Structure of populations under mixed random and sib mating

Summary The present investigation relates to various properties of population bred by mixture of breeding systems namely mixed random and sib mating. Expressions have been derived which give the genotypic frequencies in any given generation in terms of the initial values. Under the mating system considered the population will eventually become stable having a certain amount of heterozygosis depending upon the amounts of random and sib mating. The loss of heterozygosity in successive generations has been examined for varying amounts of sib mating in the population.The formulae have been derived giving the mean and genotypic variance in any given generation of continued mixed mating. The effect of the mating system considered on mean and genotypic variance in successive generations has been discussed in detail in case of (i) absence of dominance and (ii) complete dominance.

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Zusammenfassung Die vorliegende Untersuchung bezieht sich auf verschiedene Eigenheiten des Populationsverhaltens unter Einfluß einer Mischung von Paarungssystemen, nämlich der Panmixie und der Geschwisterpaarung. Es wurden Formeln abgeleitet, die die genotypischen Frequenzen in jeder beliebigen Generation in Beziehung zu den Ausgangswerten angeben. Unter dem betrachteten Paarungssystem kann die Population gegebenenfalls stabil werden mit einem bestimmten Heterozygotieanteil, der vom Ausmaß der Panmixie und der Geschwisterpaarung abhängt. Für verschiedene Anteile der Geschwisterpaarung wurde der Verlust der Heterozygotie in aufeinanderfolgenden Generationen untersucht.Die abgeleiteten Formeln liefern das Mittel und die genotypische Varianz in jeder beliebigen Generation fortgesetzter gemischter Paarung. Die Wirkung des betrachteten Paarungssystems auf Mittel und genotypische Varianz in aufeinanderfolgenden Generationen wurde (1) für den Fall der Abwesenheit der Dominanz und (2) den der vollständigen Dominanz ausführlich diskutiert.

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Journal paper No. J-6127, of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 1669. This work was supported by the National Institutes of Health, Grant No. GM-13827.     

Population genetics of autopolyploids under a mixed mating model and the estimation of selfing rate

Nowadays, the population genetics analysis of autopolyploid species faces many difficulties due to (i) limited development of population genetics tools under polysomic inheritance, (ii) difficulties to assess allelic dosage when genotyping individuals and (iii) a form of inbreeding resulting from the mechanism of ‘double reduction’. Consequently, few data analysis computer programs are applicable to autopolyploids. To contribute bridging this gap, this article first derives theoretical expectations for the inbreeding and identity disequilibrium coefficients under polysomic inheritance in a mixed mating model. Moment estimators of these coefficients are proposed when exact genotypes or just markers phenotypes (i.e. allelic dosage unknown) are available. This led to the development of estimators of the selfing rate based on adult genotypes or phenotypes and applicable to any even‐ploidy level. Their statistical performances and robustness were assessed by numerical simulations. Contrary to inbreeding‐based estimators, the identity disequilibrium‐based estimator using phenotypes is robust (absolute bias generally < 0.05), even in the presence of double reduction, null alleles or biparental inbreeding due to isolation by distance. A fairly good precision of the selfing rate estimates (root mean squared error < 0.1) is already achievable using a sample of 30–50 individuals phenotyped at 10 loci bearing 5–10 alleles each, conditions reachable using microsatellite markers. Diallelic markers (e.g. SNP) can also perform satisfactorily in diploids and tetraploids but more polymorphic markers are necessary for higher ploidy levels. The method is implemented in the software SPAGeDi and should contribute to reduce the lack of population genetics tools applicable to autopolyploids.     

