Cautions on direct gene flow estimation in plant populations

Through simulations we have investigated the statistical properties of two of the main approaches for directly estimating pollen gene flow (m) in plant populations: genotypic exclusion and mating models. When the assumptions about accurately known background pollen pool allelic frequencies are met, both methods provide unbiased results with comparable variances across a range of true m values. However, when presumed allelic frequencies differ from actual ones, which is more likely in research practice, both estimators are biased. We demonstrate that the extent and direction of bias largely depend on the difference (measured as genetic distance) between the presumed and actual pollen pools, and on the degree of genetic differentiation between the local population and the actual background pollen sources. However, one feature of the mating model is its ability to estimate pollen gene flow simultaneously with background pollen pool allelic frequencies. We have found that this approach gives nearly unbiased pollen gene flow estimates, and is practical because it eliminates the necessity of providing independent estimates of background pollen pool allelic frequencies. Violations of the mating model assumptions of random mating within local population affect the precision of the estimates only to a limited degree.     

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. I. Nonrandom associations and equilibrium structure with both maternal and paternal cytoplasmic inheritance

We determine the nuclear-dicytoplasmic effects of unidirectional gene flow via pollen and seeds upon a mixed-mating plant population, focusing on nuclear-mitochondrial-chloroplast systems where mitochondria are inherited maternally and chloroplasts paternally, as in many conifers. After first delineating the general effects of admixture (via seeds or individuals) on the nonrandom associations in such systems, we derive the full dicytonuclear equilibrium structure, including when disequilibria may be indicators of gene flow. Substantial levels of permanent two- and three-locus disequilibria can be generated in adults by (i) nonzero disequilibria in the migrant pools or (ii) intermigrant admixture effects via different chloroplast frequencies in migrant pollen and seeds. Additionally, three-locus disequilibria can be generated by higher-order intermigrant effects such as different chloroplast frequencies in migrant pollen and seeds coupled with nuclear-mitochondrial disequilibria in migrant seeds, or different nuclear frequencies in migrant pollen and seeds coupled with mitochondrial-chloroplast disequilibria in migrant seeds. Further insight is provided by considering special cases with seed or pollen migration alone, complete random mating or selfing, or migrant pollen and seeds lacking disequilibria or intermigrant admixture effects. The results complete the theoretical foundation for a new method for estimating pollen and seed migration using joint cytonuclear or dicytonuclear data.     

The diffusion model for migration and selection in a dioecious population

The diffusion approximation is derived for migration and selection at a multiallelic locus in a dioecious population subdivided into a lattice of panmictic colonies. Generations are discrete and nonoverlapping; autosomal and X-linked loci are analyzed. The relation between juvenile and adult subpopulation numbers is very general and includes both soft and hard selection; the zygotic sex ratio is the same in every colony. All the results hold for both adult and juvenile migration. If ploidy-weighted average selection, drift, and diffusion coefficients are used, then the ploidy-weighted average allelic frequencies satisfy the corresponding partial differential equation for a monoecious population. The boundary conditions and the unidimensional transition conditions for coincident discontinuities in the carrying capacity and migration rate extend identically. The previous unidimensional formulation and analysis of symmetric, nearest-neighbor migration of a monoecious population across a geographical barrier is generalized to symmetric migration of arbitrary finite range, and the transition conditions are shown to hold for a dioecious population. Thus, the entire theory of clines and of the wave of advance of favorable alleles is applicable to dioecious populations.This work was supported by National Science Foundation grant BSR-9006285     

Comparative Effects of Pollen and Seed Migration on the Cytonuclear Structure of Plant Populations. II. Paternal Cytoplasmic Inheritance

