On the prediction of simultaneous inbreeding coefficients at multiple loci

A new deterministic method for predicting simultaneous inbreeding coefficients at three and four loci is presented. The method involves calculating the conditional probability of IBD (identical by descent) at one locus given IBD at other loci, and multiplying this probability by the prior probability of the latter loci being simultaneously IBD. The conditional probability is obtained applying a novel regression model, and the prior probability from the theory of digenic measures of Weir and Cockerham. The model was validated for a finite monoecious population mating at random, with a constant effective population size, and with or without selfing, and also for an infinite population with a constant intermediate proportion of selfing. We assumed discrete generations. Deterministic predictions were very accurate when compared with simulation results, and robust to alternative forms of implementation. These simultaneous inbreeding coefficients were more sensitive to changes in effective population size than in marker spacing. Extensions to predict simultaneous inbreeding coefficients at more than four loci are now possible.     

Descent measures for two loci with some applications

For any four genes, two at each of two loci, in a population, a 15 component descent measure has been introduced. These components are the probabilities of the 15 possible arrangements on a set of initial gametes of those genes of which the four of interest are copies. Since identity by descent of genes is equivalent to their being copies of a single gene on an initial gamete, descent measures have inbreeding coefficients as special cases. The individual descent measure, defined for four genes on two uniting gametes can be evaluated for any pedigree by means of an algorithm developed here. If initial gametic frequencies are specified, descent measures allow genotypic frequencies and disequilibria functions at one and two loci to be found. The procedures are illustrated for selfing and for sib mating. Several applications of the descent measures are discussed.     

Inbreeding and inbreeding depression in endangered red wolves (Canis rufus)

In natural populations, the expression and severity of inbreeding depression can vary widely across taxa. Describing processes that influence the extent of inbreeding and inbreeding depression aid in our understanding of the evolutionary history of mating systems such as cooperative breeding and nonrandom mate selection. Such findings also help shape wildlife conservation theory because inbreeding depression reduces the viability of small populations. We evaluated the extent of inbreeding and inbreeding depression in a small, re‐introduced population of red wolves (Canis rufus) in North Carolina. Since red wolves were first re‐introduced in 1987, pedigree inbreeding coefficients (f) increased considerably and almost every wild born wolf was inbred (average f = 0.154 and max f = 0.383). The large inbreeding coefficients were due to both background relatedness associated with few founders and numerous close relative matings. Inbreeding depression was most evident for adult body size and generally absent for direct fitness measures such as reproductive success and survival; no lethal equivalents (LE = 0.00) were detected in juvenile survival. The lack of strong inbreeding depression in direct measures of fitness could be due to a founder effect or because there were no outbred individuals for comparison. Our results highlight the variable expression of inbreeding depression across traits and the need to measure a number of different traits when evaluating inbreeding depression in a wild population.     

Rank-invariant estimation of inbreeding coefficients

The two alleles an individual carries at a locus are identical by descent (ibd) if they have descended from a single ancestral allele in a reference population, and the probability of such identity is the inbreeding coefficient of the individual. Inbreeding coefficients can be predicted from pedigrees with founders constituting the reference population, but estimation from genetic data is not possible without data from the reference population. Most inbreeding estimators that make explicit use of sample allele frequencies as estimates of allele probabilities in the reference population are confounded by average kinships with other individuals. This means that the ranking of those estimates depends on the scope of the study sample and we show the variation in rankings for common estimators applied to different subdivisions of 1000 Genomes data. Allele-sharing estimators of within-population inbreeding relative to average kinship in a study sample, however, do have invariant rankings across all studies including those individuals. They are unbiased with a large number of SNPs. We discuss how allele sharing estimates are the relevant quantities for a range of empirical applications.Subject terms: Population genetics, Evolutionary biology, Molecular ecology     

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).     

