Genetic management of captive populations: the advantages of circular mating

We performed computer simulations to evaluate the effectiveness of circular mating as a genetic management option for captive populations. As a benchmark, we used the method proposed by Fernández and Caballero according to which parental contributions are set to produce minimum coancestry among the offspring and matings are performed so as to minimize mean pairwise coancestry (referred to as the Gc/mc method). In contrast to other methods, fitness does not vary with population size in the case of circular mating, and can be higher than under random mating. Whether circular mating is an effective method in conserving captive populations depends on the trade-off between different considerations. On the one hand, circular mating shows the highest allelic diversity and the lowest mean pairwise coancestry for all population sizes. It also shows a relatively higher efficiency of purging deleterious alleles. More importantly, circular mating can significantly increase the success probability of populations released to the wild relative to the Gc/mc method. On the other hand, circular mating has the drawback of showing high inbreeding rates and low fitness in early generations, which can result to an increase in the extinction probability of the captive populations. However, this increase is slight unless population size and litter size are both very low. Overall, if the slight increase in extinction probability can be tolerated then circular mating fulfils the primary goals of a captive breeding program, i.e., it maintains high levels of genetic diversity and increases the success probability of reintroduced populations.     

Interrelations between effective population size and other pedigree tools for the management of conserved populations

Genetic parameters widely used to monitor genetic variation in conservation programmes, such as effective number of founders, founder genome equivalents and effective population size, are interrelated in terms of coancestries and variances of contributions from ancestors to descendants. A new parameter, the effective number of non-founders, is introduced to describe the relation between effective number of founders and founder genome equivalents. Practical recommendations for the maintenance of genetic variation in small captive populations are discussed. To maintain genetic diversity, minimum coancestry among individuals should be sought. This minimizes the variances of contributions from ancestors to descendants in all previous generations. The method of choice of parents and the system of mating should be independent of each other because a clear-cut recommendation cannot be given on the latter.     

Using genomic tools to maintain diversity and fitness in conservation programmes

Conservation programmes aim at maximizing the survival probability of populations, by minimizing the loss of genetic diversity, which allows populations to adapt to changes, and controlling inbreeding increases. The best known strategy to achieve these goals is optimizing the contributions of the parents to minimize global coancestry in their offspring. Results on neutral scenarios showed that management based on molecular coancestry could maintain more diversity than management based on genealogical coancestry when a large number of markers were available. However, if the population has deleterious mutations, managing using optimal contributions can lead to a decrease in fitness, especially using molecular coancestry, because both beneficial and harmful alleles are maintained, compromising the long‐term viability of the population. We introduce here two strategies to avoid this problem: The first one uses molecular coancestry calculated removing markers with low minor allele frequencies, as they could be linked to selected loci. The second one uses a coancestry based on segments of identity by descent, which measures the proportion of genome segments shared by two individuals because of a common ancestor. We compare these strategies under two contrasting mutational models of fitness effects, one assuming many mutations of small effect and another with few mutations of large effect. Using markers at intermediate frequencies maintains a larger fitness than using all markers, but leads to maintaining less diversity. Using the segment‐based coancestry provides a compromise solution between maintaining diversity and fitness, especially when the population has some inbreeding load.     

An experimental evaluation with Drosophila melanogaster of a novel dynamic system for the management of subdivided populations in conservation programs

A dynamic method (DM) recently proposed for the management of captive subdivided populations was evaluated using the pilot species Drosophila melanogaster. By accounting for the particular genetic population structure, the DM determines the optimal mating pairs, their contributions to progeny and the migration pattern that minimize the overall coancestry in the population with a control of inbreeding levels. After a pre-management period such that one of the four subpopulations had higher inbreeding and differentiation than the others, three management methods were compared for 10 generations over three replicates: (1) isolated subpopulations (IS), (2) one-migrant-per-generation rule (OMPG), (3) DM aimed to produce the same or lower inbreeding coefficient than OMPG. The DM produced the lowest coancestry and equal or lower inbreeding than the OMPG method throughout the experiment. The initially lower fitness and lower variation for nine microsatellite loci of the highly inbred subpopulation were restored more quickly with the DM than with the OMPG method. We provide, therefore, an empirical illustration of the usefulness of the DM as a conservation protocol for captive subdivided populations when pedigree information is available (or can be deduced) and manipulation of breeding pairs is possible.     

