Coalescent Size Versus Coalescent Time with Strong Selection

This work studies the coalescent (ancestral pedigree, genealogy) of the entire population. The coalescent structure (topology) is robust, but selection changes the rate of coalescence (the time between branching events). The change in the rate of coalescence is not uniform, rather the reduction in the time between branching events is greatest when the coalescent is small (immediately after the common ancestor is the only member of the coalescent) with little change when the coalescent is large (immediately preceding when that common ancestor becomes fixed and the size of the coalescent is N). This provides that the reduction in the coalescent time due to selection is much greater than the reduction in the cumulative size of the coalescent (total number of ancestors of the present population after and including the most recent common ancestor) due to selection. If Ns≫1, the coalescent and fixation times are approximately equal to , which is much less than the value N which would result from neutral drift (N rather than the canonical haploid neutral fixation time 2N is the appropriate comparison for the model considered here), in particular, it is 70% less for Ns=10 and 95% less for Ns=100. However, for those values of Ns, and N ranging between 10³ and 10⁶, the reduction in the cumulative size of the coalescent of the entire population compared to the neutral case ranges from 17% to 65% (depending on the values of N and s). The coalescent time for two individuals for Ns of 10 and 100 is reduced by approximately 70% and 94%, respectively, compared with the neutral case. Because heterozygosity is proportional to the coalescent time for two individuals and the number of segregating alleles is proportional to the cumulative size of the coalescent, selection reduces heterozygosity more than it reduces the number of segregating alleles.     

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.     

A MODEL FOR THE EVOLUTION OF ASYMMETRICAL MALE HYBRID STERILITY AND ITS IMPLICATIONS FOR SPECIATION

Chromosomal analysis of several cases of asymmetrical male hybrid sterility in Drosophila has implicated the X- or the Y-chromosome and one or more autosomes. Here, I develop a model for the evolution of this phenomenon. An autosomal locus is assumed to affect viability and to interact with a Y-linked or an X-linked locus to determine male fertility. In a new environment, selection for viability favors a new allele at the autosomal locus, but incompatibility of this new allele with the sex-chromosome-linked gene generates male sterility. The incompatibility can be resolved if a new allele at the sex-linked locus invades the population. This results in nonreciprocal male hybrid sterility, the direction of the nonreciprocity being determined by the dominance or recessiveness of the new autosomal gene in its effect on fertility. It is shown that stable polymorphism for the autosomal locus is possible and that, if the equilibrium frequency of the new allele is above a critical value, the population will be constantly at the verge of speciation, “waiting” for the sex-linked mutation to occur. The appearance of this mutation causes a runaway process leading to rapid fixation of the new autosomal and sex-linked alleles. If the equilibrium frequency of the new autosomal allele is less than the critical value, deterministic speciation is impossible, but random drift may increase the frequency above the critical value and predispose the population to the invasion of the new sex-linked allele. Thus, both deterministic and stochastic modes of speciation are possible. Because deterministic speciation requires large selection coefficients, which impose a severe genetic load on the population, and because stochastic speciation requires repeated population bottlenecks, it is concluded that relative to the number of successful speciation events there will be many more events of deme extinction.     

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 stationary distribution of an asymmetrical model of selection in a random environment

The stationary distribution for the asymmetrical form of the SAS-CFF model of selection in a random environment is presented. Also presented are the conditions for the stable coexistence of K alleles. These conditions are the same as the conditions obtained from the classical constant-fitness model with the formal substitution of geometric mean fitnesses for the constant fitnesses of the classical model. Two examples are explored. In the “equally spaced” example, increases in the degree of asymmetry raise the homozygosity, which is accompanied by loss of alleles from the population. In the “best allele” example, increases in the degree of asymmetry raise the homozygosity without the loss of alleles. In both cases the frequency spectra are altered by the changes in the degree of asymmetry.     

