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A transformation is proposed of the variables used for constructing genetic selection indices, such that the reparameterized phenotypic covariance matrix is identity and the genetic covariance matrix is diagonal. These diagonal elements play the role of heritabilities of the transformed variables. The reparameterization enables sampling properties of the index weights to be easily computed and formulae are given for data from half-sib families. The index is likely to be least stable when the transformed variables of low heritability have high economic weights, and the weights will have highest sampling variance when these heritabilities are nearly equal. It is suggested that the sample roots of the determinantal equation be inspected when constructing an index in order to give some guide to its accuracy.

A simple derivation is given for a formula obtained previously for the effective size of random-mating populations with overlapping generations. The effective population size is the same as that for a population with discrete generations having the same variance of lifetime family size and the same number of individuals entering the population per generation.

Theoretical predictions of changes in variance with disruptive selection have used models of infinitely many genes so the increase in variance was necessarily due to linkage disequilibrium. With small numbers of loci, the disequilibrium is shown still to comprise the major part of the changes in variance.In a replicated experiment with Drosophila melanogaster, disruptive selection was practised for three generations, and this was followed by 5 or 7 generations of random mating. The heritability, as estimated from regression of progeny on parent, rose from 37% to 68% on selection, and subsequently declined to 45% on random mating. Changes of variance can be interpreted invoking the build up of linkage disequilibrium during selection followed by its breakdown upon relaxation. The results agree well with those obtained from Monte Carlo simulation.

The method of genotype assay, proposed by Jinks and Towey (1976) for estimating the number of effective factors in a polygenic system, assumes independent segregation if applied to number of genes. Their results are extended to include the case of linked genes, and the ratio of expected number of effective factors to number of genes is computed for a range of models. Unless all genes are on different chromosomes or many generations of inbreeding are used, the estimate of gene number is biased downwards.

To examine the effect of selection on levels of heterosis, crosses were made between three groups of six lines of mice, one group unselected (controls) and the other two selected for high (large lines) and low (small lines) 6-week body weight, respectively. The coefficient of inbreeding of each line was about 0.60. Each line was crossed reciprocally to one line from each of the parental groups, as well as producing purebred progeny. Heterosis for 3-week weight, 6-week weight and 3-6 week gain averaged 0.0%, 2.4% and 4.2%, respectively, and was higher for males than for females. Heterosis was more extensive in crosses involving large or control lines than in crosses with small lines. There was no detectable heterosis in several measures of developmental rate, such as age at vaginal opening. Food conversion efficiency and carcass composition were measured on a sample of the animals. Food consumption, gonadal fat pad weight, and hindquarters weight, each expressed as a proportion of body weight, exhibited -4.0%, 5.6%, and 2.3% heterosis, respectively.

Long-term selection response could slow down from a decline in genetic variance or in selection differential or both. A model of conflict between truncation and stabilizing selection in infinite population size is analysed in terms of the reduction in selection differential. Under the assumption of a normal phenotypic distribution, the limit to selection is found to be a function of kappa, the intensity of truncation selection, omega 2, a measure of the intensity of stabilizing selection, and sigma 2, the phenotypic variance of the character. The maintenance of genetic variation at this limit is also analyzed in terms of mutation-selection balance by the use of the "House-of-cards" approximation. It is found that truncation selection can substantially reduce the equilibrium genetic variance below that when only stabilizing selection is acting, and the proportional reduction in variance is greatest when the selection is very weak. When truncation selection is strong, any further increase in the strength of selection has little further influence on the variance. It appears that this mutation-selection balance is insufficient to account for the high levels of genetic variation observed in many long-term selection experiments.

