The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.     

Directional and Fluctuating Asymmetry in Sexual and Asexual Otiorhynchus alpicola Populations

This study concerns the contribution of directional asymmetry (DA) and fluctuating asymmetry (FA) as a characterization of variation in six sexual (diploid) and two asexual (triploid and tetraploid) populations of the weevil Otiorhynchus alpicola. It is shown that DA in sexual populations is about 1 % of the mean length of each of the seven bilateral traits and the average contribution of DA to trait variation is even lower in asexual populations (about 0.85 in triploids and 0.65 in tetraploids forms). The average contribution of FA to the total phenotypic variance is about 23 %, 12 % and 19 % in diploid, triploid and tetraploid populations, respectively. Since FA is generally regarded as a measure of developmental stability, our data indicate that triploid forms of O. alpicola are developmentally more stable than diploid and tetraploid forms. The relationship between the level of ploidy and FA is discussed.     

Tests for Parasite-mediated Frequency-dependent Selection in Natural Populations of an Asexual Plant Species

Genetic variation in plant populations for resistance to pathogens and herbivores might be maintained by parasite-mediated negative frequency-dependent selection (FDS). But it is difficult to observe the time-lagged oscillations between host and parasite genotypes that should result from FDS. To evaluate the potential for FDS, we tested for local adaptation of parasites to common clones, the role of host genetic diversity in resistance to parasites, and genetic correlations among fitness, parasitism, and the frequency of host clones. We studied three populations of Arabis holboellii, a short-lived apomictic (asexual by seed) plant attacked by rust fungi and insect herbivores. To estimate clone frequency, we used polymorphic allozyme markers on 200 individuals in each population in 1990 and in 2000. We also recorded levels of parasitism and host fitness (fruit production). Only the rust fungi showed evidence for local host adaptation; they usually increased in incidence as a function of clone frequency, and they tracked temporal change in clone frequency. In further support of FDS, parasitism was lower in populations with higher genetic diversity. However, total parasitism (herbivory and disease combined) decreased as host clone frequency and fitness increased. Thus, although the highly virulent rust pathogen showed potential for driving the cycles that result from FDS, this apparently does not occur in the populations studied because the host clones were also attacked by herbivores.Co-ordinating editor: J.F. Stuefer     

Directional Selection and Variation in Finite Populations

Predictions are made of the equilibrium genetic variances and responses in a metric trait under the joint effects of directional selection, mutation and linkage in a finite population. The "infinitesimal model" is analyzed as the limiting case of many mutants of very small effect, otherwise Monte Carlo simulation is used. If the effects of mutant genes on the trait are symmetrically distributed and they are unlinked, the variance of mutant effects is not an important parameter. If the distribution is skewed, unless effects or the population size is small, the proportion of mutants that have increasing effect is the critical parameter. With linkage the distribution of genotypic values in the population becomes skewed downward and the equilibrium genetic variance and response are smaller as disequilibrium becomes important. Linkage effects are greater when the mutational variance is contributed by many genes of small effect than few of large effect, and are greater when the majority of mutants increase rather than decrease the trait because genes that are of large effect or are deleterious do not segregate for long. The most likely conditions for "Muller's ratchet" are investigated.     

Fixation of Emerging Interviral Recombinants in Cucumber Mosaic Virus Populations

Interstrain recombinants were observed in the progenies of the Cucumber mosaic virus (CMV) reassortant L₁L₂F₃ containing RNAs 1 and 2 from LS-CMV and RNA 3 from Fny-CMV. We characterized these recombinants, and we found that their fixation was controlled by the nature of the replicating RNAs 1 and 2. We demonstrate that the 2b gene partially affects this fixation process, but only in the context of homologous RNAs 1 and 2.     