Monte carlo studies of plant mating system estimation models: the one-pollen parent and mixed mating models

Estimation of mating system parameters in plant populations typically employs family-structured samples of progeny genotypes. These estimation models postulate a mixture of self-fertilization and random outcrossing. One assumption of such models concerns the distribution of pollen genotypes among eggs within single maternal families. Previous applications of the mixed mating model to mating system estimation have assumed that pollen genotypes are sampled randomly from the total population in forming outcrossed progeny within families. In contrast, the one-pollen parent model assumes that outcrossed progeny within a family share a single-pollen parent genotype. Monte Carlo simulations of family-structured sampling were carried out to examine the consequences of violations of the different assumptions of the two models regarding the distribution of pollen genotypes among eggs. When these assumptions are violated, estimates of mating system parameters may be significantly different from their true values and may exhibit distributions which depart from normality. Monte Carlo methods were also used to examine the utility of the bootstrap resampling algorithm for estimating the variances of mating system parameters. The bootstrap method gives variance estimates that approximate empirically determined values. When applied to data from two plant populations which differ in pollen genotype distributions within families, the two estimation procedures exhibit the same behavior as that seen with the simulated data.     

The mixed mating system of the sea palm kelp Postelsia palmaeformis: few costs to selfing

Naturally isolated populations have conflicting selection pressures for successful reproduction and inbreeding avoidance. These species with limited seasonal reproductive opportunities may use selfing as a means of reproductive assurance. We quantified the frequency of selfing and the fitness consequences for inbred versus outcrossed progeny of an annual kelp, the sea palm (Postelsia palmaeformis). Using experimentally established populations and microsatellite markers to assess the extent of selfing in progeny from six founding parents, we found the frequency of selfing was higher than expected in every population, and few fitness costs were detected in selfed offspring. Despite a decline in heterozygosity of 30 per cent in the first generation of selfing, self-fertilization did not affect individual size or reproduction, and correlated only with a marginally significant decline in survival. Our results suggest both that purging of deleterious recessive alleles may have already occurred and that selfing may be key to reproductive assurance in this species with limited dispersal. Postelsia has an alteration of a free-living diploid and haploid stage, where the haploid stage may provide increased efficiency for purging the genetic load. This life history is shared by many seaweeds and may thus be an important component of mating system evolution in the sea.     

Evolution of interactions in family-structured populations: mixed mating models

THE EFFECT OF INBREEDING ON SOCIALITY IS STUDIED THEORETICALLY FOR THE EVOLUTION OF INTERACTIONS BETWEEN SIBLINGS IN CERTAIN MIXED MATING SYSTEMS THAT GIVE RISE TO INBREEDING: sib with random mating and selfing with random mating. Two approaches are taken. First, specific models of altruism are studied for the various mating systems. In the case of the additive model, inbreeding facilitates the evolution of altruistic genes. Likewise, for the multiplicative model this is usually the case, as long as the costs of altruism are not too great. Second, the case of total altruism, in which the gene has zero individual fitness but increases the fitness of associates, is studied for a general fitness formulation. In this case, inbreeding often retards the ability of such genes to increase when rare, and the equilibrium frequency of those recessive genes that can increase is totally independent of the mating system and, consequently, of the amount of inbreeding. It appears from the results presented that inbreeding facilitates most forms of altruism, but retards extreme altruism. These results stem from the fact that inbreeding increases the within-family relatedness by increasing the between-family variance in allele frequency. In most cases this facilitates altruism. However, in the case of total altruism, only heterozygotes can pass on the altruistic allele, and inbreeding tends to decrease this heterozygote class. In either case, the important effect of inbreeding lies in altering the genotypic distribution of the interactions.     

Random mating and random union of gametes models for finite dioecious populations

This paper compares a random mating model for independent trials with two random union of gametes models in the case when the population is finite and dioecious. The first random union of gametes model is examined as a two-locus dioecious model with unequal recombination values in each sex, which is a generalization of previous work. The second model can be of some biological use under certain circumstances, and it is easier to analyse (when appropriate) than the first. It is examined with and without mutation to obtain both old and new results about the fixation probabilities (in the multilocus case) and the related rates in the autosomal locus case.     

Covariances of relatives stemming from a population undergoing mixed self and random mating

We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.     

Generalized linear models with random effects; salamander mating revisited.

