Controlling genetic variability by mathematical programming in a selection scheme on an open-pollinated population in Eucalyptus globulus

A model using integer quadratic mathematical programming has been developed to control the inbreeding level (or genetic diversity) through group coancestry in a selection programme for a forestry population structured in terms of maternal families coming from different locations. A method to calculate the average group coancestry between- and within-families for these open-pollinated populations is also proposed. This model has been applied to data from a breeding programme of Australian Eucalyptus globulus. The strategy proved to be effective as reductions of up to 50% for the group coancestry of the selected individuals were reached with a loss of only 5% of the maximum attainable selection differential (corresponding to truncation selection). Received: 14 October 1999 / Accepted: 26 July 2000     

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.     

Genome-Wide Estimates of Coancestry,Inbreeding and Effective Population Size in the Spanish Holstein Population

Estimates of effective population size in the Holstein cattle breed have usually been low despite the large number of animals that constitute this breed. Effective population size is inversely related to the rates at which coancestry and inbreeding increase and these rates have been high as a consequence of intense and accurate selection. Traditionally, coancestry and inbreeding coefficients have been calculated from pedigree data. However, the development of genome-wide single nucleotide polymorphisms has increased the interest of calculating these coefficients from molecular data in order to improve their accuracy. In this study, genomic estimates of coancestry, inbreeding and effective population size were obtained in the Spanish Holstein population and then compared with pedigree-based estimates. A total of 11,135 animals genotyped with the Illumina BovineSNP50 BeadChip were available for the study. After applying filtering criteria, the final genomic dataset included 36,693 autosomal SNPs and 10,569 animals. Pedigree data from those genotyped animals included 31,203 animals. These individuals represented only the last five generations in order to homogenise the amount of pedigree information across animals. Genomic estimates of coancestry and inbreeding were obtained from identity by descent segments (coancestry) or runs of homozygosity (inbreeding). The results indicate that the percentage of variance of pedigree-based coancestry estimates explained by genomic coancestry estimates was higher than that for inbreeding. Estimates of effective population size obtained from genome-wide and pedigree information were consistent and ranged from about 66 to 79. These low values emphasize the need of controlling the rate of increase of coancestry and inbreeding in Holstein selection programmes.     

Genome-Wide Estimates of Coancestry and Inbreeding in a Closed Herd of Ancient Iberian Pigs

Maintaining genetic variation and controlling the increase in inbreeding are crucial requirements in animal conservation programs. The most widely accepted strategy for achieving these objectives is to maximize the effective population size by minimizing the global coancestry obtained from a particular pedigree. However, for most natural or captive populations genealogical information is absent. In this situation, microsatellites have been traditionally the markers of choice to characterize genetic variation, and several estimators of genealogical coefficients have been developed using marker data, with unsatisfactory results. The development of high-throughput genotyping techniques states the necessity of reviewing the paradigm that genealogical coancestry is the best parameter for measuring genetic diversity. In this study, the Illumina PorcineSNP60 BeadChip was used to obtain genome-wide estimates of rates of coancestry and inbreeding and effective population size for an ancient strain of Iberian pigs that is now in serious danger of extinction and for which very accurate genealogical information is available (the Guadyerbas strain). Genome-wide estimates were compared with those obtained from microsatellite and from pedigree data. Estimates of coancestry and inbreeding computed from the SNP chip were strongly correlated with genealogical estimates and these correlations were substantially higher than those between microsatellite and genealogical coefficients. Also, molecular coancestry computed from SNP information was a better predictor of genealogical coancestry than coancestry computed from microsatellites. Rates of change in coancestry and inbreeding and effective population size estimated from molecular data were very similar to those estimated from genealogical data. However, estimates of effective population size obtained from changes in coancestry or inbreeding differed. Our results indicate that genome-wide information represents a useful alternative to genealogical information for measuring and maintaining genetic diversity.     

A comparison of management strategies for conservation with regard to population fitness

Computer simulations have been carried out tocompare, under realistic genetic models, twomethods proposed in the literature to retaingenetic diversity in conservation programmes.In a two-step method, contributions of parentsare set up to produce minimum coancestry(kinship) among the offspring, and this isindependent of the mating system subsequentlyapplied. In a single-step method,contributions and matings are decidedsimultaneously in order to minimise coancestry.The comparison is made in terms of maintainedgenetic diversity and in terms of populationfitness. We conclude that the two methodsmaintain approximately the same geneticdiversity but the latter induces higher levelsof inbreeding, reducing the fitness of thepopulation. Avoidance of close relatives'matings improves this latter method, but thefitness levels do not reach those of thetwo-step scheme. We also investigate theperformances of different mating strategies incombination with minimum coancestry (two-stepmethod), concluding that these mating systemsdo not substantially affect the effectivenessof the management. Finally, we illustrate howminimum group coancestry can be restrictedto a minimum loss of fitness, if a measure ofthis is available for the individuals.     

