The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

EXPERIMENTAL STUDIES OF PLEIOTROPY AND EPISTASIS IN ESCHERICHIA COLI. II. COMPENSATION FOR MALADAPTIVE EFFECTS ASSOCIATED WITH RESISTANCE TO VIRUS T4

Mutations in Escherichia coli that confer resistance to virus T4 also have maladaptive effects that reduce competitive fitness. After resistant populations had evolved for 400 generations in the absence of T4, their fitness approached that of sensitive populations allowed to evolve under identical conditions. However, the resistant populations had not reverted to sensitivity. Instead, this convergence in fitness resulted from genetic changes that compensated for maladaptive pleiotropic effects of the resistance mutations. An allele selected in an evolving resistant population reduced the competitive disadvantage associated with resistance by almost half. Interestingly, this allele was also beneficial in sensitive populations, although its fitness advantage was only about one-fifth as great as it was in the resistant population. These results run counter to a commonly held view that trade-offs between components of fitness should become more pronounced as populations approach their “selective equilibria.” If a trade-off derives from some limiting energetic or material currency, then it is likely to become more pronounced as a population becomes more finely adapted. If a trade-off derives from the disruption of genetic integration, then it is likely to be diminished with further adaptation.     

A fundamental relationship between genotype frequencies and fitnesses

The set of possible postselection genotype frequencies in an infinite, randomly mating population is found. Geometric mean heterozygote frequency divided by geometric mean homozygote frequency equals two times the geometric mean heterozygote fitness divided by geometric mean homozygote fitness. The ratio of genotype frequencies provides a measure of genetic variation that is independent of allele frequencies. When this ratio does not equal two, either selection or population structure is present. Within-population HapMap data show population-specific patterns, while pooled data show an excess of homozygotes.     

Frequency-dependent selection with dominance: a window onto the behavior of the mean fitness

Selection in which fitnesses vary with the changing genetic composition of the population may facilitate the maintenance of genetic diversity in a wide range of organisms. Here, a detailed theoretical investigation is made of a frequency-dependent selection model, in which fitnesses are based on pairwise interactions between the two phenotypes at a diploid, diallelic, autosomal locus with complete dominance. The allele frequency dynamics are fully delimited analytically, along with all possible shapes of the mean fitness function in terms of where it increases or decreases as a function of the current allele frequency in the population. These results in turn allow possibly the first complete characterization of the dynamical behavior by the mean fitness through time under frequency-dependent selection. Here the mean fitness (i) monotonically increases, (ii) monotonically decreases, (iii) initially increases and then decreases, or (iv) initially decreases and then increases as equilibrium is approached. We analytically derive the exact initial and fitness conditions that produce each dynamic and how often each arises. Computer simulations with random initial conditions and fitnesses reveal that the potential decline in mean fitness is not negligible; on average a net decrease occurs 20% of the time and reduces the mean fitness by >17%.     

Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.     

Pest control by the introduction of a conditional lethal trait on multiple loci: potential, limitations, and optimal strategies

Advances in genetics have made it feasible to genetically engineer insect strains carrying a conditional lethal trait on multiple loci. We model the release into a target pest population of insects carrying a dominant and fully penetrant conditional lethal trait on 1-20 loci. Delaying the lethality for several generations after release allows the trait to become widely spread in the target population before being activated. To determine effectiveness and optimal strategies for such releases, we vary release size, number of generations until the conditional lethality, nonconditional fitness cost resulting from gene insertions, and fitness reduction associated with laboratory rearing. We show that conditional lethal releases are potentially orders of magnitude more effective than sterile male releases of equal size, and that far smaller release sizes may be required for this approach than necessary with sterile males. For example, a release of male insects carrying a conditional lethal allele that is activated in the F4 generation on 10 loci reduces the target populatioin to 10(-4) of no-release size if there are initially two released males for every wild male. We show how the effectiveness of conditional lethal releases decreases as the nonconditional fitness reduction (i.e., fitness reduction before the trait becomes lethal) associated with the conditional lethal genes increases. For example, if there is a 5% nonconditional fitness cost per conditional lethal allele, then a 2:1 (released male:wild male) release with conditional lethal alleles that are activated in the F4 generation reduces the population to 2-5% (depending on the degree of density dependence) of the no-release size. If there is a per-allele reduction in fitness, then as the number of loci is increased there is a trade-off between the fraction of offspring carrying at least one conditional lethal allele and the fitness of the released insects. We calculate the optimal number of loci on which to insert the conditional lethal gene given various conditions. In addition, we show how laboratory-rearing fitness costs, density-dependence, and all-male versus male-female releases affect the efficiency of conditional lethal releases.     