Evolution of advantageous alleles affecting population ecological characteristics in partially inbreeding populations

The fate of advantageous alleles affecting intrinsic growth rate, carrying capacity or intra-specific competitive ability was examined in a partially inbreeding population. Generally, inbreeding had an effect on the evolution of advantageous alleles affecting population ecological characteristics. For example, in a specific underdominant case the number of stable internal equilibria decreased from two to one with only a slight degree of inbreeding. Equilibrium frequencies of stable internal equilibria and stability of fixation equilibria were also affected by the degree of inbreeding. For strictly advantageous alleles, inbreeding had the same qualitative effect on the fixation probability and mean fixation time as predicted in simpler selection models.     

Pedigree and marker information requirements to monitor genetic variability

There are several measures available to describe the genetic variability of populations. The average inbreeding coefficient of a population based on pedigree information is a frequently chosen option. Due to the developments in molecular genetics it is also possible to calculate inbreeding coefficients based on genetic marker information. A simulation study was carried out involving ten sires and 50 dams. The animals were mated over a period of 20 discrete generations. The population size was kept constant. Different situations with regard to the level of polymorphism and initial allele frequencies and mating scheme (random mating, avoidance of full sib mating, avoidance of full sib and half sib mating) were considered. Pedigree inbreeding coefficients of the last generation using full pedigree or 10, 5 and 2 generations of the pedigree were calculated. Marker inbreeding coefficients based on different sets of microsatellite loci were also investigated. Under random mating, pedigree-inbreeding coefficients are clearly more closely related to true autozygosity (i.e., the actual proportion of loci with alleles identical by descent) than marker-inbreeding coefficients. If mating is not random, the demands on the quality and quantity of pedigree records increase. Greater attention must be paid to the correct parentage of the animals.     

Inbreeding and relatedness coefficients: what do they measure?

This paper reviews and discusses what is known about the relationship between identity in state, allele frequency, inbreeding coefficients, and identity by descent in various uses of these terms. Generic definitions of inbreeding coefficients are given, as ratios of differences of probabilities of identity in state. Then some of their properties are derived from an assumption in terms of differences between distributions of coalescence times of different genes. These inbreeding coefficients give an approximate measurement of how much higher the probability of recent coalescence is for some pair of genes relative to another pair. Such a measure is in general not equivalent to identity by descent; rather, it approximates a ratio of differences of probabilities of identity by descent. These results are contrasted with some other formulas relating identity, allele frequency, and inbreeding coefficients. Additional assumptions are necessary to obtain most of them, and some of these assumptions are not always correct, for example when there is localized dispersal. Therefore, definitions based on such formulas are not always well-formulated. By contrast, the generic definitions are both well-formulated and more broadly applicable.     

Coefficients of inbreeding and homozygosity in recurrent selection: The case of m linked loci

Summary Methods of calculating the coefficients of inbreeding and homozygosity in a finite population undergoing recurrent selection (self-select-intercross in succeeding generations) are investigated for the case of m linked loci and effective directional selection. These coefficients are derived in terms of vectors whose components reflect the various possible patterns of genes being identical at a given stage of the recurrent selection breeding program.For the case of two linked loci the progress of the panmictic index and/or the index of total heterozygosity through twenty-five cycles of recurrent selection is traced by means of computer-simulated populations ranging in sizes from ten through one hundred, assuming varying recombination probabilities, and assuming both minimum and maximum inbreeding selection patterns.Results indicate that the coefficient of relationship in the source population is extremely important in tracing the progress of the degree of inbreeding and/or total homozygosity, that linkage plays a major role in promoting heterozygosity in a recurrent selection system, and that careful intercrossing rather than random mating in alternate generations of the recurrent selection cycle is important in promoting maximum heterozygosity in the selected population. In the simulated populations the effect of small population sizes is observed and, in general, indications are that unless more than five complete recurrent cycles are contemplated, increasing the population size results in only relatively minor increases in panmixia, especially when linked loci are involved in the selected trait and when care is taken to avoid a maximum inbreeding selection pattern.     

