A Markov chain Monte Carlo approach for joint inference of population structure and inbreeding rates from multilocus genotype data

Nonrandom mating induces correlations in allelic states within and among loci that can be exploited to understand the genetic structure of natural populations (Wright 1965). For many species, it is of considerable interest to quantify the contribution of two forms of nonrandom mating to patterns of standing genetic variation: inbreeding (mating among relatives) and population substructure (limited dispersal of gametes). Here, we extend the popular Bayesian clustering approach STRUCTURE (Pritchard et al. 2000) for simultaneous inference of inbreeding or selfing rates and population-of-origin classification using multilocus genetic markers. This is accomplished by eliminating the assumption of Hardy-Weinberg equilibrium within clusters and, instead, calculating expected genotype frequencies on the basis of inbreeding or selfing rates. We demonstrate the need for such an extension by showing that selfing leads to spurious signals of population substructure using the standard STRUCTURE algorithm with a bias toward spurious signals of admixture. We gauge the performance of our method using extensive coalescent simulations and demonstrate that our approach can correct for this bias. We also apply our approach to understanding the population structure of the wild relative of domesticated rice, Oryza rufipogon, an important partially selfing grass species. Using a sample of n = 16 individuals sequenced at 111 random loci, we find strong evidence for existence of two subpopulations, which correlates well with geographic location of sampling, and estimate selfing rates for both groups that are consistent with estimates from experimental data (s approximately 0.48-0.70).     

Reliable selfing rate estimates from imperfect population genetic data

Genotypic frequencies at codominant marker loci in population samples convey information on mating systems. A classical way to extract this information is to measure heterozygote deficiencies (FIS) and obtain the selfing rate s from FIS = s/(2 - s), assuming inbreeding equilibrium. A major drawback is that heterozygote deficiencies are often present without selfing, owing largely to technical artefacts such as null alleles or partial dominance. We show here that, in the absence of gametic disequilibrium, the multilocus structure can be used to derive estimates of s independent of FIS and free of technical biases. Their statistical power and precision are comparable to those of FIS, although they are sensitive to certain types of gametic disequilibria, a bias shared with progeny-array methods but not FIS. We analyse four real data sets spanning a range of mating systems. In two examples, we obtain s = 0 despite positive FIS, strongly suggesting that the latter are artefactual. In the remaining examples, all estimates are consistent. All the computations have been implemented in a open-access and user-friendly software called rmes (robust multilocus estimate of selfing) available at http://ftp.cefe.cnrs.fr, and can be used on any multilocus data. Being able to extract the reliable information from imperfect data, our method opens the way to make use of the ever-growing number of published population genetic studies, in addition to the more demanding progeny-array approaches, to investigate selfing rates.     

STRONG INBREEDING DEPRESSION IN TWO SCANDINAVIAN POPULATIONS OF THE SELF‐INCOMPATIBLE PERENNIAL HERB ARABIDOPSIS LYRATA

Inbreeding depression is a key factor influencing mating system evolution in plants, but current understanding of its relationship with selfing rate is limited by a sampling bias with few estimates for self‐incompatible species. We quantified inbreeding depression (δ) over two growing seasons in two populations of the self‐incompatible perennial herb Arabidopsis lyrata ssp. petraea in Scandinavia. Inbreeding depression was strong and of similar magnitude in both populations. Inbreeding depression for overall fitness across two seasons (the product of number of seeds, offspring viability, and offspring biomass) was 81% and 78% in the two populations. Chlorophyll deficiency accounted for 81% of seedling mortality in the selfing treatment, and was not observed among offspring resulting from outcrossing. The strong reduction in both early viability and late quantitative traits suggests that inbreeding depression is due to deleterious alleles of both large and small effect, and that both populations experience strong selection against the loss of self‐incompatibility. A review of available estimates suggested that inbreeding depression tends to be stronger in self‐incompatible than in self‐compatible highly outcrossing species, implying that undersampling of self‐incompatible taxa may bias estimates of the relationship between mating system and inbreeding depression.     