Possible roles of two quinone molecules in direct and indirect proton pumps of bovine heart NADH-quinone oxidoreductase (complex I)

In many energy transducing systems which couple electron and proton transport, for example, bacterial photosynthetic reaction center, cytochrome bc₁-complex (complex III) and E. coli quinol oxidase (cytochrome bo₃ complex), two protein-associated quinone molecules are known to work together. T. Ohnishi and her collaborators reported that two distinct semiquinone species also play important roles in NADH-ubiquinone oxidoreductase (complex I). They were called SQ_Nf (fast relaxing semiquinone) and SQ_Ns (slow relaxing semiquinone). It was proposed that Q_Nf serves as a “direct” proton carrier in the semiquinone-gated proton pump (Ohnishi and Salerno, FEBS Letters 579 (2005) 4555), while Q_Ns works as a converter between one-electron and two-electron transport processes. This communication presents a revised hypothesis in which Q_Nf plays a role in a “direct” redox-driven proton pump, while Q_Ns triggers an “indirect” conformation-driven proton pump. Q_Nf and Q_Ns together serve as (1e^?/2e^?) converter, for the transfer of reducing equivalent to the Q-pool.     

EPISTASIS AND THE INCREASE IN ADDITIVE GENETIC VARIANCE: IMPLICATIONS FOR PHASE 1 OF WRIGHT'S SHIFTING-BALANCE PROCESS

Central to Wright's shifting-balance theory is the idea that genetic drift and selection in systems with gene interaction can lead to the formation of “adaptive gene complexes.” The theory of genetic drift has been well developed over the last 60 years; however, nearly all of this theory is based on the assumption that only additive gene effects are acting. Wright's theory was developed recognizing that there was a “universality of interaction effects,” which implies that additive theory may not be adequate to describe the process of differentiation that Wright was considering. The concept of an adaptive gene complex implies that an allele that is favored by individual selection in one deme may be removed by selection in another deme. In quantitative genetic terms, the average effects of an allele relative to other alleles changes from deme to deme. The model presented here examines the variance in local breeding values (LBVs) of a single individual and the covariance in the LBVs of a pair of individuals mated in the same deme relative to when they are mated in different demes. Local breeding value is a measure of the average effects of the alleles that make up that individual in a particular deme. I show that when there are only additive effects the covariance between the LBVs of individuals equals the variance in the LBV of an individual. As the amount of epistasis in the ancestral population increases, the variance in the LBV of an individual increases and the covariance between the LBVs of a pair of individuals decreases. The divergence in these two values is a measure of the extent to which the LBV of an individual varies independently of the LBVs of other individuals. When this value is large, it means that the relative ordering of the average effects of alleles will change from deme to deme. These results confirm an important component of Wright's shifting-balance theory: When there is gene interaction, genetic drift can lead to the reordering of the average effects of alleles and when coupled with selection this will lead to the formation of the adaptive gene complexes.     

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.     

Ages,extinction times,and first passage probabilities for a multiallele diffusion model with irreversible mutation

We derive some new results for diffusion models in population genetics of the “infinite-alleles” type. Assuming present allelic frequencies known, we find expected values for times which may be interpreted either as the age of the kth oldest allele or the time to extinction of the “kth-to-last” allele to be lost. We also find some conditional expectations and probabilities related to the order of extinction or creation of the alleles. Computations illustrate the dependence of the expected times on the mutation rate.     

SEXUAL SELECTION CAN INCREASE THE EFFECT OF RANDOM GENETIC DRIFT—A QUANTITATIVE GENETIC MODEL OF POLYMORPHISM IN OOPHAGA PUMILIO,THE STRAWBERRY POISON‐DART FROG

The variation in color pattern between populations of the poison‐dart frog Oophaga pumilio across the Bocas del Toro archipelago in Panama is suggested to be due to sexual selection, as two other nonsexually selecting Dendrobatid species found in the same habitat and range do not exhibit this variation. We theoretically test this assertion using a quantitative genetic sexual selection model incorporating aposematic coloration and random drift. We find that sexual selection could cause the observed variation via a novel process we call “coupled drift.” Within our model, for certain parameter values, sexual selection forces frog color to closely follow the evolution of female preference. Any between‐population variation in preference due to genetic drift is passed on to color. If female preference in O. pumilio is strongly affected by drift, whereas color in the nonsexually selecting Dendrobatid species is not, coupled drift will cause increased between‐population phenotypic variation. However, with different parameter values, coupled drift will result in between‐population variation in color being suppressed compared to its neutral value, or in little or no effect. We suggest that coupled drift is a novel theoretical process that could have a role linking sexual selection with speciation both in O. pumilio, and perhaps more generally.     

