scholarly journals A note on the diffusion approximation for the variance of the number of generations until fixation of a neutral mutant gene

1970 ◽  
Vol 15 (2) ◽  
pp. 251-255 ◽  
Author(s):  
P. Narain
Keyword(s):  
Population Size ◽  
General Expression ◽  
Diffusion Approximation ◽  
Finite Population ◽  
Mutant Gene ◽  
Matrix Methods ◽  
Neutral Gene

SUMMARYA general expression is derived for the variance of time to fixation of a neutral gene in a finite population using a diffusion approximation. The results are compared with exact values derived by matrix methods for a population size of 8.

scholarly journals The average number and the variance of generations at particular gene frequency in the course of fixation of a mutant gene in a finite population

1972 ◽  
Vol 19 (2) ◽  
pp. 109-113 ◽  
Author(s):  
Takeo Maruyama
Keyword(s):  
Computer Simulations ◽  
Diffusion Approximation ◽  
Gene Frequency ◽  
Finite Population ◽  
Mutant Gene ◽  
Gene Frequencies ◽  
Explicit Formulas ◽  
Two Generations

SUMMARYIn the case of an allele which is going to become fixed in a population, the average number of generations for which the population assumes particular gene frequencies is investigated, using the diffusion approximation. Explicit formulas were obtained and they were checked by computer simulations. As a particular case, it is shown that if a new mutant that is selectively neutral is eventually fixed in a population of size N, it spends two generations on average at each of the intermediate frequencies (1/2N, 2/2N, …, (2N−1)/2N), and the variance at each frequency is four generations.

scholarly journals Some invariant properties of a geographically structured finite population: distribution of heterozygotes under irreversible mutation

1972 ◽  
Vol 20 (2) ◽  
pp. 141-148 ◽  
Author(s):  
Takeo Maruyama
Keyword(s):  
Population Size ◽  
Total Population ◽  
Finite Population ◽  
Population Distribution ◽  
Mutant Gene ◽  
Selective Advantage ◽  
Genetic Constitution ◽  
Special Cases ◽  
And Migration ◽  
Invariant Properties

SUMMARYIt is shown that the distribution of the sum of heterozygotes, due to mutant gene(s), that appear in a finite population is invariant under geographical structure, provided that the mutant gene has additive effect on fitness and migration does not change the genetic constitution as a whole population. The expected number of heterozygotes is 2N when Ns = 0 and gradually rises to 4N as Ns increases provided s remains small, where N = the total population size and s = selective advantage of a mutant gene. The distribution of the heterozygosity summed over those generations in which the gene frequency in the entire population is specified, is also shown to be invariant. In the case of a neutral mutant, the density is equal to 4(1 − Y) where Y is the frequency of the mutant in the whole population, and in the selectively advantageous case, it is approximately equal to a constant function 4, provided that the population size times selection coefficient is sufficiently large. These quantities conditional on the fixation of the mutant are shown to be invariant and some special cases are obtained explicitly.

scholarly journals Polygenic local adaptation in metapopulations: a stochastic eco-evolutionary model

2020 ◽  
Author(s):  
Enikő Szép ◽  
Himani Sachdeva ◽  
Nick Barton
Keyword(s):  
Population Size ◽  
Local Adaptation ◽  
Genetic Drift ◽  
Diffusion Approximation ◽  
Joint Distribution ◽  
Evolutionary Model ◽  
Ecological Niches ◽  
Demographic Stochasticity ◽  
Linkage Disequilibria ◽  
Population Sizes

AbstractThis paper analyses the conditions for local adaptation in a metapopulation with infinitely many islands under a model of hard selection, where population size depends on local fitness. Each island belongs to one of two distinct ecological niches or habitats. Fitness is influenced by an additive trait which is under habitat-dependent directional selection. Our analysis is based on the diffusion approximation and accounts for both genetic drift and demographic stochasticity. By neglecting linkage disequilibria, it yields the joint distribution of allele frequencies and population size on each island. We find that under hard selection, the conditions for local adaptation in a rare habitat are more restrictive for more polygenic traits: even moderate migration load per locus at very many loci is sufficient for population sizes to decline. This further reduces the efficacy of selection at individual loci due to increased drift and because smaller populations are more prone to swamping due to migration, causing a positive feedback between increasing maladaptation and declining population sizes. Our analysis also highlights the importance of demographic stochasticity, which exacerbates the decline in numbers of maladapted populations, leading to population collapse in the rare habitat at significantly lower migration than predicted by deterministic arguments.