Multilocus estimation of selfing and its heritability

We describe a new method of estimating the selfing rate (S) in a mixed mating population based on a population structure approach that accounts for possible intergenerational correlation in selfing rate, giving rise to an estimate of the upper limit for heritability of selfing rate (h(2)). A correlation between generations in selfing rate is shown to affect one- and two-locus probabilities of identity by descent. Conventional estimates of selfing rate based on a population structure approach are positively biased by intergenerational correlation in selfing. Multilocus genotypes of individuals are used to give maximum-likelihood estimates of S and h(2) in the presence of scoring artifacts. Our multilocus estimation of selfing rate and its heritability (MESH) method was tested with simulated data for a range of conditions. Selfing rate estimates from MESH have low bias and root mean squared error, whereas estimates of the heritability of selfing rate have more uncertainty. Increasing the number of individuals in a sample helps to reduce bias and root mean squared error more than increasing the number of loci of sampled individuals. Improved estimates of selfing rate, as well as estimates of its heritability, can be obtained with this method, although a large number of loci and individuals are needed to achieve best results.     

Extensions of models for the estimation of mating systems using n independent loci

Inferences about plant mating systems increasingly use highly informative genetic markers, and investigate finer facets of the mating system. Here, four extensions of models for the estimation mating systems are described. (1) Multiallelic probabilities for the mixed selfing-random mating model are given; these are especially suitable for microsatellites; a generalized Kronecker operator is basis of this formula. (2) Multilocus probabilities for the "correlated-matings model" are given; interestingly, comparisons between single- vs multilocus estimates of correlated-paternity can provide a new measure of population substructure. (3) A measure of biparental inbreeding, the "correlation of selfing among loci", is shown to approximate the fraction of selfing due to uniparental (as opposed to biparental) inbreeding; also joint estimation of 1- 2- and 3-locus selfing rates allow separation, under a simple model, of the frequency vs the magnitude of biparental inbreeding. (4) Method-of-moments estimators for individual outcrossing rates are given. Formulae are given for both gymnosperms and angiosperms, and the computer program "MLTR" implements these methods.     

The evolution and maintenance of monoecy and dioecy in Sagittaria latifolia (Alismataceae)

Plant species rarely exhibit both monoecious and dioecious sexual systems. This limits opportunities to investigate the consequences of combined versus separate sex function on mating patterns and genetic variation and the analysis of factors responsible for the evolution and maintenance of the two sexual systems. Populations of the North American clonal aquatic Sagittaria latifolia are usually either monoecious or dioecious and often grow in close geographic proximity. We investigated mating patterns, genetic structure, and relationships between the two sexual systems using allozyme variation in populations from southern Ontario, Canada. As predicted, selfing rates in monoecious populations (n = 6, mean = 0.41) were significantly higher than in dioecious populations (n = 6, mean = 0.11). Moreover, marker-based estimates of inbreeding depression (delta) indicated strong selection against inbred offspring in both monoecious (mean delta = 0.83) and dioecious (mean delta = 0.84) populations. However, the difference in selfing rate between the sexual systems was not reflected in contrasting levels of genetic variation. Our surveys of 12 loci in 15 monoecious and 11 dioecious populations revealed no significant differences in the proportion of polymorphic loci (P), number of alleles per locus (A), or observed and expected heterozygosity (H(o) and H(e), respectively). Strong inbreeding depression favoring survival of outcrossed offspring may act to maintain similar levels of diversity between monoecious and dioecious populations. Despite geographical overlap between the two sexual systems in southern Ontario, a dendrogram of genetic relationships indicated two distinct clusters of populations largely corresponding to monoecious and dioecious populations. Reproductive isolation between monoecious and dioecious populations appears to be governed, in part, by observed differences in habitat and life history. We suggest that selfing and inbreeding depression in monoecious populations are important in the transition from monoecy to dioecy and that the maintenance of distinct sexual systems in S. latifolia is governed by interactions between ecology, life history, and mating.     