Predictions are made of the equilibrium genetic variances and responses in a metric trait under the joint effects of directional selection, mutation and linkage in a finite population. The "infinitesimal model" is analyzed as the limiting case of many mutants of very small effect, otherwise Monte Carlo simulation is used. If the effects of mutant genes on the trait are symmetrically distributed and they are unlinked, the variance of mutant effects is not an important parameter. If the distribution is skewed, unless effects or the population size is small, the proportion of mutants that have increasing effect is the critical parameter. With linkage the distribution of genotypic values in the population becomes skewed downward and the equilibrium genetic variance and response are smaller as disequilibrium becomes important. Linkage effects are greater when the mutational variance is contributed by many genes of small effect than few of large effect, and are greater when the majority of mutants increase rather than decrease the trait because genes that are of large effect or are deleterious do not segregate for long. The most likely conditions for "Muller's ratchet" are investigated.

To examine the effect of selection for body weight on levels of heterosis for reproductive traits, crosses were made between three groups of six lines of mice, one group unselected (controls) and the other two selected for high (large lines) and low (small lines) 6-week body weight, respectively. The coefficient of inbreeding of each line was about 0.60. In a comparison of purebred and crossbred progeny, both out of purebred mothers, there was on average 4% heterosis for number born, 3% for percentage weaned and 8% for numbers weaned. In a comparison of purebred and crossbred mothers, each mated to males of an unrelated strain and dissected on the 17th day of gestation, crossbreds had on average 1.6 more live embryos, which was 22% of the purebred mean. This comprised an increase of 0.6 corpora lutea, of 0.4 in survival to implantation and of 0.6 in subsequent survival to 17 days. The heterosis was similar whether the mothers had parents of the same or different size.

Estimates of the activities (Vmax) of four enzymes that generate the coenzyme NADPH, an absolute requirement for tissue fatty-acid synthesis, and of the concentration of NADP plus NADPH were made in lines of mice differing in fat content. These lines had been selected from the same base population for 20 generations, and 3 high, 3 low replicates and 1 unselected control were used. Analyses were performed on liver and gonadal fat pad (GFP) of males at 5 and 10 weeks of age. In both the liver and the GFP, measurable activities of the four enzymes: glucose-6-phosphate dehydrogenase (G6PDH), 6-phosphogluconate dehydrogenase (6PGDH), isocitrate dehydrogenase (IDH) and malic enzyme (ME) expressed per mg soluble protein were, with minor exceptions, higher in the Fat (F) than in the Lean (L) lines at both ages; the highest ratio being 2.2 for ME in the GFP. The relationships between these measurable activities (Vmax) and in vivo lipogenesis are not however known. When expressed per gram tissue, the ratios for F to L in the GFP were less than 1 in most cases, presumably because of the very different adipocyte numbers and/or sizes between the lines. There were no significant differences between the lines in the concentration of NADP plus NADPH per gram tissue in liver or GFP, suggesting that F lines converted NADP to NADPH faster than L lines. It is predicted that selection on the enzyme activities would be less efficient than direct selection at changing fat content.

Experimental checks on theoretical predictions of the build up of negative linkage disequilibrium with directional selection were made using abdominal bristle number in Drosophila melanogaster. Selection was practised for three generations before relaxation. Realized heritabilities and thus genotype variances were estimated by divergent selection. In one replicate, little change of variance occurred but in the other it increased substantially on relaxation. This result is compatible with a model of one or more genes of large effect at extreme frequencies in the base population. This is illustrated with Monte Carlo simulations. Interpretations of results was aided by considering the build up of negative disequilibrium.

A model of genetic variation of a quantitative character subject to the simultaneous effects of mutation, selection and drift is investigated. Predictions are obtained for the variance of the genetic variance among independent lines at equilibrium with stabilizing selection. These indicate that the coefficient of variation of the genetic variance among lines is relatively insensitive to the strength of stabilizing selection on the character. The effects on the genetic variance of a change of mode of selection from stabilizing to directional selection are investigated. This is intended to model directional selection of a character in a sample of individuals from a natural or long-established cage population. The pattern of change of variance from directional selection is strongly influenced by the strengths of selection at individual loci in relation to effective population size before and after the change of regime. Patterns of change of variance and selection responses from Monte Carlo simulation are compared to selection responses observed in experiments. These indicate that changes in variance with directional selection are not very different from those due to drift alone in the experiments, and do not necessarily give information on the presence of stabilizing selection or its strength.

