For a finite diploid population with no mutation, migration and selection, equations for the deviation of observed genotype frequencies from Hardy–Weinberg proportions are derived in this paper for monoecious species and for autosomal and sex-linked loci in dioecious species. It is shown that the genotype frequency deviation in finite random-mating populations results from the difference between the gene frequencies of male and female gametes, which is determined by two independent causes: the gene frequency difference between male and female parents and the sampling error due to the finite number of offspring. Previous studies have considered only one of the causes and the equations derived by previous authors are applicable only in the special case of random selection. The general equations derived here for both causes incorporate the variances and covariances of family size and thus they reduce to previous equations for random selection. Stochastic simulations are run to check the predictions from different formulae. Non-random mating and variation in census size are considered and the applications of the derived formulae are exemplified.

A model of pleiotropy with N diallelic loci contributing additively to N quantitative traits and stabilizing selection acting on each of the traits is considered. Every locus has a major contribution to one trait and a minor contribution to the rest of them, while every trait is controlled by one major locus and N−1 minor loci. It is demonstrated that a stable equilibrium with the allelic frequency equal to 0·5 in all N loci can be maintained in such a model for a wide range of parameters. Such a ‘totally polymorphic’ equilibrium is maintained for practically any strength of selection and any recombination, if the relative contribution by a minor locus to a trait is less than 20 % of the contribution by a major locus. The dynamic behaviour of the model is shown to be quite complex with a possibility under sufficiently strong selection of multiple stable equilibria and positive linkage disequilibria between loci. It is also suggested that pleiotropy among loci controlling traits experiencing direct selection can be responsible for apparent selection on neutral traits also controlled by these loci.

We have studied variability maintained in a quantitative trait by the balance between symmetric mutation and direct stabilizing selection with a fluctuating optimum. Using a simulational computer model, we have found that wide fluctuations, such that the range of the optimum changes exceeds the width of the fitness curve, increase the trait variance, often by two or three orders of magnitude, over its value under constant selection. This happens because such fluctuations cause frequent allele substitutions at the loci that control the trait. At any particular moment the variance is increased mostly due to one or several loci where more than one allele is currently common. The data on fluctuating selection in nature are reviewed

The exact test for neutrality based on the Ewens sampling distribution described previously (Slatkin, 1994) is not correct. The problem is that the test as described is based on the probability of the ordered configuration of numbers of alleles, while it should be based on the probability of the unordered configuration. The correctly implemented exact test leads to results that are similar to those from the homozygosity test proposed by Watterson (1977) for relatively small sample sizes but can still differ substantially for larger sample sizes. Programs to perform the exact test are available from the author.

Both stabilizing and directional selection acting on one or many quantitative traits usually reduce the genetic variance in a polymorphic population. Amphimixis and recombination restore the variance, pushing it closer to its value under linkage equilibrium. They thus increase the response of the population to fluctuating selection and decrease the genetic load when the mean phenotype is far from optimum. Amphimixis can have a short-term advantage over apomixis if selection fluctuates frequently and widely, so that every genotype often has a low fitness. Such selection causes high genetic variance due to frequent allele substitutions, and a high load even with amphimixis. Recombination in an amphimictic population is maintained only if selection is usually strong and effectively directional. A modifier allele causing free recombination can have a significant advantage only if fluctuations of selection are such that the load is substantial. With smaller fluctuations, an intermediate recombination rate can be established, either due to fixation of alleles that cause such a rate or due to the stable coexistence of alleles causing high and low recombination. If many traits simultaneously are under fluctuating selection, amphimixis and recombination can be maintained when selection associated with individual traits is weaker and the changes in their mean values are smaller than with a single trait. Still, the range of changes of the fitness optima in each trait must be at least of the order of the trait's standard deviation, and the total load must be high.

In plant species, typical gene mapping strategies use populations initiated from crosses between two inbred lines. However, schemes including more than two parents could be used. In this paper, a new approach is introduced which uses a four-way cross population derived from four inbred lines. The four-way cross design for mapping quantitative trait loci (QTLs) provides tests for QTL segregation in four lines simultaneously in one experiment. Therefore, it is a more economical strategy than oneusing line crosses between only two lines. The new strategy also increases the probability of detecting QTLs if they segregate in one line cross but not in the other. A multiple linear regression analysis is used for QTL detection. It is proven that the expected residual variance from the regression analysis differs from the pure environmental variance. Correction for the bias is proposed and verified by computer simulations.