Effect of linkage on the control of inbreeding in selection programmes

Selection and mating methods for controlling inbreeding in selection programmes are based on relationships obtained from pedigrees. The efficiency of these methods has always been tested by studies using genetic models of independent loci. However, under linkage the rate of inbreeding obtained from pedigrees can be different from the probability of identity by descent of genes. We simulated a quantitative trait under artificial selection controlled by a large number of genes spread on genome regions of different sizes. A method to control inbreeding based on minimising the average coancestry of selected individuals with a restriction in the loss of selection response, and a mating procedure to control inbreeding were applied. These methods, that use coancestry relationships, were not effective in controlling inbreeding when the genome sizes were smaller than five morgans or so. However, for larger genome sizes the methods were sufficiently efficient. For very tight linkage, methods that utilise molecular information from markers should be used. We finally discuss the effects of the selection of individual major genes on the neutral variability of adjacent genome regions.     

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.     

Methods to estimate effective population size using pedigree data: Examples in dog,sheep, cattle and horse

Background

Effective population sizes of 140 populations (including 60 dog breeds, 40 sheep breeds, 20 cattle breeds and 20 horse breeds) were computed using pedigree information and six different computation methods. Simple demographical information (number of breeding males and females), variance of progeny size, or evolution of identity by descent probabilities based on coancestry or inbreeding were used as well as identity by descent rate between two successive generations or individual identity by descent rate.

Results

Depending on breed and method, effective population sizes ranged from 15 to 133 056, computation method and interaction between computation method and species showing a significant effect on effective population size (P < 0.0001). On average, methods based on number of breeding males and females and variance of progeny size produced larger values (4425 and 356, respectively), than those based on identity by descent probabilities (average values between 93 and 203). Since breeding practices and genetic substructure within dog breeds increased inbreeding, methods taking into account the evolution of inbreeding produced lower effective population sizes than those taking into account evolution of coancestry. The correlation level between the simplest method (number of breeding males and females, requiring no genealogical information) and the most sophisticated one ranged from 0.44 to 0.60 according to species.

Conclusions

When choosing a method to compute effective population size, particular attention should be paid to the species and the specific genetic structure of the population studied.     

Moderate inbreeding depression in a reintroduced population of North Island robins

Reintroductions of threatened species are increasingly common in conservation. The translocation of a small subset of individuals from a genetically diverse source population could potentially lead to substantial inbreeding depression due to the high genetic load of the parent population. We analysed 12 years of data from the reintroduced population of North Island robins Petroica longipes on Tiritiri Matangi Island, New Zealand, to determine the frequency of inbreeding and magnitude of inbreeding depression. The initial breeding population consisted of 12 females and 21 males, which came from a large mainland population of robins. The frequency of mating between relatives ( f >0; 39%, n =82 pairs) and close relatives ( f =0.25; 6.1%) and the average level of inbreeding ( f =0.027) were within the range reported for other small island populations of birds. The average level of inbreeding fluctuated from year to year depending on the frequency of close inbreeding (e.g. sib–sib pairs). We found evidence for inbreeding depression in juvenile survival, with survival probability estimated to decline from 31% among non-inbred birds ( f =0) to 11% in highly inbred juveniles ( f =0.25). The estimated number of lethal equivalents based on this relationship (4.14) was moderate compared with values reported for other island populations of passerines. Given that significant loss of fitness was only evident in highly inbred individuals, and such individuals were relatively rare once the population expanded above 30 pairs, we conclude that inbreeding depression should have little influence on this robin population. Although the future fitness consequences of any loss of genetic variation due to inbreeding are uncertain, the immediate impact of inbreeding depression is likely to be low in any reintroduced population that expands relatively quickly after establishment.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

Molecular and pedigree measures of relatedness provide similar estimates of inbreeding depression in a bottlenecked population

