A mathematical approach to multiple genetic relationships

A fundamental concept in the treatment of genetic relationships is that of gene identity which first was introduced by Cotterman (1940). Based on this notion several measures of relationship evolved such as the inbreeding coefficient, the coefficient of kinship, and the identity coefficients; by means of these quantities joint and conditional phenotype probabilities could be derived. This paper is an attempt at a general mathematical treatment of genetic relationships: Identity states are defined for any number of individuals, a method is given for the calculation of the corresponding identity coefficients by means of generalized coefficients of kinship, and applications are emphasized.     

Quantifying inbreeding avoidance through extra-pair reproduction

Extra-pair reproduction is widely hypothesized to allow females to avoid inbreeding with related socially paired males. Consequently, numerous field studies have tested the key predictions that extra-pair offspring are less inbred than females’ alternative within-pair offspring, and that the probability of extra-pair reproduction increases with a female's relatedness to her socially paired male. However, such studies rarely measure inbreeding or relatedness sufficiently precisely to detect subtle effects, or consider biases stemming from failure to observe inbred offspring that die during early development. Analyses of multigenerational song sparrow (Melospiza melodia) pedigree data showed that most females had opportunity to increase or decrease the coefficient of inbreeding of their offspring through extra-pair reproduction with neighboring males. In practice, observed extra-pair offspring had lower inbreeding coefficients than females’ within-pair offspring on average, while the probability of extra-pair reproduction increased substantially with the coefficient of kinship between a female and her socially paired male. However, simulations showed that such effects could simply reflect bias stemming from inbreeding depression in early offspring survival. The null hypothesis that extra-pair reproduction is random with respect to kinship therefore cannot be definitively rejected in song sparrows, and existing general evidence that females avoid inbreeding through extra-pair reproduction requires reevaluation given such biases.     

Immigration prevents inbreeding in a growing colony of a long‐lived and philopatric seabird

Mating between close relatives can have deleterious effects on reproductive success or offspring fitness, which should favour the evolution of active or passive inbreeding avoidance mechanisms. In birds, evidence for active inbreeding avoidance by kin‐discriminative mate choice is scarce; many studies describe random mating in relation to kinship and thus support passive inbreeding avoidance by natal dispersal. However, most studies were conducted in island populations of short‐lived passerines with fast alternation of generations. In this study, we present inbreeding estimates based on pedigree data from a 16‐year study in a coastal colony of Common Terns Sterna hirundo, a long‐lived seabird with delayed sexual maturation and low rates of extra‐pair paternity. Incestuous mating was rare (four of 2387 pairs), even if partially accounting for incomplete pedigrees. Although the average relatedness of observed pairs was lower than would be expected from random pairing, the inbreeding coefficient did not differ from random mating. Hence, we found no clear evidence for active inbreeding avoidance by kin‐discriminative mate choice, and the low level of inbreeding seems to be related to the high immigration rate in the colony and thus to be maintained passively by dispersal.     

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.     

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.     

Estimation of inbreeding from ecclesiastical dispensations: application of three procedures to a Spanish case

The inbreeding coefficient of a population, estimated from ecclesiastical Roman Catholic dispensations, results from the relative contribution of different degrees of relationships (uncle-niece, first cousin, etc.). The interpopulation comparisons of consanguinity patterns may be obscured by the fact that in 1918 the Roman Catholic Church norm regulating the closest marriageable kinship was modified, limiting the application for an ecclesiastical dispensation to relatives of third degree (second cousins) or closer. Depending on the length of the period before or after the change of regulation, coefficients and rates may differ. Deviation of frequencies for multiple marriages may also occur. The aim of the present paper is to determine how the chosen procedure based on ecclesiastical dispensations may affect results, regarding the inbreeding coefficient, the consanguinity rate, the structure of consanguinity and the close/remote kinship ratio. As a sample case, information from the Gredos mountain range (central Spain) has been used.     

