Adaptive Topography in Fertility-Viability Selection Models: The Haplodiploid Case

A linear combination of partial changes of mean fitnesses from one generation to the next one is shown to be approximately equal to the additive genetic variance in fitness after enough generations and away from equilibrium in random mating haplodiploid populations under arbitrary weak frequency-dependent selection on sex-differentiated viability of individuals and sex-differentiated fertility of matings controlled at a single multiallelic locus. The result can be applied to X-linked locus models in diploid populations. The result is used to deduce approximate adaptive topographies far frequency-independent selection models in the cases of nonsex-differentiated fertilities and multiplicative sex-differentiated fertilities and for kin selection models in family-structured populations under the assumptions of single insemination and multiple insemination of females. Multiple insemination creates frequency-dependent selection regimes.     

Fundamental theorem of natural selection and frequency-dependent selection: analysis of the matrix game diploid model

Following Ewens' interpretation about Fisher's fundamental theorem of natural selection, the matrix game model for diploid populations undergoing non-overlapping, discrete generations is investigated. The total genetic variance is decomposed and it is shown that the partial change in the mean fitness, which is equal to the additive genetic variance over the mean fitness, can be thought of as a change due only to the partial changes in the phenotypic frequencies.     

Selection and mutation at a diallelic X-linked locus

Equilibria and convergence of gene frequencies are studied in the case of a diallelic X-linked locus under the influence of selection and mutation. The model used is that of an infinite diploid population with nonoverlapping discrete generations and random mating. It is proved that if the mutation rates and fitnesses are constant and the mutation rates are less than one-third, then global convergence of gene frequencies to equilibria occurs. The phase portraits of the dynamical system describing the change of allelic frequencies from one generation to the next are determined. Convergence of gene frequencies is monotone from a certain generation on if every other generation is skipped. In the case without mutation, our proof of this monotone convergence simplifies G. Palm's original proof [37].     

Some Effects of Selection When There Is Partial Full-Sib Mating

If there is selection only for viability between zygote formation and adulthood, the frequency of a particular allele changes between these two stages of life. With complete random mating this is all that happens, but if there is a positive probability that full sibs mate, there is an extra change between adulthood and the appearance of zygotes in the next generation. This occurs because there are then correlated frequencies of the alleles carried by the mates. An expression for the change in the frequency of an allele, which incorporates these two effects, is derived, and the result is found to be consistent with earlier work by the author on the probability of survival of a rare allele in a large population. The result is inconsistent with the usual expression for the change in frequency of an allele when there is partial inbreeding because that expression does not incorporate the second change in frequency within one generation.     

Prediction of additive and dominance effects in selected or unselected populations with inbreeding

Summary A genetic model with either 64 or 1,600 unlinked biallelic loci and complete dominance was used to study prediction of additive and dominance effects in selected or unselected populations with inbreeding. For each locus the initial frequency of the favourable allele was 0.2, 0.5, or 0.8 in different alternatives, while the initial narrow-sense heritability was fixed at 0.30. A population of size 40 (20 males and 20 females) was simulated 1,000 times for five generations. In each generation 5 males and 10 or 20 females were mated, with each mating producing four or two offspring, respectively. Breeding individuals were selected randomly, on own phenotypic performance or such yielding increased inbreeding levels in subsequent generations. A statistical model containing individual additive and dominance effects but ignoring changes in mean and genetic covariances associated with dominance due to inbreeding resulted in significantly biased predictions of both effects in generations with inbreeding. Bias, assessed as the average difference between predicted and simulated genetic effects in each generation, increased almost linearly with the inbreeding coefficient. In a second statistical model the average effect of inbreeding on the mean was accounted for by a regression of phenotypic value on the inbreeding coefficient. The total dominance effect of an individual in that case was the sum of the average effect of inbreeding and an individual effect of dominance. Despite a high mean inbreeding coefficient (up to 0.35), predictions of additive and dominance effects obtained with this model were empirically unbiased for each initial frequency in the absence of selection and 64 unlinked loci. With phenotypic selection of 5 males and only 10 females in each generation and 64 loci, however, predictions of additive and dominance effects were significantly biased. Observed biases disappeared with 1,600 loci for allelic frequencies at 0.2 and 0.5. Bias was due to a considerable change in allelic frequency with phenotypic selection. Ignoring both the covariance between additive and dominance effects with inbreeding and the change in dominance variance due to inbreeding did not significantly bias prediction of additive and dominance effects in selected or unselected populations with inbreeding.     