We continue our study of the effects of pollen and seed migration on the cytonuclear structure of mixed-mating plant populations by analyzing two deterministic continent-island models under the critical assumption of paternal cytoplasmic inheritance. The major results of this study that contrast with our previous conclusions based on maternal cytoplasmic inheritance are (i) pollen gene flow can significantly affect the cytonuclear structure of the island population, and in particular can help to generate cytonuclear disequilibria that greatly exceed the magnitude of those that would be produced by seed migration or mixed mating alone; (ii) with simultaneous pollen and seed migration, nonzero cytonuclear disequilibria will be maintained not only when there is disequilibrium in the immigrant pollen or seeds, but also through a variety of intermigrant admixture effects when the two pools of immigrants differ appropriately in their cytonuclear compositions; (iii) either immigrant pollen or immigrant seeds can generate disequilibria de novo in populations with initially random cytonuclear associations, but pollen migration alone generally produces lower levels of disequilibrium than does comparable seed migration, especially at high levels of self-fertilization when the overall fraction of immigrant pollen is low; (iv) the equilibrium state of the island population will be influenced by the rate of pollen gene flow whenever there is either allelic disequilibrium in the immigrant pollen or simultaneous seed migration coupled with different cytoplasmic or nuclear allele frequencies in immigrant pollen and seeds or nonzero allelic disequilibrium in either immigrant pool. The estimation of pollen migration should therefore be facilitated with paternal cytoplasmic inheritance relative to the case of maternal cytoplasmic inheritance. These basic conclusions hold whether the population is censused as seeds or as adults, but with simultaneous pollen and seed migration, the relationship between census time and the ability to detect nonrandom cytonuclear associations is complex. When migration is through pollen alone, however, the cytonuclear structure of the island population is independent of the life stage censused.     

Effective population size associated with self-fertilization: lessons from temporal changes in allele frequencies in the selfing annual Medicago truncatula

Despite its significance in evolutionary and conservation biology, few estimates of effective population size (N(e)) are available in plant species. Self-fertilization is expected to affect N(e), through both its effect on homozygosity and population dynamics. Here, we estimated N(e) using temporal variation in allele frequencies for two contrasted populations of the selfing annual Medicago truncatula: a large and continuous population and a subdivided population. Estimated N(e) values were around 5-10% of the population census size suggesting that other factors than selfing must contribute to variation in allele frequencies. Further comparisons between monolocus allelic variation and changes in the multilocus genotypic composition of the populations show that the local dynamics of inbred lines can play an important role in the fluctuations of allele frequencies. Finally, comparing N(e) estimates and levels of genetic variation suggest that H(e) is a poor estimator of the contemporaneous variance effective population size.     

The effect of selection on esterase allozymes in a barley population

Changes in gene and genotypic frequencies at four esterase loci were monitored over 25 generations in Composite Cross V, an experimental population of barley, to obtain experimental evidence concerning the balance of forces responsible for: (1) the marked differences in allelic frequencies among barleys from different ecogeographical regions of the world; and (2) the extensive allelic variation found within local populations of barley. Analyses of the highly significant changes in allelic frequencies which occurred in CCV showed they were due to directional selection favoring particular alleles and not to mutation, migration or genetic drift. The results show that intense balancing selection, featuring consistent excesses of heterozygotes, also occurred in CCV. It is concluded that among the factors of neo-Darwinian evolution, natural selection plays the predominant role in determining the observed patterns of allelic variation in the barley species as a whole.     

Selection caused by self-fertilization I. Four measures of self-fertilization and their effects on fitness

The standard models of selfing in seed plants consider only the ovules, which are assumed to have a constant selfing rate. It has recently become clear, however, that hermaphrodite or monoecious populations frequently show sexual asymmetry (nonconstant pollen:ovule fertilities among individuals). Such asymmetry usually results in pollen selfing rates which differ from those for the ovules and are frequency-dependent even for constant ovule selfing rates. A recent study of selfing rates for all gametes of an individual is extended here to include four selfing rates (for ovules, pollen, all gametes, and zygotes), and simple mathematical relationships linking the four rates are obtained. Unlike earlier models of selfing, it is not assumed that the ovule selfing rate is constant, but instead that this rate, like all the others, is determined by the mobility of the pollen, which in turn is determined by the floral biology and ecology. It is found that all four selfing rates are usually frequency-dependent. The selfing rate for all gametes (the combined selfing rate) is usually intermediate between those for the ovules and pollen, and the zygotic rate is usually the smallest of the four. The exceptions to the above statements occur for relatively extreme situations, such as complete selfing for pollen or ovules, no selfing, or sexual symmetry. Three modes of selfing are considered: prior (PS), competing (CS), and delayed (DS) self-fertilization. It is shown that if there are at least two types with different selfing rates in the population, then the ranking of their selfing rates may depend upon the frequencies of the types (for the combined and the zygotic rates), may be frequency-independent (ovule rate), or may be dependent or independent, according to the mode of selfing (pollen rate). The effects of the various influences on the amount of selfing are by no means negligible. Thus a numerical study shows pollen selfing rates for one type which vary from 0.09 to 0.96, according to its frequency. Another numerical result shows a change in combined selfing rate from 0.13 to 0.86, depending solely on the mode of selfing. Results for Scots Pine show that an ovule selfing rate of 0.5 was accompanied by a combined rate of 0.143.The population selfing rate is not the same as the mean of individual selfing rates, and can only be obtained if female fitnesses as well as ovule selfing rates are known for each type.Previous models of selfing have failed to distinguish between the effects of increased selfing and increased pollen fertility, with the result that increased selfing always resulted in greater fitness. In the present models the two effects are distinguishable, and it is found that increased selfing may result in increased or decreased fitness, depending also on population density and on a form of pollen density. Thus the old dogma that in the absence of viability and fertility selection increased selfing always results in increased fitness is finally refuted, and the importance of the influence of ecological parameters on selfing and fitness is emphasized, since population density and pollen density influence the selfing rates.     