Maintaining genetic diversity using molecular coancestry: the effect of marker density and effective population size

Background

The most efficient method to maintain genetic diversity in populations under conservation programmes is to optimize, for each potential parent, the number of offspring left to the next generation by minimizing the global coancestry. Coancestry is usually calculated from genealogical data but molecular markers can be used to replace genealogical coancestry with molecular coancestry. Recent studies showed that optimizing contributions based on coancestry calculated from a large number of SNP markers can maintain higher levels of diversity than optimizing contributions based on genealogical data. In this study, we investigated how SNP density and effective population size impact the use of molecular coancestry to maintain diversity.

Results

At low SNP densities, the genetic diversity maintained using genealogical coancestry for optimization was higher than that maintained using molecular coancestry. The performance of molecular coancestry improved with increasing marker density, and, for the scenarios evaluated, it was as efficient as genealogical coancestry if SNP density reached at least 3 times the effective population size.However, increasing SNP density resulted in reduced returns in terms of maintained diversity. While a benefit of 12% was achieved when marker density increased from 10 to 100 SNP/Morgan, the benefit was only 2% when it increased from 100 to 500 SNP/Morgan.

Conclusions

The marker density of most SNP chips already available for farm animals is sufficient for molecular coancestry to outperform genealogical coancestry in conservation programmes aimed at maintaining genetic diversity. For the purpose of effectively maintaining genetic diversity, a marker density of around 500 SNPs/Morgan can be considered as the most cost effective density when developing SNP chips for new species. Since the costs to develop SNP chips are decreasing, chips with 500 SNPs/Morgan should become available in a short-term horizon for non domestic species.     

Efficiency of the use of pedigree and molecular marker information in conservation programs

The value of molecular markers and pedigree records, separately or in combination, to assist in the management of conserved populations has been tested. The general strategy for managing the population was to optimize contributions of parents to the next generation for minimizing the global weighted coancestry. Strategies differed in the type of information used to compute global coancestries, the number and type of evaluated individuals, and the system of mating. Genealogical information proved to be very useful (at least for 10 generations of management) to arrange individuals' contributions via the minimization of global coancestry. In fact, the level of expected heterozygosity after 10 generations yielded by this strategy was 88-100% of the maximum possible improvement obtained if the genotype for all loci was known. Marker information was of very limited value if used alone. The amount and degree of polymorphism of markers to be used to compute molecular coancestry had to be high to mimic the performance of the strategy relying on pedigree, especially in the short term (for example, >10 markers per chromosome with 10 alleles each were needed if only the parents' genotype was available). When both sources of information are combined to calculate the coancestry conditional on markers, clear increases in effective population size (Ne) were found, but observed diversity levels (either gene or allelic diversity) in the early generations were quite similar to the ones obtained with pedigree alone. The advantage of including molecular information is greater when information is available on a greater number of individuals (offspring and parents vs. parents only). However, for realistic situations (i.e., large genomes) the benefits of using information on offspring are small. The same conclusions were reached when comparing the use of the different types of information (genealogical or/and molecular) to perform minimum coancestry matings.     

Mating schemes for optimum contribution selection with constrained rates of inbreeding

The effect of non-random mating on genetic response was compared for populations with discrete generations. Mating followed a selection step where the average coancestry of selected animals was constrained, while genetic response was maximised. Minimum coancestry (MC), Minimum coancestry with a maximum of one offspring per mating pair (MC1) and Minimum variance of the relationships of offspring (MVRO) mating schemes resulted in a delay in inbreeding of about two generations compared with Random, Random factorial and Compensatory mating. In these breeding schemes where selection constrains the rate of inbreeding, ΔF, the improved family structure due to non-random mating increased genetic response. For schemes with ΔF constrained to 1.0% and 100 selection candidates, genetic response was 22% higher for the MC1 and MVRO schemes compared with Random mating schemes. For schemes with a less stringent constraint on ΔF or more selection candidates, the superiority of the MC1 and MVRO schemes was smaller (5–6%). In general, MC1 seemed to be the preferred mating method, since it almost always yielded the highest genetic response. MC1 mainly achieved these high genetic responses by avoiding extreme relationships among the offspring, i.e. fullsib offspring are avoided, and by making the contributions of ancestors to offspring more equal by mating least related animals.     