The Effects of Disruptive and Stabilizing Selection on Body Size in DROSOPHILA MELANOGASTER. II. Analysis of Responses in the Thorax Selection Lines

An analysis was made of changes in mean and variance in some thorax selection lines. The decrease of mean thorax length in the stabilizing selection lines (S) was a consequence of a directional selection component, caused by the skewness of the frequency distributions. The slight or temporary increase of the phenotypic variance and the large increase of the mean value in the disruptive selection lines with random mating (D(R)) could be attributed to differences in reproduction between small and large flies (egg production and mating success). Phenotypic variability was high in two disruptive selection lines with compulsory mating of opposite extremes (D(-)). The mechanism of the change in variability was different in these replicate lines. In D(-)-1 the change was obtained by an increase of the environmental and the nonadditive genetic components of the variance. In D(-)-2 almost exclusively an increase of additive genetic variance occurred.     

Evolution under Fertility and Viability Selection

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.     

Selection in autotetraploids

Summary Theoretical studies indicated that response to selection would always be greater in diploid than in autotetraploid populations when gene frequency was the same in both, and that situations in which little or no response to selection could be expected would be more frequent in autotetraploids. Interpretation of the coefficient of selection in terms of escape from infection in a program of selection for disease or insect resistance indicated that moderate levels of escape from infection can drastically reduce response to selection in some cases.The zygotic constitution of an autotetraploid population will change as it approaches a new random mating equilibrium once selection pressure is relaxed. The changes will result in no change in the population mean if the trait under selection exhibits no dominance, but the mean will decrease slightly if there is dominance.     

Secondary theorem of natural selection in biocultural populations.

The "Secondary Theorem of Natural Selection," an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.     

Effective Size of Populations under Selection

Equations to approximate the effective size (N(e)) of populations under continued selection are obtained that include the possibility of partial full-sib mating and other systems such as assortative mating. The general equation for the case of equal number of sexes and constant number of breeding individuals (N) is N(e) = 4N/[2(1 - α(I)) + (S(k)(2) + 4Q(2)C(2)) (1 + α(I) + 2α(O))], where S(k)(2) is the variance of family size due to sampling without selection, C(2) is the variance of selective advantages among families (the squared coefficient of variation of the expected number of offspring per family), α(I) is the deviation from Hardy-Weinberg proportions, α(O) is the correlation between genes of male and female parents, and Q(2) is the term accounting for the cumulative effect of selection on an inherited trait. This is obtained as Q = 2/[2 - G(1 + r)], where G is the remaining proportion of genetic variance in selected individuals and r is the correlation of the expected selective values of male and female parents. The method is also extended to the general case of different numbers of male and female parents. The predictive value of the formulae is tested under a model of truncation selection with the infinitesimal model of gene effects, where C(2) and G are a function of the selection intensity, the heritability and the intraclass correlation of sibs. Under random mating r = α(I) = -1/(N - 1) and α(O) = 0. Under partial full-sib mating with an average proportion β of full-sib matings per generation, r & β and α(O) & α(I) & β/ (4 - 3β). The prediction equation is compared to other approximations based on the long-term contributions of ancestors to descendants. Finally, based on the approach followed, a system of mating (compensatory mating) is proposed to reduce rates of inbreeding without loss of response in selection programs in which selected individuals from the largest families are mated to those from the smallest families.     

