EVOLUTION OF DOMINANCE IN SPOROPHYTIC SELF-INCOMPATIBILITY SYSTEMS: I. GENETIC LOAD AND COEVOLUTION OF LEVELS OF DOMINANCE IN POLLEN AND PISTIL

Recent theoretical advances have suggested that various forms of balancing selection may promote the evolution of dominance through an increase of the proportion of heterozygote genotypes. We test whether dominance can evolve in the sporophytic self-incompatibility (SSI) system in plants. SSI prevents mating between individuals expressing identical SI phenotypes by recognition of pollen by pistils, which avoids selfing and inbreeding depression. SI phenotypes depend on a complex network of dominance relationships between alleles at the self-incompatibility locus ( S -locus). Empirical studies suggest that these relationships are not random, but the exact evolutionary processes shaping these relationships remain unclear. We investigate the expected patterns of dominance under the hypothesis that dominance is a direct target of natural selection. We follow the fate of a mutant allele at the S -locus whose dominance relationships are changed but whose specificity remains unaltered. We show that strict codominance is not evolutionarily stable in SSI, and that inbreeding depression due to deleterious mutations linked or unlinked to the S -locus exerts strong constraints on changes in relative levels of dominance in pollen and pistil. Our results provide a general adaptive explanation for most patterns of dominance relationships empirically observed in natural plant populations.     

The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation

We model the evolution of plant mating systems under the joint effects of pollen discounting and pollen limitation, using a dynamic model of inbreeding depression, allowing for partial purging of recessive lethal mutations by selfing. Stable mixed mating systems occur for a wide range of parameter values with pollen discounting alone. However, when typical levels of pollen limitation are combined with pollen discounting, stable selfing rates are always high but less than 1 (0.9相似文献   

EVALUATING A SIMPLE APPROXIMATION TO MODELING THE JOINT EVOLUTION OF SELF‐FERTILIZATION AND INBREEDING DEPRESSION

A comprehensive understanding of plant mating system evolution requires detailed genetic models for both the mating system and inbreeding depression, which are often intractable. A simple approximation assuming that the mating system evolves by small infrequent mutational steps has been proposed. We examine its accuracy by comparing the evolutionarily stable selfing rates it predicts to those obtained from an explicit genetic model of the selfing rate, when inbreeding depression is caused by partly recessive deleterious mutations at many loci. Both models also include pollen limitation and pollen discounting. The approximation produces reasonably accurate predictions with a low or moderate genomic mutation rate to deleterious alleles, on the order of U = 0.02–0.2. However, for high mutation rates, the predictions of the full genetic model differ substantially from those of the approximation, especially with nearly recessive lethal alleles. This occurs because when a modifier allele affecting the selfing rate is rare, homozygous modifiers are produced mainly by selfing, which enhances the opportunity for purging nearly recessive lethals and increases the marginal fitness of the allele modifying the selfing rate. Our results confirm that explicit genetic models of selfing rate and inbreeding depression are required to understand mating system evolution.     

Breeding systems in Clivia (Amaryllidaceae): late‐acting self‐incompatibility and its functional consequences

Late‐acting (ovarian) self‐incompatibility, characterized by minimal or zero seed production following self‐pollen tube growth to the ovules, is expected to show phylogenetic clustering, but can otherwise be difficult to distinguish from early‐acting inbreeding depression. In Amaryllidaceae, late‐acting self‐incompatibility has been proposed for Narcissus (Narcisseae) and Cyrtanthus (Cyrtantheae). Here, we investigate whether it occurs in the horticulturally important genus Clivia (Haemantheae) and test whether species in this genus experience ovule discounting in wild populations. Seed‐set results following controlled hand pollinations revealed that Clivia miniata and C. gardenii are largely self‐sterile. Self‐ and cross‐pollinated flowers of both species had similar proportions of pollen tubes entering the ovary, and those of C. gardenii also did not differ in the proportions of pollen tubes that penetrated ovules, thus ruling out classical gametophytic self‐incompatibility acting in the style, but not early inbreeding depression. Flowers that received equal mixtures of self‐ and cross‐pollen set fewer seeds than those that received cross‐pollen only, but it was unclear whether this effect was a result of ovule discounting or interactions on the stigma. The prevention of self‐pollination by the emasculation of either single flowers or whole inflorescences in wild populations did not affect seed set, suggesting that ovule discounting is not a major natural limitation on seed production. Flowers typically produce one to three large fleshy seeds from approximately 16 available ovules, even when supplementally hand pollinated, suggesting that fecundity is mostly resource limited. The results of this study suggest that Clivia spp. are largely self‐sterile as a result of either a late‐acting self‐incompatibility system or severe early inbreeding depression, but ovule discounting caused by self‐pollination is not a major constraint on fecundity. © 2014 The Linnean Society of London, Botanical Journal of the Linnean Society, 2014, 175 , 155–168.     