Genetic variability maintained by mutation and overdominant selection in finite populations

Mathematical properties of the overdominance model with mutation and random genetic drift are studied by using the method of stochastic differential equations (Itô and McKean 1974). It is shown that overdominant selection is very powerful in increasing the mean heterozygosity as compared with neutral mutations, and if 2Ns (N = effective population size; s = selective disadvantage for homozygotes) is larger than 10, a very low mutation rate is sufficient to explain the observed level of allozyme polymorphism. The distribution of heterozygosity for overdominant genes is considerably different from that of neutral mutations, and if the ratio of selection coefficient (s) to mutation rate (ν) is large and the mean heterozygosity (h) is lower than 0.2, single-locus heterozygosity is either approximately 0 or 0.5. If h increases further, however, heterozygosity shows a multiple-peak distribution. Reflecting this type of distribution, the relationship between the mean and variance of heterozygosity is considerably different from that for neutral genes. When s/v is large, the proportion of polymorphic loci increases approximately linearly with mean heterozygosity. The distribution of allele frequencies is also drastically different from that of neutral genes, and generally shows a peak at the intermediate gene frequency. Implications of these results on the maintenance of allozyme polymorphism are discussed.     

THREE SELECTIONS ARE BETTER THAN ONE: CLINAL VARIATION OF THERMAL QTL FROM INDEPENDENT SELECTION EXPERIMENTS IN DROSOPHILA

We report the results of two independent selection experiments that have exposed distinct populations of Drosophila melanogaster to different forms of thermal selection. A recombinant population derived from Arvin California and Zimbabwe isofemale lines was exposed to laboratory natural selection at two temperatures (T_AZ: 18°C and 28°C). Microsatellite mapping identified quantitative trait loci (QTL) on the X‐chromosome between the replicate “Hot” and “Cold” populations. In a separate experiment, disruptive selection was imposed on an outbred California population for the “knockdown” temperature (T_KD) in a thermal column. Microsatellite mapping of the “High” and “Low” populations also uncovered primarily X‐linked QTL. Notably, a marker in the shaggy locus at band 3A was significantly differentiated in both experiments. Finer scale mapping of the 3A region has narrowed the QTL to the shaggy gene region, which contains several candidate genes that function in circadian rhythms. The same allele that was increased in frequency in the High T_KD populations is significantly clinal in North America and is more common at the warm end of the cline (Florida vs. Maine; however, the cline was not apparent in Australia). Together, these studies show that independent selection experiments can uncover the same target of selection and that evolution in the laboratory can recapitulate putatively adaptive clinal variation in nature.     

On the distribution of allele frequencies in a diffusion model

An expression is derived and values tabulated for the expected allele frequencies and their variances, arranged in decreasing order in a population, from the finite and infinite alleles diffusion model in Watterson (1976). The neutral model and also a model with heterozygote selection are considered. Some observed ABO blood group allele frequencies are compared with the tabulated expected frequencies in the neutral three allele model. This extends the results of Watterson and Guess (1977) who tabulate the expected value of the most common allele. One test of neutrality previously advocated is to consider the distribution of F, the population homozygosity, conditional on G, the product of allele frequencies. However it is shown here that for a large number of alleles, F and G are asymptotically independent, the test would not be a good one in this case. A limit theorem is derived for the distribution of allele frequencies in the neutral model when the mutation rate is large. In this case F is shown to be asymptotically normal. An inequality is derived for the probability that the oldest allele in a population is amongst the r most frequent types. An inequality is also found for the probability that a sample will only contain representatives of the r most frequent allele types in the population.     

