Representation of Nonepistatic selection models and analysis of multilocus Hardy-Weinberg Equilibrium configurations

An Alternative to the Breeder’s and Lande’s Equations

The breeder’s equation is a cornerstone of quantitative genetics, widely used in evolutionary modeling. Noting the mean phenotype in parental, selected parents, and the progeny by E(Z₀), E(Z_W), and E(Z₁), this equation relates response to selection R = E(Z₁) − E(Z₀) to the selection differential S = E(Z_W) − E(Z₀) through a simple proportionality relation R = h²S, where the heritability coefficient h² is a simple function of genotype and environment factors variance. The validity of this relation relies strongly on the normal (Gaussian) distribution of the parent genotype, which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian with mean μ, an alternative, exact linear equation of the form R′ = j²S′ can be derived, regardless of the parental genotype distribution. Here R′ = E(Z₁) − μ and S′ = E(Z_W) − μ stand for the mean phenotypic lag with respect to the mean of the fitness function in the offspring and selected populations. The proportionality coefficient j² is a simple function of selection function and environment factors variance, but does not contain the genotype variance. To demonstrate this, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between them. These results generalize naturally to the concept of G matrix and the multivariate Lande’s equation Δz¯=GP−1S. The linearity coefficient of the alternative equation are not changed by Gaussian selection.

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.

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, w(rho), depends only on the vector of gene frequencies, rho; it is nondecreasing, and the single-generation change is zero only on the set, Lambda, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of w(rho). If the equilibrium points rho of rho(t) (where t denotes time in generations) are isolated, as is generic, then rho(t) converges as t-->infinity to some rho. Generically, rho(t) converges to a local maximum of w(rho). Write the vector of gametic frequencies, p, as (rho,d)(T), where d represents the vector of linkage disequilibria. If rho is a local maximum of w(rho), then the equilibrium point (rho,0)(T) is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)-->0 globally as t-->infinity. In the second case, w(rho) has a unique maximum rho and (rho,0)(T) 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 Lambda, (i) the number of demes, Gamma, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Gamma-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.

On the meaning of non-epistatic selection

In population genetics, the additive and multiplicative viability models are often used for the quantitative description of models in which the genetic contributions of several different loci are independent; that is, there is no epistasis. Non-epistasis may also be quantitatively defined in terms of measures of interaction used widely in statistics. Setting these measures of epistasis to zero yields alternative definitions of non-epistasis. We show here that these two definitions of non-epistasis are equivalent; that is, in the most general case of a multilocus, multiallele system, the additive and multiplicative viability models are unique solutions of the additive and multiplicative conditions, respectively, for non-epistasis.

On the probability of loss of new mutations in the presence of linkage disequilibrium

A new selectively neutral mutation occurs in a multilocus genetic background that has achieved a stable equilibrium at which there is a linkage disequilibrium. Perturbation techniques are applied to an extension of the branching process formulation of Fisher in order to address the question of extinction probabilities. We show that under appropriate conditions the probability of extinction of the new mutant is increased by the existence of linkage disequilibrium in the genetic background.

Multilocus autosomal sex determination

The two basic one locus sex determination models, diploid individual sex determination and parental sex determination, are generalized to the multilocus framework. As in the single locus case, it is established that there are two classes of polymorphic equilibria, equilibria with even sex ratio and equilibria with equal allele frequencies in the two sexes. The condition for external stability of this second class equilibria to invasion by a "new" mutant allele is that a "new" appropriately averaged sex ratio near the equilibrium be moved closer to the even sex ratio. However, stable polymorphisms with noneven sex ratio are not those that have a sex ratio as close as possible to 1/2, in contrast to the single locus case.

Global convergence properties in multilocus viability selection models: the additive model and the Hardy-Weinberg law

A natural coordinate system is introduced for the analysis of the global stability of the Hardy-Weinberg (HW) polymorphism under the general multilocus additive viability model. A global convergence criterion is developed and used to prove that the HW polymorphism is globally stable when each of the loci is diallelic, provided the loci are overdominant and the multilocus recombination is positive. As a corollary the multilocus Hardy-Weinberg law for neutral selection is derived.

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.

Epistasis in the multiple locus symmetric viability model

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2n gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.