Maintaining evolvability.

Although molecular methods, such as QTL mapping, have revealed a number of loci with large effects, it is still likely that the bulk of quantitative variability is due to multiple factors, each with small effect. Typically, these have a large additive component. Conventional wisdom argues that selection, natural or artificial, uses up additive variance and thus depletes its supply. Over time, the variance should be reduced, and at equilibrium be near zero. This is especially expected for fitness and traits highly correlated with it. Yet, populations typically have a great deal of additive variance, and do not seem to run out of genetic variability even after many generations of directional selection. Long-term selection experiments show that populations continue to retain seemingly undiminished additive variance despite large changes in the mean value. I propose that there are several reasons for this. (i) The environment is continually changing so that what was formerly most fit no longer is. (ii) There is an input of genetic variance from mutation, and sometimes from migration. (iii) As intermediate-frequency alleles increase in frequency towards one, producing less variance (as p --> 1, p(1 - p) --> 0), others that were originally near zero become more common and increase the variance. Thus, a roughly constant variance is maintained. (iv) There is always selection for fitness and for characters closely related to it. To the extent that the trait is heritable, later generations inherit a disproportionate number of genes acting additively on the trait, thus increasing genetic variance. For these reasons a selected population retains its ability to evolve. Of course, genes with large effect are also important. Conspicuous examples are the small number of loci that changed teosinte to maize, and major phylogenetic changes in the animal kingdom. The relative importance of these along with duplications, chromosome rearrangements, horizontal transmission and polyploidy is yet to be determined. It is likely that only a case-by-case analysis will provide the answers. Despite the difficulties that complex interactions cause for evolution in Mendelian populations, such populations nevertheless evolve very well. Longlasting species must have evolved mechanisms for coping with such problems. Since such difficulties do not arise in asexual populations, a comparison of epistatic patterns in closely related sexual and asexual species might provide some important insights.

Inferring purging from pedigree data.

The harmful effects of inbreeding can be reduced if deleterious recessive alleles were removed (purged) by selection against homozygotes in earlier generations. If only a few generations are involved, purging is due almost entirely to recessive alleles that reduce fitness to near zero. In this case the amount of purging and allele frequency change can be inferred approximately from pedigree data alone and are independent of the allele frequency. We examined pedigrees of 59,778 U.S. Jersey cows. Most of the pedigrees were for six generations, but a few went back slightly farther. Assuming recessive homozygotes have fitness 0, the reduction of total genetic load due to purging is estimated at 17%, but most of this is not expressed, being concealed by dominant alleles. Considering those alleles that are currently expressed due to inbreeding, the estimated amount of purging is such as to reduce the expressed load (inbreeding depression) in the current generation by 12.6%. That the reduction is not greater is due mainly to (1) generally low inbreeding levels because breeders in the past have tended to avoid consanguineous matings, and (2) there is essentially no information more than six generations back. The methods used here should be applicable to other populations for which there is pedigree information.

Perspective: Here's to Fisher, additive genetic variance, and the fundamental theorem of natural selection.

Fisher's fundamental theorem of natural selection, that the rate of change of fitness is given by the additive genetic variance of fitness, has generated much discussion since its appearance in 1930. Fisher tried to capture in the formula the change in population fitness attributable to changes of allele frequencies, when all else is not included. Lessard's formulation comes closest to Fisher's intention, as well as this can be judged. Additional terms can be added to account for other changes. The "theorem" as stated by Fisher is not exact, and therefore not a theorem, but it does encapsulate a great deal of evolutionary meaning in a simple statement. I also discuss the effectiveness of reproductive-value weighting and the theorem in integrated form. Finally, an optimum principle, analogous to least action and Hamilton's principle in physics, is discussed.

On the persistence and pervasiveness of a new mutation.

It has frequently been assumed that the persistence of a deleterious mutation (the average number of generations before its loss) and its pervasiveness (the average number of individuals carrying the gene before its loss) are equal. This is true for a particular simple, widely used infinite model, but this agreement is not general. If hs >> 1/(4N(e)), where hs is the selective disadvantage of mutant heterozygotes and N(e) is the effective population number, the contribution of homozygous mutants can be neglected and the simple approximate formula 1/hs gives the mean pervasiveness. But the expected persistence is usually much smaller, 2(log(e)(1/2hs) + 1 - gamma) where gamma = 0.5772. For neutral mutations, the total number of heterozygotes until fixation or loss is often the quantity of interest, and its expected value is 2N(e), with remarkable generality for various population structures. In contrast, the number of generations until fixation or loss, 2(N(e)/N)(1 + log(e)2N), is much smaller than the total number of heterozygotes. In general the number of generations is less than the number of individuals.

The Rate of Change of a Character Correlated with Fitness.

An extended form of Fisher's Fundamental Theorem of Natural Selection gives the rate of change of the mean value, [Formula: see text], of a measured character. For a character determined by multiple alleles at two loci, this is [Formula: see text] where the Newtonian superior dot means the time derivative and the circle is the time derivative of the logarithm. Covg (m, γ) is the genic (additive genetic) covariance of the character and fitness. Specifically, it is the covariance of the average excess of an allele for fitness and its average effect on the character. [Formula: see text] is the average rate of change of the value of the character for individual genotypes, weighted by their frequencies. The value could be nonzero because of changing environments or change in the age distribution of the population. The third term on the right is the average over all pairs of alleles at both loci of the product of the dominance deviation and the rate of change of ln θ(n), where θ(n) is a measure of departure from random proportions. The last term is a similar expression for epistatic interactions. If selection is much weaker than recombination, after several generations, the last two terms are much smaller than the first. When the measured character is fitness, our result reduces to Kimura's generalization of Fisher's Fundamental Theorem of Natural Selection.