scholarly journals The effect of repeated cycles of selection and regeneration in populations of finite size

1968 ◽  
Vol 11 (1) ◽  
pp. 105-112 ◽  
Author(s):  
R. N. Curnow ◽  
L. H. Baker

Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population are used to develop an iterative method for studying the effects of repeated cycles of selection and random mating. This is done by assuming a particular, but flexible and probably realistic, approximate form for the distribution of gene frequencies at each generation.The method gives for each generation the first two moments of the gene frequency distribution, the expected gain from selection, the probabilities of fixation and also the variability of gain. The variability of gain is of considerable importance in evolution, selection experiments and in plant and animal breeding programmes.Kojima's (1961) formulae have been extended to allow for differentiation between males and females. Hence different selection intensities and population sizes for the two sexes can be studied. Selfing with selection is considered separately. Extensions to cover simple examples of multiple alleles, linkage and epistasis are possible. Reference is made to previous work using transition matrices.

1961 ◽  
Vol 2 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Ken-Ichi Kojima

A theory of mass selection in a small population was developed, and the mean change in gene frequencies, the variance of gene frequency changes and the expected gain in the mean phenotypic value of an offspring population were formulated in terms of a generalized selection differential and the additive and dominance effects of genes.The magnitude of the variance of changes in gene frequency was compared with the magnitude of the variance expected from the genetic random drift in a population with the same gene frequency and of the same size in absence of selection. The former was found to be usually smaller than the latter when the gene frequency ranged from intermediate to high and when selection was directed for a high performance.The usual prediction equation for gain from selection in an infinite population was compared with the expected gain formula derived for a small population. The size of the population did not cause a serious difference between the two expected gains when there was no dominance effect of genes. Dominance alone could cause the usual prediction to be slightly more biased. The joint effects of the finite size of population and dominance gene action could amount to a considerable bias in the usual prediction equation. Such a bias can be, in the main, accounted for by the inbreeding depression.


1969 ◽  
Vol 13 (2) ◽  
pp. 117-126 ◽  
Author(s):  
Derek J. Pike

Robertson (1960) used probability transition matrices to estimate changes in gene frequency when sampling and selection are applied to a finite population. Curnow & Baker (1968) used Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population to develop an iterative procedure for studying the effects of repeated cycles of selection and regeneration. To do this they assumed a beta distribution for the unfixed gene frequencies at each generation.These two methods are discussed and a result used in Kojima's paper is proved. A number of sets of calculations are carried out using both methods and the results are compared to assess the accuracy of Curnow & Baker's method in relation to Robertson's approach.It is found that the one real fault in the Curnow-Baker method is its tendency to fix too high a proportion of the genes, particularly when the initial gene frequency is near to a fixation point. This fault is largely overcome when more individuals are selected. For selection of eight or more individuals the Curnow-Baker method is very accurate and appreciably faster than the transition matrix method.


2018 ◽  
Vol 7 (4.10) ◽  
pp. 508
Author(s):  
N. Vijayalakshmi ◽  
Dr. P.Sekhar ◽  
Dr. G.Mokesh Rayalu

Biometrics is a branch of statistics in which various mathematical and statistical techniques can be applied to biological research problems. These are two main areas of specialization of Biometry namely, Bioassays and Quantitative Genetics. Genetics concerns with Heredity and variation. Quantitative Genetics is concerned with the inheritances of quantitative differences between individuals.The essence of Quantitative Genetics is to estimate the genetic parameters such as Gene frequencies, segregation Ratios, Recombination of Genes and so on. Among them, the estimation of Gene Frequencies in the population is an important one. The proportion or percentage of genes in the population is called gene Frequency. In the present research articles, the ABO blood group system of man has been described by discussing the multiple alleles; genotypes, Frequencies and phenotypes of blood groups. The various estimation methods for estimating gene frequencies have gene presents in the present study. 


The distribution of gene frequency at a single locus in a population of diploid individuals, with two sexes, subject to mutation, non-random mating and phenotypic selection, is obtained in the case where the generations are overlapping so that individuals die one by one. This distribution is of the same form as that obtained by heuristic methods by S. Wright in a randomly mating population but the coefficients are altered both by the non-randomness of the mating and the overlapping of the generations.


1969 ◽  
Vol 13 (2) ◽  
pp. 143-163 ◽  
Author(s):  
W. G. Hill

The effect of selection on individual performance for a quantitative trait is studied theoretically for populations of finite size. The trait is assumed to be affected by environmental error and by segregation at a single locus. Exact formulae are derived to predict the change in gene frequency at this locus, initially by finding the probability distribution of the numbers of each genotype selected from a finite population of specified genotypic composition. Assuming that there is random mating and no natural selection the results are extended to describe repeated cycles of artificial selection for a monecious population. The formulae are evaluated numerically for the case of normally distributed environmental errors.Using numerical examples comparisons are made between the exact values for the predicted change in gene frequency with values obtained using approximate, but simpler, methods. Unless the gene has a large effect (α) on the quantitative trait, relative to the standard deviation of the environmental errors, the agreement between exact and approximate methods is satisfactory for most predictive purposes. The chance of fixation after repeated generations of selection is also evaluated using the exact method, and by means of a diffusion approximation and simple transition probability matrix methods. Except for very small values of population size (N) and large α the results from the diffusion equation agree closely with those from the exact method. Similar results are found from tests made of the prediction from the diffusion equation that the limit is only a function of Nα for a given intensity of selection and initial frequency, and that the rate of advance in gene frequency is proportional to 1/N for the same set of parameters.