EFFECTIVENESS OF SELECTION IN REDUCING THE GENETIC LOAD IN POPULATIONS OF PEROMYSCUS POLIONOTUS DURING GENERATIONS OF INBREEDING

It has been hypothesized that natural selection reduces the “genetic load” of deleterious alleles from populations that inbreed during bottlenecks, thereby ameliorating impacts of future inbreeding. We tested the efficiency with which natural selection purges deleterious alleles from three subspecies of Peromyscus polionotus during 10 generations of laboratory inbreeding by monitoring pairing success, litter size, viability, and growth in 3604 litters produced from 3058 pairs. In P. p. subgriseus, there was no reduction across generations in inbreeding depression in any of the fitness components. Strongly deleterious recessive alleles may have been removed previously during episodes of local inbreeding in the wild, and the residual genetic load in this population was not further reduced by selection in the lab. In P. p. rhoadsi, four of seven fitness components did show a reduction of the genetic load with continued inbreeding. The average reduction in the genetic load was as expected if inbreeding depression in this population is caused by highly deleterious recessive alleles that are efficiently removed by selection. For P. p. leucocephalus a population that experiences periodic bottlenecks in the wild, the effect of further inbreeding in the laboratory was to exacerbate rather than reduce the genetic load. Recessive deleterious alleles may have been removed from this population during repeated bottlenecks in the wild; the population may be close to a threshold level of heterozygosity below which fitness declines rapidly. Thus, the effects of selection on inbreeding depression varied substantially among populations, perhaps due to different histories of inbreeding and selection.     

Comparative analysis of N-acetylation polymorphism in humans as determined by phenotyping and genotyping

The N-acetylation polymorphisms of volunteers from the Moscow population analyzed by phenotyping and genotyping have been compared. The ratios between the proportions of fast acetylators (FAs) and slow acetylators (SAs) estimated by phenotyping and genotyping do not differ significantly from each other (47 and 44%, respectively). The absolute acetylation rate widely varies in both FAs and SAs. The NAT2 genotype and allele frequencies in the population sample have been calculated. The most frequent alleles are NAT2*4 (a “fast” allele), NAT2*5, and NAT2*6 (“slow” alleles); the most frequent genotypes are NAT2*5/*5, NAT2*4/*6, and NAT2*4/*5. Comparative analysis of N-acetylation polymorphism estimated by phenotyping and genotyping in the same subjects has shown a complete concordance between the phenotype and genotype in only 62 out of 75 subjects (87%). Comparative characteristics and presumed applications of the two approaches (quantitative estimation of acetylation rate and qualitative determination of the acetylator genotype) to the identification of individual acetylation status are presented.     

Reduced population size can induce quick evolution of inbreeding depression in the invasive ladybird <Emphasis Type="Italic">Harmonia axyridis</Emphasis>

Understanding biological invasion is currently one of the main scientific challenges for ecologists. The introduction process is crucial for the success of an invasion, especially when it involves a demographic bottleneck. A small introduced population is expected to face a higher risk of extinction before the first stage of invasion is complete if inbreeding depression, caused by the expression of deleterious alleles, is important. Changes in mating regimes or in population size can induce the evolution of deleterious allele frequencies, either by selection or by drift, possibly resulting in the purging or the fixation of such alleles within the population. The harlequin ladybird Harmonia axyridis became invasive on several continents following a scenario including at least one event of demographic bottleneck. Although native populations suffered from severe inbreeding depression, it was greatly reduced in invasive ones suggesting that deleterious alleles were purged during the invasion process. In this study, we performed an experiment designed to manipulate the effective population size of H. axyridis across successive generations to mimic contrasting introduction events. We used the measurement of two fitness-related phenotypic traits in order to test (1) if inbreeding depression can evolve at the time-scale of an invasion; and (2) if the changes in inbreeding depression following a bottleneck in laboratory conditions are compatible with the purging of deleterious alleles observed in this species. We found that two generations of very low population size are enough to induce a substantial change in inbreeding depression. Although the genetic changes mostly consisted in fixation of deleterious alleles, purging did also occur, sometimes simultaneously with fixation.     

Genetic risk assessment of the joint effect of several genes: Critical appraisal

When assessing the combined action of genes on the quantitative or qualitative phenotype we encounter a phenomenon that could be named the “paradox of the risk score summation.” It arises when the search of risk allele and assessment of their combined action are performed with the same single dataset. Too often such methodological error occurs when calculating the so called genetic risk score (GRS), which refers to the total number of alleles associated with the disease. Examples from numerous published genetic association studies are considered in which the claimed statistically significant effects can be attributed to the “risk score summation paradox.” In the second section of the review we discuss the current modifications of multiple regression analysis addressed to the so called “n ? p problem” (the number of points is much smaller than the number of possible predictors). Various algorithms for the model selection (searching the significant predictor combinations) are considered, beginning from the common marginal screening of the “top” predictors to LASSO and other modern algorithms of compressed sensing.     

