The role of finite population size and linkage in response to continued truncation selection

Summary In an attempt to analyse long-term response in finite dioecious populations, selection processes are simulated on a computer with situations of parental population size, linkages between loci, selection intensity, and heritability, specified in a 3⁴ factorial design. A diploid polygenic system of 40 loci on 4 chromosomes is considered for additive genes. Linkage levels are specified as free recombinations, adjacent loci 5 map units apart, and as clusters on chromosomes with a distance of only .5 units between adjacent loci. Parental populations of 8, 16, and 64, truncation selection of 1/2, 1/4, and 1/8 of the progeny each generation, and initial heritability of 1, 1/3, and 1/9 are simulated for various populations.For these populations, which are initially samples from a theoreticalHardy-Weinberg situation, it is shown that an initial linear phase of response, which may last for only 2 or 3 generations in some cases, depends on the intensity of selection alone. The effects and interactions of all the above factors on the curvilinearity of response in later generations are analysed. It appears that linkages between loci have a strong influence in reducing the rate of response and the total response. In the extreme cases of gene clusters in a parental population size of 8 with low heritability, truncation selection is relatively almost completely ineffective in causing change in the mean over generations. The effect of tight linkage is also exhibited in causing more reduction in genotypic variance than can be accounted for by corresponding response.The depressing effect of finiteness of population size on the rate of response and the total response appears to increase in geometric proportion with linkages between loci. The number of generations to fixation appears to be reduced in a similar manner. A strong interaction between population size and linkage is thereby found in various analyses. With parental populations as large as 64, linkage effects on response are negligible when recombinations between adjacent loci are .05 or more. In such situations there is a slower rate of response in later generations with linkage but the total response attained and the rate of fixation of inferior genes is about the same as for free recombinations. Increase in the intensity of selection appears to augment the effects of linkage in reducing the rate of response in later generations. This type of interaction is attributed to the accumulation of gametic disequilibria due to selection which are retained in the population over generations with linkage.

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Zusammenfassung In der Absicht, das Verhalten einer begrenzten diözischen Population über einen langen Zeitraum zu analysicren, wurden Selektionsvorgänge auf einem Computer simuliert. Hierbei wurden die Größe der Elterpopulation, die Koppelung zwischen den Loci, die Selektionsintensität und die Heritabilität in einem 3⁴-faktoriellen Versuch variiert. Es wird ein diploides polygenes System mit vierzig Loci auf vier Chromosomen mit additiver Genwirkung zugrunde gelegt. Für die Koppelungsbeziehungen werden freie Rekombination, ein Abstand von fünf Rekombinationseinheiten zwischen benachbarten Loci und die Bildung von Genclustern auf den Chromosomen mit jeweils nur 0,5Morgan-Einheiten Abstand zwischen benachbarten Loci angenommen. Es werden elterliche Populationen des Umfanges 8, 16 und 64, trunkierende (stutzende) Selektion mit einer Fraktion von 1/2, 1/4 und 1/8 der Nachkommen je Generation und eine ursprüngliche Heritabilität von 1, 1/3 und 1/9 für verschiedene Populationen simuliert.Für alle jene Populationen, die ursprünglich als Stichproben aus einer theoretischenHardy-Weinberg-Situation stammen, kann gezeigt werden, daß eine anfänglich lineare Phase der Reaktion, die in einigen Fällen nur über zwei bis drei Generationen anhält, allein von der Selektionsintensität abhängt. Die Wirkungen und Wechselwirkungen aller oben genannten Faktoren auf die Nichtlinearität der Reaktion in späteren Generationen wird untersucht. Es zeigt sich, daß Koppelung zwischen den Loci einen starken Einfluß auf die Reduktion der Reaktionsgeschwindigkeit und auf die Endreaktion ausübt. In dem extremen Fall der Gencluster in einer Ausgangspopulation des Umfanges 8 mit geringer Heritabilität ist die trunkierende Selektion hinsichtlich der Änderung des Mittels über Generationen hinweg praktisch völlig unwirksam. Die Wirkung enger Koppelung manifestiert sich außerdem in einer stärkeren Reduktion der genotypischen Varianz, als sie auf Grund der entsprechenden Reaktion erklärt werden kann. Der reduzierende Effekt der Begrenzung des Populationsumfanges auf die Reaktionsgeschwindigkeit und die Endreaktion erweist sich als geometrisch proportional zur Koppelung zwischen den Loci. Die Zahl der Generationen bis zur Fixierung wird in ähnlicher Weise reduziert. Hierbei wird eine starke Wechselwirkung zwischen der Populationsgröße und der Koppelung in den verschiedenen Untersuchungen beobachtet. Der Einfluß der Koppelung auf die Reaktion der Populationen kann vernachlässigt werden, wenn die elterliche Population den Umfang 64 hat und die Rekombination zwischen benachbarten Loci 0,05 übersteigt. In derartigen Situationen gibt es zwar eine langsamere Antwortrate in späteren Generationen mit Koppelung, jedoch ist die Endreaktion, die erreicht wird, und die Fixierungsrate überlegener Gene etwa die gleiche wie bei freier Spaltung. Eine Zunahme der Selektionsintensität scheint die Wirkung der Koppelung hinsichtlfch der Reduktion der Reaktionsgeschwindigkeit in späteren Generationen zu vergrößern. Dieser Typ der Wechselwirkung wird der Häufung gametischer Ungleichgewichte, die infolge der Selektion über Generationen in der Population erhalten werden, zugeschrieben.

