On the Theory of Partially Inbreeding Finite Populations. I. Partial Selfing

Some stochastic theory is developed for monoecious populations of size N in which there are probabilities beta and 1 - beta of reproduction by selfing and by random mating. It is assumed that beta much greater than N-1. Expressions are derived for the inbreeding coefficient of one random individual and the coefficient of kinship of two random separate individuals at time t. The mean and between-lines variance of the fraction of copies of a locus that are identical in two random separate individuals in an equilibrium population are obtained under the assumption that there is an infinite number of possible alleles. It is found that the theory for random mating populations holds if the effective population number is Ne = N'/(1 + FIS), where FIS is the inbreeding coefficient at equilibrium when N is infinite and N' is the reciprocal of the probability that two gametes contributing to random separate adults come from the same parent. When there is a binomial distribution of successful gametes emanating from each adult, N' = N. An approximation to the probability that an allele A survives if it is originally present in one AA heterozygote is found to be 2(N'/N)(FISS1 + (1 - FIS)S2), where S1 and S2 are the selective advantages of AA and AA in comparison with AA. In the last section it is shown that if there is partial full sib mating and binomial offspring distributions Ne = N/(1 + 3FIS).     

Effects of Partial Inbreeding on Fixation Rates and Variation of Mutant Genes

Diffusion methods were used to investigate the fixation probability, average time until fixation and extinction, and cumulative heterozygosity and genetic variance for single mutant genes in finite populations with partial inbreeding. The critical parameters in the approximation are the coefficient of inbreeding due to nonrandom mating (F) and the effective population size (Ne), which also depends on F and the variance of family size. For large Ns, the fixation probability (u) is u = 2(Ne/N)s (F + h - Fh), where N is the population census, s is the coefficient of selection of the mutant homozygote and h is the coefficient of dominance. For Poisson family size (independent Poisson distributions of selfed and nonselfed offspring with partial selfing, and independent Poisson distributions of male and female numbers with partial sib mating), Ne = N/(1 + F), and the time until fixation is approximately equal to Ne/N times the time to fixation with random mating, but this relation does not hold, however, for other distributions of family size. The cumulative nonadditive variance until fixation or loss for dominant genes is reduced with increasing F while for recessive genes it is increased with intermediate values of F. The average time until extinction of deleterious mutations is reduced by increasing F. This reduction, when expressed as a proportion, is approximately independent of the initial gene frequency as well as the selective disadvantage if this is large.     

Effective Size of Nonrandom Mating Populations

Nonrandom mating whereby parents are related is expected to cause a reduction in effective population size because their gene frequencies are correlated and this will increase the genetic drift. The published equation for the variance effective size, Ne, which includes the possibility of nonrandom mating, does not take into account such a correlation, however. Further, previous equations to predict effective sizes in populations with partial sib mating are shown to be different, but also incorrect. In this paper, a corrected form of these equations is derived and checked by stochastic simulation. For the case of stable census number, N, and equal progeny distributions for each sex, the equation is [formula: see text], where Sk2 is the variance of family size and alpha is the departure from Hardy-Weinberg proportions. For a Poisson distribution of family size (Sk2 = 2), it reduces to Ne = N/(1 + alpha), as when inbreeding is due to selfing. When nonrandom mating occurs because there is a specified system of partial inbreeding every generation, alpha can be substituted by Wright's FIS statistic, to give the effective size as a function of the proportion of inbred mates.     

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.     

Strong inbreeding depression in male mating behaviour in a poeciliid fish

The magnitude of inbreeding depression is often larger in traits closely related to fitness, such as survival and fecundity, compared to morphological traits. Reproductive behaviour is also closely associated with fitness, and therefore expected to show strong inbreeding depression. Despite this, little is known about how reproductive behaviour is affected by inbreeding. Here we show that one generation of full‐sib mating results in a decrease in male reproductive performance in the least killifish (Heterandria formosa). Inbred males performed less gonopodial thrusts and thrust attempts than outbred males (δ = 0.38). We show that this behaviour is closely linked with fitness as gonopodial performance correlates with paternity success. Other traits that show inbreeding depression are offspring viability (δ = 0.06) and maturation time of males (δ = 0.19) and females (δ = 0.14). Outbred matings produced a female biased sex ratio whereas inbred matings produced an even sex ratio.     

