Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations

Summary The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

Extending the coalescent to multilocus systems: the case of balancing selection

Natural populations are structured spatially into local populations and genetically into diverse 'genetic backgrounds' defined by different combinations of selected alleles. If selection maintains genetic backgrounds at constant frequency then neutral diversity is enhanced. By contrast, if background frequencies fluctuate then diversity is reduced. Provided that the population size of each background is large enough, these effects can be described by the structured coalescent process. Almost all the extant results based on the coalescent deal with a single selected locus. Yet we know that very large numbers of genes are under selection and that any substantial effects are likely to be due to the cumulative effects of many loci. Here, we set up a general framework for the extension of the coalescent to multilocus scenarios and we use it to study the simplest model, where strong balancing selection acting on a set of n loci maintains 2n backgrounds at constant frequencies and at linkage equilibrium. Analytical results show that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. However, simulation results reveal that the structured coalescent approach breaks down when the number of backgrounds approaches the population size, because of stochastic fluctuations in background frequencies. A new method is needed to extend the structured coalescent to cases with large numbers of backgrounds.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Controlling the false-positive rate in multilocus genome scans for selection

Rapid typing of genetic variation at many regions of the genome is an efficient way to survey variability in natural populations in an effort to identify segments of the genome that have experienced recent natural selection. Following such a genome scan, individual regions may be chosen for further sequencing and a more detailed analysis of patterns of variability, often to perform a parametric test for selection and to estimate the strength of a recent selective sweep. We show here that not accounting for the ascertainment of loci in such analyses leads to false inference of natural selection when the true model is selective neutrality, because the procedure of choosing unusual loci (in comparison to the rest of the genome-scan data) selects regions of the genome with genealogies similar to those expected under models of recent directional selection. We describe a simple and efficient correction for this ascertainment bias, which restores the false-positive rate to near-nominal levels. For the parameters considered here, we find that obtaining a test with the expected distribution of P-values depends on accurately accounting both for ascertainment of regions and for demography. Finally, we use simulations to explore the utility of relying on outlier loci to detect recent selective sweeps. We find that measures of diversity and of population differentiation are more effective than summaries of the site-frequency spectrum and that sequencing larger regions (2.5 kbp) in genome-scan studies leads to more power to detect recent selective sweeps.     

Dynamic regimes in a model of single-locus density-dependent selection

A model of density-dependent selection in a Mendelian single-locus population was analyzed in the case where the fitnesses of genotypic forms are exponential functions of the population size. Analytical and numerical studies of the model were performed for a diallelic locus, and parametric regions were established for different dynamic behaviors of the model. The diallelic model of density-dependent selection was generalized to a multiallelic locus; the results of its analysis are described.     

Variable selection in regression-based estimation of dynamic treatment regimes

Dynamic treatment regimes (DTRs) consist of a sequence of decision rules, one per stage of intervention, that aim to recommend effective treatments for individual patients according to patient information history. DTRs can be estimated from models which include interactions between treatment and a (typically small) number of covariates which are often chosen a priori. However, with increasingly large and complex data being collected, it can be difficult to know which prognostic factors might be relevant in the treatment rule. Therefore, a more data-driven approach to select these covariates might improve the estimated decision rules and simplify models to make them easier to interpret. We propose a variable selection method for DTR estimation using penalized dynamic weighted least squares. Our method has the strong heredity property, that is, an interaction term can be included in the model only if the corresponding main terms have also been selected. We show our method has both the double robustness property and the oracle property theoretically; and the newly proposed method compares favorably with other variable selection approaches in numerical studies. We further illustrate the proposed method on data from the Sequenced Treatment Alternatives to Relieve Depression study.     

Central Equilibria in Multilocus Systems. II. Bisexual Generalized Nonepistatic Selection Models

This paper is a continuation of the paper "Central Equilibria in Multilocus Systems I," concentrating on existence and stability properties accruing to central H-W type equilibria in multilocus bisexual systems acted on by generalized nonepistatic selection forces coupled to recombination events. The stability conditions are discussed and interpreted in three perspectives, and the influence of sexual differences in linkage relationships together with sex-dependent selection is appraised. In this case we deduce that the stability conditions of the H-W polymorphism in the bisexual model coincide exactly with the conditions for the corresponding monoecious model, provided that the recombination distribution imposed is that of the arithmetic mean of the male and female recombination distributions. A second concern has the same recombination distribution for both sexes, but contrasting selection regimes between sexes. It is then established that, with respect to discerning the relevance of the H-W equilibrium, there is an equivalent monoecious selection regime which is an appropriate "weighted combination" of the male and female selection forms. Finally, in the case where the selection and recombination structures are both sex dependent, a hierarchy of comparisons is elaborated, seeking to unravel the nature of selection-recombination interaction for monoecious versus diocecious systems.     

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.     

Population equilibria and stability in plant-animal pollination systems

Differential equation models of the change in population size with change in time have provided valuable insight into nature and have served as tools for management of both predator-prey and competition systems. Presented here is an extension of the differential equation models for application to pollination ecology plant-animal interactions. Analyses define conditions for equilibria and for equilibrium stability for a one-plant one-animal pollinator system. Predictions about plant speciation and pollinator foraging behavior are discussed.     

Protein salting-out: phase equilibria in two-protein systems

The phase behavior of two aqueous binary protein mixtures, lysozyme-chymotrypsin and lysozyme-ovalbumin, was determined in ammonium sulfate solutions. Protein concentrations were determined in both phases as a function of pH and ionic strength. For lysozyme-chymotrypsin mixtures, the observed phase behavior was similar to that for each individual protein; the presence of the second protein had little influence. The phase behavior of lysozyme-ovalbumin mixtures, however, was different from that of the respective single-protein systems. Lysozyme and ovalbumin are found together in egg whites; their association is both pH and ionic-strength dependent. The association of proteins is a key determinant of protein solubility in salt solutions. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 53: 567-574, 1997.