Individual Variation in Inbreeding Depression: The Roles of Inbreeding History and Mutation

We use mutation-selection recursion models to evaluate the relative contributions of mutation and inbreeding history to variation among individuals in inbreeding depression and the ability of experiments to detect associations between individual inbreeding depression and mating system genotypes within populations. Poisson mutation to deleterious additive or recessive alleles generally produces far more variation among individuals in inbreeding depression than variation in history of inbreeding, regardless of selfing rate. Moreover, variation in inbreeding depression can be higher in a completely outcrossing or selfing population than in a mixed-mating population. In an initially random mating population, the spread of a dominant selfing modifier with no pleiotropic effects on male outcross success causes a measurable increase in inbreeding depression variation if its selfing rate is large and inbreeding depression is caused by recessive lethals. This increase is observable during a short period as the modifier spreads rapidly to fixation. If the modifier alters selfing rate only slightly, it fails to spread or causes no measurable increase in inbreeding depression variance. These results suggest that genetic associations between mating loci and inbreeding depression loci could be difficult to demonstrate within populations and observable only transiently during rapid evolution to a substantially new selfing rate.     

Estimating selfing rates from reconstructed pedigrees using multilocus genotype data

Several methods have been developed to estimate the selfing rate of a population from a sample of individuals genotyped for several marker loci. These methods can be based on homozygosity excess (or inbreeding), identity disequilibrium, progeny array (PA) segregation or population assignment incorporating partial selfing. Progeny array-based method is generally the best because it is not subject to some assumptions made by other methods (such as lack of misgenotyping, absence of biparental inbreeding and presence of inbreeding equilibrium), and it can reveal other facets of a mixed-mating system such as patterns of shared paternity. However, in practice, it is often difficult to obtain PAs, especially for animal species. In this study, we propose a method to reconstruct the pedigree of a sample of individuals taken from a monoecious diploid population practicing mixed mating, using multilocus genotypic data. Selfing and outcrossing events are then detected when an individual derives from identical parents and from two distinct parents, respectively. Selfing rate is estimated by the proportion of selfed offspring in the reconstructed pedigree of a sample of individuals. The method enjoys many advantages of the PA method, but without the need of a priori family structure, although such information, if available, can be utilized to improve the inference. Furthermore, the new method accommodates genotyping errors, estimates allele frequencies jointly and is robust to the presence of biparental inbreeding and inbreeding disequilibrium. Both simulated and empirical data were analysed by the new and previous methods to compare their statistical properties and accuracies.     

INBREEDING DEPRESSION IN A SELF-COMPATIBLE,ANDRODIOECIOUS CRUSTACEAN,EULIMNADIA TEXANA

The observation that offspring produced by the mating of close relatives are often less fit than those produced by matings between unrelated individuals (i.e., inbreeding depression) has commonly been explained in terms of the increased probability of expressing deleterious recessive alleles among inbred offspring (the partial dominance model). This model predicts that inbreeding depression should be limited in regularly inbreeding populations because the deleterious alleles that cause inbreeding depression (i.e., the genetic load) should be purged by regularly exposing these alleles to natural selection. We indirectly test the partial dominance model using four highly inbred populations of an androdioecious crustacean, the clam shrimp Eulimnadia texana. These shrimp are comprised of males and hermaphrodites, the latter capable of either self-fertilizing or mating with a male (i.e., outcrossing between hermaphrodites is impossible). Hermaphrodites are further subdivided into monogenics (produced via self-fertilization) and amphigenics (produced via self-fertilization or outcrossing). Electrophoretic evidence suggests significant differences in heterozygosity among populations, but that selfing rates were not statistically different (average s = 0.67). Additional electrophoretic analyses reveal that three previously described sex-linked loci (Fum, Idh-1, and Idh-2) are all tightly linked to each other, with crossing over on the order of 1% per generation. Although selfing rates are clearly high, we present evidence that early inbreeding depression (hatching rates, juvenile survival, and age at sexual maturity) exists in all four populations. For all of these factors, inbreeding depression was inferred by the positive correlation of multilocus heterozygosity and fitness. Cumulative inbreeding depression (8) is between 0.41 and 0.47 across all populations, which appears to be too low to limit the effects of purging via identity disequilibrium. Instead, we suggest that the maintenance of inbreeding depression in these populations is due to the observed linkage group, which we suggest contains a large number of genes including many related to fitness. Segregation of such a large linkage group would explain our observations of the predominance of amphigenic hermaphrodites in our field samples and of survival differences between monogenics and amphigenics within selfed clutches. We propose that a modified form of the overdominance model for inbreeding depression operating at the level of linkage groups maintains the observed levels of inbreeding depression in these populations even in the face of high rates of selfing.     

