On the theory of artificial selection in finite populations

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.

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

Genetics Research ◽

10.1017/s001667230001123x ◽

1968 ◽

Vol 11 (1) ◽

pp. 105-112 ◽

Cited By ~ 5

Author(s):

R. N. Curnow ◽

L. H. Baker

Keyword(s):

Gene Frequency ◽

Random Mating ◽

Finite Size ◽

Transition Matrices ◽

Gene Frequencies ◽

Multiple Alleles ◽

Expected Gain ◽

Approximate Form ◽

Breeding Programmes ◽

Selection Intensities

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.

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GENETIC CHANGES WITH GENERATIONS OF ARTIFICIAL SELECTION

Genetics ◽

10.1093/genetics/95.3.769 ◽

1980 ◽

Vol 95 (3) ◽

pp. 769-782 ◽

Cited By ~ 1

Author(s):

Luis Silvela

Keyword(s):

Artificial Selection ◽

Gene Frequency ◽

Recurrent Selection ◽

Finite Size ◽

Conditional Probabilities ◽

Algebraic Equations ◽

Genetic Changes ◽

Population Mean ◽

Quantitative Estimates

ABSTRACT Using conditional probabilities and moment-generating matrices, I derived approximate algebraic equations that give expectations of gene frequency, population mean, gene frequency variance within lines, or heterozygosity, and gene frequency variance between lines, or drift, for repeated cycles of recurrent selection in populations of finite size. For genes of large effect, the responses to selection differ substantially from the classical expectations, and equations are derived that give quantitative estimates of asymmetry of response when selection is done in opposite directions. Particular cases of the derived formulae yield equations given by other authors. The error involved in the approximations is discussed in the APPENDIX.

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THE VARIANCE OF THE NUMBER OF LOCI HAVING A GIVEN GENE FREQUENCY

Genetics ◽

10.1093/genetics/73.2.361 ◽

1973 ◽

Vol 73 (2) ◽

pp. 361-366

Author(s):

Takeo Maruyama

Keyword(s):

Population Size ◽

Gene Frequency ◽

Random Mating ◽

Finite Size ◽

Selection Coefficient ◽

Average Heterozygosity ◽

Random Mating Population ◽

Mating Population

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.

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A note on the theory of artificial selection in finite populations and application to QTL detection by bulk segregant analysis

Genetics Research ◽

10.1017/s0016672398003334 ◽

1998 ◽

Vol 72 (1) ◽

pp. 55-58 ◽

Cited By ~ 2

Author(s):

WILLIAM G. HILL

Keyword(s):

Quantitative Trait Locus ◽

Artificial Selection ◽

Quantitative Trait ◽

Gene Frequency ◽

Bulk Segregant Analysis ◽

Finite Population ◽

Qtl Detection ◽

Selection For ◽

Trait Locus ◽

Almost All

Formulae are given for computing the distribution of numbers of selected individuals of each genotype and thus change in gene frequency at a locus with a large effect on a quantitative trait under truncation selection in a finite population. Results are illustrated with respect to use of selection for quantitative trait locus (QTL) detection, specifically by bulk segregant analysis with linked markers, for which probabilities that selected samples will comprise almost all one genotypic class are computed.

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A SIMULATION STUDY OF TRUNCATION SELECTION FOR A QUANTITATIVE TRAIT OPPOSED BY NATURAL SELECTION

Genetics ◽

10.1093/genetics/94.4.989 ◽

1980 ◽

Vol 94 (4) ◽

pp. 989-1000

Author(s):

Francis Minvielle

Keyword(s):

Natural Selection ◽

Artificial Selection ◽

Random Mating ◽

Selection Response ◽

Quantitative Character ◽

Mass Selection ◽

Frequency Selection ◽

Selection Experiments ◽

Selection For ◽

Selective Process

ABSTRACT A quantitative character controlled at one locus with two alleles was submitted to artificial (mass) selection and to three modes of opposing natural selection (directional selection, overdominance and underdominance) in a large random-mating population. The selection response and the limits of the selective process were studied by deterministic simulation. The lifetime of the process was generally between 20 and 100 generations and did not appear to depend on the mode of natural selection. However, depending on the values of the parameters (initial gene frequency, selection intensity, ratio of the effect of the gene to the environmental standard deviation, fitness values) the following outcomes of selection were observed: fixation of the allele favored by artificial selection, stable nontrivial equilibrium, unstable equilibrium and loss of the allele favored by artificial selection. Finally, the results of the simulation were compared to the results of selection experiments.

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The spread of genes in random mating control populations

Genetics Research ◽

10.1017/s0016672300003074 ◽

1962 ◽

Vol 3 (1) ◽

pp. 1-10 ◽

Cited By ~ 12

Author(s):

J. W. James

Keyword(s):

Genetic Drift ◽

Gene Frequency ◽

Random Mating ◽

Effective Size ◽

Relative Importance ◽

Genetic Sampling ◽

Under Sampling ◽

Poultry Flock ◽

The University

1. The effect of genetic sampling, when this sampling is without replacement, on variation in gene frequency is studied, and equations describing the genetic drift are derived. The effective size turns out to be about one greater than under sampling with replacement.2. The relation between ‘spread of genes’ and genetic drift is worked out.3. The University of Queensland control poultry flock is analysed by these methods.4. The design of control populations is discussed with particular reference to the relative importance of genetic drift and phenotypic sampling.