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Journal Paper No. 5872, Iowa Agriculture and Home Economics Experiment Station, Ames, supported by National Science Foundation Grant 19218 and National Institute of Health Grant GM-13827.

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On leave fromWest Pakistan Agricultural University Lyallpur.

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Statistical Laboratory and Department of Animal Science, respectively.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.     

Population dynamics of the additive polygenic system under truncation selection]

Common features of the equations describing dynamics of the additive polygenic system under truncation selection are summarized. A combination of parameters playing the role of the effective selective pressure on the ith polygenic locus was revealed. The product of mean relative fitnesses of the individual polygenic loci, [formula: see text], was shown to play the role of relative mean fitness of the polygenic population. This value depends on the measurable parameters of the character distribution in the population: [formula: see text]. It was shown that under the constant population number during truncation selection, the characteristic of the best genotype increases, [formula: see text]; which is also a product of the frequencies of preferable genotypes at individual polygenic loci. This value plays the role of the proportion of the number of the best ("champion") genotype in the population. In fact, this is the champion genotype polygene consensus pattern frequency, which a priori indicates the possibility of the champion pattern fixation. The analogue of Haldane's dilemma for the polygenic system which restrict the number of polygenes simultaneously subjected to adaptive evolution [formula: see text] was obtained for the case of constant effective population number (Ne = const).     

On the effects of background selection in small populations on comparisons of molecular variation

Deleterious mutations affect genetic variation at linked neutral loci. Neutral variation can be reduced due to background selection, but in small population and with tight linkage such variation may increase due to associative overdominance. Here I report the results of computer simulations of diploid genotypes in small populations, where I look at the effect of deleterious mutations and linkage on comparisons of intra- and interspecific variation. Each chromosome consisted of 2000 loci where deleterious and neutral mutations occurred. The ratio of nonsynonymous to synonymous substitution rates (Ka/Ks) either increases with tight linkage or is unaffected, depending on the strength of selection. The ratio of the numbers of segregating mutations to the number of fixed mutations decreases under the conditions leading to background selection but can increase at tight linkage. Numbers of segregating sites (Sn) are less affected than nucleotide site diversity (pi), pi reduces more than Sn at intermediate linkage, but pi increases more than Sn when linkage is tight. Similar effects as found for Sn and pi are observed for heterozygosity and variance in allele size of tandem repeat loci.     

On the probability of correct selection for large k populations, with application to microarray data

One frontier of modern statistical research is the problems arising from data sets with extremely large k (>1000) populations, e.g. microarray and neuroimaging data. For many such problems the focus shifts from testing for significance to selecting, filtering, or screening. Classical Ranking and Selection Methodology (RSM) studied the probability of correct selection (PCS). PCS is the probability that the "best" (t = 1) of k populations is truly selected, according to some specified criteria of best. This paper extends and adapts two selection goals from the RSM literature that are suitable for large k problems (d-best and G-best selection). It is then shown how estimation of PCS for selecting multiple (t > 1) populations with d-best and G-best selection can be implemented to provide a useful measure of the quality of a given selection. A simulation study and the application of the proposed method to a benchmark microarray data set show it is an effective and versatile tool for assessing the probability that a particular gene selection or gene filtering step truly obtains the best genes. Moreover, the proposed method is fully general and may be applied to any such extremely large k problem.     

Variance in quantitative traits due to linked dominant genes and variance in heterozygosity in small populations

The influence of small population size (N) on the genetic variance within and between randomly bred unselected lines, with selfing permitted, is investigated for a model of a quantitative trait determined by linked genes that show dominance within loci but are additive over loci. Formulae for within-line variance include terms in linkage disequilibrum, which occurs by chance in the lines and these are evaluated in terms of N, map length and gene number.—The expected variance within lines is increased by this disequilibrium, quite substantially if there are many loci, with most of the increase being between or within full-sib families and almost no change expected between half-sib families or in the covariance of offspring and parent. If all loci are unlinked, there is no increase in variance within full-sib families. The variance between lines is little affected by disequilibrum generated by chance.—Expressions for the variance between individuals in heterozygosity over the whole genome are special cases of those for the variance due to linked dominated genes, and formulae are given and evaluated. The coefficient of variation of heterozygosity is at least (see PDF) and can be much higher for species with few chromosomes.     