Why do females mate multiply? A review of the genetic benefits

The aim of this review is to consider the potential benefits that females may gain from mating more than once in a single reproductive cycle. The relationship between non-genetic and genetic benefits is briefly explored. We suggest that multiple mating for purely non-genetic benefits is unlikely as it invariably leads to the possibility of genetic benefits as well. We begin by briefly reviewing the main models for genetic benefits to mate choice, and the supporting evidence that choice can increase offspring performance and the sexual attractiveness of sons. We then explain how multiple mating can elevate offspring fitness by increasing the number of potential sires that compete, when this occurs in conjunction with mechanisms of paternity biasing that function in copula or post-copulation. We begin by identifying cases where females use pre-copulatory cues to identify mates prior to remating. In the simplest case, females remate because they identify a superior mate and 'trade up' genetically. The main evidence for this process comes from extra-pair copulation in birds. Second, we note other cases where pre-copulatory cues may be less reliable and females mate with several males to promote post-copulatory mechanisms that bias paternity. Although a distinction is drawn between sperm competition and cryptic female choice, we point out that the genetic benefits to polyandry in terms of producing more viable or sexually attractive offspring do not depend on the exact mechanism that leads to biased paternity. Post-copulatory mechanisms of paternity biasing may: (1) reduce genetic incompatibility between male and female genetic contributions to offspring; (2) increase offspring viability if there is a positive correlation between traits favoured post-copulation and those that improve performance under natural selection; (3) increase the ability of sons to gain paternity when they mate with polyandrous females. A third possibility is that genetic diversity among offspring is directly favoured. This can be due to bet-hedging (due to mate assessment errors or temporal fluctuations in the environment), beneficial interactions between less related siblings or the opportunity to preferentially fertilise eggs with sperm of a specific genotype drawn from a range of stored sperm depending on prevailing environmental conditions. We use case studies from the social insects to provide some concrete examples of the role of genetic diversity among progeny in elevating fitness. We conclude that post-copulatory mechanisms provide a more reliable way of selecting a genetically compatible mate than pre-copulatory mate choice. Some of the best evidence for cryptic female choice by sperm selection is due to selection of more compatible sperm. Two future areas of research seem likely to be profitable. First, more experimental evidence is needed demonstrating that multiple mating increases offspring fitness via genetic gains. Second, the role of multiple mating in promoting assortative fertilization and increasing reproductive isolation between populations may help us to understand sympatric speciation.     

Polyandry in coal tits Parus ater: fitness consequences of putting eggs into multiple genetic baskets

Females of many species mate with multiple males within a single reproductive cycle. One hypothesis to explain polyandry postulates that females benefit from increasing within-brood genetic diversity. Two mechanisms may render sire genetic diversity beneficial for females, genetic bet-hedging vs. non-bet-hedging. We analysed whether females of the socially monogamous coal tit (Parus ater) benefit via either of these mechanisms when engaging in extra-pair (i.e. polyandrous) mating. To obtain a measure of within-brood genetic diversity as a function of paternal genetic contributions, we calculated a sire diversity index based on the established Shannon-Wiener Index. In 246 broods from two consecutive years, sire genetic diversity had no effect on either the mean or the variance in brood fitness measured as offspring recruitment within 4 years after birth. The hypothesis that benefits of increasing sire diversity contribute to selection for female extra-pair mating behaviour in P. ater was therefore not supported.     

Effect of non-random mating on genomic and BLUP selection schemes

Background

The risk of long-term unequal contribution of mating pairs to the gene pool is that deleterious recessive genes can be expressed. Such consequences could be alleviated by appropriately designing and optimizing breeding schemes i.e. by improving selection and mating procedures.