DOES MATE LIMITATION IN SELF‐INCOMPATIBLE SPECIES PROMOTE THE EVOLUTION OF SELFING? THE CASE OF LEAVENWORTHIA ALABAMICA

Genetic diversity at the S‐locus controlling self‐incompatibility (SI) is often high because of negative frequency‐dependent selection. In species with highly patchy spatial distributions, genetic drift can overwhelm balancing selection and cause stochastic loss of S‐alleles. Natural selection may favor the breakdown of SI in populations with few S‐alleles because low S‐allele diversity constrains the seed production of self‐incompatible plants. We estimated S‐allele diversity, effective population sizes, and migration rates in Leavenworthia alabamica, a self‐incompatible mustard species restricted to discrete habitat patches in rocky glades. Patterns of polymorphism were investigated at the S‐locus and 15 neutral microsatellites in three large and three small populations with 100‐fold variation in glade size. Populations on larger glades maintained more S‐alleles, but all populations were estimated to harbor at least 20 S‐alleles, and mate availabilities typically exceeded 0.80, which is consistent with little mate limitation in nature. Estimates of the effective size (N_e) in each population ranged from 600 to 1600, and estimated rates of migration (m) ranged from 3 × 10⁻⁴ to nearly 1 × 10⁻³. According to theoretical models, there is limited opportunity for genetic drift to reduce S‐allele diversity in populations with these attributes. Although pollinators or resources limit seed production in small glades, limited S‐allele diversity does not appear to be a factor promoting the incipient breakdown of SI in populations of this species that were studied.     

Statistical Mechanics and the Evolution of Polygenic Quantitative Traits

The evolution of quantitative characters depends on the frequencies of the alleles involved, yet these frequencies cannot usually be measured. Previous groups have proposed an approximation to the dynamics of quantitative traits, based on an analogy with statistical mechanics. We present a modified version of that approach, which makes the analogy more precise and applies quite generally to describe the evolution of allele frequencies. We calculate explicitly how the macroscopic quantities (i.e., quantities that depend on the quantitative trait) depend on evolutionary forces, in a way that is independent of the microscopic details. We first show that the stationary distribution of allele frequencies under drift, selection, and mutation maximizes a certain measure of entropy, subject to constraints on the expectation of observable quantities. We then approximate the dynamical changes in these expectations, assuming that the distribution of allele frequencies always maximizes entropy, conditional on the expected values. When applied to directional selection on an additive trait, this gives a very good approximation to the evolution of the trait mean and the genetic variance, when the number of mutations per generation is sufficiently high (4Nμ > 1). We show how the method can be modified for small mutation rates (4Nμ → 0). We outline how this method describes epistatic interactions as, for example, with stabilizing selection.     

The Evolution of Canalization and Evolvability in Stable and Fluctuating Environments

Using a multilinear model of epistasis we explore the evolution of canalization (reduced mutational effects) and evolvability (levels of additive genetic variance) under different forms of stabilizing and fluctuating selection. We show that the total selection acting on an allele can be divided into a component deriving from adaptation of the trait mean, a component of canalizing selection favoring alleles that epistatically reduce the effects of other allele substitutions, and a component of conservative selection disfavoring rare alleles. While canalizing selection operates in both stable and fluctuating environments, it may not typically maximize canalization, because it gets less efficient with increasing canalization, and reaches a balance with drift, mutation and indirect selection. Fluctuating selection leads to less canalized equilibria than stabilizing selection of comparable strength, because canalization then becomes influenced by erratic correlated responses to shifting trait adaptation. We conclude that epistatic systems under bounded fluctuating selection will become less canalized than under stabilizing selection and may support moderately increased evolvability if the amplitude of fluctuations is large, but canalization is still stronger and evolvability lower than expected under neutral evolution or under patterns of selection that shift the trait in directions of positive (reinforcing) epistasis.