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.     

Fixed contributions designs vs. minimization of global coancestry to control inbreeding in small populations

Populations with small census sizes are at risk because of the loss of genetic variability and the increase of inbreeding and its harmful consequences. For situations with different numbers of males and females, several hierarchical designs have been proposed to control inbreeding through the fixation of individuals' contributions. An alternative method, based on the minimization of global coancestry, has been proposed to determine contributions as to yield of the lowest levels of inbreeding in the population. We use computer simulations to assess the relative efficiency of the different methods. The results show that minimizing the global coancestry leads to equal or lower levels of inbreeding in the short and medium term, although one of the hierarchical designs provides lower asymptotic inbreeding rates and, thus, less net inbreeding in the long term. We also investigate the performance of the alternative methods against departures from the ideal conditions, such as inbred or differentially related base individuals and random failures in the expected contributions. The method of minimization of global coancestry turns out to be more flexible and robust under these realistic situations.     

Management of subdivided populations in conservation programs: development of a novel dynamic system

Within the context of a conservation program the management of subdivided populations implies a compromise between the control of the global genetic diversity, the avoidance of high inbreeding levels, and, sometimes, the maintenance of a certain degree of differentiation between subpopulations. We present a dynamic and flexible methodology, based on genealogical information, for the maximization of the genetic diversity (measured through the global population coancestry) in captive subdivided populations while controlling/restricting the levels of inbreeding. The method is able to implement specific restrictions on the desired relative levels of coancestry between and within subpopulations. By accounting for the particular genetic population structure, the method determines the optimal contributions (i.e., number of offspring) of each individual, the number of migrants, and the particular subpopulations involved in the exchange of individuals. Computer simulations are used to illustrate the procedure and its performance in a range of reasonable scenarios. The method performs well in most situations and is shown to be more efficient than the commonly accepted one-migrant-per-generation strategy.     

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.     

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.     

Removing exogenous information using pedigree data

Management of certain populations requires the preservation of its pure genetic background. When, for different reasons, undesired alleles are introduced, the original genetic conformation must be recovered. The present study tested, through computer simulations, the power of recovery (the ability for removing the foreign information) from genealogical data. Simulated scenarios comprised different numbers of exogenous individuals taking part of the founder population and different numbers of unmanaged generations before the removal program started. Strategies were based on variables arising from classical pedigree analyses such as founders’ contribution and partial coancestry. The efficiency of the different strategies was measured as the proportion of native genetic information remaining in the population. Consequences on the inbreeding and coancestry levels of the population were also evaluated. Minimisation of the exogenous founders’ contributions was the most powerful method, removing the largest amount of genetic information in just one generation. However, as a side effect, it led to the highest values of inbreeding. Scenarios with a large amount of initial exogenous alleles (i.e. high percentage of non native founders), or many generations of mixing became very difficult to recover, pointing out the importance of being careful about introgression events in populations where these are undesired.     

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.     

Advantages of using molecular coancestry in the removal of introgressed genetic material

Background

When introgression of undesired exogenous genetic material occurs in a population intended to remain pure, actions are necessary to recover the original background. It has been shown that genome-wide information can replace pedigree information for different objectives and is a valuable tool in the fields of genetic conservation and breeding. In this simulation study, molecular information provided by 50 000 SNP was used to minimise the molecular coancestry between individuals of an admixed population and the foreign individuals that originally introgressed a native population in order to remove the exogenous DNA.

Results

This management method, which detects the ‘purest’ individuals to be used as parents for the next generation, allowed recovery of the native genetic background to a great extent in all simulated scenarios. However, it also caused an increase in inbreeding larger than expected because of the lower number of individuals selected as parents and the higher coancestry between them. In scenarios involving several introgression events the method was more efficient than in those involving a single introgression event because part of the genetic information was mixed with the native genetic material for a shorter period.