Evolutionary Relationship of DNA Sequences in Finite Populations

With the aim of analyzing and interpreting data on DNA polymorphism obtained by DNA sequencing or restriction enzyme technique, a mathematical theory on the expected evolutionary relationship among DNA sequences (nucleons) sampled is developed under the assumption that the evolutionary change of nucleons is determined solely by mutation and random genetic drift. The statistical property of the number of nucleotide differences between randomly chosen nucleons and that of heterozygosity or nucleon diversity is investigated using this theory. These studies indicate that the estimates of the average number of nucleotide differences and nucleon diversity have a large variance, and a large part of this variance is due to stochastic factors. Therefore, increasing sample size does not help reduce the variance significantly. The distribution of sample allele (nucleomorph) frequencies is also studied, and it is shown that a small number of samples are sufficient in order to know the distribution pattern.     

The Evolution of Selectively Similar Electro-Phoretically Detectable Alleles in Finite Natural Populations

Most of the models of population genetics are not realistic when applied to data on electrophoretic variants of proteins because the same net charge may result from any of several amino acid combinations. In the absence of realistic models they have, however, been widely used to test competing hypotheses about the origin and maintenance of genetic variation in populations. In this paper I present a general method for determining probability generating functions for electrophoretic state differences. Then I use the method to find allelic state difference distributions for selectively similar electrophoretically detectable alleles in finite natural populations.Predicted patterns of genetic variation, both within and among species, are in reasonable accord with those found in the Drosophila willistoni group by Ayala et al. (1972) and by Ayala and Tracey (1974).     

On the Theory of Partially Inbreeding Finite Populations. III. Fixation Probabilities under Partial Selfing When Heterozygotes Are Intermediate in Viability

In a previous paper by the senior author, an approximation to the probability of survival was given for a mutant, which is originally present in a single heterozygote, in a population that reproduces partly by selfing and partly by random mating. The population was assumed to be very large, but the result obtained is general with regard to the level of dominance in viability. In this paper two errors which were made in that earlier work are corrected. A general approximate expression is then derived for the probability that an allele A is fixed in a partially self fertilizing population of size N, if its initial frequency is p, selection is weak and heterozygotes with the allele are exactly intermediate in viability compared with genotypes AA and AA. A rigorous proof is given for a special case that is a generalization of the classical binomial sampling model. In this case, but not in general, the approximate fixation probability is independent of the probability of reproduction by selfing. Some implications are discussed.     

Estimation of a Difference in Finite Populations with Missing Observations

Sampling strategies for the difference are constructed when missing observations are present. Two different situations are analyzed. One of them is related with a non random device settled by the statistician for reducing costs. The other is a non response problem. An unbiased minimum variance estimator is obtained in the first case and an approximation to it is deduced. The unbiased estimation in the second is associated with subsampling tactics.     

Variance of Gene Frequencies from Recurrent Selection in Finite Populations

An equation was derived for estimation of the variance of gene frequencies due to drift in the presence of recurrent selection. The equation assumes knowledge of gene frequencies in successive generations. These can be approximated in at least three ways. Simulation data demonstrated that when satisfactory approximations are used, the variances supplied by the derived formula agree well with observed variances.     

Influences of Dominance and Evolution of Sex in Finite Diploid Populations

Most eukaryotes reproduce sexually. Although the benefits of sex in diploids mainly stem from recombination and segregation, the relative effects of recombination and segregation are relatively less known. In this study, we adopt an infinite loci model to illustrate how dominance coefficient of mutations affects the above-mentioned genetic events. However, we assume mutational effects to be independent and also ignore the effects of epistasis within loci. Our simulations show that with different levels of dominance, segregation and recombination may play different roles. In particular, recombination more commonly has a major impact on the evolution of sex when deleterious mutations are partially recessive. In contrast, when deleterious mutations are dominant, segregation becomes more important than recombination, a finding that is consistent with previous studies stating that segregation, rather than recombination, is more likely to drive the evolution of sex. Moreover, beneficial mutations alone remarkably increases the effects of recombination. We also note that populations favor sexual reproduction when deleterious mutations become more dominant or beneficial mutations become more recessive. Overall, these results illustrate that the existence of dominance is an important mechanism that affects the evolution of sex.     

Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

Evolutionary game dynamics of two strategies in finite population is studied by continuous probabilistic approach. Besides frequency dependent selection, mutation was also included in this study. The equilibrium probability density functions of abundance, expected time to extinction or fixation were derived and their numerical solutions are calculated as illustrations. Meanwhile, individual-based computer simulations are also done. A comparison reveals the consistency between theoretical analysis and simulations.     

Selection in Finite Populations. III. an Experimental Examination

The theory determining the probability of fixation of new mutants in a population that allows increased adaptation has been developed by Kimura. Two experimental systems were used to examine this theory. First, alleles were introduced at low frequencies into populations and their fate observed. The second approach was to follow populations with closely linked mutants that were introduced on different chromosomes and to follow the fate of favorable recombinants. These experiments allowed the investigation of the appropriateness of the mathematical model for the particular biological system. In the experiments reported here using two X-linked Drosophila mutants, y and w, the predictions of the theory were reasonably fulfilled in the first experimental system (introduction of new alleles), but not in the second system (favorable recombinants). However, the theoretical framework seems quite robust in that it allowed a satisfactory explanation of the experimental results for the second system as well. The probability of, and the time until, production of a favorable recombinant is discussed.     

On the Divergence of Alleles in Nested Subsamples from Finite Populations

Within-population variation at the DNA level will rarely be studied by sequencing of loci of randomly chosen individuals. Instead, individuals will usually be chosen for sequencing based on some knowledge of their genotype. Data collected in this way require new sampling theory. Motivated by these observations, we have examined the sampling properties of a finite population model with two mutation processes and with no selection or recombination. One mutation process generates new alleles according to an infinite-alleles model, and the other generates polymorphisms at sites according to an infinite-sites model. A sample of n genes is considered. The stationary distribution of the number of segregating sites in a subsample from one of the allelic classes in the sample conditional on the allelic configuration of the sample is studied. A recursive scheme is developed to compute the moments of this distribution, and it is shown that the distribution is functionally independent of the number of additional alleles in the sample and their respective frequencies in the sample. For the case in which the sample contains only two alleles, the distribution of the number of segregating sites in a subsample containing both alleles conditional on the sample frequencies of the alleles is studied. The results are applied to the analysis of DNA sequences of two alleles found at the Adh locus of Drosophila melanogaster. No significant departure from the neutral model is detected.     

A Study on a Nearly Neutral Mutation Model in Finite Populations

As a nearly neutral mutation model, the house-of-cards model is studied in finite populations using computer simulations. The distribution of the mutant effect is assumed to be normal. The behavior is mainly determined by the product of the population size, N, and the standard deviation, sigma, of the distribution of the mutant effect. If 4N sigma is large compared to one, a few advantageous mutants are quickly fixed in early generations. Then most mutation becomes deleterious and very slow increase of the average selection coefficient follows. It takes very long for the population to reach the equilibrium state. Substitutions of alleles occur very infrequently in the later stage. If 4N sigma is the order of one or less, the behavior is qualitatively similar to that of the strict neutral case. Gradual increase of the average selection coefficient occurs and in generations of several times the inverse of the mutation rate the population almost reaches the equilibrium state. Both advantageous and neutral (including slightly deleterious) mutations are fixed. Except in the early stage, an increase of the standard deviation of the distribution of the mutant effect decreases the average heterozygosity. The substitution rate is reduced as 4N sigma is increased. Three tests of neutrality, one using the relationship between the average and the variance of heterozygosity, another using the relationship between the average heterozygosity and the average number of substitutions and Watterson's homozygosity test are applied to the consequences of the present model. It is found that deviation from the neutral expectation becomes apparent only when 4N sigma is more than two. Also a simple approximation for the model is developed which works well when the mutation rate is very small.