Uniqueness of polymorphic equilibria under soft selection

Any two allele polymorphic equilibrium of a subdivided haploid population subject to soft selection is stable. This provides that for a two allele system in a subdivided haploid population there is a globally attracting equilibrium which is polymorphic if a polymorphic equilibrium exists, otherwise monomorphic. These results extend to diploid populations if within each habitat the heterozygote viability is greater than or equal to the geometric mean of the homozygote viabilities.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

Selection-migration regimes characterized by a globally stable equilibrium

The principle that a subdivided population subject to overdominance viability selection in each habitat will manifest a unique, globally attractng polymorphic equilibrium is posited. This follows as a corollary to the stronger principle that, if haploid selection or submultiplicative diploid selection (definition: the geometric mean of the homozygote viabilities is less than or equal to the heterozygote viability) is operating in each habitat,there is a unique, globally attracting stable equilibrium that may be monomorphic or polymorphic. These principles are proven for a broad spectrum of migration patterns. In all such migration selection systems, multiple fixation states cannot be simultaneously stable under submultiplicative viability regimes. Contrasting examples where submultiplicative viabilities are not in force are given.     

Evolution under multiallelic migration-selection models

The loss of a specified allele and the convergence of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection are investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Sufficient conditions are established for global fixation and for global loss of a particular allele. When migration is either sufficiently weak or sufficiently strong relative to selection, the equilibria are described, convergence of the gene frequencies is demonstrated, and sufficient conditions for the increase of a suitably defined mean fitness are offered. If the selection pattern is the same in every colony and such that in a panmictic population there is a globally asymptotically stable, internal (i.e., completely polymorphic) equilibrium point, then under certain weak assumptions on migration, the gene frequencies in the subdivided population converge globally to that equilibrium point. Thus, in this case, the ultimate state of the population is unaffected by geographical structure.     

The evolutionary effect of endemic infectious disease: Continuous models for an invariant pathogen

Continuous deterministic models are used to investigate the relationship between the epidemiology of endemic infectious disease and the genetics of natural selection in the host population when a specific genetic locus controls susceptibility to disease under a variety of circumstances. One locus, two allele genes are considered in the contexts of haploid and diploid host populations while the agent of infection is assumed to be invariant. It is found that polymorphic equilibria exist and are stable for certain parameter combinations in each of the cases studied. The equilibrium levels of gene frequencies and disease prevalence depend on both genetic and epidemic factors.     

Evolution under the multiallelic Levene model

The evolution of the multiallelic Levene model is investigated. New sufficient conditions for nonexistence of a completely polymorphic equilibrium and for global loss of an allele and information on which allele(s) will be lost are deduced from some new results on multidimensional recursion relations. In the absence of dominance, a more detailed analysis is presented. Sufficient conditions for global fixation of a particular allele are established. When the number of alleles equals the number of demes, necessary and sufficient conditions for the existence of an isolated, globally asymptotically stable, completely polymorphic equilibrium point are derived, and this equilibrium is explicitly determined. Three examples, one with arbitrarily many alleles and two with three alleles, illustrate the theory.     

Population structure and Mdh-1 locus variation in Apis mellifera ligustica

In a wide area of the Piedmont of Italy the apiaries of Apis mellifera ligustica Spin., (the Italian bee) show homogeneous allelic frequency distributions at the Mdh-1 locus, the only one known to be polymorphic in worker bees. This can be explained by considering that an apiary is not a closed genetic system and that among apiaries gene flow is sufficient to overcome the different forces of inbreeding and random genetic drift. Nevertheless there is some evidence for partial subdivision because the pooled samples show a weak Wahlund effect. Moreover, the M allele at the same locus can be used as a diagnostic marker to distinguish A. m. ligustica populations (M absent or at very low frequencies) from A. m. mellifera French populations (monomorphic for M). The two honey-bee varieties, almost entirely separated by the Alps, hybridize with each other in very limited alpine areas. Hybrid populations show intermediate M frequencies.     

Genetic characterization and diversity among avocado (<Emphasis Type="Italic">Persea americana</Emphasis> Mill.) genotypes from INIA-CENIAP,Venezuela

Studies on Persea americana have been addressed in different ways with biochemical and molecular techniques. Microsatellites are able to detect multiple alleles for particular loci and are therefore a useful tool to study genealogical relationships, population structures and genetic mapping. Ninety-six samples from 49 cultivars including three horticultural groups and hybrids were collected from the avocado germplasm bank at INIA-CENIAP (Venezuela). A modified DNA extraction protocol was performed. Forty microsatellites were selected from previous references, PCR amplifications were performed, and presence/absence, size, and number of alleles were evaluated on polyacrylamide gels. Attributes for polymorphic alleles were analyzed with POPGENE, and genetic diversity was calculated by effective sample size, number of alleles per locus (Na), effective number of alleles (Ne), Shannon information index (In), observed heterozygosis (H), expected heterozygosity (He), Wright’s fixation index (Fis), and allele frequencies. Only 14 primers were amplified, and AVT106 primer resulted monomorphic. Unique genotypes for each sample were obtained. Nine loci showed allele patterns that can be useful for taxonomic identification of cultivars or varieties. Comparing values of Fis with Ho and He, we found a direct relationship where low heterozygosis alleles identified in the population may affect the expected level. Allele frequencies ranged from 0.5632 to 0.0105. For all loci, at least one rare allele was observed. With the available information from genetic analysis, an identifying system was implemented for selected avocado cultivars maintained at the INIA-CENIAP Venezuelan germplasm bank on the basis of molecular data.     