Conclusions

Genome-wide information can be used to identify the purest individuals via the minimisation of molecular coancestry between individuals of the admixed and exogenous populations. Removal of the undesired genetic material is more efficient with a molecular-based approach than with a pedigree-based approach.     

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.     

Marker-based estimation of the coefficient of coancestry in hybrid breeding programmes

Molecular markers allow to estimate the pairwise relatedness between the members of a breeding pool when their selection history is no longer available or has become too complex for a classical pedigree analysis. The field of population genetics has several estimation procedures at its disposal, but when the genotyped individuals are highly selected inbred lines, their application is not warranted as the theoretical assumptions on which these estimators were built, usually linkage equilibrium between marker loci or even Hardy–Weinberg equilibrium, are not met. An alternative approach requires the availability of a genotyped reference set of inbred lines, which allows to correct the observed marker similarities for their inherent upward bias when used as a coancestry measure. However, this approach does not guarantee that the resulting coancestry matrix is at least positive semi-definite (psd), a necessary condition for its use as a covariance matrix. In this paper we present the weighted alikeness in state (WAIS) estimator. This marker-based coancestry estimator is compared to several other commonly applied relatedness estimators under realistic hybrid breeding conditions in a number of simulations. We also fit a linear mixed model to phenotypical data from a commercial maize breeding programme and compare the likelihood of the different variance structures. WAIS is shown to be psd which makes it suitable for modelling the covariance between genetic components in linear mixed models involved in breeding value estimation or association studies. Results indicate that it generally produces a low root mean squared error under different breeding circumstances and provides a fit to the data that is comparable to that of several other marker-based alternatives. Recommendations for each of the examined coancestry measures are provided.     

Effective Size and F-Statistics of Subdivided Populations. I. Monoecious Species with Partial Selfing

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Effective Size and F-Statistics of Subdivided Populations. II. Dioecious Species

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Optimum contribution selection in large general tree breeding populations with an application to Scots pine

Development of selection methods that optimises selection differential subject to a constraint on the increase of inbreeding (or coancestry) in a population is an important part of breeding programmes. One such method that has received much attention in animal breeding is the optimum contribution (OC) dynamic selection method. We implemented the OC algorithm and applied it to a diallel progeny trial of Pinus sylvestris L. (Scots pine) focussing on two traits (total tree height and stem diameter). The OC method resulted in a higher increase in genetic gain (8–30%) compared to the genetic gain achieved using standard restricted selection method at the same level of coancestry constraint. Genetic merit obtained at two different levels of restriction on coancestry showed that the benefit of OC was highest when restriction was strict. At the same level of genetic merit, OC decreased coancestry with 56 and 39% for diameter and height, respectively, compared to the level of coancestry obtained using unrestricted truncation selection. Inclusion of a dominance term in the statistical model resulted in changes in contribution rank of trees with 7 and 13% for diameter and height, respectively, compared to results achieved by using a pure additive model. However, the genetic gain was higher for the pure additive model than for the model including dominance for both traits.     

Effects of coancestry on accuracy of individual assignments to population of origin: examples using Great Lakes lake trout (Salvelinus namaycush)

Methods for assigning individuals to population of origin are widely used in ecological genetics, resources management, and forensics. Characteristics of genetic data obtained from putative source populations that enhance accuracy of assignment are well established. How non-independence within and among unknown individuals to be classified [i.e., gene correlations within individual (inbreeding) and gene correlations among individuals within group (coancestry)] affect assignment accuracy is poorly understood. We used empirical data for six microsatellite loci and offspring from full-sib crosses of hatchery strains of lake trout (Salvelinus namaycush; Salmonidae) representing known levels of coancestry (mean θ = 0.006 and 0.06) within families to investigate how gene correlations can affect assignment. Additional simulations were conducted to further investigating the influence of allelic diversity (2, 6 or 10 alleles per locus) and inbreeding (F = 0.00, 0.05, and 0.15) on assignment accuracy for cases of low and high inter-population variance in allele frequency (mean F _st = 0.01 and 0.1, respectively). Inbreeding had no effect on accuracy of assignments. In contrast, variance in assignment accuracy across replicated simulations, and for each empirical case study increased with increasing coancestry, reflecting non-independence of probabilities of correct assignment among members of kin groups. Empirical estimates of assignment error rates should be interpreted with caution if appreciable levels of coancestry are suspected. Additional emphasis should be placed on sampling designs (spatially and temporally) that define or minimize the potential for sampling related individuals.     

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.