On the evolution of genetic incompatibility systems. IV. Modification of response to an existing antigen polymorphism under partial selfing

A 2-locus model of the evolution of self-incompatibility in a population practicing partial selfing is presented. An allele is introduced at a modifier locus which influences the strength of the rejection reaction expressed by the style in response to antigens recognized in pollen. Two causes of inbreeding depression are investigated. First, offspring viability depends solely on the source (self or non-self) of the fertilizing pollen. Second, offspring viability declines with the expression of recessive deleterious alleles, segregating at a third (disease) locus, which exhibit an imperfect association with antigen alleles. Evolutionary changes occurring at the disease locus are not considered in this study. The condition under which a modifier allele that intensifies the incompatibility reaction increases when rare depends upon the number of antigens, the frequency of recessive deleterious alleles at the disease locus, and the level of association between the antigen locus and the disease locus. It is the improvement of viability among offspring derived by outcrossing, rather than the prevention of self-fertilization, that may represent the primary evolutionary function of genetic incompatibility systems.     

On the Evolution of Genetic Incompatibility Systems. VI. a Three-Locus Modifier Model for the Origin of Gametophytic Self-Incompatibility

Recent genetic analyses have demonstrated that self-incompatibility in flowering plants derives from the coordinated expression of a system of loci. To address the selective mechanisms through which a genetic system of this kind evolves, I present a three-locus model for the origin of gametophytic self-incompatibility. Conventional models assume that a single locus encodes all physiological effects associated with self-incompatibility and that the viability of offspring depends only on whether they were derived by selfing or outcrossing. My model explicitly represents the genetic determination of offspring viability by a locus subject to symmetrically overdominant selection. Initially, the level of expression of the proto-S locus is insufficient to induce self-incompatibility. Weak gametophytic self-incompatibility arises upon the introduction of a rare allele at an unlinked modifier locus which enhances the expression of the proto-S locus. While conventional models predict that the origin of self-incompatibility requires at least two- to threefold levels of inbreeding depression, I find that the comparatively low levels of inbreeding depression generated by a single overdominant locus can ensure the invasion of an enhancer of self-incompatibility under sufficiently high rates of receipt of self-pollen. Associations among components of the incompatibility system promote the origin of self-incompatibility. Enhancement of heterozygosity at the initially neutral proto-S locus improves offspring viability through associative overdominance. Further, the modifier that enhances the expression of self-incompatibility develops a direct association with heterozygosity at the overdominant viability locus. These results suggest that the evolutionary processes by which incompatibility systems originate may differ significantly from those associated with their breakdown. The genetic mechanism explored here may apply to the evolution of other systems that restrict reproduction, including maternal-fetal incompatibility in mammals.     

ABSENCE OF POLLEN DISCOUNTING IN A GENOTYPE OF IPOMOEA PURPUREA EXHIBITING INCREASED SELFING

Throughout southeastern North America, the annual morning glory Ipomoea purpurea exhibits a polymorphism at a locus that influences the intensity of floral pigmentation. Previous studies have shown that when rare, the homozygous white genotype has a greater selfing rate than the homozygous dark genotype. In the absence of pollen discounting (a reduction in transmission of pollen to other plants by genotypes that exhibit increased selfing) and inbreeding depression, this increased selfing rate should favor the white allele. Experiments reported here confirm that the white genotype has elevated selfing rates when rare but indicate pollen discounting is not associated with elevated selfing. Rather, white genotypes contribute more pollen to the outcross pollen pool. The disparity between genotypes in both selfing rates and success at pollen contribution to other plants disappears at intermediate to high frequencies of the white allele. Pollinator movements are consistent with the pattern of selfing. These results suggest that elevated selfing and enhanced success at pollen donation contribute to maintenance of the white allele in natural populations of morning glories.     