Methods

We studied the effect of mating designs, random, minimum coancestry and minimum covariance of ancestral contributions on rate of inbreeding and genetic gain for schemes with different information sources, i.e. sib test or own performance records, different genetic evaluation methods, i.e. BLUP or genomic selection, and different family structures, i.e. factorial or pair-wise.

Results

Results showed that substantial differences in rates of inbreeding due to mating design were present under schemes with a pair-wise family structure, for which minimum coancestry turned out to be more effective to generate lower rates of inbreeding. Specifically, substantial reductions in rates of inbreeding were observed in schemes using sib test records and BLUP evaluation. However, with a factorial family structure, differences in rates of inbreeding due mating designs were minor. Moreover, non-random mating had only a small effect in breeding schemes that used genomic evaluation, regardless of the information source.

Conclusions

It was concluded that minimum coancestry remains an efficient mating design when BLUP is used for genetic evaluation or when the size of the population is small, whereas the effect of non-random mating is smaller in schemes using genomic evaluation.     

Inbreeding depression from selfing and mating between relatives in the Neotropical tree <Emphasis Type="Italic">Cariniana legalis</Emphasis> Mart. Kuntze

Selfing or mating between related individuals in self-compatible hermaphroditic tree species may lead to inbreeding depression (ID) due to homozygosis in recessive, identical by descent alleles. In general, studies of ID in tree species have been based on comparisons of selfed individuals (produced by controlled pollination) with outcrossed individuals for quantitative traits in progeny tests. However, this approach requires a long time to quantify the extent of ID. Thus, we used an approach based on genetic markers to estimate coancestry coefficients between assigned parents from paternity analysis in two populations of the Neotropical tree Cariniana legalis. Using this method, we were able to determine which seedlings in a nursery trial originated from; (i) outcrossing between un-related trees, (ii) mating between related trees and (iii) selfing. We detected a low selfing rate (<10 %), but a substantial quantity of seedlings from mating between related parents (minimum of 35.7 %). In general, the outcrossed seedlings from unrelated parents exhibited significantly greater genetic diversity than those resulting from selfing and mating among relatives. The extent of ID varied among traits and populations. Outcrossed seedlings originating from unrelated trees generally showed greater survival than seedlings originating from selfing and related parents. Inbreeding depression was greater in the selfed seedlings than in those from mating among related parents. The results are discussed in terms of implications for genetic conservation, breeding and environmental restoration using the species.     

Management of genetic diversity in small farm animal populations

Many local breeds of farm animals have small populations and, consequently, are highly endangered. The correct genetic management of such populations is crucial for their survival. Managing an animal population involves two steps: first, the individuals who will be permitted to leave descendants are to be chosen and the number offspring they will be permitted to produce has to be determined; second, the mating scheme has to be identified. Strategies dealing with the first step are directed towards the maximisation of effective population size and, therefore, act jointly on the reduction in the loss of genetic variation and in the increase of inbreeding. In this paper, the most relevant methods are summarised, including the so-called 'Optimum Contribution' methodology (contributions are proportional to the coancestry of each individual with the rest), which has been shown to be the best. Typically, this method is applied to pedigree information, but molecular marker data can be used to complete or replace the genealogy. When the population is subjected to explicit selection on any trait, the above methodology can be used by balancing the response to selection and the increase in coancestry/inbreeding. Different mating strategies also exist. Some of the mating schemes try to reduce the level of inbreeding in the short term by preventing mating between relatives. Others involve regular (circular) schemes that imply higher levels of inbreeding within populations in the short term, but demonstrate better performance in the long term. In addition, other tools such as cryopreservation and reproductive techniques aid in the management of small populations. In the future, genomic marker panels may replace the pedigree information in measuring the coancestry. The paper also includes the results of several experiments and field studies on the effectiveness and on the consequences of the use of the different strategies.     