Evolution of altruism under group selection in large and small populations in fluctuating environments

A continuous, graded form of group selection which does not involve extinction of demes can effectively oppose selection on the individual level against an altruistic allele under fluctuating environments in infinitely large demes among which uniform mixing occurs every generation. Although group selection cannot alter the conditions necessary for the initial increase of altruistic alleles, group selection can significantly influence the stationary distribution of gene frequency which is attained once stochastic forces have allowed theirintroduction. Drift is a more effective source of variation than fluctuations in selection when the variance in selection is moderate to small. High numbers of demes promote polymorphism under both graded group selection and extinction group selection.     

Extended linkage disequilibrium surrounding the hemoglobin E variant due to malarial selection

The hemoglobin E variant (HbE; ( beta )26Glu-->Lys) is concentrated in parts of Southeast Asia where malaria is endemic, and HbE carrier status has been shown to confer some protection against Plasmodium falciparum malaria. To examine the effect of natural selection on the pattern of linkage disequilibrium (LD) and to infer the evolutionary history of the HbE variant, we analyzed biallelic markers surrounding the HbE variant in a Thai population. Pairwise LD analysis of HbE and 43 surrounding biallelic markers revealed LD of HbE extending beyond 100 kb, whereas no LD was observed between non-HbE variants and the same markers. The inferred haplotype network suggests a single origin of the HbE variant in the Thai population. Forward-in-time computer simulations under a variety of selection models indicate that the HbE variant arose 1,240-4,440 years ago. These results support the conjecture that the HbE mutation occurred recently, and the allele frequency has increased rapidly. Our study provides another clear demonstration that a high-resolution LD map across the human genome can detect recent variants that have been subjected to positive selection.     

Optimal marker-assisted selection to increase the effective size of small populations

An approach to the optimal utilization of marker and pedigree information in minimizing the rates of inbreeding and genetic drift at the average locus of the genome (not just the marked loci) in a small diploid population is proposed, and its efficiency is investigated by stochastic simulations. The approach is based on estimating the expected pedigree of each chromosome by using marker and individual pedigree information and minimizing the average coancestry of selected chromosomes by quadratic integer programming. It is shown that the approach is much more effective and much less computer demanding in implementation than previous ones. For pigs with 10 offspring per mother genotyped for two markers (each with four alleles at equal initial frequency) per chromosome of 100 cM, the approach can increase the average effective size for the whole genome by approximately 40 and 55% if mating ratios (the number of females mated with a male) are 3 and 12, respectively, compared with the corresponding values obtained by optimizing between-family selection using pedigree information only. The efficiency of the marker-assisted selection method increases with increasing amount of marker information (number of markers per chromosome, heterozygosity per marker) and family size, but decreases with increasing genome size. For less prolific species, the approach is still effective if the mating ratio is large so that a high marker-assisted selection pressure on the rarer sex can be maintained.     

Natural selection promotes the conservation of linkage of co-expressed genes

Although there is increasing evidence that eukaryotic gene order is not always random, there is no evidence that putatively favourable gene arrangements are preserved by selection more than expected by chance. In yeast (Saccharomyces cerevisiae), for example, co-expressed genes tend to be linked, but whether such gene pairs tend to remain linked more often than expected under null neutral expectations is not known. We show using gene pairs in the S. cerevisiae–Candida albicans comparison that highly co-expressed gene pairs are conserved as pairs at about twice the average rate. However, co-expressed genes also tend to be in close physical proximity and, as expected from a null neutral model, genes (be they co-expressed or not) that are physically close together tend to be retained more often. This physical proximity, however, only accounts for a small proportion of the enhanced degree of conservation of co-expressed gene pairs. These results demonstrate that purely neutralist models of gene order evolution are not realistic.     

On estimating the linkage of marker genes to viability genes controlling inbreeding depression

Statistical properties and extensions of Hedrick and Muona's method for mapping viability alleles causing inbreeding depression are discussed in this paper. Their method uses the segregation ratios among selfed progeny of a marker-locus heterozygote to estimate the viability reduction, s, of an allele and its recombination fraction, c, with the marker. Explicit estimators are derived for c and s, including expressions for their variances. The degree of estimation bias is examined for cases when (1) the viability allele is partially recessive and (2) the marker locus is linked to two viability loci. If linkage or viability reduction is moderate, very large sample sizes are required to obtain reliable estimates of c and s, in part because these estimates show a statistical correlation close to unity. Power is further reduced because alleles causing viability reduction often occur at low frequency at specific loci in a population. To increase power, we present a statistical model for the joint analysis of several selfed progeny arrays selected at random from a population. Assuming a fixed total number of progeny, we determine the optimal number of progeny arrays versus number of progeny per array under this model. We also examine the increase of information provided by a second, flanking marker. Two flanking markers provide vastly superior estimation properties, reducing sample sizes by approximately 95% from those required by a single marker.     