In recent years much effort has been devoted to extending regression methodology to non-Gaussian data, where responses are not independent. These methods for dependent responses are suitable for data from longitudinal studies or nested designs. However, use of these methods for crossed designs seems to have serious limitations due to the intensive computations involved because of the intractable nature of the joint distribution. In this paper, we cast the problem in a Bayesian framework and use a Monte Carlo method, the Gibbs sampler, to avoid current computational limitations. The flexibility of this approach is illustrated by analyzing the interesting salamander mating data reported by McCullagh and Nelder (1989, Generalized Linear Models, 2nd edition, London: Chapman and Hall).     

Evolutionary rescue in randomly mating,selfing, and clonal populations

Severe environmental change can drive a population extinct unless the population adapts in time to the new conditions (“evolutionary rescue”). How does biparental sexual reproduction influence the chances of population persistence compared to clonal reproduction or selfing? In this article, we set up a one‐locus two‐allele model for adaptation in diploid species, where rescue is contingent on the establishment of the mutant homozygote. Reproduction can occur by random mating, selfing, or clonally. Random mating generates and destroys the rescue mutant; selfing is efficient at generating it but at the same time depletes the heterozygote, which can lead to a low mutant frequency in the standing genetic variation. Due to these (and other) antagonistic effects, we find a nontrivial dependence of population survival on the rate of sex/selfing, which is strongly influenced by the dominance coefficient of the mutation before and after the environmental change. Importantly, since mating with the wild‐type breaks the mutant homozygote up, a slow decay of the wild‐type population size can impede rescue in randomly mating populations.     

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.     

Polymorphism maintenance in populations with mixed random mating and apomixis subjected to stabilizing and cyclical selection

We analysed a diploid population model with a mixed breeding system that includes panmixia and apomixis. Each individual produces a part (ss) of its progeny by random mating, the remainder (1-ss) being a result of precise copying (vegetative reproduction or apomixis) of the parental genotype. Both constant and periodically varying selection regimes were considered. In the main model, the selected trait was controlled by two diallelic additive or semidominant loci, A/a and B/b, whereas the parameter of breeding system (ss) was genotype-independent. A numerical iteration of the evolutionary equations were used to evaluate the proportion (V) of population trajectories converging to internal (polymorphic) fixed points. The results were the following. (a) A complex pattern of dependence of polymorphism stability on interaction among the breeding system, recombination rate, and the genetic architecture of the selected trait emerged. (b) The recombination provided some advantage to sex at intermediate period lengths and strong-to-moderate selection intensities. (c) The complex limiting behavior (CLB) was quite compatible with sexual reproduction, at least within the framework of pure genetic (not including variations in population density) models of multilocus varying selection.     

Linkage disequilibrium in two-locus,finite, random mating models without selection or mutation

Linkage disequilibrium is discussed for three two-locus, finite, random mating models in genetics, two models being Markov chains and the third a diffusion process. The expected value of the square of the disequilibrium function at any time is computed for the Markov chains. There is discussion of the relationships between the models and of the influence of finite population size on correlation between loci. It is suggested that there may be danger in assuming too simple a relationship between population size and degree of disequilibrium.     

Continuous-genotype models and assortative mating

Feldman and Cavalli-Sforza (1979a,b) have argued that the convergence properties of classical models of assortative mating are not known, and that these models involve arbitrary assumptions which assume rather than derive the achievement of equilibrium. A careful consideration of all models shows that the classical models are well defined and seem to achieve their equilibria. The model used by Feldman and Cavalli-Sforza involves an arbitrary assumption. Consideration of the models of Wright, Fisher, Bulmer, and Lande in the context of assortative mating or of selection versus mutation shows that these models are consistent with each other. The treatment of the balance between mutation and normalizing selection by Cavalli-Sforza and Feldman comes to conclusions sharply different from those of other authors, apparently as a result of this same arbitrary assumption.     

Linear mixed models with flexible distributions of random effects for longitudinal data

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.