Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

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 1/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 1/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.     

Effect of Mating Structure on Variation in Linkage Disequilibrium

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (N_e) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium.——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N_e, mating structure, n, and linkage or recombination parameter.——The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.     

Interchromosomal Biased Gene Conversion, Mutation and Selection in a Multigene Family

A mathematical model of the effects of interchromosomal biased gene conversion, mutation and natural selection on a multigene family is developed and analyzed. The model assumes two allelic states at each of n loci. The effects of genetic drift are ignored. The model is developed under the assumption of no recombination, but the analysis shows that, at equilibrium, there is no linkage disequilibrium, which implies that the conclusions are valid for arbitrary recombination among loci. At equilibrium, the balance between mutation, gene conversion and selection depends on the ratio of the mutation rates to the quantity [s + g(2α - 1)/ n], where s is the increment or decrement in relative fitness with each additional copy of one of the alleles, g is the conversion rate, and α is a measure of the bias in favor of one of the alleles. When this quantity is large relative to the mutation rates, the allele that has the net advantage, combining the effects of selection and conversion, will be nearly fixed in the multigene family. A comparison of these results with those from a comparable model of intrachromosomal biased conversion shows that biased interchromosomal conversion leads to approximately the same equilibrium copy number as does intrachromosomal conversion of the same strength. Interchromosomal conversion is much more effective in causing the substitution of one allele by another. The relative frequencies of interchromosomal and intrachromosomal conversion is indicated by the extent of the linkage disequilibrium among the loci in a multigene family.     

Genetic Diversity of Plasmodium falciparum in Haiti: Insights from Microsatellite Markers

Hispaniola, comprising Haiti and the Dominican Republic, has been identified as a candidate for malaria elimination. However, incomplete surveillance data in Haiti hamper efforts to assess the impact of ongoing malaria control interventions. Characteristics of the genetic diversity of Plasmodium falciparum populations can be used to assess parasite transmission, which is information vital to evaluating malaria elimination efforts. Here we characterize the genetic diversity of P. falciparum samples collected from patients at seven sites in Haiti using 12 microsatellite markers previously employed in population genetic analyses of global P. falciparum populations. We measured multiplicity of infections, level of genetic diversity, degree of population geographic substructure, and linkage disequilibrium (defined as non-random association of alleles from different loci). For low transmission populations like Haiti, we expect to see few multiple infections, low levels of genetic diversity, high degree of population structure, and high linkage disequilibrium. In Haiti, we found low levels of multiple infections (12.9%), moderate to high levels of genetic diversity (mean number of alleles per locus = 4.9, heterozygosity = 0.61), low levels of population structure (highest pairwise F_st = 0.09 and no clustering in principal components analysis), and moderate linkage disequilibrium (ISA = 0.05, P<0.0001). In addition, population bottleneck analysis revealed no evidence for a reduction in the P. falciparum population size in Haiti. We conclude that the high level of genetic diversity and lack of evidence for a population bottleneck may suggest that Haiti’s P. falciparum population has been stable and discuss the implications of our results for understanding the impact of malaria control interventions. We also discuss the relevance of parasite population history and other host and vector factors when assessing transmission intensity from genetic diversity data.     

Linkage disequilibrium in a finite population that is partially selfing

The linkage disequilibrium expected in a finite, partially selfing population is analyzed, assuming the infinite allele model. Formulas for the expected sum of squares of the linkage disequilibria and the squared standard linkage disequilibrium are derived from the equilibrium values of sixteen inbreeding coefficients required to describe the behavior of the system. These formulas are identical to those obtained with random mating if the effective population size N_e = (1-½S)N and the effective recombination value r_e = (1-S)r/(1-½S), where S is the proportion of selfing, are substituted for the population size and the recombination value. Therefore, the effect of partial selfing at equilibrium is to reduce the population size by a factor 1-½S and the recombination value by a factor (1-S)/(1-½S).     

The estimates of effective population size based on linkage disequilibrium are virtually unaffected by natural selection

The effective population size (N_e) is a key parameter to quantify the magnitude of genetic drift and inbreeding, with important implications in human evolution. The increasing availability of high-density genetic markers allows the estimation of historical changes in N_e across time using measures of genome diversity or linkage disequilibrium between markers. Directional selection is expected to reduce diversity and N_e, and this reduction is modulated by the heterogeneity of the genome in terms of recombination rate. Here we investigate by computer simulations the consequences of selection (both positive and negative) and recombination rate heterogeneity in the estimation of historical N_e. We also investigate the relationship between diversity parameters and N_e across the different regions of the genome using human marker data. We show that the estimates of historical N_e obtained from linkage disequilibrium between markers (N_eLD) are virtually unaffected by selection. In contrast, those estimates obtained by coalescence mutation-recombination-based methods can be strongly affected by it, which could have important consequences for the estimation of human demography. The simulation results are supported by the analysis of human data. The estimates of N_eLD obtained for particular genomic regions do not correlate, or they do it very weakly, with recombination rate, nucleotide diversity, proportion of polymorphic sites, background selection statistic, minor allele frequency of SNPs, loss of function and missense variants and gene density. This suggests that N_eLD measures mainly reflect demographic changes in population size across generations.     