On the utility of linkage disequilibrium as a statistic for identifying targets of positive selection in nonequilibrium populations

A critically important challenge in empirical population genetics is distinguishing neutral nonequilibrium processes from selective forces that produce similar patterns of variation. We here examine the extent to which linkage disequilibrium (i.e., nonrandom associations between markers) improves this discrimination. We show that patterns of linkage disequilibrium recently proposed to be unique to hitchhiking models are replicated under nonequilibrium neutral models. We also demonstrate that jointly considering spatial patterns of association among variants alongside the site-frequency spectrum is nonetheless of value. Through a comparison of models of equilibrium neutrality, nonequilibrium neutrality, equilibrium hitchhiking, nonequilibrium hitchhiking, and recurrent hitchhiking, we evaluate a linkage disequilibrium (LD) statistic (omega(max)) that appears to have power to identify regions recently shaped by positive selection. Most notably, for demographic parameters relevant to non-African populations of Drosophila melanogaster, we demonstrate that selected loci are distinguishable from neutral loci using this statistic.     

Effect of linkage on behavior of mutant genes in finite populations

In this paper, the effectiveness of the apparent or “associated” overdominance caused by linked selected loci on maintaining genetic variability was investigated using diffusion models. It was shown that when a population is subdivided, the amount of linkage disequilibrium is determined by the local population size if the recombination fraction is much larger than the migration rates. Also, it was shown that the associative overdominance temporarily creates a sort of inertia against perturbation in gene frequencies even if it cannot endow any definite equilibrium values. Hence, it has no long term effect on the rate of mutant substitution in evolution.     

含有母性影响遗传基因的连锁定位系的构建

母性影响遗传基因由于其杂交后代的表型受母本基因型的影响, 而不能直接反映当代个体的基因型, 这给连锁定位测交亲本(三隐性或双隐性系统)的培育带来困难, 从而影响这类基因的定位研究。设计了一套杂交培育方案, 其核心是使母性影响遗传基因先纯合, 再使非母性影响遗传基因纯合。采用所设计的方案, 成功培育了家蚕第13连锁群的赤蚁(ch)、无鳞毛翅(nlw, 新突变)和褐色卵t (b-t, 母性影响遗传)的三隐性系统, 并培育了第19连锁群的狭胸(nb)和第二肾形卵(ki-2, 母性影响遗传, 待定位突变基因)的双隐性系统。     

Optimum contribution selection in large general tree breeding populations with an application to Scots pine

Development of selection methods that optimises selection differential subject to a constraint on the increase of inbreeding (or coancestry) in a population is an important part of breeding programmes. One such method that has received much attention in animal breeding is the optimum contribution (OC) dynamic selection method. We implemented the OC algorithm and applied it to a diallel progeny trial of Pinus sylvestris L. (Scots pine) focussing on two traits (total tree height and stem diameter). The OC method resulted in a higher increase in genetic gain (8–30%) compared to the genetic gain achieved using standard restricted selection method at the same level of coancestry constraint. Genetic merit obtained at two different levels of restriction on coancestry showed that the benefit of OC was highest when restriction was strict. At the same level of genetic merit, OC decreased coancestry with 56 and 39% for diameter and height, respectively, compared to the level of coancestry obtained using unrestricted truncation selection. Inclusion of a dominance term in the statistical model resulted in changes in contribution rank of trees with 7 and 13% for diameter and height, respectively, compared to results achieved by using a pure additive model. However, the genetic gain was higher for the pure additive model than for the model including dominance for both traits.     

Habitat selection reduces extinction of populations subject to Allee effects

Theoretical studies indicate that a single population under an Allee effect will decline to extinction if reduced below a particular threshold, but the existence of multiple local populations connected by random dispersal improves persistence of the global population. An additional process that can facilitate persistence is the existence of habitat selection by dispersers. Using analytic and simulation models of population change, I found that when habitat patches exhibiting Allee effects are connected by dispersing individuals, habitat selection by these dispersers increases the likelihood that patches persist at high densities, relative to results expected by random settlement. Populations exhibiting habitat selection also attain equilibrium more quickly than randomly dispersing populations. These effects are particularly important when Allee effects are large and more than two patches exist. Integrating habitat selection into population dynamics may help address why some studies have failed to find extinction thresholds in populations, despite well-known Allee effects in many species.