher in sexual populations than in asexual populations due to Mendelian segregation (for example, Figure 5; Chasnov, 2000; Kirkpatrick and Jenkins, 1989). The relative fitness between individuals produced by sexual and asexual reproduction was influenced very little by partial asexuality in a self-compatible population (Figures 7b and d). However, in small self-incompatible populations, the asexual reproduction rate increased the advantage of asexual compared with sexual reproduction (Figures 7a and c). These populations consisted mainly of heterozygous individuals (Figure 3). Therefore, asexual reproduction produced mainly Aa individuals, with only slightly lower fitness than the fittest AA haplotype. However, sexual reproduction produced a high proportion of aa individuals, reducing the average fitness of offspring, a reduction that was not compensated by the production of the fittest AA individuals. This could potentially lead to a positive feedback effect on the asexual reproduction rate, but this was not studied in our model where the asexual reproduction rate was a fixed parameter.

Figure 7
figure 7

Fitness ratio between sexually and asexually produced individuals at increasing rates of asexual reproduction under two different selection regimes according to reproductive system. Box plots represent median (black line), first and third quartiles (box) and 5 and 95% percentiles (whiskers) of 1000 observations from simulations performed at seven rates of asexual reproduction (c=0, 0.5, 0.8, 0.9, 0.99, 0.999 and 1) and four population sizes (N=25, 50, 100 and 1000). (a) Self-incompatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (b) self-compatible population with a viability locus with mildly deleterious allele (s=0.1, h=0.2), (c) self-incompatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02) and (d) self-compatible population with a viability locus with highly recessive lethal allele (s=1, h=0.02).

Full size image

Discussion

The number of S-alleles and inbreeding depression are considered to be the main factors that influence the maintenance or breakdown of SI systems (Charlesworth and Charlesworth, 1979). A decrease in the number of S-alleles is expected to favor the breakdown of SI because self-compatible mutants can invade the population, whereas the maintenance of inbreeding depression within populations prevents the breakdown of SI. In this study, we examined how drift and partial asexuality may modify the dynamics of these two key parameters in a self-incompatible population.

Drift, combined with partial asexuality, dramatically reduced the number of S-alleles

As already predicted by Yokoyama and Hetherington (1982), we showed that the number of S-alleles decreases with the effective size of the population. Thus, in small populations, repeated bottlenecks or founder events may lead to the breakdown of the SI system. However, Karron (1987) found no differences in mating systems between rare and widespread congeners across several families (but see also a study on some Brassicaceae species, Kunin and Shmida (1997)). As for the impact of asexuality, our simulation study confirmed a recent result shown analytically by Vallejo-Marín and Uyenoyama (2008). The number of S-alleles decreased with asexuality due to the weakened influence of balancing selection in a partially asexual population (Figure 2). In contrast with other effects of asexuality that were only observed for very high rates of asexuality, intermediate asexual reproduction rates were sufficient to significantly reduce the number of S-alleles. High rates of asexual reproduction combined with drift produced extremely low numbers of S-alleles. This effect was strong enough to even reduce the number of S-alleles to two (that is, fixation of an S-locus genotype, Table 1), which stopped all possibility of sexual reproduction until a new S-allele arose from mutation or migration. Therefore, in such extreme cases, this drastic effect of asexuality on the number of S-alleles should favor the breakdown of SI. However, paradoxically, the number of S-alleles may not be very relevant in a system where asexuality serves to provide reproductive assurance.

Drift, combined with partial asexuality and SI, led to an increase in mutation load

Drift influenced the level of mutation load. As population size decreased, the frequency of the deleterious allele decreased and mutation load decreased due to more effective purging. We also observed a much greater variance in mutation load in very small populations (Table 2 and Figure 5, N=25). In this case, selection was overwhelmed by drift and this caused a higher probability of fixation of the deleterious allele and an increase in mutation load. Similar results have been described by Bataillon and Kirkpatrick (2000), Glémin (2003) and Haag and Roze (2007).

Drift is also a key component of mutation load in asexual populations (Haag and Roze, 2007). Indeed, large asexual and sexual populations showed comparable frequencies of the deleterious allele and comparable mutation loads (Figure 5; see also Haag and Roze (2007)). However, mutation load was greater in small asexual populations than in small sexual ones. This could be explained by the absence of segregation in asexual populations, in which heterozygous individuals are present in high frequencies (Table 2 and Figure 5; see also Haag and Roze (2007)).

In this study, we additionally showed that SI had a function in increasing mutation load in partially asexual, self-incompatible populations. Partial asexual reproduction and small population sizes produced strong associations (that is, linkage disequilibrium) between deleterious alleles (on any locus of the genome) and S-alleles. Glémin et al. (2001) have already shown that deleterious mutations on loci linked to the S-locus may be sheltered by balancing selection acting on rare S-alleles. Asexuality and SI both favored the increase of the heterozygote genotype Aa within the population, leading to an increase of mutation load in small self-incompatible, partially asexual populations.

Inbreeding depression remained low in small, partially asexual, self-incompatible populations

Inbreeding depression followed different patterns than the ones observed for mutation load. For very small populations, inbreeding depression is expected to decrease (Bataillon and Kirkpatrick, 2000), due to the absence of polymorphism at the viability locus. This was observed in small self-compatible and self-incompatible populations (Glémin et al. (2001), and our own simulations, Figure 6). As for the impact of asexuality, our results showed no large differences between the self-compatible and the self-incompatible cases for small populations. The effect of strong drift, in the presence of strong asexuality, was to reduce genotype diversity. In most cases, populations were fixed at one allele, but sometimes the population was fixed at the heterozygote genotype Aa. This mainly occurred in self-incompatible populations (see Table 2). With no or very low genotype diversity, inbreeding depression had very low values, even if deleterious alleles were present at high frequencies in the population.