Frequency-Dependent Selection at the PGM-1 Locus of DROSOPHILA PSEUDOOBSCURA

Frequency-dependent fitness was studied at the Pgm-1 locus of Drosophila pseudoobscura with respect to two fitness components: rate of development and larva-to-adult survival. The Pgm-1 locus is very polymorphic with only two alleles, Pgm-1¹⁰⁰ and Pgm-1¹⁰⁴, occurring at high frequencies. For each of these two alleles, 20 homozygous strains were obtained from a sample of 1,140 wild-inseminated females. First-instar larvae of the two genotypes were combined in a set of eight different frequencies: 0.0, 0.10, 0.25, 0.40, 0.60, 0.75, 0.90, and 1.0. Frequency-dependent fitness effects were observed for the two survival-related fitness components examined: larvae of the less common genotype develop faster and have a higher probability of survival than larvae of the more common genotype. The rate of survival at intermediate genotypic frequencies is similar to that in pure cultures. If selection acted solely as frequency-dependent effects on survival-related components of fitness, the equilibrium frequency of the Pgm-1¹⁰⁰ allele would be 0.615 for a two-genotype system, which fits an observed frequency range for this allele in nature between 0.55 and 0.71. Experimentally created linkage disequilibrium was excluded from the experiment by using a large number of independent strains. It is nevertheless possible that the frequency-dependent selection may not affect the Pgm-1 locus per se, but may reflect a linkage disequilibrium present in the natural population. Even if this were the case, the frequency-dependent selection could affect the frequency of the Pgm-1 alleles in nature.     

Selection at the Esterase-2 Locus of Drosophila buzzatii? Perturbation-Reperturbation Experiments

Apparent selection affecting starch gel electrophoretic alleles at the Esterase-2 locus of Drosophila buzzatii has been detected in laboratory and natural populations. Perturbation-reperturbation of allele frequencies in replicated laboratory populations attempts to test direct selective effects at the locus versus effects of linked loci. Sequential gel electrophoresis has identified more alleles within starch classes, and three of these alleles (within the a, b and c starch alleles) were used in cage population experiments. Allele a/1.00/1.00/1.00 was set up in 10 replicate populations with allele c/1.00/1.00/1.00, and in an independent 10 replicate populations with allele b/0.99/1.01/1.00. For each set, three reperturbations were done. Replicate populations generally showed similar patterns of allele frequency change and clear directionality: effects of selection, not drift. However, four populations deviated from their replicates, indicating dissipation of linkage disequilibrium. Estimates of pre-adult viability in the F₂ of pair-wise crosses among 12 sequential gel electrophoretic alleles showed very variable modes of inheritance and relative viability fitnesses. Together with the diversity of patterns of allele frequency change in the cage populations, these results suggest a gene complex, with selection acting on an interacting set of loci which may include Esterase-2.     

The Sampling Distribution of Linkage Disequilibrium under an Infinite Allele Model without Selection

The sampling distributions of several statistics that measure the association of alleles on gametes (linkage disequilibrium) are estimated under a two-locus neutral infinite allele model using an efficient Monte Carlo method. An often used approximation for the mean squared linkage disequilibrium is shown to be inaccurate unless the proper statistical conditioning is used. The joint distribution of linkage disequilibrium and the allele frequencies in the sample is studied. This estimated joint distribution is sufficient for obtaining an approximate maximum likelihood estimate of C = 4Nc, where N is the population size and c is the recombination rate. It has been suggested that observations of high linkage disequilibrium might be a good basis for rejecting a neutral model in favor of a model in which natural selection maintains genetic variation. It is found that a single sample of chromosomes, examined at two loci cannot provide sufficient information for such a test if C less than 10, because with C this small, very high levels of linkage disequilibrium are not unexpected under the neutral model. In samples of size 50, it is found that, even when C is as large as 50, the distribution of linkage disequilibrium conditional on the allele frequencies is substantially different from the distribution when there is no linkage between the loci. When conditioned on the number of alleles at each locus in the sample, all of the sample statistics examined are nearly independent of theta = 4N mu, where mu is the neutral mutation rate.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