Polyandry restores female fertility and paternal effects diminished by inbreeding in Hippodamia convergens

1. The fertility restoration hypothesis posits that polyandry can evolve when female fertility is reduced by matings with related males, but restored by matings with unrelated ones. 2. Using a promiscuous ladybird, this hypothesis was tested by mating mature, virgin females twice with sib and non‐sib males in all four permutations and observing female fertility. The development of progeny from the first and 10th clutches was also followed to test for differences in paternal effects. 3. Mating treatment did not affect fecundity, but egg viability was reduced by sib matings, and restored by non‐sib matings, regardless of mating sequence. In addition, negative paternal effects of sib matings on progeny (lower survival) were compensated by non‐sib matings. 4. The survival of offspring in first clutches was sensitive to the paternal mating sequence and was higher if a sibling male was preceded by a non‐sib male, compared with the reverse, consistent with a time lag in response to male epigenetic signals. Offspring survival did not differ between these two treatments in the 10th clutch, suggesting a blending of paternal effects over time. 5. The results are indicative of interactions between the paternal effects contributed by different males, and between paternal and maternal effects, the latter causing faster development in later clutches. 6. Thus, the reproductive benefits of polyandry for H. convergens females are potentially both genetic and epigenetic, as both egg fertility and beneficial paternal effects were diminished by sib matings, but restored by matings with unrelated males.     

Lack of behavioural evidence for kin avoidance in mate choice in a hymenopteran parasitoid (Hymenoptera: Braconidae)

Mechanisms for inbreeding avoidance should be prevalent in insects that reproduce by arrhenotokous haplodiploidy because of the higher potential production of unviable diploid males in inbred matings. Few studies have focused on mating strategies in insect parasitoids and even less on kinship relationships during mate choice. In this study we tested avoidance of kin as mate in the parasitic wasp Aphidius matricariae (Hymenoptera: Braconidae) using an ethological approach. Key mating parameters, such as male wing fanning, latent period before genitalia contact and duration of copulation were measured. No evidence for kin avoidance in mate choice in both A. matricariae males and females was observed in our behaviour (no choice or choice tests) tests. This lack of ethological sib mating avoidance could be due to different factors such as sex determination rule different than the single locus complementary sex determination, making lower the proportion of diploid males in case of sib matings and thus its negative consequence. The existence of other inbreeding avoidance strategies and mechanisms that reduce the probability of 2 receptive relatives meeting in nature may be common, for example, inbred mating may be rare through differential dispersal, delayed maturation, or protandry.     

Similar Performance of Diploid and Haploid Males in an Ant Species without Inbreeding Avoidance

Under haplodiploidy, a characteristic trait of all Hymenoptera, females develop from fertilised eggs, and males from unfertilised ones. Males are therefore typically haploid. Yet, inbreeding can lead to the production of diploid males that often fail in development, are sterile or are of lower fertility. In most Hymenoptera, inbreeding is avoided by dispersal flights of one or both sexes, leading to low diploid male loads. We investigated causes for the production of diploid males and their performance in a highly inbred social Hymenopteran species. In the ant Hypoponera opacior, inbreeding occurs between wingless sexuals, which mate within the mother nest, whereas winged sexuals outbreed during mating flights earlier in the season. Wingless males mate with queen pupae and guard their mating partners. We found that they mated randomly with respect to relatedness, indicating that males do not avoid mating with close kin. These frequent sib‐matings lead to the production of diploid males, which are able to sire sterile triploid offspring. We compared mating activity and lifespan of haploid and diploid wingless males. As sexual selection acts on the time of emergence and body size in this species, we also investigated these traits. Diploid males resembled haploid ones in all investigated traits. Hence, albeit diploid males cannot produce fertile offspring, they keep up with haploid males in their lifetime mating success. Moreover, by fathering viable triploid workers, they contribute to the colonies' work force. In conclusion, the lack of inbreeding avoidance led to frequent sib‐matings of wingless sexuals, which in turn resulted in the regular production of diploid males. However, in contrast to many other Hymenopteran species, diploid males exhibit normal sexual behaviour and sire viable, albeit sterile daughters.     