How geitonogamous selfing affects sex allocation in hermaphrodite plants

Does the mode of self-pollination affect the evolutionarily stable allocation to male vs. female function? We distinguish the following scenarios. (1) An ‘autogamous’ species, in which selfing occurs within the flower prior to opening. The pollen used in selfing is a constant fraction of all pollen grains produced. (2) A species with ‘abiotic pollination’, in which selfing occurs when pollen dispersed in one flower lands on the stigma of a nearby flower on the same plant (geitonogamy). The selfing rate increases with male allocation but a higher selfing rate does not mean a reduced export of pollen. (3) An ‘animal-pollinated’ species with geitonogamous selfing. Here the selfing rate also increases with male allocation, but pollen export to other plants in the population is a decelerating function of the number of simultaneously open flowers. In all three models selfing selects for increased female allocation. For model 3 this contradicts the general opinion that geitonogamous selfing does not affect evolutionarily stable allocations. In all models, the parent benefits more from a female-biased allocation than any other individual in the population. In addition, in models 2 and 3, greater male allocation results in more local mate competition. In model 3 and in model 2 with low levels of inbreeding depression, hermaphroditism is evolutionarily stable. In model 2 with high inbreeding depression, the population converges to a fitness minimum for the relative allocation to male function. In this case the fitness set is bowed inwards, corresponding with accelerating fitness gain curves. If the selfing rate increases with plant size, this is a sufficient condition for size-dependent sex allocation (more allocation towards seeds in large plants) to evolve. We discuss our results in relation to size-dependent sex allocation in plants and in relation to the evolution of dioecy.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

Potential bias in inbreeding depression estimates when using pedigree relationships to assess the degree of homozygosity for loci under selection

A potential bias in estimation of inbreeding depression when using pedigree relationships to assess the degree of homozygosity for loci under selection is indicated. A comparison of inbreeding coefficients based on either pedigree or genotypic frequencies indicated that, as a result of selection, the inbreeding coefficient based on pedigree might not correspond with the random drift of allelic frequencies. Apparent differences in average levels of both inbreeding coefficients were obtained depending on the genetic model (additive versus dominance, initial allelic frequencies, heritability) and the selection system assumed (no versus mass selection). In the absence of selection, allelic frequencies within a small population change over generations due to random drift, and the pedigree-based inbreeding coefficient gives a proper assessment of the accompanying probability of increased homozygosity within a replicate by indicating the variance of allelic frequencies over replicates. With selection, in addition to random drift, directional change in allelic frequencies is not accounted for by the pedigree-based inbreeding coefficient. This result implies that estimation of inbreeding depression for traits under either direct or indirect selection, estimated by a regression of performance on pedigree-based coefficients, should be carefully interpreted.Deceased     

On migration load of seeds and pollen grains in a local population

We have extended Wright's model of migration load to hermaphrodite plants showing variation at a single locus with two alleles. The model incorporates independent migration of seeds and pollen grains, the selection at both the haploid gametophyte and the diploid sporophyte stages, and a mixed mating system. The analytical relations between migration load and migration rate of seeds and pollen grains are explicitly formulated. The results show that under certain conditions, seed flow can have a more effect on migration load than pollen flow. Pollen selection at the gametophyte stage cannot substantially affect the migration load at the sporophyte stage. Selection at the diploid sporophyte stage is critical in determining the migration load of pollen grains. The relative migration loads of pollen versus seeds can be approximately estimated in predominantly outcrossing populations by the ratio of pollen flow to twice the seed flow, when the selection coefficient (s(T)) is greater than, or approximately equal to, the migration rate (m).     

