Releve ranking based on a sum of squares criterion

Summary When a very large number of phytosociological types have to be compared, a reduction of the number of relevés is desirable. In this paper a method of relevé selection from given phytosociological tables is suggested. The method is based on a sum of squares criterion. The advantage, in comparison with other selection procedures, is that this method provides a means on the basis of which the efficiency of a relevé selection can be objectively measured.Contribution from the Working Group for Data-Processing in Phytosociology, International Society for Vegetation Science.The work was completed at the Department of Plant Sciences of the University of Western Ontario, London, Canada. We wish to thank Prof. L. Orlóci for the hospitality and the helpful discussions. The work was supported by Italian C.N.R., within the project Promozione qualità dell'ambiente subproject Metodologie matematiche e basi di dati.     

Size of breeding populations required for selection programs

The minimum population size required for selection in order to reduce the effect of genetic drift to a particular level has been considered. The model of Nicholas was extended to include the measurement-error variance in the response variance. Situations where the sex ratios among scored and breeding individuals are unequal are also considered. When the duration of a selection experiment is relatively long, Nicholas' approximation (i.e., assuming that measurement error is negligible relative to drift) is useful in determining the minimum effective population size required. However, the measurement-error variance becomes an important source of variation in short-term ( 5 generations) selection experiments, and should not be ignored.     

The conflict between natural and artificial selection in finite populations

Summary A single locus model of the interaction between natural selection and artificial selection for a quantitative character in a finite population, assuming heterozygote superiority in natural fitness but additive action on the character, has been studied using transition probability matrices.If natural selection is strong enough to create a selection plateau in which genetic variance declines relatively slowly, then the total response to artificial selection prior to the plateau will be much less than that expected in the absence of natural selection, and the half-life of response will be shorter. Such a plateau is likely to have a large proportion, if not all, of the original genetic variance still present. In selection programmes using laboratory animals, it seems likely that the homozygote favoured by artificial selection must be very unfit before such a plateau will occur. A significant decrease in population fitness as a result of artificial selection does not necessarily imply that the metric character is an important adaptive character.These implications of this model of natural selection are very similar to those derived by James (1962) for the optimum model of natural selection. In fact, there seems to be no aspect of the observable response to artificial selection that would enable anyone to distinguish between these two models of natural selection.     

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Early generation selection for agronomic characters in a potato breeding programme

Summary A random sample of seedlings representing high, medium and poor vigour was studied for tuber colour, tuber shape, eye depth, tuber cracking, tuber yield per plant, average tuber weight and number of tubers per plant in four successive generations (F₁, F₁, F₁C₂, and F₁C₃). Based on the performance of vigour groups in various generations and inter-generation correlation coefficients, we propose a procedure for the elimination of unproductive genotypes early in the breeding programme. The data indicates that seedlings of poor vigour can be discarded at the seedling stage prior to transplantation in the field. The rejection of clones on the basis of tuber colour, tuber shape, eye depth and tuber cracking can also be initiated at the seedling stage. For tuber yield and average tuber weight a negative selection (rejection of poor phenotypes) is suggested from the first clonal generation and for number of tubers, from second clonal generation, until statistically sound replicated trials can be conducted for carrying out positive selection.     

Efficiency of including first-generation information in second-generation ranking and selection: results of computer simulation

Using computer simulation, we evaluated the impact of using first-generation information to increase selection efficiency in a second-generation breeding program. Selection efficiency was compared in terms of increase in rank correlation between estimated and true breeding values (i.e., ranking accuracy), reduction in coefficient of variation of correlation coefficients (i.e., ranking reliability), and increase in realized gain, with best linear unbiased prediction (BLUP). The test populations were generated with varying parameters: selection strategy (forward vs backward selection of parents); number of parents (24∼96); number of crosses per parent (1∼8); heritability (0.05∼0.35); ratio of dominance to additive variance (0∼3); ratio of additive-by-site to additive variance (0∼3); and ratio of dominance-by-site to additive variance (0∼3). The two selection strategies gave distinct results. When parents of the second-generation crosses had been selected via backward selection, adding first-generation information markedly increased selection efficiency. Conversely, when parents had been selected via forward selection, first-generation information provided little increase in efficiency. The amount of increase depended more on heritabilities in both generations and less on dominance and genotype–by–environment effects. Including first-generation information helped more when there were many parents and few crosses per parent in the second generation. Only in the case of extremely low first-generation heritabilities was there no benefit to adding first-generation information in terms of improved ranking reliability and accuracy.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.     

