Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist-generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.     

Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, , depends only on the vector of gene frequencies, ρ; it is nondecreasing, and the single-generation change is zero only on the set, Λ, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of . If the equilibrium points of ρ(t) (where t denotes time in generations) are isolated, as is generic, then ρ(t) converges as t→∞ to some . Generically, ρ(t) converges to a local maximum of . Write the vector of gametic frequencies, p, as , where d represents the vector of linkage disequilibria. If is a local maximum of , then the equilibrium point is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)→0 globally as t→∞. In the second case, has a unique maximum and is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ, (i) the number of demes, Γ, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ−1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.     

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ₀ (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ₀ is an increasing function of the panmictic rate and to deduce the effect on λ₀ of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ₀; bounds on λ₀ are established; and λ₀ is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.     

Clines with partial panmixia in an unbounded unidimensional habitat

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an unbounded unidimensional habitat is investigated. Migration and selection are both weak. The former is homogenous and isotropic; the latter has no dominance. The population density is uniform. A simple, explicit formula is derived for the maximum value β₀ of the scaled panmictic rate β for which a cline exists. The former depends only on the asymptotic values of the scaled selection coefficient. If the two alleles have the same average selection coefficient, there exists a unique, globally asymptotically stable cline for every β≥0. Otherwise, if β≥β₀, the allele with the greater average selection coefficient is ultimately fixed. If β<β₀, there exists a unique, globally asymptotically stable cline, and some polymorphism is retained even infinitely far from its center. The gene frequencies at infinity are determined by a continuous-time, two-deme migration-selection model. An explicit expression is deduced for the monotone cline in a step-environment. These results differ fundamentally from those for the classical cline without panmixia.     

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.     

Patterns of multiallelic polymorphism maintained by migration and selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.     

Evolution at a multiallelic locus under migration and uniform selection

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The drift and diffusion coefficients may depend on position, but the selection coefficients do not. It is established that if p is a uniform equilibrium point under pure selection, then p is a migration-selection equilibrium, and that generically the introduction of migration does not change the stability of p. It is also proved that if p is a uniform, globally asymptotically stable, internal equilibrium point under pure selection, then the gene frequencies converge to p when both migration and selection are present. Thus, in this case, after a sufficiently long time, there is no genetic indication of the spatial distribution of the population.     

Three components of genetic drift in subdivided populations

Wright's metaphor of sampling is extended to consider three components of genetic drift: those occurring before, during, and after migration. To the extent that drift at each stage behaves like an independent random sample, the order of events does not matter. When sampling is not random, the order does matter, and the effect of population size is confounded with that of mobility. The widely cited result that genetic differentiation of local groups depends only on the product of group size and migration rate holds only when nonrandom sampling does not occur prior to migration in the life cycle.     

The genetic architecture of adaptation under migration-selection balance

Many ecologically important traits have a complex genetic basis, with the potential for mutations at many different genes to shape the phenotype. Even so, studies of local adaptation in heterogeneous environments sometimes find that just a few quantitative trait loci (QTL) of large effect can explain a large percentage of observed differences between phenotypically divergent populations. As high levels of gene flow can swamp divergence at weakly selected alleles, migration-selection-drift balance may play an important role in shaping the genetic architecture of local adaptation. Here, we use analytical approximations and individual-based simulations to explore how genetic architecture evolves when two populations connected by migration experience stabilizing selection toward different optima. In contrast to the exponential distribution of allele effect sizes expected under adaptation without migration (Orr 1998), we find that adaptation with migration tends to result in concentrated genetic architectures with fewer, larger, and more tightly linked divergent alleles. Even if many small alleles contribute to adaptation at the outset, they tend to be replaced by a few large alleles under prolonged bouts of stabilizing selection with migration. All else being equal, we also find that stronger selection can maintain linked clusters of locally adapted alleles over much greater map distances than weaker selection. The common empirical finding of QTL of large effect is shown to be expected with migration in a heterogeneous landscape, and these QTL may often be composed of several tightly linked alleles of smaller effect.     

How to compute the effective size of spatiotemporally structured populations using separation of time scales

Calculations to derive effective population size become highly complicated when complex population structure is considered. We provide an easy method of computing the effective size of a subdivided population with overlapping generations (a spatiotemporally structured population) using an approximation based on separation of time scales. We also numerically compute the effective size to verify the accuracy of the derived formula. Various interesting quantities, including moments of coalescent time, are readily derived using this approach.     