Population dynamics of a gametophytic factor controlling selective fertilization

The population behavior of a gametophytic factor (Ga) which involves gametic selection due to failure ofga pollen onGa Ga orGa ga styles in competition withGa pollen, was investigated by computer simulation. A constant versus randomly varying gametic selection parameter (k) and four different schemes of zygotic selection were introduced in this model for analyzing conditions favorable for the maintenance of locusGa polymorphic in a large, mixed selfing and random mating population. Stable polymorphism was obtained only with rather substantial heterozygote advantage at locusGa whereas the opposing pressures of gametic and zygotic selection yielded fixation of alleleGa orga around a critical value of k instead of a range of k-values allowing nontrivial equilibria. With weak selection and stochastic k, however, very slow rates of change in the genotypic proportions allowed transient polymorphism. In these cases, the rate of outcrossing (t) and the initial frequency ofGa were critical in determining the rate of allelic substitution. Moreover, low values of t allowed the replacement of alleleGa byga even with rather weak zygotic selection. These findings on the balance between gametic and zygotic selection and a markedly frequency-dependent process are briefly discussed in relation to the dynamics of similar factors involving the mating system.     

Inbreeding and outbreeding effects on pollen fitness and zygote survival in Silene nutans (Caryophyllaceae)

In plants, selfing and outcrossing may be affected by maternal mate choice and competition among pollen and zygotes. To evaluate this in Silene nutans, we pollinated plants with mixtures of (1) self‐ and outcross pollen and (2) pollen from within a population and from another population. Pollen fitness and zygote survival was estimated from the zygote survival and paternity of seeds. Self pollen had a lower fitness than outcross pollen, and selfed zygotes were less likely, or as likely, to develop into seeds. Hybrid zygotes survived as frequently or more than local zygotes, and pollen from one of the populations fertilized most ovules in both populations. Our results thus indicate strong maternal discrimination against selfing, whereas the success of outbreeding seems mostly affected by divergent pollen performance. The implications for the evolution of maternal mate choice are discussed.     

Genotypic variation for reproductive characters,and the influence of pollen-ovule ratio on selfing rate in rape seed (Brassica napus)

The variance component for genotypic effects and covariable plant vigor were estimated for pollen production, number of ovules, the weight of the flower without anther sacs and pistil, and the gender of the whole plant, in the rape seed variety Topas. There was a significant effect of the genotype for all measured characters. We did not find any trade-off between the male and female functions. Small plants were relatively more male-biased than large plants. The weight of the flower without anther sacs or pistil was positively correlated with pollen production. The selfing rates of the lines were positively correlated with the pollen-ovule ratio, suggesting that the selfing rate of individual plants is positively correlated with pollen-ovule ratio, thus supporting the “mass-action” model. It is proposed that the observed increase in the population selfing rate during the flowering season for some species, including rape seed, can be caused by an overall decrease in male investment.     

A theory for exaggerated secondary sexual traits in animal-pollinated plants

We analyze two mathematical models of adaptive investment in rewarding plant traits. In both models, the attractiveness of a particular trait value declines as the mean value in the population increases (asymmetric competition), giving relatively rewarding traits a competitive advantage. Including this competition for pollinator visits in a standard model of hermaphroditic sex allocation shifts additional allocation to pollinator rewards at the expense of allocation to pollen and seeds. In the second model, plants can invest additional resources in pollinator rewards but suffer reduced viability and rising costs due to excess pollen removal and within-plant selfing (geitonogamy). Despite these accumulating costs, increasing the magnitude of asymmetric competition exaggerates the ESS investment in rewards beyond the equilibrium in cases where attractiveness depends only on a plant’s absolute reward value. We suggest that the type of frequency dependent selection modeled here is fundamentally equivalent to sexual selection in animal populations (with some unique exceptions). Testing the main assumptions of our models may reveal whether seemingly “extravagant” floral traits are strictly analogous to the exaggerated secondary sexual traits of animals. An erratum to this article can be found at     

PHYSIOLOGICAL AND BEHAVIORAL ADAPTATION TO VARYING ENVIRONMENTS: A MATHEMATICAL MODEL

We develop a mathematical model to explore the evolution of habitat selection and physiological adaptation in a heterogeneous environment. The model assumes the following conditions: 1) a panmictic population of infinite size; 2) prereproductive individuals mobile enough to move between patches; 3) alleles at one locus code for absence or presence of adaptation to detrimental patches; 4) alleles at a second locus code for absence or presence of behavior(s) that cause avoidance of the detrimental patches; 5) additive effects of alleles controlling physiology and behavior; 6) frequency-independent fitness. Results of the model indicate that nontrivial, polymorphic equilibria do not exist. The pattern of genotypic fitnesses and the initial allelic frequencies can influence whether the population adapts by physiological or behavioral mechanisms, or by both. Linkage between the two loci can alter the outcome of evolution, given specified genotypic fitness values and initial allelic frequencies.     