Potential for improvement by selection for reducing sugar content after cold storage for three potato populations

The objectives of this study were to examine the expected response to selection for reducing-sugar content after cold storage in three hybrid populations, to determine whether these populations included clones low in reducing sugars, and to investigate the effectiveness of indirect selection for chip colour based on selection of sugar content after cold storage. The three hybrid populations included: a random sample of 39 clones of Population 1, which was derived from crossing ND860-2 (a clone low in reducing sugars) with F58089 (a clone intermediate in reducing sugars); 40 clones of Population 2, which was obtained from crossing ND860-2 with Russette (a clone high in reducing sugars); and 40 clones of Population 3, which was derived from crossing Russette with F58089. Sugar content and chip colour were assessed in tubers stored for 2 months at 4 °C at Cambridge, Ontario, and at 3 °C at Benton Ridge, New Brunswick. Population 1 had a slightly greater predicted response to selection for reduction in glucose and total reducing sugars than the other two populations. This could be attributed to higher heritability estimates for Population 1, which was a reflection of smaller clone × environment interaction mean squares. The greater potential advance by selection for fructose, glucose, and total reducing sugars, was a direct consequence of its lower means for these traits. Low reducing-sugar clones were found in all three populations, indicating their potential use for the selection of low reducing sugars. Populations 2 and 3, however, would require stronger selection pressures and, therefore, large population sizes. Expected correlated responses for chip colour by selection for fructose and glucose were similar to, and sometimes exceeded, the expected direct responses in all three populations. Indirect responses for chip colour by selection for sucrose, however, were lower than direct selection responses. These results indicate that indirect selection for chip colour, by selection for either fructose or glucose content after cold storage, is as effective as direct selection for chip colour.     

Effect of mass selection on the within-family genetic variance in finite populations

Summary The adequacy of an expression for the withinfamily genetic variance under pure random drift in an additive infinitesimal model was tested via simulation in populations undergoing mass selection. Two hundred or one thousand unlinked loci with two alleles at initial frequencies of 1/2 were considered. The size of the population was 100 (50 males and 50 females). Full-sib matings were carried out for 15 generations with only one male and one female chosen as parents each generation, either randomly or on an individual phenotypic value. In the unselected population, results obtained from 200 replicates were in agreement with predictions. With mass selection, within-family genetic variance was overpredicted by theory from the 12th and 4th generations for the 1,000 and 200 loci cases, respectively. Taking into account the observed change in gene frequencies in the algorithm led to a much better agreement with observed values. Results for the distribution of gene frequencies and the withinlocus genetic covariance are presented. It is concluded that the expression for the within-family genetic variance derived for pure random drift holds well for mass selection within the limits of an additive infinitesimal model.     

The non-existence of a principle of natural selection

The theory of natural selection is a rich systematization of biological knowledge without a first principle. When formulations of a proposed principle of natural selection are examined carefully, each is seen to be exhaustively analyzable into a proposition about sources of fitness and a proposition about consequences of fitness. But whenever the fitness of an organic variety is well defined in a given biological situation, its sources are local contingencies together with the background of laws from disciplines other than the theory of natural selection; and the consequences of fitness for the long range fate of organic varieties are essentially applications of probability theory. Hence there is no role and no need for a principle of the theory of natural selection, and any generalities that may hold in that theory are derivative rather than fundamental.     

The value of indirect selection

Summary Conditions are developed under which progeny-testing using indirect selection can give more rapid genetic improvement than using direct selection. Analogous conditions for mass selection are given in Searle [1954].Paper No. BU-335 in the Biometrics Unit, Cornell University.