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.

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.

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.

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 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.

scholarly journals Deviation from Hardy–Weinberg proportions in finite populations

1996 ◽  
Vol 68 (3) ◽  
pp. 249-257 ◽  
Author(s):  
Jinliang Wang
Keyword(s):  
Sampling Error ◽  
Random Mating ◽  
Random Selection ◽  
Stochastic Simulations ◽  
Gene Frequencies ◽  
Dioecious Species ◽  
Male And Female ◽  
Census Size ◽  
Genotype Frequencies ◽  
Gene Frequency Difference

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.

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 Some observations on asymmetrical correlated responses to selection

1966 ◽  
Vol 7 (1) ◽  
pp. 44-57 ◽  
Author(s):  
B. B. Bohren ◽  
W. G. Hill ◽  
A. Robertson
Keyword(s):  
Model Selection ◽  
Gene Frequency ◽  
Genetic Parameters ◽  
Correlated Response ◽  
Correlated Responses ◽  
Gene Frequencies ◽  
Genetic Covariance ◽  
Gene Effects ◽  
Selection Experiments ◽  
Relative Change

The pattern of changes of the genetic covariance between two characters on selection was examined in an effort to explain the asymmetry of correlated responses in two traits, or of the same trait in two environments, frequently observed in experimental results.The algebraic conclusions were further examined by model selection experiments using a computer. The computer was programmed to calculate the change in gene frequency from generation to generation and to calculate from it the expected changes in genetic variances and covariance as selection proceeded. This procedure was carried out with several models of gene effects and gene frequencies.Asymmetry of the genetic covariance, and consequently of the correlated responses, resulted when the relative change in gene frequency at the loci contributing positively and negatively to the covariance depended on the trait selected. The conditions necessary for the development of asymmetry were examined and the results suggest that any symmetry found in an experiment is perhaps more surprising than asymmetry. Probably the most frequent contribution to asymmetry in practice will be from loci contributing negatively to the covariance and having frequencies other than 0·5.Accurate prediction of correlated response over many generations is therefore not possible without prior knowledge of the composition of the genetic covariance, as well as its magnitude. The validity of existing theory for the prediction of correlated responses is likely to be much poorer than for the prediction of direct responses. Predictions would then have to be based on the genetic parameters estimated in each generation.

scholarly journals DYNAMICS OF THE LINKAGE DISEQUILIBRIUM FUNCTION UNDER MODELS OF GENE-FREQUENCY HITCHHIKING

Genetics ◽  
1981 ◽  
Vol 99 (2) ◽  
pp. 337-356
Author(s):  
Marjorie A Asmussen ◽  
Michael T Clegg
Keyword(s):  
Linkage Disequilibrium ◽  
Gene Frequency ◽  
Initial Conditions ◽  
Recombination Fraction ◽  
Qualitative Description ◽  
Selection Intensity ◽  
Gene Frequencies ◽  
Dependent Behavior ◽  
Wide Range ◽  
Over Time

ABSTRACT The dynamic behavior of the linkage disequilibrium (D) between a neutral and a selected locus is analyzed for a variety of deterministic selection models. The time-dependent behavior of D is governed by the gene frequency at the selected locus (p) and by the selection (s) and recombination (r) parameters. Thomson (1977) showed numerically that D may increase under certain initial conditions. We give exact conditions for D to increase in time, which require that the selection intensity exceed the recombination fraction (s > r) and that p be near zero or one. We conclude from this result that gene frequency hitchhiking is most likely to be important when a new favorable mutant enters a population. We also show that, for what can be a wide range of gene frequencies, D will decay at a faster rate than the neutral rate. Consequently, the hitchhiking effect may quickly diminish as the selected gene becomes more common.—The method of analysis allows a complete qualitative description of the dynamics of D as a function of s and r. Two major findings concern the range of gene frequencies at the selected locus for which D either increases over time or decays at a faster rate than under neutrality. For all models considered, the region where D increases (i) first enlarges then shrinks as selection intensifies, and (ii) steadily shrinks as r increases. In contrast, the region of accelerated decay constantly enlarges as the selection intensity increases. This region will either shrink or enlarge as r increases, depending upon the form of selection in force.

Zufallspaarung mit teilweiser Selbstung unter verschiedenen Modellen von häufigkeitsabhängiger Selektion /Mixed Random Mating and Selfing with Different Models of Frequency-Dependent Selection

1978 ◽  
Vol 33 (9-10) ◽  
pp. 755-768 ◽  
Author(s):  
M. Hühn
Keyword(s):  
Genetic Polymorphisms ◽  
Population Genetic ◽  
Random Mating ◽  
Large Population ◽  
Decisive Role ◽  
Initial Population ◽  
Special Importance ◽  
Gene Frequencies ◽  
Frequency Dependent ◽  
Frequency Dependent Selection

Abstract Given a large population with mixed random mating and selfing (one locus-two alleles) different models of frequency-dependent selection were discussed - including a simple biometrical model for considering and analysing the competitive effects between neighbouring individuals in plant populations. For each model there were studied: changes in gene frequencies, population genetic equilibria, times until reaching these equilibria etc. - in dependence of the different parameters used: composition of the initial population, probability of selfing, selection-coefficients, competition-parameters.Apart from only few differing results it follows from the studies performed in these investigations, that the different composition of the initial population is of no particular importance as well for the gene frequencies p̂ at equilibrium as for the time t̂ until reaching these equilibria. This result is especially right for p̂.Different probabilities of selfing and different degrees of dominance in the selection coefficients are indeed of some influence on the existence and location of the population genetic equilibria, but here too we find an disproportionately stronger dependence with the time t̂ until reaching the equilibrium than with the gene frequency p̂ at equilibrium . The special importance of overdominance for the maintenance of genetic polymorphisms, which is well known in the case of non­ frequency-dependent selection (see: model 1 of the present studies) turn out to be of some other meaning in the models of frequency-dependent selection, which were analysed in the present paper: Depart from only few special situations (model 2 ; complete self-fertilization in models 5 and 6 ; extremely high probabilities of selfing in model 7) nontrivial equilibria are reached for all degrees of dominance. Therefore, the special importance of overdominance mentioned above, not proves right in the case of frequency-dependent selection.The investigations of the present paper have shown, that existence and location of the non­ trivial population genetic equilibria are determined not so much by degree of dominance and probability of selfing, but the equilibria are mainly determined by the model of the investigation used in the concerning studies.In the case of frequency-dependent selection, therefore, the explicit form of the fitness values as functions of the frequencies plays the decisive role in maintaining genetic polymorphisms.

scholarly journals The spread of genes in random mating control populations

1962 ◽  
Vol 3 (1) ◽  
pp. 1-10 ◽  
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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