Improved dairy cattle mating plans at herd level using genomic information

From 2012 to 2018, 223 180 Montbéliarde females were genotyped in France and the number of newly genotyped females increased at a rate of about 33% each year. With female genotyping information, farmers have access to the genomic estimated breeding values of the females in their herd and to their carrier status for genetic defects or major genes segregating in the breed. This information, combined with genomic coancestry, can be used when planning matings in order to maximize the expected on-farm profit of future female offspring. We compared different mating allocation approaches for their capacity to maximize the expected genetic gain while limiting expected progeny inbreeding and the probability to conceive an offspring homozygous for a lethal recessive allele. Three mate allocation strategies (random mating (RAND), sequential mating (gSEQ€) and linear programing mating (gLP€)) were compared on 160 actual Montbéliarde herds using male and female genomic information. Then, we assessed the benefit of using female genomic information by comparing matings planned using only female pedigree information with the equivalent strategy using genomic information. We measured the benefit of adding genomic expected inbreeding and risk of conception of an offspring homozygous for a lethal recessive allele to Net merit in mating plans. The influence of three constraints was tested: by relaxing the constraint on availability of a particular semen type (sexed or conventional) for bulls, by adding an upper limit of 8.5% coancestry between mate pairs or by using a more stringent maximum use of a bull in a herd (5% vs 10%). The use of genomic information instead of pedigree information improved the mate allocation method in terms of progeny expected genetic merit, genetic diversity and risk to conceive an offspring homozygous for a lethal recessive allele. Optimizing mate allocation using linear programming and constraining coancestry to a maximum of 8.5% per mate pair reduced the average coancestry with a small impact on expected Net Merit. In summary, for male and female selection pathways, using genomic information is more efficient than using pedigree information to maximize genetic gain while constraining the expected inbreeding of the progeny and the risk to conceive an offspring homozygous for a lethal recessive allele. This study also underlines the key role of semen type (sexed vs conventional) and the associated constraints on the mate allocation algorithm to maximize genetic gain while maintaining genetic diversity and limiting the risk to conceive an offspring homozygous for a lethal recessive allele.     

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.     

Genetic Diversity and the Survival of Populations

Abstract: In this comprehensive review, a range of factors is considered that may influence the significance of genetic diversity for the survival of a population. Genetic variation is essential for the adaptability of a population in which quantitatively inherited, fitness-related traits are crucial. Therefore, the relationship between genetic diversity and fitness should be studied in order to make predictions on the importance of genetic diversity for a specific population. The level of genetic diversity found in a population highly depends on the mating system, the evolutionary history of a species and the population history (the latter is usually unknown), and on the level of environmental heterogeneity. An accurate estimation of fitness remains complex, despite the availability of a range of direct and indirect fitness parameters. There is no general relationship between genetic diversity and various fitness components. However, if a lower level of heterozygosity represents an increased level of inbreeding, a reduction in fitness can be expected. Molecular markers can be used to study adaptability or fitness, provided that they represent a quantitative trait locus (QTL) or are themselves functional genes involved in these processes. Next to a genetic response of a population to environmental change, phenotypic plasticity in a genotype can affect fitness. The relative importance of plasticity to genetic diversity depends on the species and population under study and on the environmental conditions. The possibilities for application of current knowledge on genetic diversity and population survival for the management of natural populations are discussed.     

Methods for minimizing the loss of genetic diversity in conserved populations with overlapping generations

Minimization of the average coancestry in a population has been theoretically proven to be the most efficient method to preserve genetic diversity. In the present study, based on a population genetic model, two methods to minimize the average coancestry in populations with overlapping generations were developed. For a given parental coancestry structure, the first method (OG) minimizes the average coancestry in the next generation, and the second method (LT) is designed to minimize the long-term accumulation of coancestry. The efficiencies of the two methods were examined by stochastic simulation. Compared to random choice of parents, the annual effective population sizes under the two proposed methods increased 2–3 folds. The difference among the two methods was small in a population with short generation interval. For populations with long generation intervals, the OG method showed a slightly larger annual effective size in an initial few years. However, in the subsequent years, the LT method gave a 5–15% larger annual effective size than the OG method. From these results, it is suggested that the LT method would be preferred to the OG method in most practical situations.     

