Regular Systems Of Inbreeding Research Articles

Prediction of fitness under different breeding designs in conservation programs

AbstractThe expected change in fitness under inbreeding due to deleterious recessive alleles depends on the amount of inbreeding load harbored by a population, that is, the load of deleterious recessive mutations concealed in the heterozygous state, and the opposing effect of genetic purging to remove such a load. This change in fitness can be thus predicted if an estimate of the inbreeding load and the purging coefficient (the parameter that quantifies the amount of purging) are available. These two parameters can be estimated in pedigreed populations, as has been shown for populations under random mating. A question arises whether these parameters can also be estimated under other breeding systems as well as whether they allow accurate prediction of the corresponding expected change in fitness. In conservation programs, it is usually recommended to preserve genetic diversity by equalizing contributions from parents to progeny and avoiding inbred matings. Regular systems of inbreeding have also been proposed as breeding conservation strategies to purge the inbreeding load. Using computer simulations, we first test a method to jointly estimate the initial inbreeding load and the purging coefficient in populations subjected to equalization of parental contributions, circular mating, and partial full‐sib mating. Then, using the expected values of the inbreeding coefficient and the variance of family size, as well as the estimates of the inbreeding load and the purging coefficient, we make simple predictions of the change in fitness over generations under these breeding systems and compare them with simulation results. We discuss how these fitness predictions can help undertaking conservation designs under different breeding scenarios.

Time Since Common Pedigree Ancestors with Two Progeny per Individual

Constraining individuals to two progeny (versus Poisson distribution) increases the time since a pedigree (nongenetic) common ancestor, but the time still increases logarithmically in the population size. This is confirmed by simulations for discrete generations and rigorously for expected time with a modification of the Moran model. Selfing increases the expected time since a common ancestor with both the Poisson progeny distribution and two progeny per individual. As selfing approaches one, the time since a common ancestor asymptotically approaches infinity with two progeny per individual, but only twice the population size with the Poisson progeny distribution. Regular systems of inbreeding with two progeny per individual can either increase or decrease the time since a common ancestor as contrasted with random mating with two progeny per individual.

The Fundamental Theorem of Natural Selection in Ewens' sense (case of fertility selection).

We show that the Fundamental Theorem of Natural Selection in Ewens' sense is valid in the case of fertility selection: the additive genetic variance in fertility divided by the mean fertility is exactly equal to the partial change in the mean fertility from the current generation to the next. This partial change is the increase in the mean additive value caused by frequency changes from on generation to the next. This partial change is the increase in the mean additive value caused by frequency changes from one generation to the next but keeping unchanged the additive values. The only hypothesis on mating is that it does not affect the allelic frequencies in the sense that these are the same before and after mating in the parental generation, which occurs for a wide range of mating patterns going from random mating to several regular systems of inbreeding and cases of assortative mating. The fertility of couples is determined by the genes at an arbitrary number of loci, and the additive (average) allelic allelic effects are defined by a linear system of equations, which is used to extend Ewens' optimality principle to the case of fertility selection.

The Importance of Mating Structure Versus Progeny Distribution for Genetic Identity under Mutation

The relative importance of mating structure versus the distribution of progeny number for the genetic identity in a population is investigated. Both homozygosity and identity between individuals in a population at equilibrium with mutation are used as measures. Several regular systems of inbreeding, as well as random mating, are contrasted subject to the constraint of exactly two progeny per individual. Conversely, different distributions of progeny number are contrasted under both random mating and half-sib mating. Both the mating system and the progeny distribution are significant determinants of genetic identity, but sometimes these factors are interrelated so that causality cannot be assigned to either factor. When the mating structure or progeny distribution is changed, identity within and between individuals may both increase (or both decrease), but they may also change in opposite directions.

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.

On the robustness of regular systems of inbreeding

Half-sib, first cousin, half nth cousin, and nth cousin mating systems are robust in that small deviations from the mating structure will not significantly alter the levels of genetic identity or effect qualitative distinctions between the models. Substitution of nearest kin in matings may either increase or decrease the level of homozygosity depending on the mating structure; the effect of a single error in the mating structure is not an accurate indicator of the equilibrium resulting from recurrent errors. Models of mixed half nth cousin or nth cousin mating show that the relative frequency of the lowest order inbred mating essentially determines the level of homozygosity. Any positive relative frequency of more distant matings will reduce the probability of identity by descent under half-sib or first cousin mating to less than 1.