Amino Acid Covariation in a Functionally Important Human Immunodeficiency Virus Type 1 Protein Region Is Associated With Population Subdivision

The frequently reported amino acid covariation of the highly polymorphic human immunodeficiency virus type 1 (HIV-1) exterior envelope glycoprotein V3 region has been assumed to reflect fitness epistasis between residues. However, nonrandom association of amino acids, or linkage disequilibrium, has many possible causes, including population subdivision. If the amino acids at a set of sequence sites differ in frequencies between subpopulations, then analysis of the whole population may reveal linkage disequilibrium even if it does not exist in any subpopulation. HIV-1 has a complex population structure, and the effects of this structure on linkage disequilibrium were investigated by estimating within- and among-subpopulation components of variance in linkage disequilibrium. The amino acid covariation previously reported is explained by differences in amino acid frequencies among virus subpopulations in different patients and by nonsystematic disequilibrium among patients. Disequilibrium within patients appears to be entirely due to differences in amino acid frequencies among sampling time points and among chemokine coreceptor usage phenotypes of virus particles, but not source tissues. Positive selection explains differences in allele frequencies among time points and phenotypes, indicating that these differences are adaptive rather than due to genetic drift. However, the absence of a correlation between linkage disequilibrium and phenotype suggests that fitness epistasis is an unlikely cause of disequilibrium. Indeed, when population structure is removed by analyzing sequences from a single time point and phenotype, no disequilibrium is detectable within patients. These results caution against interpreting amino acid covariation and coevolution as evidence for fitness epistasis.     

A slow selection analysis of two locus, two allele traits.

A deterministic, continuous time model for the dynamics of two locus, two allele Mendelian traits in a large randomly mating diploid population is derived. The model allows for frequency and time dependent birth and death rates. It is analyzed under the assumption that the selective forces acting in the population are small. Slow selection approximations to the system's solution are then constructed. Two particular cases are studied. First, when linkage between loci is tight, the population is shown to rapidly approach Hardy-Weinberg proportions, which then may vary on a (slow) time scale determined by differential fitness. In the case of constant birth and death rates, a measure of the population's fitness is shown to increase on the slow time scale after an initial rapid adjustment. The second case considered is for loose linkage; a population near linkage equilibrium is studied. It is shown that the epistatic parameters cancel and that the results agree with the tight linkage case to leading order. The linkage disequlibrium is described in both cases.     

Pest control by the release of insects carrying a female-killing allele on multiple loci

With recent advances in genetics, many new strategies for pest control have become feasible. This is the second article in which we model new techniques for pest control based on the mass release of genetically modified insects. In this article we model the release of insects carrying a dominant and redundant female killing or sterilizing (FK) allele on multiple genetic loci. If such insects are released into a target population, the FK allele can become widely spread in the population through the males while reducing the population each generation by killing females. We allow the number of loci used to vary from 1 to 20. We also allow the FK allele to carry a fitness cost in males due to the gene insertions. Using a model, we explore the effectiveness and optimal strategies for such releases. In the most ideal circumstances (no density-dependence and released insects equal in fitness to wild ones), FK releases are several orders of magnitude more effective than equal sized sterile male releases. For example, a single release of 19 FK-bearing males for every two wild males, with the released males carrying the FK allele on 10 loci, reduces the target population to 0.002% of no-release size. An equal sized sterile release reduces the target population to 5% of no-release size. We also show how the effectiveness of the technique decreases as the fitness cost of the FK alleles in males increases. For example, the above mentioned release reduces the target population to 0.7% of no-release size if each FK allele carries a fitness cost in males of 5%. Adding a simple model for density-dependence and assuming that each of the released males carries the FK allele on six loci, we show that the release size necessary to reduce the target population to 1/100 of no-release size in 10 generations of releases varies from 0.44:1 to 4:1 (depending on parameter values). We also calculate the optimal number of loci on which to put the FK allele under various circumstances.     