Sex differences in linkage between autosomal loci

In the case of conventional selection theory with multiplicative gene action between loci and no sex differences except in crossover frequencies, it is shown that the usual conditions for stability hold when the mean of the recombination frequencies for the two sexes is used. For additive gene action between loci, it is shown that, after one generation of random mating, the gene frequencies of male and female origin are the same. This equality implies the nondecreasing property of the mean fitness function. Some attention is also given to neutral loci.     

Effects of Population Size and Selection Intensity on Responses to Disruptive Selection in DROSOPHILA MELANOGASTER

Disruptive selection for sternopleural bristle number with opportunity for random mating was done in the four treatment combinations of two population sizes (40 pairs and 8 pairs of selected parents) and two selection intensities (1 in 40 and 1 in 2). In each generation, matings among selected parents were observed in a mating chamber, and progeny collected separately from each female parent. In the high number, high selection intensity treatment, divergence between the high and low parts ceased about generation 11. The isolation index increased rapidly to generation 3, but then fluctuated to termination of the population at generation 17. The overall isolation index was significant, indicating a real tendency to assortative mating. The failure of the isolation index to increase after generation 3 was attributed to lower average mating fitness of high males (due to inbreeding) and reduced receptivity of low females (due to a homozygous lethal gene with a large effect on sternopleural bristle number in heterozygotes). In the two low number treatments, isolation indices fluctuated from generation to generation with no obvious trends, and none of the overall isolation indices were significantly different from zero. The high number, low selection intensity treatment showed very little divergence, and one of the replicates showed, in contrast with expectation and the high number, high selection intensity treatment, a significant tendency to disassortative mating. Intense disruptive selection may lead to assortative mating.     

Joint effect of selection, linkage and partial inbreeding on additive genetic variance in an infinite population

Under the inifinitesimal model of gene effects, selection reduces the additive genetic variance by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of additive genetic variance is enhanced. Inbreeding in an infinite population also alters the additive genetic variance through the generation of positive association among genes within a locus. In the present study, the joint effect of selection, linkage and partial inbreeding (partial selfing or partial full-sib mating) on the additive genetic variance was modeled. The recurrence relations of the additive genetic variance between successive generations and the prediction equation of the asymptotic additive genetic variance were derived. Numerical computation showed that although partially inbred populations initially maintain larger genetic variances, the accumulated effect of selection overrides the effect of inbreeding. Stochastic simulation was carried out to check the precision of prediction, showing that the obtained equations give a satisfactory prediction during initial generations. However, the predicted values always overestimate the simulated values, especially in later generations. Based on these results, possible extensions and perspectives of the assumed model were discussed.     

Selection with Partial Selfing. I. Mass Selection

The expected responses to mass selection carried out before or after reproduction in a population whose members all have a fixed probability of self pollination (s) are formulated using covariances of relatives and their component quadratic functions for a model with arbitrary additive and dominance effects. The response measured in the first generation offspring after selection (immediate gain) can differ from that retained when the population has regained equilibrium (permanent gain). The population mean behaves in a predictable manner during the return to equilibrium, and its value at any time can be predicted from earlier generations. The permanent gain from selection after reproduction is always (1 + s)/2 times as large as that from selection before reproduction, but the relationship of the immediate gains depends on the genetic model assumed. Numerical analysis applied to a model with two alleles per locus and varying allele frequencies, dominance ratios and numbers of loci showed that the proportion of the immediate gain retained at equilibrium was reduced with the large inbreeding depression associated with increasing dominance levels and numbers of loci and was generally lower for selection after reproduction than before. In the absence of information as to the magnitude of genetic variances and inbreeding depression in species reproducing by partial selfing, the importance of this phenomenon is unknown.     

The distribution of allelic effects under mutation and selection

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.     

Selection for Increased Mutation Rates with Fertility Differences between Matings

Previous studies of mutation modification have considered models in which selection is a result of viability differences that are sex symmetric. The results of a numerical study of a model in which selection is a result of fertility differences between mated pairs demonstrate that the type of selection to which a population is subject can have a significant impact on the evolution of various parameters of the genetic system. When the fertility of matings between individuals with different genotypes exceeds the fertility of at least some of the matings between individuals with the same genotype, selection may favor increased rates of mutation, in contrast to the results from all existing constant viability models with random mating and infinite population size. Increased mutation rates are most frequently favored when forward and back mutation occur at approximately equal rates and when the modifying locus is loosely linked to the selected locus. We present one example in which selection favors increased rates of mutation even though the selection scheme is reducible to one of differential viability between the sexes.     