Increased inbreeding and strong kinship structure in Taxus baccata estimated from both AFLP and SSR data

Habitat fragmentation can have severe genetic consequences for trees, such as increased inbreeding and decreased effective population size. In effect, local populations suffer from reduction of genetic variation, and thus loss of adaptive capacity, which consequently increases their risk of extinction. In Europe, Taxus baccata is among a number of tree species experiencing strong habitat fragmentation. However, there is little empirical data on the population genetic consequences of fragmentation for this species. This study aimed to characterize local genetic structure in two natural remnants of English yew in Poland based on both amplified fragment length polymorphism (AFLP) and microsatellite (SSR) markers. We introduced a Bayesian approach that estimates the average inbreeding coefficient using AFLP (dominant) markers. Results showed that, in spite of high dispersal potential (bird-mediated seed dispersal and wind-mediated pollen dispersal), English yew populations show strong kinship structure, with a spatial extent of 50–100 m, depending on the population. The estimated inbreeding levels ranged from 0.016 to 0.063, depending on the population and marker used. Several patterns were evident: (1) AFLP markers showed stronger kinship structure than SSRs; (2) AFLP markers provided higher inbreeding estimates than SSRs; and (3) kinship structure and inbreeding were more pronounced in denser populations regardless of the marker used. Our results suggest that, because both kinship structure and (bi-parental) inbreeding exist in populations of English yew, gene dispersal can be fairly limited in this species. Furthermore, at a local scale, gene dispersal intensity can be more limited in a dense population.     

Effects of population structures and selection strategies on the purging of inbreeding depression due to deleterious mutations

Stochastic simulations were run to compare the effects of nine breeding schemes, using full-sib mating, on the rate of purging of inbreeding depression due to mutations with equal deleterious effect on viability at unlinked loci in an outbred population. A number of full-sib mating lines were initiated from a large outbred population and maintained for 20 generations (if not extinct). Selection against deleterious mutations was allowed to occur within lines only, between lines or equal within and between lines, and surviving lines were either not crossed or crossed following every one or three generations of full-sib mating. The effectiveness of purging was indicated by the decreased number of lethal equivalents and the increased fitness of the purged population formed from crossing surviving lines after 20 generations under a given breeding scheme. The results show that the effectiveness of purging, the survival of the inbred lines and the inbreeding level attained are generally highest with between-line selection and lowest with within-line selection. Compared with no crossing, line crossing could lower the risk of extinction and the inbreeding coefficient of the purged population substantially with little loss of the effectiveness of purging. Compromising between the effectiveness of purging, and the risk of extinction and inbreeding coefficient, the breeding scheme with equal within- and between-line selection and crossing alternatively with full-sib mating is generally the most desirable scheme for purging deleterious mutations. Unless most deleterious mutations have relatively large effects on fitness in species with reproductive ability high enough to cope with the depressed fitness and thus increased risk of extinction with inbreeding, it is not justified to apply a breeding programme aimed at purging inbreeding depression by inbreeding and selection to a population of conservation concern.     

Royal dynasties as human inbreeding laboratories: the Habsburgs

The European royal dynasties of the Early Modern Age provide a useful framework for human inbreeding research. In this article, consanguineous marriage, inbreeding depression and the purging of deleterious alleles within a consanguineous population are investigated in the Habsburgs, a royal dynasty with a long history of consanguinity over generations. Genealogical information from a number of historical sources was used to compute kinship and inbreeding coefficients for the Habsburgs. The marriages contracted by the Habsburgs from 1450 to 1750 presented an extremely high mean kinship (0.0628±0.009), which was the result of the matrimonial policy conducted by the dynasty to establish political alliances through marriage. A strong inbreeding depression for both infant and child survival was detected in the progeny of 71 Habsburg marriages in the period 1450–1800. The inbreeding load for child survival experienced a pronounced decrease from 3.98±0.87 in the period 1450–1600 to 0.93±0.62 in the period 1600–1800, but temporal changes in the inbreeding depression for infant survival were not detected. Such a reduction of inbreeding depression for child survival in a relatively small number of generations could be caused by elimination of deleterious alleles of a large effect according with predictions from purging models. The differential purging of the infant and child inbreeding loads suggest that the genetic basis of inbreeding depression was probably very different for infant and child survival in the Habsburg lineage. Our findings provide empirical support that human inbreeding depression for some fitness components might be purged by selection within consanguineous populations.     