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Effects of dominance and size of population on response to mass selection

Genetics Research ◽

10.1017/s0016672300000689 ◽

1961 ◽

Vol 2 (2) ◽

pp. 177-188 ◽

Cited By ~ 15

Author(s):

Ken-Ichi Kojima

Keyword(s):

Gene Frequency ◽

Finite Size ◽

Prediction Equation ◽

Small Population ◽

Mass Selection ◽

Gene Frequencies ◽

Random Drift ◽

Expected Gain ◽

The Mean ◽

Frequency Changes

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.

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The contribution of quantitative trait loci and neutral marker loci to the genetic variances and covariances among quantitative traits in random mating populations.

Genetics ◽

10.1093/genetics/139.1.445 ◽

1995 ◽

Vol 139 (1) ◽

pp. 445-455 ◽

Cited By ~ 1

Author(s):

A Ruiz ◽

A Barbadilla

Keyword(s):

Quantitative Trait Loci ◽

Quantitative Trait ◽

Quantitative Traits ◽

Random Mating ◽

Unbiased Estimation ◽

Limited Information ◽

Trait Loci ◽

Marker Loci ◽

Neutral Marker ◽

Variance Explained

Abstract Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.

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PERFORMANCE ANALYSIS OF OPTIMIZATION METHODS FOR SOLVING TRAVELING SALESMAN PROBLEM

Innovative technologies and scientific solutions for industries ◽

10.30837/itssi.2021.15.069 ◽

2021 ◽

pp. 69-75

Author(s):

Chandra Agung ◽

Natalia Christine

Keyword(s):

Approximate Method ◽

Traveling Salesman Problem ◽

Optimal Solution ◽

Optimization Methods ◽

Traveling Salesman ◽

Exact Method ◽

Problem Size ◽

Exact Methods ◽

Approximate Methods ◽

Bee Colony

The subject of this research is distance and time of several city tour problems which known as traveling salesman problem (tsp). The goal is to find out the gaps of distance and time between two types of optimization methods in traveling salesman problem: exact and approximate. Exact method yields optimal solution but spends more time when the number of cities is increasing and approximate method yields near optimal solution even optimal but spends less time than exact methods. The task in this study is to identify and formulate each algorithm for each method, then to run each algorithm with the same input and to get the research output: total distance, and the last to compare both methods: advantage and limitation. Methods used are Brute Force (BF) and Branch and Bound (B&B) algorithms which are categorized as exact methods are compared with Artificial Bee Colony (ABC), Tabu Search (TS) and Simulated Annealing (SA) algorithms which are categorized as approximate methods or known as a heuristics method. These three approximate methods are chosen because they are effective algorithms, easy to implement and provide good solutions for combinatorial optimization problems. Exact and approximate algorithms are tested in several sizes of city tour problems: 6, 9, 10, 16, 17, 25, 42, and 58 cities. 17, 42 and 58 cities are derived from tsplib: a library of sample instances for tsp; and others are taken from big cities in Java (West, Central, East) island. All of the algorithms are run by MATLAB program. The results show that exact method is better in time performance for problem size less than 25 cities and both exact and approximate methods yield optimal solution. For problem sizes that have more than 25 cities, approximate method – Artificial Bee Colony (ABC) yields better time which is approximately 37% less than exact and deviates 0.0197% for distance from exact method. The conclusion is to apply exact method for problem size that is less than 25 cities and approximate method for problem size that is more than 25 cities. The gap of time will be increasing between two methods when sample size becomes larger.

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NATURAL SELECTION OPPOSING ARTIFICIAL SELECTION: A TWO-LOCUS DETERMINISTIC SIMULATION

Genetics ◽

10.1093/genetics/98.1.231a ◽

1981 ◽

Vol 98 (1) ◽

pp. 231-238

Author(s):

Francis Minvielle

Keyword(s):

Numerical Simulation ◽

Linkage Disequilibrium ◽

Natural Selection ◽

Artificial Selection ◽

Random Mating ◽

Base Population ◽

Truncation Selection ◽

Starting Point ◽

Mating Population ◽

Central Equilibrium

ABSTRACT A two-locus, two-allele metric trait was submitted to artificial truncation selection and to three types of opposing natural selection (two-locus extensions of directional selection, overdominance and underdominance) by numerical simulation in a large random-mating population. Limits to selection were generally reached by generation 100. Intermediate selection plateaus were found, with minor genes, for all three modes of opposing natural selection, but they were least frequent with underdominance. Multiple outcomes were common. In particular, fixation of the genotype favored by artificial selection was often associated with fixation of another genotype and/or with a central equilibrium; the end point actually reached depended on the genetic starting point of the simulation. In general, when the alleles favored by truncation selection were combined (positive linkage disequilibrium) in the base population, or when the trait was determined by major genes, artificial selection would prevail. Limitations inherent to this type of work are discussed, and possible avenues for further work on the antagonism between artificial and natural selection are proposed.

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