Partial male sterility and the evolution of nuclear gynodioecy in plants

Gynodioecy, a genetic dimorphism of females and hermaphrodites, is pertinent to an understanding of the evolution of plant gender, mating and genetic variability. Classical models of nuclear gynodioecy attribute the maintenance of the dimorphism to frequency-dependent selection in which the female phenotype has a fitness advantage at low frequency owing to a doubled ovule fertility. Here, I analyse explicit genetic models of nuclear gynodioecy that expand on previous work by allowing partial male sterility in combination with either fixed or dynamically evolving mutational inbreeding depression. These models demonstrate that partial male sterility causes fitness underdominance at the mating locus, which can prevent the spread of females. However, if partial male sterility is compensated by a change in selfing rate, overdominance at the mating locus can cause the spread of females. Overdominance at introduction of the male sterility allele can be caused by high inbreeding depression and a lower selfing rate in the heterozygote, by purging of mutations by a higher selfing rate in the heterozygote, and by low inbreeding depression and a higher selfing rate in the heterozygote. These processes might be of general importance in the maintenance of mating polymorphisms in plants.     

Effect of Selection on the Failure Rate Function

Effect of selection procedures such as truncation and genotypic selection, are studied using selection differentials. These studies are mainly restricted to the study of mean or percentiles of the characteristics of interest. However, in situations where improvement on longetivity is itself of interest, in addition to studying mean or percentiles, one may study the reliability characteristics such as Failure Rate, Mean Residual Life etc., of the lifetimes of individuals. In this paper, we study the effect of genotypic selection when the distribution function of the lifetime is governed by a single locus with 2 alleles. We compute the difference in the failure rate function from one generation to the next generation for different values of n, the number of generations. We also obtain the limiting values of the failure rate function as n and compute the number of generations required for the failure rate to reach a value close enough to the limiting failure rate function. We also consider an example in a two-loci case and study the effect of selection on the failure rate function.     

Simulation of X-Linked Selection in Drosophila

The change in gene frequency for two X-linked mutants, y and w, in a number of experiments was compared to that predicted from a genetic simulation program which utilized estimated differences in relative mating ability, fecundity, and viability. The simulation gave excellent predictions of gene frequency change even when experiments were started with different initial gene frequencies in the males and females or when the two loci were segregating simultaneously. The rate of elimination was slower when there were unequal initial gene frequencies than when males and females had equal initial gene frequencies. Simulation demonstrated that this was a general phenomenon when there is strong selection but that the opposite is true for weak selection. In two other experiments, the mating advantage of wild-type males was balanced by a fecundity advantage in mutant females. In all four replicates of both experiments, the mutant was maintained for several generations at the high initial frequency but then decreased quickly and was eliminated. Results obtained restarting one of these experiments with flies from a generation after the decline in gene frequency indicated that a linked gene and not frequency-dependent selection was responsible for the unpredictable gene-frequency change in the mutant. Using a least squares technique, it was found that a recessive fecundity locus 15 map units from the w locus gave the best fit for both experiments.     

选育过程中的粤黄鸡血液淀粉酶（Amy—1）基因频率的世代变化

本文调查了粤黄鸡两个选育方向和交配方式均不同的品系的血液淀粉酶(Amy-1)各世代的分布情况,并以不同Amy-1类型的粤黄鸡作了交配试验。发现两个品系各世代都始终保持Amy-1分布不平衡状态,这种分布不平衡可能由各种Amy-1基因型生活力不同引起;另外,人工选择也可能造成Amy-1基因频率的改变。交配方式是否对Amy-1基因频率的改变起作用值得进一步研究。