Numerical solution of structured population models

Numerical methods are presented for a general mass-structured population model with demographic rates that depend on individual mass, time, and total population mass. Several types of numerical methods are described, Eulerian methods, implicit methods, and the method of characteristics. These methods are compared for a sample problem with an exact solution. The preferred numerical technique combines the method of characteristics with an adaptive grid. Numerical solution of model equations developed for mosquitofish illustrates this method and demonstrates how seasons can play a dominant role in shaping population development.     

Size-structured demographic models of coral populations

The demographic processes of growth, mortality, and the recruitment of young individuals, are the major organizing forces regulating communities in open systems. Here we present a size-structured (rather than age-structured) population model to examine the role of these different processes in space-limited open systems, taking coral reefs as an example. In this flux-diffusion model the growth rate of corals depends both on the available free-space (i.e. density-dependence) and on the particular size of the coral. In our analysis we progressively study several different forms of growth rate functions to disentangle the effects of free space and size-dependence on the model's stability. Unlike Roughgarden et al. [1985. Demographic theory for an open marine population space-limited recruitment. Ecology 66(1), 54-67], whose principal result is that the growth of settled organisms is destabilizing, we find that size-dependent growth rate often has the potential to endow stability. This is particularly true, if the growth rate is dependent on available free space (i.e. density dependent), but examples are given for growth rates that even lack this property. Further insights into reef system fragility are found through studying the sensitivity of the model steady state to changes in recruitment.     

Autosome STRs in native South America-Testing models of association with geography and language

Information on one Ecuadorian and three Peruvian Amerindian populations for 11 autosomal short tandem repeat (STR) loci is presented and incorporated in analyses that includes 26 other Native groups spread all over South America. Although in comparison with other studies we used a reduced number of markers, the number of populations included in our analyses is currently unmatched by any genome-wide dataset. The genetic polymorphisms indicate a clear division of the populations into three broad geographical areas: Andes, Amazonia, and the Southeast, which includes the Chaco and southern Brazil. The data also show good agreement with proposed hypotheses of splitting and dispersion of major language groups over the last 3,000 years. Therefore, relevant aspects of Native American history can be traced using as few as 11 STR autosomal markers coupled with a broad geographic distribution of sampled populations.     

Effective population size,genetic diversity,and coalescence time in subdivided populations

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.     

Viability analyses with habitat-based metapopulation models

Population viability analysis (PVA) models incorporate spatial dynamics in different ways. At one extreme are the occupancy models that are based on the number of occupied populations. The simplest occupancy models ignore the location of populations. At the other extreme are individual-based models, which describe the spatial structure with the location of each individual in the population, or the location of territories or home ranges. In between these are spatially structured metapopulation models that describe the dynamics of each population with structured demographic models and incorporate spatial dynamics by modeling dispersal and temporal correlation among populations. Both dispersal and correlation between each pair of populations depend on the location of the populations, making these models spatially structured. In this article, I describe a method that expands spatially structured metapopulation models by incorporating information about habitat relationships of the species and the characteristics of the landscape in which the metapopulation exists. This method uses a habitat suitability map to determine the spatial structure of the metapopulation, including the number, size, and location of habitat patches in which subpopulations of the metapopulation live. The habitat suitability map can be calculated in a number of different ways, including statistical analyses (such as logistic regression) that find the relationship between the occurrence (or, density) of the species and independent variables which describe its habitat requirements. The habitat suitability map is then used to calculate the spatial structure of the metapopulation, based on species-specific characteristics such as the home range size, dispersal distance, and minimum habitat suitability for reproduction. Received: April 1, 1999 / Accepted: October 29, 1999     

Constraints on mutability in a multiallelic gene family

A highly variable family of related DNA sequences was examined in order to determine the effect of local sequence environment on substitution mutation; 29 sequences from the Brassica self-incompatibility gene family, which possess a high level of nonsynonymous mutations, were aligned and grouped according to their similarity and function. The level and distribution of substitution mutations were calculated. A nonrandom distribution of sequence variation was observed along the sequences. The effect of neighbor biases and structural and thermodynamic measures were then compared in the absence of strong codon conservation. Biases were observed in the rates of substitution of the same base pair in different local sequence environments. The effect of the 5 neighbor was such that nucleotide A or C was associated with more mutations than G or T. There were significant interactions of certain dinucleotides with the frequency of mutation. Sequence-dependent measures of helical stability, intrinsic curvature, components of curvature, and stacking interactions were calculated for each sequence. Decreased helical stability was found to be associated with increased mutation. The compound measure of curvature, calculated according to the wedge model, showed little association with mutation. However, the components of increased wedge angle and decreased twist both showed an association with increased mutation. A small effect of A-type DNA stacking was found to be associated with mutated bases. Correspondence to: G.J. King     

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.