Positive assortative mating with selection restrictions on group coancestry enhances gain while conserving genetic diversity in long-term forest tree breeding

Selection and mating principles in a closed breeding population (BP) were studied by computer simulation. The BP was advanced, either by random assortment of mates (RAM), or by positive assortative mating (PAM). Selection was done with high precision using clonal testing. Selection considered both genetic gain and gene diversity by "group-merit selection", i.e. selection for breeding value weighted by group coancestry of the selected individuals. A range of weights on group coancestry was applied during selection to vary parent contributions and thereby adjust the balance between gain and diversity. This resulted in a series of scenarios with low to high effective population sizes measured by status effective number. Production populations (PP) were selected only for gain, as a subset of the BP. PAM improved gain in the PP substantially, by increasing the additive variance (i.e. the gain potential) of the BP. This effect was more pronounced under restricted selection when parent contributions to the next generation were more balanced with within-family selection as the extreme, i.e. when a higher status effective number was maintained in the BP. In that case, the additional gain over the BP mean for the clone PP and seed PPs was 32 and 84% higher, respectively, for PAM than for RAM in generation 5. PAM did not reduce gene diversity of the BP but increased inbreeding, and in that way caused a departure from Hardy-Weinberg equilibrium. The effect of inbreeding was eliminated by recombination during the production of seed orchard progeny. Also, for a given level of inbreeding in the seed orchard progeny or in a mixture of genotypes selected for clonal deployment, gain was higher for PAM than for RAM. After including inbreeding depression in the simulation, inbreeding was counteracted by selection, and the enhancement of PAM on production population gain was slightly reduced. In the presence of inbreeding depression the greatest PP gain was achieved at still higher levels of status effective number, i.e. when more gene diversity was conserved in the BP. Thus, the combination of precise selection and PAM resulted in close to maximal short-term PP gain, while conserving maximal gene diversity in the BP.Communicated by O. Savolainen     

Multiple paternity in wild house mice (Mus musculus musculus): effects on offspring genetic diversity and body mass

Multiple mating is common in many species, but it is unclear whether multiple paternity enhances offspring genetic diversity or fitness. We conducted a survey on wild house mice (Mus musculus musculus), and we found that in 73 pregnant females, 29% of litters had multiple sires, which is remarkably similar to the 23–26% found in feral populations of Mus musculus domesticus in the USA and Australia, respectively. The question is: How has selection maintained multiple mating in these subspecies since the evolutionary divergence, ca. 2800–6000 years ago? We found no evidence that multiple paternity enhanced females’ litter size, contrary to the fertility assurance or genetic benefits hypotheses. Multiple paternity was associated with reduced mean and variance in offspring body mass, which suggests that females allocate fewer resources or that there is increased intrauterine conflict in multiple-versus single-sired litters. We found increased allelic diversity (though not heterozygosity) in multiple-sired litters, as predicted by the genetic diversity hypothesis. Finally, we found that the dams’ heterozygosity was correlated with the mean heterozygosity of their offspring in single-and multiple-sired litters, suggesting that outbred, heterozygous females were more likely to avoid inbreeding than inbred, homozygous females. Future studies are needed to examine how increased genetic diversity of litters and smaller mean (and variance) offspring body mass associated with multiple paternity affect offspring fitness.     

Mating patterns and pollinator mobility are critical traits in forest fragmentation genetics

Most woody plants are animal-pollinated, but the global problem of habitat fragmentation is changing the pollination dynamics. Consequently, the genetic diversity and fitness of the progeny of animal-pollinated woody plants sired in fragmented landscapes tend to decline due to shifts in plant-mating patterns (for example, reduced outcrossing rate, pollen diversity). However, the magnitude of this mating-pattern shift should theoretically be a function of pollinator mobility. We first test this hypothesis by exploring the mating patterns of three ecologically divergent eucalypts sampled across a habitat fragmentation gradient in southern Australia. We demonstrate increased selfing and decreased pollen diversity with increased fragmentation for two small-insect-pollinated eucalypts, but no such relationship for the mobile-bird-pollinated eucalypt. In a meta-analysis, we then show that fragmentation generally does increase selfing rates and decrease pollen diversity, and that more mobile pollinators tended to dampen these mating-pattern shifts. Together, our findings support the premise that variation in pollinator form contributes to the diversity of mating-pattern responses to habitat fragmentation.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.