Individual‐based estimates of the degree of inbreeding or parental relatedness from pedigrees provide a critical starting point for studies of inbreeding depression, but in practice wild pedigrees are difficult to obtain. Because inbreeding increases the proportion of genomewide loci that are identical by descent, inbreeding variation within populations has the potential to generate observable correlations between heterozygosity measured using molecular markers and a variety of fitness related traits. Termed heterozygosity‐fitness correlations (HFCs), these correlations have been observed in a wide variety of taxa. The difficulty of obtaining wild pedigree data, however, means that empirical investigations of how pedigree inbreeding influences HFCs are rare. Here, we assess evidence for inbreeding depression in three life‐history traits (hatching and fledging success and juvenile survival) in an isolated population of Stewart Island robins using both pedigree‐ and molecular‐derived measures of relatedness. We found results from the two measures were highly correlated and supported evidence for significant but weak inbreeding depression. However, standardized effect sizes for inbreeding depression based on the pedigree‐based kin coefficients (k) were greater and had smaller standard errors than those based on molecular genetic measures of relatedness (RI), particularly for hatching and fledging success. Nevertheless, the results presented here support the use of molecular‐based measures of relatedness in bottlenecked populations when information regarding inbreeding depression is desired but pedigree data on relatedness are unavailable.     

Pedigrees and microsatellites among endangered ungulates: what do they tell us?

Relationships between pedigree coefficients of inbreeding and molecular metrics are generally weak, suggesting that measures of heterozygosity estimated using microsatellites may be poor surrogates of genome-wide inbreeding. We compare three endangered species of gazelles ( Gazella ) with different degrees of threat in their natural habitats, for which captive breeding programmes exist. For G. dorcas, the species with the largest founding population, the highest and most recent number of founding events, the correlation between pedigree coefficient of inbreeding and molecular metrics was higher than for outbred populations of mammals, probably because it has both higher mean f and variance. For the two species with smaller founding populations, conventional assumptions about founders, i.e. outbred and unrelated, are unrealistic. When realistic assumptions about the founders were made, clear relationships between pedigree coefficients of inbreeding and molecular metrics were revealed for G. cuvieri. This population had a small founding population, but it did experience admixture years later; thus, the relationship between inbreeding and molecular metrics in G. cuvieri is very similar to the expected values but lower than in G. dorcas . In contrast, no relationship was found for G. dama mhorr which had a much smaller founding population than had been previously assumed, which probably had high levels of inbreeding and low levels of genetic variability, and no admixture. In conclusion, the strength of the association between pedigree coefficient of inbreeding and molecular metrics among endangered species depends on the level of inbreeding and genetic variability present in the founding population, its size and its history.     

QTL mapping of inbreeding-related cold sensitivity and conditional lethality in Drosophila melanogaster

Inbreeding depression is a central theme within genetics, and is of specific interest for researchers within evolutionary and conservation genetics and animal and plant breeding. Inbreeding effects are thought to be caused by the joint expression of conditional and unconditional deleterious alleles. Whenever the expression of deleterious alleles is conditional, this can result in extreme environmental sensitivity in certain inbred lineages. Analysis of conditional lethal effects can reveal some of the loci that are sensitive to inbreeding. We performed a QTL (quantitative trait locus) mapping study of inbreeding-related and conditionally expressed lethality in Drosophila melanogaster. The lethal effect was triggered by exposure to a cold shock. We used a North Carolina crossing Design 3 to establish the mapping population, as well as to estimate the average dominance ratio and heritability. We found two QTL on the major autosomes carrying recessive lethals that caused male mortality, one of which also affected female mortality. More detailed study of these loci will provide information on the mechanistic basis and environmental sensitivity of inbreeding depression.     