ADAPTIVE DYNAMICS IN ALLELE SPACE: EVOLUTION OF GENETIC POLYMORPHISM BY SMALL MUTATIONS IN A HETEROGENEOUS ENVIRONMENT

We demonstrate how a genetic polymorphism of distinctly different alleles can develop during long-term frequency-dependent evolution in an initially monomorphic diploid population, if mutations have only small phenotypic effect. As a specific example, we use a version of Levene's (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than by the two homozygotes. Unlike in the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence. Our results differ from the population genetic models of short-term evolution in two aspects: (1) a polymorphism that is population genetically stable may be invaded by a new mutant allele and, as a consequence, the population may fall back to monomorphism, (2) long-term evolution by allele substitutions may lead from a population where polymorphism is not possible into one where polymorphism is possible.     

Polymorphic equilibria with assortative mating and selection in subdivided populations

Two modes of assortative mating, partial selfing and assorting by phenotypic classes, are studied in a subdivided population. Differential viability is allowed and the selection intensities and assorting tendencies are permitted to vary among the habitats. There exists a symmetric polymorphism; we delimit its level of heterozygosity and stability nature (dependent on the selection intensities and assorting propensities). This complements studies of the fixation states and thereby provides further insight into the global equilibrium structure in subdivided populations. Circumstances are given where the fixation states and symmetric polymorphism comprise the global equilibrium structure. Examples are also given where migration engenders stable polymorphic equilibria and stable polymorphic equilibrium cycles which are absent in single demes without migration.     

Enzyme polymorphism in Drosophila melanogaster populations in Iraq

Electrophoretic studies of the degree and pattern of polymorphism at two third-chromosome loci, esterase-6 (Est-6) and phosphoglucomutase (PGM), were carried out in three Drosophila melanogaster populations collected from different localities in Iraq: Mosul, Tuwaitha, and Basrah. The results show that only the Tuwaitha population was polymorphic for both loci; the other two populations were polymorphic for Est-6 and monomorphic for PGM. The allele frequency changes at both loci were followed for 20 generations in an experimental cage derived from the Tuwaitha population; it was found that there is a deviation from Hardy-Weinberg equilibrium at both loci toward the homozygote.     

A general approach to genetic equilibria with an uneven sex ratio.

It is shown that in a diploid population at a genetic equilibrium with an uneven sex ratio the distribution of all (inertial) autosomal genes is symmetric between the two sexes. If several allelic genes occur in the population then the sex ratios of the various genotypes uniquely determine the uneven population sex ratio. The equilibrium is unstable to invasion by new genes which are relatively more frequent among the less numerous sex. A new proof is given to the fact that if genotype sex ratios do not change then, following an invasion by a new allele, a new stable polymorphic equilibrium can only correspond to a sex ratio which is more even than the old one. Application of these general results to a model of offspring sex determination by parents' autosomal genes shows the equilibrium to be unstable to invasion by new genes which tend to increase the production of offspring of the minority sex, as proposed by Fisher. The relevance of the Shaw-Mohler formula in this context is shown to derive from its role in describing the dynamics of autosomal genes in general. In connection with a model for offspring sex determination by one parent, a new proof is given to the fact that, following an invasion by a new allele, a new stable polymorphic equilibrium can only correspond to a sex ratio which is more even than the old one.     

Enzyme polymorphism in Schistosoma mattheei from cattle in the Eastern Transvaal Lowveld

Enzyme electrophoresis was conducted on 10 Schistosoma mattheei adult worm samples, comprising 270 individuals, collected from cattle in the Eastern Transvaal Lowveld. Glucose-6-phosphate dehydrogenase (G6PDH) was studied in all the samples and phosphoglucomutase (PGM) and malate dehydrogenase (MDH) in five populations each. Only one population was polymorphic for G6PDH. In this population, in addition to the allele found in all the other samples, a second allele occurred with a similar Rf value to S. haematobium. The two alleles were in Hardy-Weinberg equilibrium. MDH-1 exhibited two alleles. However, these alleles were not in equilibrium. In certain populations, heterozygotes occurred together with homozygotes of one of the alleles only. PGM was monomorphic in all the populations studied.     