Effective size of populations under partial inbreeding and selection on a set of linked additive genes

The prediction theory of effective population size (Ne) is extended to cover selection on a set of linked additive genes and partial inbreeding (partial selfing or partial full-sib mating). Ne under selection is generally expressed as a function of the cumulative change in frequency of a neutral gene due to the random association between the neutral and selected genes generated by finite sampling. In this study, the association under partial selfing was classified into two types, the association between the neutral and selected genes on the same gamete, and the association between the neutral and selected genes each on the different gametes in the same parent. For partial full-sib mating, an additional association, i.e., the association between the neutral and selected genes each in the different parents in the same family, was included in the model. According to this classification of the association, the coefficient accounting for the cumulative change in frequency of the neutral gene was partitioned into two or three components. A method for computing the partitioned coefficients was obtained from the transition matrix approach, in which the joint effect of linkage, selection and partial inbreeding was taken into account. To assess the joint effects of linkage, selection and partial inbreeding on Ne, numerical computations with the obtained expressions were carried out. The effect of linkage on Ne was generally small, except for an extremely small genome size, while the partial inbreeding resulted in a drastic reduction in Ne. For a given genome size, Ne was essentially independent of the length and number of chromosomes. Some of these results were verified by stochastic simulations.     

Comparaison des croisements au hasard,prère-sœur et demi frère-sœur par le coefficient de dépression

The chromosomal theory of inbreeding based on a gametic interaction system lead us to define a depression coefficientD. Comparison of random, sib and half-sib matings (with inbreeding coefficientF=0, 1/4 and 1/8) shows thatD depends on the structure of the starting population and on values of the model parameters. This result accounts for responses of lines whose depression does not depend directly on the inbreeding coefficient and which theories of inbreeding based on increasing homozygosity fail to explain.     

The Effects of Inbreeding on Male Courtship Behaviour and Coloration in Guppies

Despite an extensive literature on inbreeding depression in a variety of traits, relatively little is known about the effects of inbreeding on patterns of sexual behaviour. In this study, we use the guppy Poecilia reticulata to investigate the influence of parental relatedness (full‐sib. vs. unrelated matings) on sexual behaviour and colour pattern variation in adult male offspring. Our results revealed that one generation of full‐sibling mating resulted in a strong decline in male sexual motivation (courtship intensity and following behaviour) and mating success (number of successful copulations and insemination success – as estimated by the frequency of post‐copulatory jerks after mating), but we detected no significant influence of inbreeding on colour pattern variation. Our finding that sexual ornamentation (colour) did not show similar inbreeding depression to courtship behaviour may be due to differences in the genetic basis of these traits. Our findings indicate that courtship can be a sensitive indicator of genetic stress and add to an emerging picture that sexually selected traits can be severely affected by inbreeding depression.     

Genetic components of variation in Nemophila menziesii undergoing inbreeding: morphology and flowering time.

The standard approaches to estimation of quantitative genetic parameters and prediction of response to selection on quantitative traits are based on theory derived for populations undergoing random mating. Many studies demonstrate, however, that mating systems in natural populations often involve inbreeding in various degrees (i.e. , self matings and matings between relatives). Here we apply theory developed for estimating quantitative genetic parameters for partially inbreeding populations to a population of Nemophila menziesii recently obtained from nature and experimentally inbred. Two measures of overall plant size and two of floral size expressed highly significant inbreeding depression. Of three dominance components of phenotypic variance that are defined under partial inbreeding, one was found to contribute significantly to phenotypic variance in flower size and flowering time, while the remaining two components contributed only negligibly to variation in each of the five traits considered. Computer simulations investigating selection response under the more complete genetic model for populations undergoing mixed mating indicate that, for parameter values estimated in this study, selection response can be substantially slowed relative to predictions for a random mating population. Moreover, inbreeding depression alone does not generally account for the reduction in selection response.     

Single-locus sex determination in the parasitoid wasp Cotesia glomerata (Hymenoptera: Braconidae)

The parasitoid Cotesia glomerata usually produces female-biased sex ratios in the field, which are presumably caused by inbreeding and local mate competition (LMC); yet, sibling mating increases the production of males, leading to the male-biased sex ratio of broods in the laboratory. Previous studies have suggested that the sex allocation strategy of C. glomerata is based on both partial LMC in males and inbreeding avoidance in females. The current study investigated the presence of single-locus complementary sex determination (sl-CSD) as a sex-determining mechanism in this species through inbreeding experiment, cytological examination and microsatellite analysis. Cytological examination detected diploid males in nine of 17 single pairs of sibling mating, thus in agreement with the proportion of matched matings predicted by the sl-CSD model. Sex ratio shifts in these matched sibling matings were consistent with the sl-CSD model with less viable diploid males. The haploid males have a single set of maternal chromosomes (n = 10), whereas diploid males possess a double set of chromosomes (2n = 20). Microsatellite analyses confirmed that diploid males produced from the matched matings inherited segregating genetic materials from both parents. Thus, this study provides the first solid evidence for the presence of sl-CSD as a sex-determining mechanism in the braconid genus Cotesia.     