Estimating effective population size from linkage disequilibrium: severe bias in small samples

Effective population size (N _e) is a central concept in evolutionary biology and conservation genetics. It predicts rates of loss of neutral genetic variation, fixation of deleterious and favourable alleles, and the increase of inbreeding experienced by a population. A method exists for the estimation of N _e from the observed linkage disequilibrium between unlinked loci in a population sample. While an increasing number of studies have applied this method in natural and managed populations, its reliability has not yet been evaluated. We developed a computer program to calculate this estimator of N _e using the most widely used linkage disequilibrium algorithm and used simulations to show that this estimator is strongly biased when the sample size is small (<‰100) and below the true N _e. This is probably due to the linkage disequilibrium generated by the sampling process itself and the inadequate correction for this phenomenon in the method. Results suggest that N _e estimates derived using this method should be regarded with caution in many cases. To improve the method’s reliability and usefulness we propose a way to determine whether a given sample size exceeds the population N _e and can therefore be used for the computation of an unbiased estimate.     

Evolution and intraspecific exploitative competition I. One-locus theory for small additive gene effects

Using the model of exploitative competition of R. H. MacArthur and R. Levins (1967, Amer. Natur. 101, 377–385), evolution at a gene locus which influences the niche position is considered. The locus has multiple alleles, and the contributions of the alleles to the genotypic value are additive. The resource spectrum and the utilization functions of the genotypes are assumed to be Gaussian. Evolution will make the mode of the niche converge to the resource optimum, as long as the allele contributions are small compared to the distance between the mode of the niche and the resource optimum. When this distance is of the same order of magnitude as the allele contributions, then the globally stable equilibrium will maintain at most two alleles in the population, unless the allele contributions are large. Classical overdominance is not needed to maintain polymorphism. This result predicts high linkage disequilibrium in similar multilocus models. It is concluded that intraspecific competition can be a powerful force in maintaining two-allele polymorphisms, and that it can maintain high linkage disequilibrium among closely linked loci.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.     

Selection never dominates drift (nor vice versa)

The probability that the fitter of two alleles will increase in frequency in a population goes up as the product of N (the effective population size) and s (the selection coefficient) increases. Discovering the distribution of values for this product across different alleles in different populations is a very important biological task. However, biologists often use the product Ns to define a different concept; they say that drift “dominates” selection or that drift is “stronger than” selection when Ns is much smaller than some threshold quantity (e.g., ½) and that the reverse is true when Ns is much larger than that threshold. We argue that the question of whether drift dominates selection for a single allele in a single population makes no sense. Selection and drift are causes of evolution, but there is no fact of the matter as to which cause is stronger in the evolution of any given allele.     

Two-locus,fourth-order gene frequency moments: Implications for the variance of squared linkage disequilibrium and the variance of homozygosity

Identity coefficients are used to construct a sufficient set of equations to determine the fourth-order moments of gene frequencies for two linked loci. This allows the variance of the expected squared linkage disequilibrium to be found. It is shown that the coefficient of variation is generally greater than one and if the mutation rate is small, the standard deviation is more than four times the size of the mean. This demonstrates that squared linkage disequilibrium is a highly variable quantity. The variance of homozygosity for a gene which consists of two sites can also be obtained. Recombination between these sites increases the variance of homozygosity, suggesting that intragenic recombination significantly changes all the expected moments of gene frequencies if 4Nμ > 1.0 and r > μ (where N is the population size, μ is the mutation rate of the gene to neutral alleles, and r is the recombination rate between two sites within the gene).     

The Sampling Distribution of Linkage Disequilibrium

The probabilities of obtaining particular samples of gametes with two completely linked loci are derived. It is assumed that the population consists of N diploid, randomly mating individuals, that each of the two loci mutate according to the infinite allele model at a rate µ and that the population is at equilibrium. When 4Nµ is small, the most probable samples of gametes are those that segregate only two alleles at either locus. The probabilities of various samples of gametes are discussed. The results show that most samples with completely linked loci have either a very small or a very large association between the alleles of each locus. This causes the distribution of linkage disequilibrium to be skewed and the distribution of the correlation coefficient to be bimodal. The correlation coefficient is commonly used as a test statistic with a chi square distribution and yet has a bimodal distribution when the loci are completely linked. Thus, such a test is not likely to be accurate unless the rate of recombination between the loci and/or the effective population size are sufficiently large enough so that the loci can be treated as unlinked.     