Frequency-dependent selection in logistic growth models

This paper describes the dynamics of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. We modify the basic density-dependent logistic growth model by inserting three possible types of frequency dependence in the fitness functions. These models are analyzed and contrasted with the purely density-dependent situation. Examples are given of periodic fluctuations in allele frequency and population density, which would be impossible for purely density-dependent fitness functions.     

Dynamics of molecular markers linked to the resistance loci in a mosquito-Plasmodium system

Models on the evolution of resistance to parasitism generally assume fitness tradeoffs between the costs of being parasitized and the costs associated with resistance. This study tested this assumption using the yellow fever mosquito Aedes aegypti and malaria parasite Plasmodium gallinaceum system. Experimental mosquito populations were created by mixing susceptible and resistant strains in equal proportions, and then the dynamics of markers linked to loci for Plasmodium resistance and other unlinked neutral markers were determined over 12 generations. We found that when the mixed population was maintained under parasite-free conditions, the frequencies of alleles specific to the susceptible strain at markers closely linked to the loci for resistance (QTL markers) as well as other unlinked markers increased significantly in the first generation and then fluctuated around equilibrium frequencies for all six markers. However, when the mixed population was exposed to an infected blood meal every generation, allele frequencies at the QTL markers for resistance were not significantly changed. Small population size caused significant random fluctuations of allele frequencies at all marker loci. Consistent allele frequency changes in the QTL markers and other unlinked markers suggest that the reduced fitness in the resistant population has a genome-wide effect on the genetic makeup of the mixed population. Continuous exposure to parasites promoted the maintenance of alleles from the resistant Moyo-R strain in the mixed population. The results are discussed in relation to the proposed malaria control strategy through genetic disruption of vector competence.     

PARASITES AND THE EVOLUTION OF SELF-FERTILIZATION

Abstract.— Assuming all else is equal, an allele for selfing should spread when rare in an outcrossing population and rapidly reach fixation. Such an allele will not spread, however, if self‐fertilization results in inbreeding depression so severe that the fitness of selfed offspring is less that half that of outcrossed offspring. Here we consider an ecological force that may also counter the spread of a selfing allele: coevolution with parasites. Computer simulations were conducted for four different genetic models governing the details of infection. Within each of these models, we varied both the level of selfing in the parasite and the level of male‐gamete discounting in the host (i.e., the reduction in outcrossing fitness through male function due to the selfing allele). We then sought the equilibrium level of host selfing under the different conditions. The results show that, over a wide range of conditions, parasites can select for host reproductive strategies in which both selfed and outcrossed progeny are produced (mixed mating). In addition, mixed mating, where it exits, tends to be biased toward selfing.     

The Effects of Density Dependence and Immigration on Local Adaptation and Niche Evolution in a Black-Hole Sink Environment

We examine the effects of density dependence and immigration on local adaptation in a "black-hole sink" habitat, i.e., a habitat in which isolated populations of a species would tend to extinction but where a population is demographically maintained by recurrent one-way migration from a separate source habitat in which the species persists. Using a diploid, one-locus model of a discrete-generation sink population maintained by immigration from a fixed source population, we show that a locally favored allele will spread when rare in the sink if the absolute fitness (or, in some cases, the geometric-mean absolute fitness) of heterozygotes with the favored allele is above one in the sink habitat. With density dependence, the criterion for spread can depend on the rate of immigration, because immigration affects local densities and, hence, absolute fitness. Given the successful establishment of a locally favored allele, it will be maintained by a migration-selection balance and the resulting polymorphic population will be sustained deterministically with either stable or unstable dynamics. The densities of stable polymorphic populations tend to exceed densities that would be maintained in the absence of the favored allele. With strong density regulation, spread of the favored allele may destabilize population dynamics. Our analyses show that polymorphic populations which form subsequent to the establishment of favorable alleles have the capacity to persist deterministically without immigration. Finally, we examined the probabilistic rate at which new favored alleles arise and become established in a sink population. Our results suggest that favored alleles are established most readily at intermediate levels of immigration.     