Sib mating designs for mapping quantitative trait loci

The power to separate the variance of a quantitative trait locus (QTL) from the polygenic variance is determined by the variability of genes identical by descent (IBD) at the QTL. This variability may increase with inbreeding. Selfing, the most extreme form of inbreeding, increases the variability of the IBD value shared by siblings, and thus has a higher efficiency for QTL mapping than random mating. In self-incompatible organisms, sib mating is the closest form of inbreeding. Similar to selfing, sib mating may also increase the power of QTL detection relative to random mating. In this study, we develop an IBD-based method under sib mating designs for QTL mapping. The efficiency of sib mating is then compared with random mating. Monte Carlo simulations show that sib mating designs notably increase the power for QTL detection. When power is intermediate, the power to detect a QTL using full-sib mating is, on average, 7% higher than under random mating. In addition, the IBD-based method proposed in this paper can be used to combine data from multiple families. As a result, the estimated QTL parameters can be applied to a wide statistical inference space relating to the entire reference population. This revised version was published online in July 2006 with corrections to the Cover Date.     

Half-sib mating structures

All ways in which all matings in a population can be between half-sibs under a generalization of regular systems of inbreeding are characterized for both finite and infinite populations. A model of random half-sib mating is developed and analyzed, and the asymptotic configuration of populations subject to it is described. The classical model of half-sib mating which ensues from the standard definition of regular systems of inbreeding is only one of many ways a population can propagate by half-sib mating, and a wide range of genetic identity is possible dependent on which half-sib mating structure governs a population.     

The transition to social inbred mating systems in spiders: role of inbreeding tolerance in a subsocial predecessor

The social spiders are unusual among cooperatively breeding animals in being highly inbred. In contrast, most other social organisms are outbred owing to inbreeding avoidance mechanisms. The social spiders appear to originate from solitary subsocial ancestors, implying a transition from outbreeding to inbreeding mating systems. Such a transition may be constrained by inbreeding avoidance tactics or fitness loss due to inbreeding depression. We examined whether the mating system of a subsocial spider, in a genus with three social congeners, is likely to facilitate or hinder the transition to inbreeding social systems. Populations of subsocial Stegodyphus lineatus are substructured and spiders occur in patches, which may consist of kin groups. We investigated whether male mating dispersal prevents matings within kin groups in natural populations. Approximately half of the marked males that were recovered made short moves (< 5m) and mated within their natal patch. This potential for inbreeding was counterbalanced by a relatively high proportion of immigrant males. In mating experiments, we tested whether inbreeding actually results in lower offspring fitness. Two levels of inbreeding were tested: full sibling versus non-sib matings and matings of individuals within and between naturally occurring patches of spiders. Neither full siblings nor patch mates were discriminated against as mates. Sibling matings had no effect on direct fitness traits such as fecundity, hatching success, time to hatching and survival of the offspring, but negatively affected offspring growth rates and adult body size of both males and females. Neither direct nor indirect fitness measures differed significantly between within patch and between-patch pairs. We tested the relatedness between patch mates and nonpatch mates using DNA fingerprinting (TE-AFLP). Kinship explained 30% of the genetic variation among patches, confirming that patches are often composed of kin. Overall, we found limited male dispersal, lack of kin discrimination, and tolerance to low levels of inbreeding. These results suggest a history of inbreeding which may reduce the frequency of deleterious recessive alleles in the population and promote the evolution of inbreeding tolerance. It is likely that the lack of inbreeding avoidance in subsocial predecessors has facilitated the transition to regular inbreeding social systems.     