Role of parasite transmission in promoting inbreeding: II. Pedigree reconstruction reveals sib‐transmission and consequent kin‐mating

Even though parasitic flatworms are one of the most species‐rich groups of hermaphroditic organisms, we know virtually nothing of their mating systems (selfing or kin‐mating rates) in nature. Hence, we lack an understanding of the role of inbreeding in parasite evolution. The natural mating systems of parasitic flatworms have remained elusive due to the inherent difficulty in generating progeny‐array data in many parasite systems. New developments in pedigree reconstruction allow direct inference of realized selfing rates in nature by simply using a sample of genotyped individuals. We built upon this advancement by utilizing the closed mating systems, that is, individual hosts, of endoparasites. In particular, we created a novel means to use pedigree reconstruction data to estimate potential kin‐mating rates. With data from natural populations of a tapeworm, we demonstrated how our newly developed methods can be used to test for cosibling transmission and inbreeding depression. We then showed how independent estimates of the two mating system components, selfing and kin‐mating rates, account for the observed levels of inbreeding in the populations. Thus, our results suggest that these natural parasite populations are in inbreeding equilibrium. Pedigree reconstruction analyses along with the new companion methods we developed will be broadly applicable across a myriad of parasite species. As such, we foresee that a new frontier will emerge wherein the diverse life histories of flatworm parasites could be utilized in comparative evolutionary studies to broadly address ecological factors or life history traits that drive mating systems and hence inbreeding in natural populations.     

Characterization of deleterious mutations in outcrossing populations.

Deng and Lynch recently proposed estimating the rate and effects of deleterious genomic mutations from changes in the mean and genetic variance of fitness upon selfing/outcrossing in outcrossing/highly selfing populations. The utility of our original estimation approach is limited in outcrossing populations, since selfing may not always be feasible. Here we extend the approach to any form of inbreeding in outcrossing populations. By simulations, the statistical properties of the estimation under a common form of inbreeding (sib mating) are investigated under a range of biologically plausible situations. The efficiencies of different degrees of inbreeding and two different experimental designs of estimation are also investigated. We found that estimation using the total genetic variation in the inbred generation is generally more efficient than employing the genetic variation among the mean of inbred families, and that higher degree of inbreeding employed in experiments yields higher power for estimation. The simulation results of the magnitude and direction of estimation bias under variable or epistatic mutation effects may provide a basis for accurate inferences of deleterious mutations. Simulations accounting for environmental variance of fitness suggest that, under full-sib mating, our extension can achieve reasonably well an estimation with sample sizes of only approximately 2000-3000.     

Effect of selection against deleterious mutations on the decline in heterozygosity at neutral loci in closely inbreeding populations.

Transition matrices for selfing and full-sib mating were derived to investigate the effect of selection against deleterious mutations on the process of inbreeding at a linked neutral locus. Selection was allowed to act within lines only (selection type I) or equally within and between lines (type II). For selfing lines under selection type I, inbreeding is always retarded, the retardation being determined by the recombination fraction between the neutral and selected loci and the inbreeding depression from the selected locus, irrespective of the selection coefficient (s) and dominance coefficient (h) of the mutant allele. For selfing under selection type II or full-sib mating under both selection types, inbreeding is delayed by weak selection (small s and sh), due to the associative overdominance created at the neutral locus, and accelerated by strong selection, due to the elevated differential contributions between alternative alleles at the neutral locus within individuals and between lines (for selection type II). For multiple fitness loci under selection, stochastic simulations were run for populations with selfing, full-sib mating, and random mating, using empirical estimates of mutation parameters and inbreeding load in Drosophila. The simulations results are in general compatible with empirical observations.     