Evolution at a multiallelic locus under migration and uniform selection

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The drift and diffusion coefficients may depend on position, but the selection coefficients do not. It is established that if p is a uniform equilibrium point under pure selection, then p is a migration-selection equilibrium, and that generically the introduction of migration does not change the stability of p. It is also proved that if p is a uniform, globally asymptotically stable, internal equilibrium point under pure selection, then the gene frequencies converge to p when both migration and selection are present. Thus, in this case, after a sufficiently long time, there is no genetic indication of the spatial distribution of the population.     

How multilocus genotypic pattern helps to understand the history of selfing populations: a case study in Medicago truncatula

The occurrence of populations exhibiting high genetic diversity in predominantly selfing species remains a puzzling question, since under regular selfing genetic diversity is expected to be depleted at a faster rate than under outcrossing. Fine-scale population genetics approaches may help to answer this question. Here we study a natural population of the legume Medicago truncatula in which both the fine-scale spatial structure and the selfing rate are characterized using three different methods. Selfing rate estimates were very high ( approximately 99%) irrespective of the method used. A clear pattern of isolation by distance reflecting small seed dispersal distances was detected. Combining genotypic data over loci, we could define 34 multilocus genotypes. Among those, six highly inbred genotypes (lines) represented more than 75% of the individuals studied and harboured all the allelic variation present in the population. We also detected a large set of multilocus genotypes resembling recombinant inbred lines between the most frequent lines occurring in the population. This finding illustrates the importance of rare recombination in redistributing available allelic diversity into new genotypic combinations. This study shows how multilocus and fine-scale spatial analyses may help to understand the population history of self-fertilizing species, especially to make inferences about the relative role of foundation/migration and recombination events in such populations.     

An experimental study of the S-Allee effect in the self-incompatible plant <Emphasis Type="Italic">Biscutella neustriaca</Emphasis>

Homomorphic self-incompatibility (SI) evolved in many plant families to enforce selfing avoidance, and is controlled by a single multiallelic locus (the S-locus). In a fragmented landscape, strong variation in population size and in local density is expected to cause strong variation in allelic diversity at the S-locus, which could generate an Allee effect on female reproductive success by constraining compatible pollen availability. In this experimental study, we aimed at detecting this SI-specific Allee effect (or S-Allee effect) in the endangered species Biscutella neustriaca. We demonstrated the occurrence of a SI mating system in the species and determined compatibility relationships among genotypes through a large set of controlled pollinations. For the experiment, we chose three different pollen receptor genotypes, each compatible with respectively 100, 75 and 25% of four other genotypes, which constituted the pollen sources. We placed different ramets of each receptor at different distances from the pollen sources to control for pollen limitation due to low local density, and we measured the seed set on each receptor plant three times consecutively. Analyses performed with generalized linear mixed models showed that both the distance to the pollen sources and the mate availability due to SI had a significant effect on seed set, with a strong reduction observed when mate availability was limited to 25%. Our results suggest that pollen limitation due to a restriction in compatible mate availability could occur in small or scattered populations exhibiting low allelic diversity at the S-locus.     

Delayed autonomous self-pollination in Hibiscus laevis (Malvaceae)

Delayed autonomous self-pollination allows outcrossing to occur while also ensuring that seeds are produced in the absence of pollen vectors. We investigated variation in the efficacy of this pollination mechanism in populations of Hibiscus laevis. Recurvature of stylar branches occurred after 1 d of anthesis, and in plants from Ohio, Illinois, Arkansas, and Oklahoma this behavior resulted in autonomous selfing (surprisingly, stylar movement was facultative in that it did not take place when the stigmas were already pollinated). In contrast to these more northern populations, the distance between anthers and stigmas was too great to allow autonomous selfing in plants from Texas, Mississippi, and Louisiana. Greenhouse studies of plants from Ohio demonstrated that autonomous selfing resulted in an average of 29.5 seeds per flower, as compared to 59.9 seeds per flower from hand-pollination of stigmas with self pollen. In an assessment of the possible significance of this selfing mode, emasculated flowers did not set significantly fewer seeds in a natural stand in Ohio, suggesting that few seeds resulted from autonomous selfing at that site. Modest inbreeding depression was detected at this population. Our results suggest that delayed autonomous selfing is more common in northern populations, where it may facilitate population establishment and persistence at times when pollinators are scarce.