Natural selection, fitness entropy, and the dynamics of coevolution

The coevolutionary dynamics of interacting populations were studied by combining continuous time Lotka-Volterra models of population growth with single-locus genetic models of weak selection. The effects of natural selection on population growth were evaluated using Ginzburg's fitness entropy function as a measure of the deviation of a population's initial allele frequencies from their polymorphic equilibrium values. This entropy measure was used to relate the dynamics of a community composed of evolving populations to the dynamics of a "reference community" whose populations are initially in genetic equilibrium. Specifically, a quantity called the "selective difference area" was defined as the total difference between the population size trajectories of a reference and evolving population. The selective difference area represents the amount of extra life a species would realize if the entire community were at genetic equilibrium. It was shown that this selective difference area is a simple linear function of the initial fitness entropies of each species. This prediction is independent of the strength of selection and holds for any arbitrary set of initial population densities. Numerical examples were presented to illustrate the results. Under the assumption of weak selection, a generalization for arbitrary population growth models was outlined.     

Local population structure and sex ratio: evolution in gynodioecious plants

Although the influence of population structure on evolution has been explored previously in a variety of theoretical studies, there are few examples of specific traits whose fitness is likely to be modified by the local structure. Here we focus on a specific trait, sex expression in gynodioecious plants, and derive a model in which the fitness of females and hermaphrodites is a function of the local sex ratio. By using the concept d genes. As a consequence, when local demes vary in sex ratio, a polymorphism for a cytoplasmic male sterility (CMS) allele can be maintained in the absence of nuclear alleles that restore male function. When of subjective frequencies, it is shown that among-deme variance in the local sex ratio reduces the average fitness of females when pollen availability limits fertility. In contrast, sex ratio variance increases the fitness of hermaphrodites from the perspective of maternally inherited genes and lessens the negative impact of pollen limitation on hermaphrodite fitness when it is measured from the perspective of biparentally inheriterestorer alleles are introduced into the model, polymorphism cannot be maintained simultaneously at both the cytoplasmic and nuclear loci. In that case, the CMS allele spreads to fixation, and the equilibrium frequency of females is an inverse function of the equilibrium frequency of the restorer allele, which increases with increased structure. The results exemplify how population structure can greatly alter the fitness and evolution of a frequency-dependent trait.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

An inclusive fitness analysis of synergistic interactions in structured populations

We study the evolution of a pair of competing behavioural alleles in a structured population when there are non-additive or ‘synergistic’ fitness effects. Under a form of weak selection and with a simple symmetry condition between a pair of competing alleles, Tarnita et al. provide a surprisingly simple condition for one allele to dominate the other. Their condition can be obtained from an analysis of a corresponding simpler model in which fitness effects are additive. Their result uses an average measure of selective advantage where the average is taken over the long-term—that is, over all possible allele frequencies—and this precludes consideration of any frequency dependence the allelic fitness might exhibit. However, in a considerable body of work with non-additive fitness effects—for example, hawk–dove and prisoner''s dilemma games—frequency dependence plays an essential role in the establishment of conditions for a stable allele-frequency equilibrium. Here, we present a frequency-dependent generalization of their result that provides an expression for allelic fitness at any given allele frequency p. We use an inclusive fitness approach and provide two examples for an infinite structured population. We illustrate our results with an analysis of the hawk–dove game.     

论替换负荷与Haldane进退维谷

在等位基因A优于等位基因α的选择下,种群中不存在有现实意义的H－W平衡;用平衡频率导出的Haldane进退维谷只是一种数学佯谬;替换负荷是选择下的非平衡种群的遗传负荷,且亲代负荷最大,随着世代的增加,后代负荷越来越小,种群对选择越来越适应。     

Medea selfish genetic elements as tools for altering traits of wild populations: a theoretical analysis

One strategy for controlling transmission of insect-borne disease involves replacing the native insect population with transgenic animals unable to transmit disease. Population replacement requires a drive mechanism to ensure the rapid spread of linked transgenes, the presence of which may result in a fitness cost to carriers. Medea selfish genetic elements have the feature that when present in a female, only offspring that inherit the element survive, a behavior that can lead to spread. Here, we derive equations that describe the conditions under which Medea elements with a fitness cost will spread, and the equilibrium allele frequencies are achieved. Of particular importance, we show that whenever Medea spreads, the non-Medea genotype is driven out of the population, and we estimate the number of generations required to achieve this goal for Medea elements with different fitness costs and male-only introduction frequencies. Finally, we characterize two contexts in which Medea elements with fitness costs drive the non-Medea allele from the population: an autosomal element in which not all Medea-bearing progeny of a Medea-bearing mother survive, and an X-linked element in species in which X/Y individuals are male. Our results suggest that Medea elements can drive population replacement under a wide range of conditions.     

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.