Inferring recent outcrossing rates using multilocus individual heterozygosity: application to evolving wheat populations

Using multilocus individual heterozygosity, a method is developed to estimate the outcrossing rates of a population over a few previous generations. Considering that individuals originate either from outcrossing or from n successive selfing generations from an outbred ancestor, a maximum-likelihood (ML) estimator is described that gives estimates of past outcrossing rates in terms of proportions of individuals with different n values. Heterozygosities at several unlinked codominant loci are used to assign n values to each individual. This method also allows a test of whether populations are in inbreeding equilibrium. The estimator's reliability was checked using simulations for different mating histories. We show that this ML estimator can provide estimates of outcrossing rates for the final generation outcrossing rate (t(0)) and a mean of the preceding rates (t(p)) and can detect major temporal variation in the mating system. The method is most efficient for low to intermediate outcrossing levels. Applied to nine populations of wheat, this method gave estimates of t(0) and t(p). These estimates confirmed the absence of outcrossing t(0) = 0 in the two populations subjected to manual selfing. For free-mating wheat populations, it detected lower final generation outcrossing rates t(0) = 0-0.06 than those expected from global heterozygosity t = 0.02-0.09. This estimator appears to be a new and efficient way to describe the multilocus heterozygosity of a population, complementary to Fis and progeny analysis approaches.     

THE EVOLUTION OF SELF-FERTILIZATION AND INBREEDING DEPRESSION IN PLANTS. I. GENETIC MODELS

The amounts of inbreeding depression upon selfing and of heterosis upon outcrossing determine the strength of selection on the selfing rate in a population when this evolves polygenically by small steps. Genetic models are constructed which allow inbreeding depression to change with the mean selfing rate in a population by incorporating both mutation to recessive and partially dominant lethal and sublethal alleles at many loci and mutation in quantitative characters under stabilizing selection. The models help to explain observations of high inbreeding depression (> 50%) upon selfing in primarily outcrossing populations, as well as considerable heterosis upon outcrossing in primarily selfing populations. Predominant selfing and predominant outcrossing are found to be alternative stable states of the mating system in most plant populations. Which of these stable states a species approaches depends on the history of its population structure and the magnitude of effect of genes influencing the selfing rate.     

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.     

INBREEDING DEPRESSION IN TWO MIMULUS TAXA MEASURED BY MULTIGENERATIONAL CHANGES IN THE INBREEDING COEFFICIENT

In mixed-mating plant populations, one can estimate the relative fitness of selfed progeny w by measuring the inbreeding coefficient F and selfing rate s of adults of one generation, together with F of adults in the following generation (after selection). In the first application of this multigenerational method, we estimated F and s for adults over three consecutive generations in adjacent populations of two annual Mimulus taxa: the outbreeding M. guttatus and the inbreeding M. platycalyx. This gave estimates of w for the last two generations. Although average multilocus selfing rates were high in both taxa (0.63 in M. guttatus; 0.84 in M. platycalyx), the relative fitness of selfed progeny averaged only 0.19 in M. guttatus and 0.32 in M. platycalyx. An alternative estimator for w that incorporates biparental inbreeding gave even lower estimates of w. These values are significantly below the 0.5 threshold thought to favor selfing, and show that partially selfing populations can harbor substantial genetic load. In accordance with the purging hypothesis, the more highly selfing M. platycalyx showed marginally lower inbreeding depression than M. guttatus in both years (P = 0.08). Inbreeding depression and selfing rates also varied among years in concert among taxa. Several sources of bias are discussed, but computer simulations indicate it is unlikely that w is biased downwards by linkage of marker loci to load loci.     