Pedigree analysis of the closed nucleus of Iranian Baluchi sheep

A study was conducted to characterise genetic diversity in the closed nucleus of Baluchi sheep using pedigree analysis. Herdbook information collected between 1979 and 2008, including pedigree records on 21,721 animals, was used to compute inbreeding and average generation intervals. Effective population size and parameters derived from probability of gene origin were computed for ewes born between 2005 and 2008 with both parents known (female reference population). The average complete generation equivalent of the female reference population was 5.47. The mean generation interval was 3.33 years in the studied period. From 1983 to 1994, the rate of increase in inbreeding was approximately 0.2% per year, but, after 1994, inbreeding did not increase as in the preceding years and had an approximately flat trend over time. The mean relationship coefficients among rams, among ewes and between rams and ewes in active animals were calculated to predict the future level of inbreeding. The effective number of founders, effective number of ancestors and founder genome equivalent of the reference population were 80, 47 and 19.5, respectively. The realised effective population size was 134 animals. The results of this study indicated that the population under study has fairly good genetic variability.     

Measures of Autozygosity in Decline: Globalization, Urbanization, and Its Implications for Medical Genetics

This research investigates the influence of demographic factors on human genetic sub-structure. In our discovery cohort, we show significant demographic trends for decreasing autozygosity associated with population variation in chronological age. Autozygosity, the genomic signature of consanguinity, is identifiable on a genome-wide level as extended tracts of homozygosity. We identified an average of 28.6 tracts of extended homozygosity greater than 1 Mb in length in a representative population of 809 unrelated North Americans of European descent ranging in chronological age from 19–99 years old. These homozygous tracts made up a population average of 42 Mb of the genome corresponding to 1.6% of the entire genome, with each homozygous tract an average of 1.5 Mb in length. Runs of homozygosity are steadily decreasing in size and frequency as time progresses (linear regression, p<0.05). We also calculated inbreeding coefficients and showed a significant trend for population-wide increasing heterozygosity outside of linkage disequilibrium. We successfully replicated these associations in a demographically similar cohort comprised of a subgroup of 477 Baltimore Longitudinal Study of Aging participants. We also constructed statistical models showing predicted declining rates of autozygosity spanning the 20th century. These predictive models suggest a 14.0% decrease in the frequency of these runs of homozygosity and a 24.3% decrease in the percent of the genome in runs of homozygosity, as well as a 30.5% decrease in excess homozygosity based on the linkage pruned inbreeding coefficients. The trend for decreasing autozygosity due to panmixia and larger effective population sizes will likely affect the frequency of rare recessive genetic diseases in the future. Autozygosity has declined, and it seems it will continue doing so.     

Quantifying gene flow from spatial genetic structure data in a metapopulation of Chamaecrista fasciculata (Leguminosae)

Abstract An extensive allozyme survey was conducted within a natural "meta" population of the native North American annual legume, Chamaecrista fasciculata (Leguminosae) to quantify genetic structure at different spatial scales. Gene flow was then estimated by a recently developed indirect method based on a continuous population model, using pairwise kinship coefficients between individuals. The indirect estimates of gene flow, quantified in terms of neighborhood size, with an average value on the order of 150 individuals, were concordant among different spatial scales (subpopulation, population, metapopulation). This gene-flow value lies within the range of direct estimates previously documented from observations of pollen and seed dispersal for the same metapopulation. Monte Carlo simulations using the direct measures of gene flow as parameters further demonstrated that the observed spatial pattern of allozyme variation was congruent with a model of isolation by distance. Combining previously published estimates of pollen dispersal distances with kinship coefficients from this study, we quantified biparental inbreeding relative to either a single subpopulation or the whole metapopulation. At the level of a neighborhood, little biparental inbreeding was observed and most departure from Hardy-Weinberg genotypic proportions was explained by self-fertilization, whereas both selfing and biparental inbreeding contributed to nonrandom mating at the metapopulation level. Gene flow was also estimated from indirect methods based on a discontinuous population structure model. We discuss these results with respect to the effect of a patchy population structure on estimation of gene flow.