The Price (1970, 1972) equation is applied to the problem of describing the changes in the moments of allelic effects caused by selection, mutation and recombination at loci governing a quantitative genetic character. For comparable assumptions the resulting equations are the same as those obtained by different means by Barton & Turelli (1987; Turelli & Barton, 1989). The Price equation provides a natural framework within which to examine certain kinds of non-additive allelic effects, recombination and assortative mating. The use of the Price equation is illustrated by finding the equilibrium genetic variance under multiplicative dominance and epistasis and under assortative mating at an additive locus. The limitations of the use of recursion equations for the moments of allelic effects are also discussed.

Maternal inheritance is the non-Mendelian transmission of traits, from mothers to their offspring. Despite its presence in virtually all organisms, acting through a variety of mechanisms, the evolutionary consequences of maternal inheritance are not well understood. Here we review and extend a model of the inheritance and evolution of multiple quantitative characters with complex pathways of maternal effects. Extensions of the earlier model include common family environmental effects not associated with maternal phenotype, sexual dimorphism, and paternal effects (non-Mendelian influence of the father on offspring traits). We find that, in contrast to simple Mendelian inheritance, maternal inheritance produces qualitatively different evolutionary dynamics for two reasons: (1) the response to selection on a set of characters depends not only on their additive genetic variances and covariances, but also on maternal characters that influence them, and (2) time lags in the response to selection create a form of evolutionary momentum. These results have important implications for evolution in natural populations and practical applications in the economic improvement of domesticated species. We derive selection indices that maximize either the economic improvement in a single generation of artificial selection or the asymptotic rate of improvement in long-term selection programmes, based on individual merit or a combination of individual and family merit. Numerical examples show that accounting for maternal inheritance can lead to considerable increases in the efficiency of artificial selection.

Estimates of the activities (Vmax) of six enzymes involved in de novo fat synthesis were made in replicated lines of mice differing in fat content. These lines had been selected high and low for 20 generations with three replicates each of Fat, Control and Lean lines and for a further eight generations high and low as an unreplicated line. The activities of ATP-citrate lyase (ACL), acetyl-CoA carboxylase (ACC), fatty acid synthetase (FAS), cytoplasmic malate dehydrogenase (MDH), malic enzyme (ME) and pyruvate kinase (PK) were determined in vitro in both liver and gonadal fatpad tissues taken at ages five and ten weeks. The activities of ACL, ACC, FAS and ME were significantly higher in the Fat than the Lean lines, and the differences were more pronounced at the earlier age and in the gonadal fatpad where activities in the Fat lines were higher by factors of 3·5, 2·4, 2·5 and 3·5 respectively. The activity of PK was unchanged in each tissue. MDH activity was significantly lower in adipose tissue in the Fat lines than the Lean lines at age ten weeks but not at age five weeks or in liver tissue. Results from replicates indicated that random genetic drift affected enzyme activities but nevertheless significant changes in activity were associated with the direction of selection. The changes in enzyme activity reported here are similar to those known to be associated with major mutations causing obesity in mice.

Mutation-selection balance in a multi-locus system is investigated theoretically, using a modification of Bulmer's infinitesimal model of selection on a normally-distributed quantitative character, taking the number of mutations per individual (n) to represent the character value. The logarithm of the fitness of an individual with n mutations is assumed to be a quadratic, decreasing function of n. The equilibrium properties of infinitely large asexual populations, random-mating populations lacking genetic recombination, and random-mating populations with arbitrary recombination frequencies are investigated. With ‘synergistic’ epistasis on the scale of log fitness, such that log fitness declines more steeply as n increases, it is shown that equilibrium mean fitness is least for asexual populations. In sexual populations, mean fitness increases with the number of chromosomes and with the map length per chromosome. With ‘diminishing returns’ epistasis, such that log fitness declines less steeply as n increases, mean fitness behaves in the opposite way. Selection on asexual variants and genes affecting the rate of genetic recombination in random-mating populations was also studied. With synergistic epistasis, zero recombination always appears to be disfavoured, but free recombination is disfavoured when the mutation rate per genome is sufficiently small, leading to evolutionary stability of maps of intermediate length. With synergistic epistasis, an asexual mutant is unlikely to invade a sexual population if the mutation rate per diploid genome greatly exceeds unity. Recombination is selectively disadvantageous when there is diminishing returns epistasis. These results are compared with the results of previous theoretical studies of this problem, and with experimental data.