On the classification of regular systems of inbreeding

Regular systems of inbreeding are defined as those with discrete, nonoverlapping generations and with the same number of individuals and mating pattern in every generation. Given the number of individuals in a generation, there are many possible regular mating systems. A notion of when two such mating systems are equivalent is introduced, and several necessary conditions are given for such an equivalence. The use of these conditions is illustrated for N = 2, 3, 4 and 5 individuals, and a complete enumeration has been found for these cases: the numbers of inequivalent mating systems are 1, 5, 57 and 858, respectively. The maximal eigenvalue of the matrix q that specifies the recursion relations satisfied by the probabilities of identity have also been found for these cases. For N = 3 and 4 (and 2 trivially), circular mating gives the slowest rate of approach to genetic uniformity of those systems that do evolve to uniformity, but for N = 5 there are two other mating systems that have a slower rate of convergence, and for N = 6 partial results show that there are many such examples.

Regular systems of inbreeding with mutation

Probability of identity by type is studied for regular systems of inbreeding in the presence of mutation. Analytic results are presented for half-sib mating, first cousin mating, and half nth cousin mating under both infinite allele and two allele (back mutation) models. Reasonable rates of mutation do not provide significantly different results from probability of identity by descent in the absence of mutation. Homozygosity is higher under half-sib mating than under first cousin mating, but the expected number of copies of a gene in the population is higher under first cousin mating than under half-sib mating.

Regular systems of inbreeding.

Regular systems of inbreeding with discrete, nonoverlapping generations and the same number of individuals and mating pattern in every generation are studied. The matrix Q that specifies the recursion relations satisfied by the probabilities of identity is expressed in terms of the matrix M that describes the mating system. Necessary and sufficient conditions for convergence to genetic uniformity are given, and it is determined which probabilities of identity approach one. If the mating system has certain symmetries and these are imposed initially, then a matrix R, of lower dimension than Q, specifies the recursion relations. For such a mating system, for generic initial conditions, the condensed matrix R suffices for determining whether convergence to uniformity occurs and which probabilities of identity approach one. If Q is irreducible, the maximal eigenvalue of R is the same as that of Q. If Q is also aperiodic, this implies that the asymptotic rate of convergence to homogeneity of the condensed system is the same as that of the complete one. The above results apply to autosomal loci in monoecious (with or without selfing) and dioecious populations and to X-linked loci. As an example, all the eigenvalues and right and left eigenvectors of Q for circular mating are found.

Covariances among Relatives in Populations under Mixed Self-Fertilization and Random Mating

S UMMARY The study deals with a population which is reproducing partly due to random mating and partly due to selfing. The effect of such a combination of mating systems in varying degrees, on the genetic variability in the population and also on the covariances of relatives, is considered. The covariances of relatives are determined in terms of the additive and dominance components of variance, and the amount of selfing. These covariances could be used for estimation of the two components of variance. The results are compared with the corresponding results for a random-mating population. Several research workers have considered the variance, and covariances of relatives, in a diploid population where the mating process is either random mating or some regular system of inbreeding such as selfing or full-sib mating. Fisher (1918) considered a random-mating population, obtained the variance of individuals and developed formulas for the correlation between various types of relatives for autosomal loci for the cases of no epistasis and dual epistasis. These results have been extended to cover more complex situations in random-mating populations. Kempthorne (1955) and Horner (1956) have studied covariances of relatives under certain regular systems of inbreeding. Cockerham (1963) has described several mating designs and their analyses for the estimation of genetic variances based on covariances of relatives. The aim of the present paper is to investigate similar properties when the progeny is produced partly by outcrossing and partly by selfing. Such a situation is of interest because there are crops such as wheat, rice, barley and lima beans, which are largely, but not wholly, selfed, and crops like cotton which are moderately self-fertilized. We shall assume that the population is large and that it reproduces in nonoverlapping generations with constant probabilities e of selfing and y ( = 1- ) of random mating. We shall discuss the nature of the variance and of covariances of relatives under such a mixture of mating systems.

Analysis of some regular systems of inbreeding

A probabilistic method is used to calculate the asymptotic rate of loss of genetic variability for half-sib and first-cousin mating of an infinite number of individuals. For both systems of inbreeding, all probabilities of nonidentity by descent converge to zero at ultimate rate proportional to t-1/2, where t is the time in generations. The rate of decay is faster for half-sibs than for first cousins, the respective asymptotic allozygosities being 4(pit)-1/2 and 8(pi t)-1/2.

An extension of the probability approach to genetic relationships: One locus

A path-counting method has been presented for calculating the nine phylotype pair probabilities ( k-coefficients of this paper) for a diploid autosomal locus. Cockerham (1971) has provided a somewhat more elegant method for arriving at these probabilities. The procedure explained herein is more akin to the familiar calculation of Wright's inbreeding coefficient and is easier to apply to specific complex pedigrees such as arise in human genetics problems. Cockerham's more general formulation is superior for the investigation of regular systems of inbreeding.