POPULATION DYNAMICS OF SEX-DETERMINING ALLELES IN HONEY BEES AND SELF-INCOMPATIBILITY ALLELES IN PLANTS

Genetics ◽  
1979 ◽  
Vol 91 (3) ◽  
pp. 609-626 ◽  
Author(s):  
Shozo Yokoyama ◽  
Masatoshi Nei
Keyword(s):  
Population Dynamics ◽  
Population Size ◽  
Honey Bee ◽  
Honey Bees ◽  
Finite Population ◽  
Large Population ◽  
Allele Frequencies ◽  
Self Incompatibility ◽  
Overdominant Selection ◽  
Incompatibility Alleles

ABSTRACT Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is l/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from l/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.

DIFFUSION OF A POPULATION OF INTERACTING REPLICATORS IN SEQUENCE SPACE

2002 ◽  
Vol 05 (04) ◽  
pp. 457-461 ◽  
Author(s):  
BÄRBEL M. R. STADLER
Keyword(s):  
Simple Model ◽  
Population Size ◽  
Computer Simulations ◽  
Mutation Rate ◽  
Sequence Space ◽  
Finite Population ◽  
Fitness Landscape ◽  
Diffusion Constant ◽  
Sequence Length

We consider a simple model for catalyzed replication. Computer simulations show that a finite population moves in sequence space by diffusion analogous to the behavior of a quasispecies on a flat fitness landscape. The diffusion constant depends linearly on the per position mutation rate and the ratio of sequence length and population size.

The supersession of one rumour by another

1977 ◽  
Vol 14 (1) ◽  
pp. 127-134 ◽  
Author(s):  
G. K. Osei ◽  
J. W. Thompson
Keyword(s):  
Population Size ◽  
Finite Population ◽  
Closed Population ◽  
Asymptotic Case ◽  
Distribution Of The Maximum ◽  
Maximum Value

A model is considered for a situation in which one rumour suppresses another in a closed population. The distribution of the maximum value attained by the proportion spreading the weaker rumour is obtained in the asymptotic case, and this is compared with some actual distributions for finite population size. Closer approximations to the latter distributions are obtained.

scholarly journals THE EVOLUTION OF MULTIGENE FAMILIES UNDER INTRACHROMOSOMAL GENE CONVERSION

Genetics ◽  
1984 ◽  
Vol 106 (3) ◽  
pp. 529-548
Author(s):  
Thomas Nagylaki
Keyword(s):  
Population Size ◽  
Gene Conversion ◽  
Finite Population ◽  
Convergence Time ◽  
Crossing Over ◽  
Cubic Equation ◽  
Crossover Rate ◽  
Monoecious Population ◽  
New Alleles ◽  
Repeated Genes

ABSTRACT A model for the evolution of the probabilities of genetic identity within and between loci of a multigene family in a finite population is formulated and investigated. Unbiased intrachromosomal gene conversion, equal crossing over between tandemly repeated genes, random genetic drift and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Formulas for the equilibrium values of the probabilities of identity and a cubic equation for the rate of convergence are deduced. Numerical examples indicate the following. The amount of homology at equilibrium generally decreases as the mutation rate, the population size and the number of repeats increase; it may increase or decrease with increasing crossover rate. The intralocus homology has an intermediate minimum, whereas the interlocus homology increases, as the rate of gene conversion increases. The intralocus homology decreases, whereas the interlocus homology increases, as the proportion of symmetric heteroduplexes increases. The characteristic convergence time can be sufficiently short to imply that intrachromosomal gene conversion may be an important mechanism for maintaining sequence homogeneity among repeated genes. The convergence time decreases as the conversion rate and the proportion of symmetric heteroduplexes increase; although exceptions occur, it generally increases as the population size and the number of repeats increase; it may increase or decrease with increasing crossover rate.

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