Estimating Long-Term Mating Systems Using DNA Sequences

Plant mating systems often involve a mixture of self fertilizations and outcross fertilizations. The degree of selfing has a large impact on the genetic composition of natural populations and on the evolution of the mating system itself in response to such factors as inbreeding depression. This paper describes a means of estimating the long-term rate of self-fertilization from samples of alleles taken from individuals in a population. Use is made of the genealogy of pairs of alleles at a locus within individuals and pairs between individuals. The degree of selfing is closely related to the extent to which the number of nucleotide sites differing within an individual is reduced relative to the number differing between individuals. Importantly, the estimate of long-term selfing is largely independent of population size and is not affected by historical fluctuations in population size; instead it responds directly to the mating system itself. The approach outlined here is most appropriate to evolutionary problems in which the long-term nature of the mating system is of interest, such as to determine the relationship between prior inbreeding and inbreeding depression.     

Effects of Interference Between Selected Loci on the Mutation Load,Inbreeding Depression,and Heterosis

A classical prediction from single-locus models is that inbreeding increases the efficiency of selection against partially recessive deleterious alleles (purging), thereby decreasing the mutation load and level of inbreeding depression. However, previous multilocus simulation studies found that increasing the rate of self-fertilization of individuals may not lead to purging and argued that selective interference among loci causes this effect. In this article, I derive simple analytical approximations for the mutation load and inbreeding depression, taking into account the effects of interference between pairs of loci. I consider two classical scenarios of nonrandomly mating populations: a single population undergoing partial selfing and a subdivided population with limited dispersal. In the first case, correlations in homozygosity between loci tend to reduce mean fitness and increase inbreeding depression. These effects are stronger when deleterious alleles are more recessive, but only weakly depend on the strength of selection against deleterious alleles and on recombination rates. In subdivided populations, interference increases inbreeding depression within demes, but decreases heterosis between demes. Comparisons with multilocus, individual-based simulations show that these analytical approximations are accurate as long as the effects of interference stay moderate, but fail for high deleterious mutation rates and low dominance coefficients of deleterious alleles.     

Microsatellite variation and evolution in the Mimulus guttatus species complex with contrasting mating systems

Mutational variability at microsatellite loci is shaped by both population history and the mating system. In turn, alternate mating systems in flowering plants can resolve aspects of microsatellite loci evolution. Five species of yellow monkeyflowers (Mimulus sect. Simiolis) differing for historical rates of inbreeding were surveyed for variation at six microsatellite loci. High levels of diversity at these loci were found in both outcrossing and selfing taxa. In line with allozyme studies, inbreeders showed more partitioning of diversity among populations, and diversity in selfing taxa was lower than expected from reductions in effective population size due to selfing alone, suggesting the presence of either population bottlenecks or background selection in selfers. Evaluation of the stepwise mutation model (a model of DNA replication slippage) suggests that these loci evolve in a stepwise fashion. Inferred coalescent times of microsatellite alleles indicate that past bottlenecks of population size or colonization events were important in reducing diversity in the inbreeding taxon.      

Mating patterns and genetic diversity in the wild daffodil Narcissus longispathus (Amaryllidaceae)

Despite the importance of Narcissus to ornamental horticulture, there have been no population genetic studies of wild species, many of which have narrow distributions. Here, we measure selfing rates and levels of genetic diversity at allozyme loci in six populations of Narcissus longispathus, a self-compatible daffodil endemic to a few mountain ranges in southeastern Spain. The populations were distributed among four distinct river valleys encompassing two main watersheds in the Sierra de Cazorla mountains. Selfing rates averaged 0.37 (range 0.23-0.46), resulting in significant inbreeding coefficients for the progeny (f = 0.324). In contrast, estimates of inbreeding in parental genotypes were not significantly different from zero (f = 0.001), indicating that few selfed offspring survive to maturity because of inbreeding depression. Species-wide estimates of genetic diversity for the six populations were P(s) = 0.38, H(es) = 0.119 and A(s) = 1.27 with significant genetic differentiation among populations theta = 0.15. The observed patterns of genetic differentiation among populations are likely influenced by the mating system, and a combination of local topography, watershed affinities and gene flow.