On the evolution of genetic incompatibility systems. III. Introduction of weak gametophytic self-incompatibility under partial inbreeding

I explore the proposition that genetic incompatibility systems serve as a means for parents to evaluate and discriminate among their own offspring. Conditions for the initial increase of gametophytic self-incompatibility in a self-compatible population undergoing selfing, sibmating, and random outcrossing are reported. The adaptive value of reducing the concordance between offspring and maternal genotypes depends upon the relative changes in the numbers of offspring derived by the three modes, parent-offspring relatedness, and the magnitude of distortion of transmission ratios through pollen. Recessivity of stylar expression and low rates of receipt of pollen from related individuals facilitate the evolution of self-incompatibility. Viewed as a means of preferential maternal investment in offspring of high quality, self-incompatibility may be regarded as serving a function in common with diverse phenomena, including sexual selection, brood reduction, and other forms of prezygotic and postzygotic incompatibility. Associations between incompatibility loci and loci expressing inbreeding depression are expected to improve the reliability of the level of concordance at incompatibility loci as a measure of genomic homozygosity and offspring quality.     

Functional pleiotropy and mating system evolution in plants: frequency-independent mating

Mutations that alter the morphology of floral displays (e.g., flower size) or plant development can change multiple functions simultaneously, such as pollen export and selfing rate. Given the effect of these various traits on fitness, pleiotropy may alter the evolution of both mating systems and floral displays, two characters with high diversity among angiosperms. The influence of viability selection on mating system evolution has not been studied theoretically. We model plant mating system evolution when a single locus simultaneously affects the selfing rate, pollen export, and viability. We assume frequency-independent mating, so our model characterizes prior selfing. Pleiotropy between increased viability and selfing rate reduces opportunities for the evolution of pure outcrossing, can favor complete selfing despite high inbreeding depression, and notably, can cause the evolution of mixed mating despite very high inbreeding depression. These results highlight the importance of pleiotropy for mating system evolution and suggest that selection by nonpollinating agents may help explain mixed mating, particularly in species with very high inbreeding depression.     

Inbreeding depression in Campanula rapunculoides L. I. A comparison of inbreeding depression in plants derived from strong and weak self-incompatibility phenotypes

The evolution of selfing taxa from outcrossing ancestors has occurred repeatedly and is the subject of many theoretical models, yet few empirical studies have examined the immediate consequences of inbreeding in a population with variable expression of self-incompatibility. Because self-incompatibility breaks down with floral age in Campanula rapunculoides, we were able to mate outbred and selfed maternal plants in a crossing design which produced progeny with inbreeding coefficients of 0, 0.25, 0.50 and 0.75. Cumulative inbreeding depression in plants that were selfed for one generation was very high in families derived from strongly self-incompatible plants (average δ = 0.98), and somewhat lower in families derived from plants with weaker expression of self-incompatibility (average δ = 0.90). Relative to outbred progeny, inbred progeny produced fewer seeds, had lower rates of germination, less vegetative growth and fewer flowers per plant. Inbred progeny also took longer to germinate, and longer to produce a first leaf and to flower. Interestingly, inbred plants also produced 40% fewer seeds than outcrossed plants (t-test P < 0.001) even when mated to the same, unrelated pollen donor, suggesting that inbreeding can produce profound maternal effects. Most importantly, our results demonstrate that progeny derived from plants with stronger expression of self-incompatibility exhibited greater levels of inbreeding depression than progeny from plants with weaker expression of self-incompatibility. Moreover, the decline in fitness (cumulative, ln-transformed) over the four inbreeding levels was steeper for the progeny of the strongly self-incompatible lineages. These empirical results suggest that inbreeding depression and mating system phenotype have the potential to coevolve.     

