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.     

Selection Response in Finite Populations

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (σ(AW0)(2)) and mutational (σ(M)(2)) variance. The cumulative response to selection until generation t(CR(t)) can be approximated as & where N(e) is the effective population size, σ(AW &)(2) = N(e)σ(M)(2) is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R(0) is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. β is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R(0) is the major determinant of the short term selection response, but σ(M)(2), N(e) and β are also important for the long term. A selection method of high accuracy using family information gives a small N(e) and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.     

Probability of Fixation and Mean Fixation Time of an Overdominant Mutation

The probability of fixation of an overdominant mutation in a finite population depends on the equilibrium gene frequency in an infinite population (m) and the product (A) of population size and selection intensity. If m < 0.5 (disadvantageous overdominant genes), the probability is generally much lower than that of neutral genes; but if m is close to 0.5 and A is relatively small, it becomes higher. If m > 0.5 (advantageous overdominant genes), the probability is largely determined by the fitness of heterozygotes rather than that of mutant homozygotes. Thus, overdominance enhances the probability of fixation of advantageous mutations. The average number of generations until fixation of an overdominant mutation also depends on m and A. This average time is long when m is close to 0.5 but short when m is close to 0 or 1. This dependence on m and A is similar to that of Robertson's retardation factor.     

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     

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.     

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.     

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).     

The Effects of Environmental Heterogeneity on the Genetics of Finite Populations

The effects on a panmictic population of a patchy environment and stochastic selection were investigated by computer simulation. The model allowed for different patterns and intensities of selection and various densities of distribution of the offspring. It delineated the effects of these variables on the correlation between the genotype and the environment, the level of heterozygosity, the change in gene frequency from generation to generation and the divergence between similar populations due to random effects.     

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.     

Probability Distribution of Drug-Resistant Mutants in Unselected Populations of Mycobacterium tuberculosis

The fluctuation test shows that Mycobacterium tuberculosis mutates to resistance to isoniazid, streptomycin, ethambutol and rifampin spontaneously and at random. The average mutation rates for the drugs, in the same order, were calculated to be 2.56 × 10⁻⁸, 2.95 × 10⁻⁸, 10⁻⁷, and 2.25 × 10⁻¹⁰ mutation per bacterium per generation. The relatively high mutation rate to ethambutol resistance and the low mutation rate to rifampin resistance were confirmed by analyzing the increase in the proportion of mutants with time in a growing population of the tubercle bacilli. The highest proportions of mutants to be expected in unselected populations of the tubercle bacilli were calculated from the results of fluctuation tests.     

Hidden Genetic Variability within Electromorphs in Finite Populations

The amount of hidden genetic variability within electromorphs in finite populations is studied by using the infinite site model and stepwise mutation model simultaneously. A formula is developed for the bivariate probability generating function for the number of codon differences and the number of electromorph state differences between two randomly chosen cistrons. Using this formula, the distribution as well as the mean and variance of the number of codon differences between two identical or nonidentical electromorphs are studied. The distribution of the number of codon differences between two randomly chosen identical electromorphs is similar to the geometric distribution but more leptokurtic. Studies are also made on the number of codon differences between two electromorphs chosen at random one from each of two populations which have been separated for an arbitrary number of generations. It is shown that the amount of hidden genetic variability is very large if the product of effective population size and mutation rate is large.     

The Utility of an Online Convenience Panel for Reaching Rare and Dispersed Populations

Gaps in data collection systems, as well as challenges associated with gathering data from rare and dispersed populations, render current health surveillance systems inadequate to identify and monitor efforts to reduce health disparities. Using sexual and gender minorities we investigated the utility of using a large nonprobability online panel to conduct rapid population assessments of such populations using brief surveys. Surveys of the Google Android Panel (four assessing sexual orientation and one assessing gender identity and sex assigned at birth) were conducted resulting in invitation of 53,739 application users (37,505 of whom viewed the invitation) to generate a total of 34,759 who completed screening questions indicating their sexual orientation, or gender identity and sex at birth. Where possible we make comparisons to similar data from two population-based surveys (NHIS and NESARC). We found that 99.4% to 100.0% of respondents across our Google Android panel samples completed the screening questions and 97.8% to 99.2% of those that consented to participate in our surveys indicated they were “OK” with the content of surveys that assessed sexual orientation and sex/gender. In our Google Android panel samples there was a higher percentage of sexual minority respondents than in either NHIS or NESARC with 7.4% of men and 12.4% of women reporting gay, lesbian or bisexual identities. The proportion sexual minority was 2.8 to 5.6 times higher in the Google Android panel samples than was found in the 2012 NHIS sample, for men and women, respectively. The percentage of “transgender” identified individuals in the Google sample was 0.7%, which is similar to 0.5% transgender identified through the Massachusetts BRFSS, and using a transgender status item we found that 2.0% of the overall sample fit could be classified as transgender. The Google samples sometimes more closely approximated national averages for ethnicity and race than NHIS.     

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.