Paternal leakage sustains the cytoplasmic polymorphism underlying gynodioecy but remains invasible by nuclear restorers

Cytoplasmic male sterility (CMS) in plants often results in gynodioecious populations, composed of hermaphrodites and male-sterile females. All models of gynodioecy assume maternal inheritance of the cytoplasmic alleles and postulate a variety of negatively frequency-dependent mechanisms to maintain the cytoplasmic polymorphisms observed in many natural populations. However, in some plant species, mitochondria are transmitted at least occasionally by pollen, a process called paternal leakage. We show that even a small amount of paternal leakage is sufficient to sustain a permanent, stable cytoplasmic polymorphism. Because only hermaphrodites provide pollen in gynodioecious species, the effects of paternal leakage are biased and occur more often from the non-CMS male-fertile haplotype to the CMS male-sterile haplotype. We also show that a nuclear restorer disrupts the polymorphic cytoplasmic equilibrium, leading to fixation of both the CMS allele and the restorer. Although a dominant nuclear restorer fixes, it fixes much more slowly than in the standard CMS models. Although a stable cytonuclear polymorphism is possible with "matching alleles" nuclear restoration, oscillations to low frequencies present a risk of loss by drift. Paternal leakage enhances the stability of joint cytonuclear polymorphism by reducing the chance that a CMS allele is lost by drift.     

The genetic consequences of spatially varying selection in the panmictic American eel (Anguilla rostrata)

Our understanding of the genetic basis of local adaptation has recently benefited from the increased power to identify functional variants associated with environmental variables at the genome scale. However, it often remains challenging to determine whether locally adaptive alleles are actively maintained at intermediate frequencies by spatially varying selection. Here, we evaluate the extent to which this particular type of balancing selection explains the retention of adaptive genetic variation in the extreme situation of perfect panmixia, using the American eel (Anguilla rostrata) as a model. We first conducted a genome scan between two samples from opposite ends of a latitudinal environmental gradient using 454 sequencing of individually tagged cDNA libraries. Candidate SNPs were then genotyped in 992 individuals from 16 sampling sites at different life stages of the same cohort (including larvae from the Sargasso Sea, glass eels, and 1-year-old individuals) as well as in glass eels of the following cohort. Evidence for spatially varying selection was found at 13 loci showing correlations between allele frequencies and environmental variables across the entire species range. Simulations under a multiple-niche Levene's model using estimated relative fitness values among genotypes rarely predicted a stable polymorphic equilibrium at these loci. Our results suggest that some genetic-by-environment interactions detected in our study arise during the progress toward fixation of a globally advantageous allele with spatially variable effects on fitness.     

Temporal variation in the genetic structure of host-associated populations of the small ermine moth Yponomeuta padellus (Lepidoptera, Yponomeutidae)

Temporal changes in allele frequencies were studied in host-associated populations of the small ermine moth Yponomeuta padellus. At one site, populations from three host plants (Sorbus aucuparia, Amelanchier larnarckii , and Crataegus spp.) were sampled annually during a four-year-period and analysed with 20 polymorphic allozyme markers. At two other sites, allele frequencies at 5- 6 enzyme loci of Y. padellus populations from two different host plants were also tested for consistency over a 13-year-pcriod. Significant allele frequency changes occurred in the short-term analysis, whereas allele frequencies remained relatively stable through time in the long-term analyses. Furthermore, allele frequencies of Y. padellus populations from Crataegus spp. were relatively stable compared to the other host populations. The role of the agents responsible for the observed patterns is discussed.     

Misconceptions and false expectations in neutral evolution

Neutral evolution results from random recurrent mutation and genetic drift. A small part of random evolution, that which is related to protein or DNA polymorphisms, is the subject of the Neutral Theory of Evolution. One of the foundations of this theory is the demonstration that the mutation rate (m) is equal to the substitution rate. Since both rates are independent of population size, they are independent of drift, which is dependent upon population size. Neutralists have erroneously equated the substitution rate with the fixation rate, despite the fact that they are antithetical conceptions. The neutralists then applied the random walk stochastic model to justify alleles or bases that were fixated or eliminated. In this model, once the allele or base frequencies reach the monomorphic states (values of 1.0 or 0.0), the absorbing barriers, they can no longer return to the polymorphic state. This operates in a pure mathematical model. If recurrent mutation occurs (as in biotic real systems) fixation and elimination are impossible. A population of bacteria in which m = 10(-8) base mutation (or substitution)/site/generation and the reproduction rate is 1000 cell cycle/year should replace all its genome bases in approximately 100,000 years. The expected situation for all sites is polymorphism for the four bases rather than monomorphism at 1.0 or 0.0 frequencies. If fixation and elimination of a base for more than 500,000 years are impossible, then most of the neutral theory is untenable. A new complete neutral model, which allows for recurrent substitutions, is proposed here based on recurrent mutation or substitution and drift alone. The model fits a binomial or Poisson distribution and not a geometric one, as does neutral theory.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Polymorphism in a mangrove snail in Papua New Guinea

Snails in the genus Littorina (Gastropoda, Prosobranchia) were examined in an area of mangroves in Papua New Guinea. It is shown that these may be separated into three species belonging to the L. scabra complex. Two of the species live on the roots only while the third is restricted to the leaves. The root species are monomorphic and cryptic and co-exist with three other monomorphic cryptic species of snails. The leaf species is polymorphic. This is a particularly clear-cut case of the tendency seen in gastropods for species of dark, uniform habitats to be monomorphic, while those on bright, patterned foliage backgrounds are polymorphic. It is suggested that the polymorphism is maintained by resemblance of the morphs to different elements in the mangrove foliage background, which are present at rather constant frequencies over a wide area.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.