Coevolution of self-fertilization and inbreeding depression. I. Mutation-selection balance at one and two loci

Simple theories for the evolution of breeding systems suggest that the fate of an allele that modifies the rate of self-fertilization hinges only on the degree to which selfing reduces opportunities for outcrossing ("pollen discounting") and the extent of inbreeding depression. These theories predict that outcrossing evolves whenever deleterious mutations have a more severe effect in combination than expected from their individual effects. We study the evolutionary dynamics of a modifier of the rate of self-fertilization in populations subject to complete pollen discounting and recurrent mutations which impair viability at a single locus in diploids and at two loci in haploids. Our analysis indicates that genetic associations arising immediately upon the introduction of a rare modifier allele generate substantial quantitative and qualitative departures from expectation. Higher rates of segregation under selfing in our one-locus diploid model generate positive associations between enhancers of selfing and wild-type viability alleles, which in turn favor the evolution of selfing under a wider range of conditions than expected. Greater opportunities for recombination under outcrossing in our two-locus haploid model generate positive associations between enhancers of outcrossing and wild-type viability alleles. These associations favor the evolution of outcrossing under a wider range of conditions, and introduce the possibility of stable mixed mating systems involving both selfing and outcrossing. Our explicit analysis of genetic associations between loci affecting viability and the rate of self-fertilization indicates that modifiers that enhance the production of offspring with very high (and very low) viability by promoting segregation or recombination develop positive associations with high viability. This advantage of producing extremes can compensate for an initial disadvantage in offspring number.     

HOW DEPRESSED? ESTIMATES OF INBREEDING EFFECTS DURING SEED DEVELOPMENT DEPEND ON REPRODUCTIVE CONDITIONS

Inbreeding depression can reduce the performance of offspring produced by mating between relatives, with consequences for population dynamics and sexual-system evolution. In flowering plants, inbreeding depression commonly acts most intensely during seed development. This predispersal component is typically estimated by comparing seed production following exclusive self- and cross-pollination, but such estimates are unbiased only if seed production is limited by ovule availability, rather than by pollen receipt or seed-development resources. To overcome this problem, we propose experimental and statistical methods based on a model of ovule fertilization and seed development that accounts for differential fertilization by self- and cross-pollen, limited ovule viability or receptivity, differential survival of self- and cross-zygotes and limited resource availability. Simulations illustrate that the proposed methods eliminate bias in estimated predispersal inbreeding depression caused by pollen limitation and can improve estimates under resource limitation. Application of these methods to two orchid species further demonstrates their utility in identifying and estimating diverse influences on reproductive performance under typical conditions. Although our theoretical results raise questions about the reported intensity of predispersal inbreeding depression, our proposed methods guard against bias while also providing insight into plant reproduction.     

The avoidance of self-interference in the endemic daffodil Narcissus cyclamineus (Amaryllidaceae)

Hermaphrodite flowers usually possess floral traits to avoid the negative effects derived from inbreeding depression and/or self-interference between pollen export and reception, both acting as the main selective pressures on those floral traits. The avoidance of self-interference is widely accepted as the primary force promoting the separation between sexes within the flowers in time (dichogamy) and/or space (herkogamy) for self-incompatible species, which are already protected from the negative effects of inbreeding depression by the incompatibility system. Different degrees of incompatibility, herkogamy, and dichogamy have been reported for the genus Narcissus. However, the only mechanism for the separation of sexes reported up to date for Narcissus cyclamineus is herkogamy, while the presence of dichogamy and the type of incompatibility in this species remain uncertain. In this study, we analyze the patterns of sexual reproduction in N. cyclamineus to ascertain whether there is any selective pressure favouring sexual segregation or its maintenance and their mechanisms. N. cyclamineus is self-incompatible and dichogamy can be rejected for this species. Even though the species is self-incompatible, when cross-pollination is preceded by self-pollination the number of ovules available for legitimate crosses is diminished (ovule discounting). Pollinators are scarce during the flowering period, resulting in pollen limitation. It is suggested that both the scarcity of pollinators and ovule discounting may be acting synergically to promote herkogamy or its maintenance in this species.     

Influence of pollen limitation and inbreeding depression in the maintenance of incomplete dichogamy in Salvia elegans

The widespread presence of incomplete dichogamy (i.e., partial separation in time between male and female phases) in flowering plants is a long‐standing question in floral evolution. In this study, we proposed four scenarios in which depending on the particular combination of pollen limitation and inbreeding depression, the presence of complete dichogamy, incomplete dichogamy, or adichogamy may be favored. Moreover, we evaluated the role of pollen limitation and inbreeding depression in a natural population of Salvia elegans to test the validity of our predicted scenarios. Our results indicate that S. elegans is partially protandrous as pollen viability and stigma receptivity overlap in the last days of life of the flower. Furthermore, through pollination treatments, we found no evidence of pollen limitation or inbreeding depression in any of the evaluated fitness components. As expected by one of the proposed scenarios, incomplete dichogamy seems to be favored in plants with absence of inbreeding depression and pollen limitation as a way to diminish interference between male and female functions.     