A precise theorem is given for the increase in fitness due to natural selection on diploids subject to random mating, non-overlapping generations and not more than two loci; the method of extension to more loci is given by Kojima & Kelleher, and a precise theorem is given here for any number of loci when there is no recombination. The increase is equal to the haploid (or genic) variance in fitness, multiplied by a factor which is equal to two in the absence of dominance, but which otherwise is a function of gene frequency and dominance. The theorem is compared with that of Kimura, which is more general but harder to apply, and to those of Kojima & Kelleher and Fisher, which are respectively restricted to slow selection and absence of epistasis. The new theorem is used to predict the equilibria in populations polymorphic for two loci, and to deal especially with the quasi-stable equilibrium, for which the critical value of recombination is formulated, and the through point, at which a stable and unstable equilibrium meet and annihilate each other. The effect of this in space is to produce a stepped cline, in which gene frequencies and gametic excess change suddenly over a short distance; in time, the through point brings a new slant to Wright’s multiple peak theory of evolution, as populations can move precipitately from peak to peak without the help of random processes. Mean fitness is related only indirectly to population density. By distinguishing carefully between mean absolute fitness (which is the rate of population growth) and mean relative fitness (which is more useful than the absolute parameter for predicting genetical equilibria) we can show the effects of various types of density control on the genetical composition of the population; density dependent selection may appear to be gene-frequency dependent. The fundamental law of evolution is probably a thermodynamic law of increasing matter energy, which is related only tenuously to the law of increasing genetical fitness.


Genetics ◽  
1973 ◽  
Vol 73 (2) ◽  
pp. 361-366
Author(s):  
Takeo Maruyama

ABSTRACT Considering a random mating population of finite size, the variance of the number of loci having a given gene frequency was derived under the assumption of a steady flux of mutations. The variance of average heterozygosity among populations was also derived under the same assumption. It was shown that these variances are proportional to the population size if the mutants are selectively neutral, and they are inversely proportional to the selection coefficient if the mutants are selectively advantageous and additive in their fitness.


The distribution of gene frequency at a single locus in a population of diploid individuals with two sexes, subject to mutation, non-random mating and phenotypic selection, is obtained in the case where all the population dies at the same time and is replaced by a new generation. The distribution is similar to that obtained by Wright with a correction due to the non-randomness of the mating.


1996 ◽  
Vol 68 (3) ◽  
pp. 249-257 ◽  
Author(s):  
Jinliang Wang

SummaryFor 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.


2018 ◽  
Vol 124 (4) ◽  
pp. 521-529 ◽  
Author(s):  
Gancho T Slavov ◽  
Christopher L Davey ◽  
Maurice Bosch ◽  
Paul R H Robson ◽  
Iain S Donnison ◽  
...  

Abstract Background Miscanthus has potential as a biomass crop but the development of varieties that are consistently superior to the natural hybrid M. × giganteus has been challenging, presumably because of strong G × E interactions and poor knowledge of the complex genetic architectures of traits underlying biomass productivity and climatic adaptation. While linkage and association mapping studies are starting to generate long lists of candidate regions and even individual genes, it seems unlikely that this information can be translated into effective marker-assisted selection for the needs of breeding programmes. Genomic selection has emerged as a viable alternative, and prediction accuracies are moderate across a range of phenological and morphometric traits in Miscanthus, though relatively low for biomass yield per se. Methods We have previously proposed a combination of index selection and genomic prediction as a way of overcoming the limitations imposed by the inherent complexity of biomass yield. Here we extend this approach and illustrate its potential to achieve multiple breeding targets simultaneously, in the absence of a priori knowledge about their relative economic importance, while also monitoring correlated selection responses for non-target traits. We evaluate two hypothetical scenarios of increasing biomass yield by 20 % within a single round of selection. In the first scenario, this is achieved in combination with delaying flowering by 44 d (roughly 20 %), whereas, in the second, increased yield is targeted jointly with reduced lignin (–5 %) and increased cellulose (+5 %) content, relative to current average levels in the breeding population. Key Results In both scenarios, the objectives were achieved efficiently (selection intensities corresponding to keeping the best 20 and 4 % of genotypes, respectively). However, the outcomes were strikingly different in terms of correlated responses, and the relative economic values (i.e. value per unit of change in each trait compared with that for biomass yield) of secondary traits included in selection indices varied considerably. Conclusions Although these calculations rely on multiple assumptions, they highlight the need to evaluate breeding objectives and explicitly consider correlated responses in silico, prior to committing extensive resources. The proposed approach is broadly applicable for this purpose and can readily incorporate high-throughput phenotyping data as part of integrated breeding platforms.


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