Selection was practised for improved feed efficiency (gain/feed intake) of mice on two alternative feeding regimes. In one set of lines animals were fed ad libitum, in the other set they were individually fed a fixed amount of feed (about 10% below the control ad libitum intake) which was not changed over generations. For each treatment, a pair of replicate lines (E) were selected on efficiency from 3-5 weeks of age for 8 generations and another pair (L) from 5-7 weeks for 7 generations. A control line was maintained for both E and L lines. In terminal generations mice from each line were tested on each feeding regime, and carcasses of ad libitum fed mice were analysed.The realized heritability (within families) for efficiency averaged 13%, without much variation over treatments. In the E lines efficiency increased by about 18% of the control mean and in the L lines by about 60%, although absolute changes were small, and responses were similar on the two feeding regimes. Weights at the start of test decreased in the E lines and increased in the L lines; weights at the end of test increased in both.When tested on the alternative regimes, no interactions were detected for live weights, weight gains or efficiency; selection under fixed intake led to the same increase in appetite as did that under ad libitum.There were no interactions for carcass composition. Selection for efficiency led to an increase in fatness on both selection regimes and both weight ranges.

Methods are outlined for analyzing data on genotype frequencies at several codominant loci in random mating diploid populations. Maximum likelihood (ML) methods are given for estimating chromosomal frequencies. Using these, a succession of models of assumed independence of gene frequency are fitted. These are based on those used in multi-dimensional contigency tables, and tests for association (linkage disequilibrium), made using likelihood ratios. The methods are illustrated with an example.

Analysis of linkage disequilibrium D among restriction sites or bases in DNA sequences, arising from mutations in finite populations, depends on a knowledge of the variance-covariance structure of measures such as D2 between different pairs of sites. This requires evaluation of the eighth moments of gene frequencies among two, three, and four loci, and the necessary methodology is derived here and results are computed. While primary emphasis is placed on disequilibrium arising from mutation or gene conversion, the methodology also allows for the joint effects of only drift and recombination. Numerical results confirm that squared linkage disequilibria can have high variances and covariances.

Some aspects of Wright's great contribution to quantitative genetics and animal breeding are reviewed in relation to current research and practice. Particular aspects discussed are as follows: the utility of his definition of inbreeding coefficient in terms of the correlation of uniting gametes; the maintenance of genetic variation in the optimum model; the inter-relations between past and present animal-breeding practice and the shifting-balance theory of evolution.

The effects of inbreeding on heterozygosity, mean performance, variation within and between lines, and fixation or loss of genes are discussed. Theoretical predictions for degree of pedigree inbreeding and consequences for genes neutral with respect to fitness are outlined first, and then the likely modifications necessary to account for natural selection during inbred line development are considered. For illustration, some results from experiments with laboratory animals are reviewed. Inbred lines or F1 crosses between them are special genotypes, not representative of the population from which they were derived. Similarly, outbred stocks of laboratory animals that have a long history in the laboratory, or narrow base, or are maintained in small numbers, will lack variation and become like inbred lines. The efficacy of using either inbreds or such outbred stocks in toxicological screening is questioned on the ground that the range of genotypes is inadequate.

A theoretical investigation has been made of the influence of population size (N) and recombination fraction (c) on linkage disequilibrium (D) between a pair of loci. Two situations were studied: (i) where both loci had no effect on fitness and (ii) where they showed heterozygote superiority, but no epistacy.If the populations are initially in linkage equilibrium, then the mean value ofD remains zero with inbreeding, but the mean ofD (2) increases to a maximum value and decreases until fixation is reached at both loci. The tighter the linkage and the greater the selection, then the later is the maximum in the mean ofD (2) reached, and the larger its value. The correlation of gene frequencies,r, in the population of gametes within segregating lines was also studied. It was found that, for a range of selection intensities and initial gene frequencies, the mean value ofr (2) was determined almost entirely byN c and time, measured proportional toN.The implication of these results on observations of linkage disequilibrium in natural populations is discussed.