Phenotype‐associated inbreeding biases estimates of inbreeding depression in a wild bird population

Inbreeding depression is usually quantified by regressing individual phenotypic values on inbreeding coefficients, implicitly assuming there is no correlation between an individual's phenotype and the kinship coefficient to its mate. If such an association between parental phenotype and parental kinship exists, and if the trait of interest is heritable, estimates of inbreeding depression can be biased. Here we first derive the expected bias as a function of the covariance between mean parental breeding value and parental kinship. Subsequently, we use simulated data to confirm the existence of this bias, and show that it can be accounted for in a quantitative genetic animal model. Finally, we use long‐term individual‐based data for white‐throated dippers (Cinclus cinclus), a bird species in which inbreeding is relatively common, to obtain an empirical estimate of this bias. We show that during part of the study period, parents of inbred birds had shorter wings than those of outbred birds, and as wing length is heritable, inbred individuals were smaller, independent of any inbreeding effects. This resulted in the overestimation of inbreeding effects. Similarly, during a period when parents of inbred birds had longer wings, we found that inbreeding effects were underestimated. We discuss how such associations may have arisen in this system, and why they are likely to occur in others, too. Overall, we demonstrate how less biased estimates of inbreeding depression can be obtained within a quantitative genetic framework, and suggest that inbreeding and additive genetic effects should be accounted for simultaneously whenever possible.     

Prediction of rates of inbreeding in selected populations

A method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into 'contribution' matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, Ne approximately 2N/(mu 2r + sigma 2r), where N is the number of individuals per generation and mu r and sigma 2r are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing 'competitiveness' of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.     

A comparison of strategies for selecting breeding pairs to maximize genetic diversity retention in managed populations

Captive breeding programs aim to maintain populations that are demographically self-sustaining and genetically healthy. It has been well documented that the best way for managed breeding programs to retain gene diversity (GD) and limit inbreeding is to select breeding pairs that minimize a population's average kinship. We used a series of computer simulations to test 4 methods of minimizing average kinship across a variety of scenarios with varying generation lengths, mortality rates, reproductive rates, and rates of breeding pair success. "Static MK Selection" and "Dynamic MK Selection" are 2 methods for iteratively selecting genetically underrepresented individuals for breeding, whereas "Ranked MK Selection" and "Simultaneous MK Selection" are 2 methods for concurrently selecting the group of breeding individuals that produce offspring with the lowest average kinship. For populations with discrete generations (24 tested scenarios), we found that the Simultaneous and Ranked MK Selection methods were generally the best, nearly equivalent methods for selecting breeding pairs that retained GD and limited inbreeding. For populations with overlapping generations (198 tested scenarios), we found that Dynamic MK Selection was the most robust method for selecting breeding pairs. We used these results to provide guidelines for identifying which method of minimizing average kinship was most appropriate for various breeding program scenarios.     

Efficient path-based computations on pedigree graphs with compact encodings

A pedigree is a diagram of family relationships, and it is often used to determine the mode of inheritance (dominant, recessive, etc.) of genetic diseases. Along with rapidly growing knowledge of genetics and accumulation of genealogy information, pedigree data is becoming increasingly important. In large pedigree graphs, path-based methods for efficiently computing genealogical measurements, such as inbreeding and kinship coefficients of individuals, depend on efficient identification and processing of paths. In this paper, we propose a new compact path encoding scheme on large pedigrees, accompanied by an efficient algorithm for identifying paths. We demonstrate the utilization of our proposed method by applying it to the inbreeding coefficient computation. We present time and space complexity analysis, and also manifest the efficiency of our method for evaluating inbreeding coefficients as compared to previous methods by experimental results using pedigree graphs with real and synthetic data. Both theoretical and experimental results demonstrate that our method is more scalable and efficient than previous methods in terms of time and space requirements.     

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     

Modeling the effect of inbreeding among founders in linkage analysis

In this paper, we present a unified mathematical model for linkage analysis that allows for inbreeding among founders in all families. The identical by descent (IBD) configuration of each pedigree is modeled as a Markov process containing two parameters; the inverse inbreeding and kinship coefficient and a rate parameter proportional to the inverse expected length of chromosome segments shared IBD by two different founder haplotypes. We use hidden Markov models and define a forward-backward algorithm for computing the conditional IBD-distribution given marker data, thereby extending the multipoint method of Lander and Green [1987. Construction of multilocus genetic maps in humans, Proc. Natl. Acad. Sci. USA 84, 2363-2367] to situations where founders are inbred. Our methodology is valid for arbitrary pedigree structures. Simulation and theoretical approximations for nonparametric linkage (NPL) analysis based on affected sib pairs reveal that NPL scores are inflated and type 1 errors increased when the inbreeding coefficient or rate parameter is underestimated. When the parents are genotyped, we present a general way of modifying the score function to drastically reduce this effect.     