Embryonic inbreeding depression varies among populations and by mating system in Witheringia solanacea (Solanaceae)

? Premise of the study: Embryonic inbreeding depression is a key influence on mating system evolution and can be difficult to estimate in self-incompatible species. A pollen chase experiment was used to estimate the magnitude of embryonic inbreeding depression in Costa Rican Witheringia solanacea, a species polymorphic for self-incompatibility (SI). In a pollen chase experiment, bud self-pollinations are followed after anthesis by outcross pollinations, with a comparable pair of outcross pollinations used as a control. Lowered seed set for the self-precedence treatment indicates embryonic inbreeding depression. ? Methods: Embryonic inbreeding depression was assayed for self-compatible (SC) individuals and for SI plants from two populations that differ quantitatively in the onset and enzymatic activity of their SI response. Microsatellite markers were used to assay the selfing rate of a sample of surviving progeny from the prior self-pollination treatment. ? Key results: SC individuals showed no evidence of embryonic inbreeding depression. In SI plants, prior self-pollination reduced seed number by 28-70%, depending on population. Microsatellite genotyping revealed that embryonic inbreeding depression was even more severe than estimated by the phenotypic data: for mature fruits resulting from self-pollination precedence, the majority of the progeny were the result of outcross fertilization. ? Conclusions: Lineage-specific purging of recessive lethals has accompanied the evolution of SC in this species. SI populations show contrasting levels of embryonic inbreeding depression, with nearly complete embryonic lethality upon selfing in the Monteverde population. In the face of high embryonic inbreeding depression, an increase in selfing rate can evidently occur only under severe pollen limitation.     

Inbreeding depression and mixed mating in Leptosiphon jepsonii: a comparison of three populations

BACKGROUND AND AIMS: Inbreeding depression is thought to play a central role in the evolution and maintenance of cross-fertilization. Theory indicates that inbreeding depression can be purged with self-fertilization, resulting in positive feedback for the selection of selfing. Variation among populations of Leptosiphon jepsonii in the timing and rate of self-fertilization provides an opportunity to study the evolution of inbreeding depression and mating systems. In addition, the hypothesis that differences in inbreeding depression for male and female fitness can stabilize mixed mating in L. jepsonii is tested. METHODS: In a growth room experiment, inbreeding depression was measured in three populations with mean outcrossing rates ranging from 0.06 to 0.69. The performance of selfed and outcrossed progeny is compared at five life history stages. To distinguish between self-incompatibility and early inbreeding depression, aborted seeds and unfertilized ovules were counted in selfed and outcrossed fruits. In one population, pollen and ovule production was quantified to estimate inbreeding depression for male and female fitness. KEY RESULTS: Both prezygotic barriers and inbreeding depression limited self seed set in the most outcrossing population. Cumulative inbreeding depression ranged from 0.297 to 0.501, with the lowest value found in the most selfing population. Significant inbreeding depression for early life stages was found only in the more outcrossing populations. Inbreeding depression was not significant for pollen or ovule production. CONCLUSIONS: The results provide modest support for the hypothesized relationship between inbreeding depression and mating systems. The absence of early inbreeding depression in the more selfing populations is consistent with theory on purging. Differences in male and female expression of inbreeding depression do not appear to stabilize mixed mating in L. jepsonii. The current estimates of inbreeding depression for L. jepsonii differ from those of previous studies, underscoring the effects of environmental variation on its expression.     

繁殖保障和延迟自交的研究进展

尽管植物在进化过程中面临不利自花授粉的选择,但许多植物仍维持混合的授粉机制。繁殖保障假说是解释自交进化的最重要因子之一,一直是植物生殖生态学和进化生物学关注的焦点之一。概述了近年来的主要研究热点及其进展,包括自交进化的遗传和生态机制及理论模型探讨、繁殖保障假说的提出、验证自交能否提供繁殖保障的例证、延迟自交的类型及延迟自交能否提供繁殖保障的例证等方面。介绍了我国在繁殖保障和延迟自交方面研究的现状和不足之处,结合国际上研究繁殖保障假说的发展趋势已由单季节、单种群、单因子的研究阶段过渡到多季节、多种群、多因子（自交方式及其所占比例、花粉折损、种子折损、自交率和近交衰退）的综合研究阶段,及由传统的、经典的研究方法过渡到应用现代实验手段（如SSR、SNP等分子标记）和先进仪器设备的研究阶段,提出今后研究中应注意的问题。有必要借用多学科（植物学、生态学和分子生物学）的方法及手段进行不同物种的对比和综合细致的研究。     

