A general theory of the distribution of gene frequencies - I. Overlapping generations

1958 ◽  
Vol 149 (934) ◽  
pp. 102-112 ◽  
Keyword(s):  
General Theory ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Random Mating ◽  
Phenotypic Selection ◽  
Single Locus ◽  
Heuristic Methods ◽  
Gene Frequencies ◽  
Mating Population

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.

A general theory of the distribution of gene frequencies - II. Non-overlapping generations

1958 ◽  
Vol 149 (934) ◽  
pp. 113-116 ◽  
Keyword(s):  
General Theory ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Random Mating ◽  
Phenotypic Selection ◽  
Single Locus ◽  
Gene Frequencies ◽  
New Generation

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.

scholarly journals On sojourn times at particular gene frequencies

1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold
Keyword(s):  
Markov Chains ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Finite Population ◽  
Diffusion Method ◽  
Gene Frequencies ◽  
Sojourn Times ◽  
The Mean ◽  
Number Of Visits ◽  
Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

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
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.

Mean fitness and the equilibria in multilocus polymorphisms

1967 ◽  
Vol 169 (1014) ◽  
pp. 31-58 ◽  
Keyword(s):  
Population Density ◽  
Gene Frequency ◽  
Random Mating ◽  
Relative Fitness ◽  
Gene Frequencies ◽  
Theory Of Evolution ◽  
Density Control ◽  
Mean Fitness ◽  
Fundamental Law ◽  
Matter Energy

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.

THE ESTIMATION OF GENE FREQUENCIES IN A RANDOM-MATING POPULATION

1955 ◽  
Vol 20 (2) ◽  
pp. 97-115 ◽  
Author(s):  
By R. CEPPELLINI ◽  
M. SINISCALCO ◽  
C. A. B. SMITH
Keyword(s):  
Random Mating ◽  
Gene Frequencies ◽  
Random Mating Population ◽  
Mating Population

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

Genetics ◽  
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.

scholarly journals Predicting Rates of Inbreeding in Populations Undergoing Selection

Genetics ◽  
2000 ◽  
Vol 154 (4) ◽  
pp. 1851-1864 ◽  
Author(s):  
John A Woolliams ◽  
Piter Bijma
Keyword(s):  
Overlapping Generations ◽  
Linear Models ◽  
Selection Process ◽  
Correction Term ◽  
Random Mating ◽  
Nonrandom Mating ◽  
Selection Processes ◽  
Genetic Contributions ◽  
The Relationship

AbstractTractable forms of predicting rates of inbreeding (ΔF) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. ΔF was shown to be ~¼(1 − ω) times the expected sum of squared lifetime contributions, where ω is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express ΔF in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing ¼ (since ω = 0) was increased to ½. Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting ΔF with sib indices in discrete generations since previously published solutions had proved complex.

scholarly journals Average Number of Nucleotide Differences in a Sample From a Single Subpopulation: A Test for Population Subdivision

Genetics ◽  
1987 ◽  
Vol 117 (1) ◽  
pp. 149-153
Author(s):  
Curtis Strobeck
Keyword(s):  
Monte Carlo Simulation ◽  
Population Structure ◽  
Random Mating ◽  
Population Subdivision ◽  
Expected Number ◽  
Migration Rates ◽  
Mating Population ◽  
Unbiased Estimates ◽  
Increasing Function ◽  
Stepping Stone Model

ABSTRACT Unbiased estimates of θ = 4Nµ in a random mating population can be based on either the number of alleles or the average number of nucleotide differences in a sample. However, if there is population structure and the sample is drawn from a single subpopulation, these two estimates of θ behave differently. The expected number of alleles in a sample is an increasing function of the migration rates, whereas the expected average number of nucleotide differences is shown to be independent of the migration rates and equal to 4N  Tµ for a general model of population structure which includes both the island model and the circular stepping-stone model. This contrast in the behavior of these two estimates of θ is used as the basis of a test for population subdivision. Using a Monte-Carlo simulation developed so that independent samples from a single subpopulation could be obtained quickly, this test is shown to be a useful method to determine if there is population subdivision.

scholarly journals A comparison of two methods for predicting changes in the distribution of gene frequency when selection is applied repeatedly to a finite population

1969 ◽  
Vol 13 (2) ◽  
pp. 117-126 ◽  
Author(s):  
Derek J. Pike
Keyword(s):  
Iterative Procedure ◽  
Gene Frequency ◽  
Finite Population ◽  
Transition Matrices ◽  
Single Cycle ◽  
Gene Frequencies ◽  
Mean And Variance ◽  
The Mean ◽  
The One ◽  
Selection Of

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.

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