Evolution of recombination due to random drift

In finite populations subject to selection, genetic drift generates negative linkage disequilibrium, on average, even if selection acts independently (i.e., multiplicatively) upon all loci. Negative disequilibrium reduces the variance in fitness and hence, by Fisher's (1930) fundamental theorem, slows the rate of increase in mean fitness. Modifiers that increase recombination eliminate the negative disequilibria that impede selection and consequently increase in frequency by "hitchhiking." Thus, stochastic fluctuations in linkage disequilibrium in finite populations favor the evolution of increased rates of recombination, even in the absence of epistatic interactions among loci and even when disequilibrium is initially absent. The method developed within this article allows us to quantify the strength of selection acting on a modifier allele that increases recombination in a finite population. The analysis indicates that stochastically generated linkage disequilibria do select for increased recombination, a result that is confirmed by Monte Carlo simulations. Selection for a modifier that increases recombination is highest when linkage among loci is tight, when beneficial alleles rise from low to high frequency, and when the population size is small.     

Smaller Effective Population Sizes Evidenced by Loss of Microsatellite Alleles in Tributary-Spawning Populations of Sockeye Salmon from the Kvichak River, Alaska drainage

We tested signals of historical reductions in effective population size within populations of sockeye salmon Oncorhynchus nerka returning to Bristol Bay, Alaska, to examine the roles that ecotype, migration obstacles, and drainage might play in the highly variable production of the Kvichak River drainage. We collected data for eight microsatellite loci from ～100 fish at each of 16 locations within the Kvichak River drainage and five locations within the more productively stable Naknek River drainage. Pair-wise exact tests were used to group similar collections within ecotype, within drainage, and above and below migration obstacles. After grouping, collections represented independent populations for further analyses. We examined the number of alleles per locus, mean ratio of the number of alleles to the range in allele size, heterozygosity excess, and gametic disequilibrium as measures of reduction-in-population-size events. Number of alleles per locus revealed the largest number of significant differences. Tributary populations showed a stronger signal consistent with reduced effective population size than did beach populations within the Kvichak River drainage. Kvichak River drainage populations showed a stronger signal consistent with reduced effective population size than did the Naknek River drainage populations. Populations above migration obstacles showed signals consistent with reduction in historical population sizes in multiple measures indicating some of these reductions may be severe enough to qualify as demographic bottlenecks.     

A comparison of single‐sample effective size estimators using empirical toad (Bufo calamita) population data: genetic compensation and population size‐genetic diversity correlations

The accuracy and precision of four single‐sample estimators of effective population size, N_e (heterozygote excess, linkage disequilibrium, Bayesian partial likelihood and sibship analysis) were compared using empirical data (microsatellite genotypes) from multiple natterjack toad (Bufo calamita) populations in Britain (n = 16) and elsewhere in Europe (n = 10). Census size data were available for the British populations. Because toads have overlapping generations, all of these methods estimated the number of effective breeders N_b rather than N_e. The heterozygote excess method only provided results, without confidence limits, for nine of the British populations. Linkage disequilibrium gave estimates for 10 British populations, but only six had finite confidence limits. The Bayesian and sibship methods both produced estimates with finite confidence limits for all the populations. Although the Bayesian method was the most precise, on most criteria (insensitivity to locus number, correlation with other effective and census size estimates and correlation with genetic diversity) the sibship method performed best. The results also provided evidence of genetic compensation in natterjack toads, and highlighted how the relationship between effective size and genetic diversity can vary as a function of geographical scale.     

Relationship of runs of homozygosity with adaptive and production traits in a paternal broiler line

Genomic regions under high selective pressure present specific runs of homozygosity (ROH), which provide valuable information on the genetic mechanisms underlying the adaptation to environment imposed challenges. In broiler chickens, the adaptation to conventional production systems in tropical environments lead the animals with favorable genotypes to be naturally selected, increasing the frequency of these alleles in the next generations. In this study, ~1400 chickens from a paternal broiler line were genotyped with the 600 K Affymetrix® Axiom® high-density (HD) genotyping array for estimation of linkage disequilibrium (LD), effective population size (N_e), inbreeding and ROH. The average LD between adjacent single nucleotide polymorphisms (SNPs) in all autosomes was 0.37, and the LD decay was higher in microchromosomes followed by intermediate and macrochromosomes. The N_e of the ancestral population was high and declined over time maintaining a sufficient number of animals to keep the inbreeding coefficient of this population at low levels. The ROH analysis revealed genomic regions that harbor genes associated with homeostasis maintenance and immune system mechanisms, which may have been selected in response to heat stress. Our results give a comprehensive insight into the relationship between shared ROH regions and putative regions related to survival and production traits in a paternal broiler line selected for over 20 years. These findings contribute to the understanding of the effects of environmental and artificial selection in shaping the distribution of functional variants in the chicken genome.