A Temporal Dimension to the Influence of Pollen Rewards on Bee Behaviour and Fecundity in Aloe tenuior

The net effect of pollen production on fecundity in plants can range from negative – when self-pollen interferes with fecundity due to incompatibility mechanisms, to positive – when pollen availability is associated with increased pollinator visitation and fecundity due to its utilization as a reward. We investigated the responses of bees to pollen and nectar rewards, and the effects of these rewards on pollen deposition and fecundity in the hermaphroditic succulent shrub Aloe tenuior. Self-pollinated plants failed to set fruit, but their ovules were regularly penetrated by self-pollen tubes, which uniformly failed to develop into seeds as expected from ovarian self-incompatibility (or strong early inbreeding depression). Bees consistently foraged for pollen during the morning and early afternoon, but switched to nectar in the late afternoon. As a consequence of this differential foraging, we were able to test the relative contribution to fecundity of pollen- versus nectar-collecting flower visitors. We exposed emasculated and intact flowers in either the morning or late afternoon to foraging bees and showed that emasculation reduced pollen deposition by insects in the morning, but had little effect in the afternoon. Despite the potential for self-pollination to result in ovule discounting due to late-acting self-sterility, fecundity was severely reduced in artificially emasculated plants. Although there were temporal fluctuations in reward preference, most bee visits were for pollen rewards. Therefore the benefit of providing pollen that is accessible to bee foragers outweighs any potential costs to fitness in terms of gender interference in this species.     

Loss of gametophytic self-incompatibility with evolution of inbreeding depression

Gametophytic self-incompatibility (SI) in plants is a widespread mechanism preventing self-fertilization and the ensuing inbreeding depression, but it often evolves to self-compatibility. We analyze genetic mechanisms for the breakdown of gametophytic SI, incorporating a dynamic model for the evolution of inbreeding depression allowing for partial purging of nearly recessive lethal mutations by selfing, and accounting for pollen limitation and sheltered load linked to the S-locus. We consider two mechanisms for the breakdown of gametophytic SI: a nonfunctional S-allele and an unlinked modifier locus that inactivates the S-locus. We show that, under a wide range of conditions, self-compatible alleles can invade a self-incompatible population. Conditions for invasion are always less stringent for a nonfunctional S-allele than for a modifier locus. The spread of self-compatible genotypes is favored by extremely high or low selfing rates, a small number of S-alleles, and pollen limitation. Observed parameter values suggest that the maintenance of gametophytic SI is caused by a combination of high inbreeding depression in self-incompatible populations coupled with intermediate selfing rates of the self-compatible genotypes and sheltered load linked to the S-locus.     

Evolutionary Dynamics of Sporophytic Self-Incompatibility Alleles in Plants

The stationary frequency distribution and allelic dynamics in finite populations are analyzed through stochastic simulations in three models of single-locus, multi-allelic sporophytic self-incompatibility. The models differ in the dominance relationships among alleles. In one model, alleles act codominantly in both pollen and style (SSIcod), in the second, alleles form a dominance hierarchy in pollen and style (SSIdom). In the third model, alleles interact codominantly in the style and form a dominance hierarchy in the pollen (SSIdomcod). The SSIcod model behaves similarly to the model of gametophytic self-incompatibility, but the selection intensity is stronger. With dominance, dominant alleles invade the population more easily than recessive alleles and have a lower frequency at equilibrium. In the SSIdom model, recessive alleles have both a higher allele frequency and higher expected life span. In the SSIdomcod model, however, loss due to drift occurs more easily for pollen-recessive than for pollen-dominant alleles, and therefore, dominant alleles have a higher expected life span than the more recessive alleles. The process of allelic turnover in the SSIdomcod and SSIdom models is closely approximated by a random walk on a dominance ladder. Implications of the results for experimental studies of sporophytic self-incompatibility in natural populations are discussed.