Chen et al. (1997) proposed that asexuality would favor the maintenance of SI. Their reasoning was that asexual reproduction would increase a population's genetic load due to lethal recessive mutations, which was confirmed by our simulation results. However, mutation load and inbreeding depression did not show parallel trends, and the high level of mutation load due to the combination of drift, asexuality and SI was accompanied by low levels of inbreeding depression. Nevertheless, inbreeding depression generally tended to increase with asexuality due to the accumulation of deleterious alleles and this should favor the maintenance of SI.

Evolution of partially asexual, self-incompatible populations: a third possible outcome

Until now, the evolutionary interest of partially asexual, self-incompatible populations has been focused on two possible outcomes: (1) the system is stable and SI is maintained or (2) the system is unstable and SI disappears (Chen et al., 1997; Young et al., 2002; Vallejo-Marín, 2007; Vallejo-Marín and O′Brien, 2007). However, a third possible outcome should be considered: being the unstable system, sexual reproduction completely disappears. This type of event was observed in our simulations due to a reduction in the number of S-alleles to two (see above). However, we also studied how selection on the viability locus could contribute to the potential loss of sexuality. Our results showed that the fitness of individuals produced by asexual reproduction (Wa) was greater than that of individuals produced by random sexual mating (Wo) in small populations (Figure 7). Fitness of individuals (potentially) produced by selfing (Ws) was also lower than Wa in small self-incompatible populations, suggesting that asexuality may have an immediate advantage over selfing under such circumstances.

Therefore, we speculate that asexual reproduction could take over sexual reproduction more easily than self-compatibility over SI when low numbers of S-alleles reduce mate availability. Despite the effect of asexuality on inbreeding depression and S-locus diversity, asexual reproduction may offer an alternative means of reproduction under these adverse conditions and SI may be temporarily dormant instead of breaking down. Asexual populations could subsequently recover a sexual self-incompatible reproductive system when new S-alleles are introduced in the population (for instance, by migration). However, prolonged periods of asexual reproduction may facilitate the accumulation of mutations affecting sexual traits, including SI, because selective pressures are ineffective for these traits (Eckert, 2001). Thus, alternative periods of sexual and asexual reproduction may be necessary for such dynamics, which is probably the case for plants combining asexual and sexual reproduction (Bengtsson and Ceplitis, 2000). Nevertheless, the scenarios described here need to be confirmed by further and more complex simulations and other factors may favor self-compatible sexual reproduction, such as the more effective purging of deleterious mutations.

Perspectives

In this study, we used a model with physically unlinked loci and a single viability locus. However, because the linkage disequilibrium produced by asexuality affected the whole-genome, population genomic models may be necessary to fully study the effect of deleterious alleles on the S-locus for small, partially asexual populations. In addition, physically linked loci with deleterious alleles (which were not studied here) may reduce the number of S-alleles (Uyenoyama, 2003) and the strength of that effect may be influenced by the rate of asexuality as discussed by Vallejo-Marín and Uyenoyama (2008). Moreover, in this study, it was not possible to investigate the evolutionary dynamics of mating system and asexuality modifiers because these were assumed to be fixed parameters. It would be then interesting to develop a model to test the invasion propensity of self-compatible or asexual mutants. Finally, models incorporating migration or population structure are necessary to study more realistic dynamics of partially asexual, self-incompatible populations. Population structure is expected to affect the genetic diversity of viability and the S-locus; for instance, by introducing a third allele by migration within subpopulations fixed at two S-alleles. These models should also take into account the specificities of the different types of asexual reproduction where, for instance, the asexual propagules may have a lower dispersal capacity (for example, vegetative reproduction) and may create spatial clusters of individuals with the same genotype.

Recent theoretical work (Vallejo-Marín and Uyenoyama, 2008) and our own model lead to several predictions about S-locus diversity, mutation load and inbreeding depression within partially asexual self-incompatible populations. However, we need more experimental observations to test and validate these theoretical advances. First, as outlined by Vallejo-Marín and O′Brien (2007), if asexuality relieves selective pressures favoring the breakdown of SI, the co-occurrence of asexuality and SI should be frequent. Association of SI and asexuality has been studied in only a few monospecific studies (R. leiolepis, Young et al., 2002; Eucalyptus morrisbyi, Jones et al., 2005; Prunus avium, Stoeckel et al., 2006). The association was found at a multispecific level by Chen et al. (1997) in Australian Droseraceae and by Vallejo-Marín and O′Brien (2007) in Solanaceae. More reports on species combining SI and asexuality would enhance our understanding of pressures explaining their maintenance. Second, experimental measurements of asexual reproduction rates and number of S-alleles in species showing various levels of asexual reproduction would be useful to test predictions from our simulations. Although experimental measurements of inbreeding depression in self-incompatible species are available (Mena-Alí et al., 2008), studies comparing inbreeding depression from different populations with contrasting sizes and rates of asexual reproduction are lacking.

Conclusions

This study investigated for the first time the effect of partial asexual reproduction on the fitness of self-incompatible populations. For loci unlinked to the S-locus and large populations, fitness values were similar to those of a self-compatible population. However, in small populations, the combination of drift, asexuality and SI increased mutation load due to the accumulation of deleterious alleles. This increase in the mutation load was accompanied by a selective advantage of asexually produced offspring compared to sexually produced offspring. Therefore, future studies addressing the maintenance of SI in partially asexual populations (by studying the invasibility of self-compatible genotypes) will need to (1) account for whole-genome fitness and (2) acknowledge that the process may not be independent of the maintenance of sex itself (that is, will need to establish whether modifiers of asexual reproduction rates invade more easily than self-compatible alleles).