An Analysis of Heterosis VS. Inbreeding Effects with an Autotetraploid Cross-Fertilized Plant: MEDICAGO SATIVA L

Self-fertilization and crossing were combined to produce a large number of levels of inbreeding and of degrees of kinship. The inbreeding effect increases with the complexity of the character and with its supposed relationship with fitness. A certain amount of heterozygosity appears to be necessary for the expression of variability. With crossing of unrelated noninbred plants, genetic variance is mainly additive, but with inbreeding its major part is nonadditive. High additivity in crossing, therefore, coexists with strong inbreeding depression. However, even in inbreeding the genetic coefficient of covariation among relatives appears to be strongly and linearly related to the classical coefficient of kinship. This means that deviations from the additive model with inbreeding could be partly due to an effect of inbreeding on variances through an effect on means. An attempt to analyze genetic effects from a theoretical model, based upon the identity by descent relationship at the level of means and of covariances between relatives, tends to show that allelic interactions are more important and nonallelic interactions are less important for a character closely related to fitness. For a complex character, these results lead to the conception of a genome organized in polygenic complementary blocks integrating epistasis and dominance. Some consequences for plant breeding are also discussed.     

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.     

Medico-genetic study of the population of Kostroma Province. VI. The parameters of isolation by distance in the populations of the Bui and Shar'ya Districts of Kostroma Province

A model of isolation by distance proposed by Malécot and developed by Morton is applied to the data on marriage distances collected in two regions of Kostroma Province. There is good agreement between the estimates of local inbreeding when using the isonymy method and the model of isolation by distance. Interpopulation kinship approaches 0 at the distance 700 km. The mean coefficient of kinship for parents in the families with autosomal-recessive pathology is 20 times higher than mean coefficient of kinship in the population.     

Clastogenic effect of the psychotropic drug thioridazine on human chromosomes in vivo

Summary This paper deals with some other population genetic aspects associated with the incidence of a type of primary congenital glaucoma that occurs very frequently in the Gypsy population of Slovakia. In addition to the decreased fertility of affected individuals of Gypsy origin being determined, the relative reproduction fitness and the selection coefficient against this disease were estimated. An increased number of kinship intermarriages in parents of the patients were recorded, namely in the Gypsy group (45.6%). The average inbreeding coefficient for the Gypsy group (F=0.0091) and the non-Gypsy group (F=0.0030) was calculated. Based on the high frequency of primary congenital glaucoma in a relatively small Gypsy subpopulation and on data about their origin, immigration, and settlements in the territory of Slovakia, the authors consider a special case of gene drift—the founder effect—to be the most plausible explanation of the given fact.     

INBREEDING IN DARWIN'S MEDIUM GROUND FINCHES (GEOSPIZA FORTIS)

We studied the frequency and causes of inbreeding and its effect on reproductive success in a population of Darwin's Medium Ground Finches (Geospiza fortis) on Isla Daphne Major, Galápagos, during four breeding seasons (1981, 1983, 1984, and 1987). Pedigree analysis showed that levels of inbreeding were low but comparable with those observed in other passerine birds. For pairs with at least half of their grandparents known, approximately 20% of all pairings were between detectably related birds. The frequency of pairings between closely related birds (coefficient of kinship [φ] ≥ 0.250) among all pairs was 0.6%. We detected no effect of inbreeding on reproductive success, although sample sizes were small. The observed reproductive output of related pairs was not significantly different from the output of unrelated pairs, and there was no correlation between a pair's kinship coefficient and an estimate of the potential magnitude of inbreeding depression. Comparisons with a study of Great Tits (Parus major) by van Noordwijk and Scharloo (1981) suggest that, even if present, the fitness costs of inbreeding in this population of G. fortis would be low. Observed levels of inbreeding in each breeding episode were accurately predicted by simulations of random mating in which relatedness had no influence on pairing between individuals. This result suggests that levels of inbreeding in this population are determined more by demographic factors than by behavioral avoidance of mating with kin.