INBREEDING AND VARIANCE EFFECTIVE SIZES FOR NONRANDOM MATING POPULATIONS

Following an inbreeding approach and assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, I have derived expressions for the inbreeding effective size, N_eI, for a finite diploid population with variable census sizes for three cases: monoecious populations with partial selfing; dioecious populations of equal numbers of males and females with partial sib mating; and unequal numbers of males and females with random mating. For the first two cases, recurrence equations for the inbreeding coefficient are also obtained, which allow inbreeding coefficients to be predicted exactly in both early and late generations. Following the variance of change in gene frequency approach, a general expression for variance effective size, N_eV, is obtained for a population with unequal numbers of male and female individuals, arbitrary family size distribution, and nonrandom mating. All the parameters involved are allowed to change over generations. For some special cases, the equation reduces to the simple expressions approximately as derived by previous authors. Comparisons are made between equations derived by the present study and those obtained by previous authors. Some of the published equations for N_eI and N_eV are shown to be incomplete or incorrect. Stochastic simulations are run to check the results where disagreements with others are involved.     

The Effect of Inbreeding on Mating Behaviour of West Indian Sweet Potato Weevil Euscepes postfasciatus

When there is an inbreeding depression, mating with a kin individual is generally considered maladaptive behaviour. However, in some conditions, the inclusive fitness benefits from inbreeding may outweigh the costs of inbreeding depression, and thus, inbreeding tolerance is often adaptive. Inbreeding depression and the effect of relatedness on mating behaviour in the West Indian sweet potato weevil Euscepes postfasciatus were examined. No significant inbreeding depression was detected as indicated by body weight and number of progeny emerging from sweet potato roots. Male mating performance (i.e. number of mating occurrences per night) was adversely affected by inbreeding depression, but the effect was low (fitness loss was 6.3%). Although there were no significant differences in latency to mounting, pre‐copulatory guarding, copulation and post‐copulatory guarding duration between full‐sib and non‐kin pairs, the copulation rate of full‐sib pairs was significantly higher than that of non‐kin pairs. These results support the theoretical prediction that when inbreeding depression is weak, copulation with close relative individuals is favoured.     

Theory of inbreeding and covariances between relatives under full-sib mating in diploids.

A generation matrix theory of full-sib mating is developed in which 13 mating "classes" are distinguished according to identity of genes in individuals mated and identity of genotypes as belonging to homozygous, parental, or offspring sets. The 13 times 13 matrix reveals some properties of the full-sib mating system not shown by previous work. The eigenvalues and a set of eigenvectors for the generation matrix, and the general solution for the frequencies of mating classes among descendants of an original mating of genotypes ab times cd, are given. The genotypic array of descendants in an arbitrary generation is also given. A new formula is derived for the coefficient of inbreeding in generation n + m in terms of coefficients of inbreeding in earlier generations. An algorithm is presented for calculating the probability of a given situation of identity of alleles carried by two individuals given only the indices of their own respective generations and the generation of their most recent common ancestor. The application of such probabilities to obtaining covariances between relatives in a full-sib mating system, under the assumptions of independence and non-interaction among loci, is illustrated. All results are shown to agree with previous work in special cases. All possible full sib, generation n - 1 parent-generation n + m offspring, and generation n uncle-generation n + m nephew covariances for 1 less than n + m less than or equal to 8 are obtained using the given algorithm.     

Inbreeding depression in a critically endangered carnivore

Harmful effects arising from matings between relatives (inbreeding) is a long‐standing observation that is well founded in theory. Empirical evidence for inbreeding depression in natural populations is however rare because of the challenges of assembling pedigrees supplemented with fitness traits. We examined the occurrence of inbreeding and subsequent inbreeding depression using a unique data set containing a genetically verified pedigree with individual fitness traits for a critically endangered arctic fox (Vulpes lagopus) population. The study covered nine years and was comprised of 33 litters with a total of 205 individuals. We recorded that the present population was founded by only five individuals. Over the study period, the population exhibited a tenfold increase in average inbreeding coefficient with a final level corresponding to half‐sib matings. Inbreeding mainly occurred between cousins, but we also observed two cases of full‐sib matings. The pedigree data demonstrated clear evidence of inbreeding depression on traditional fitness traits where inbred individuals displayed reduced survival and reproduction. Fitness traits were however differently affected by the fluctuating resource abundande. Inbred individuals born at low‐quality years displayed reduced first‐year survival, while inbred individuals born at high‐quality years were less likely to reproduce. The documentation of inbreeding depression in fundamental fitness traits suggests that inbreeding depression can limit population recovery. Introducing new genetic material to promote a genetic rescue effect may thus be necessary for population long‐term persistence.