The Evolution of Multilocus Systems under Weak Selection

The evolution of multilocus systems under weak selection 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. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << c(min), where s and c(min) denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t(1) ~ (ln s)/ln(1 - c(min)) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s -> 0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s(2)). Then after a time t(2) ~ 2t(1), the linkage disequilibria are nearly constant, their rate of change being O(s(2)). Furthermore, with an error of O(s(2)), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s(3)), then the single-generation change in the mean fitness is ΔW = W(-1)V(g) + O(s(3)), where V(g) designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s(2)) evolves at the rate ΔZ = W(-1)C(g) + O(s(2)), where C(g) represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t(2), the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.     

Selection in Complex Genetic Systems I. the Symmetric Equilibria of the Three-Locus Symmetric Viability Model

The symmetric equilibria of the three-locus symmetric viability model are determined and their stability analyzed. For tight linkage there may be four stable equilibria, each characterized by having one pair of complementary chromosomes in high frequencies, with all others low. For looser linkage the only stable symmetric equilibrium is that with complete linkage equilibrium. For intermediate recombination values both types of equilibria may be stable. A new class of equilibria with all pairwise linkage disequilibria zero, but with third order linkage disequilibrium, has been discovered. It may be stable for tight linkage.     

Multilocus Behavior in Random Environments I. Random Levene Models

In this paper the consequences of natural selection acting on several loci simultaneously in a spatially fluctuating environment are described. The fitnesses of the genotypes are assumed to be additive both within and between loci. The environment is assumed to be made up of a very large (effectively infinite) number of patches in which fitnesses are assigned at random. The resulting deterministic model is called a Random Levene Model and its properties are approximate by a system of differential equations. The main equilibrium properites are that (1) the linkage disequilibrium is zero and (2) the correlations in fitness between alleles at different loci are the principle determinants of the dynamic inter-locus interactions. Although there is no epistasis as conventionally defined, the equilibrium state at the two loci are highly interdependent, the governing principle being that two alleles at different loci whose fitness are negatively correlated across environments have a higher overall fitness due to the reduction in their variance in fitness through the negative correlation. When a large number of loci are considered, they naturally fall into correlation groupings which lead to an enhanced likelihood for polymorphism over that predicted by single-locus theory.     

Natural Selection for within-Generation Variance in Offspring Number II. Discrete Haploid Models

In the classical model of genetic drift in population genetics theory, use is made of a hypothetical "infinite-gametic pool". If, instead, the gametic pool is determined by the random number of offspring per individual, a new form of natural selection acting on the variance in offspring number occurs. A diffusion model of this selection process is derived and some of its properties are explored. It is shown that, independent of the sampling scheme used, the diffusion equation has the drift coefficient M(p) = p(1-p) (mul--mu2 + sigma2e2--sigma2el) and the diffusion coefficient v(p) equals p(1-p) [psigma2e2 + (l--p)sigma2el]. It is also pointed out that the Direct Product Branching process model of genetic drift introduces a non-biological interaction between individuals and is thus inappropriate for modeling natural selection.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

Multilocus model of sympatric speciation II. Two characters

The method used in the previous paper (Kondrashov, 1983Theor. Pop. Biol.27, 000-000) is applied to population polymorphic at two quantitative characters. Sympatric speciation is found to be possible in the case when difference in two characters is necessary for reproductive isolation. The influence of various factors on the process of sympatric speciation is studied and selection forces necessary for its completion are found. Speciation occurs more readily under the action both of disruptive selection in separate characters and of selection against individuals with “unbalanced” phenotypes. This type of selection is also most realistic when various phenotypes make use of different niches. The results obtained allow the supposition that the possibility of sympatric speciation is not reduced to a few cases when reproductive isolation between the forming species develops due to minor genetic differences. It is also shown that if one of the characters is not directly involved in the processes concerning speciation then the forming species do not differ in the character. Relative frequencies of the intermediate phenotypes are found for the terminal stage of speciation.     

Concentrating Engines and the Kidney: II. Multisolute Central Core Systems

The analysis of the central core model of the renal medulla is extended to multisolute systems. It is shown that total solute concentration obeys the same differential equations for core and ascending limb as in a single solute system. Equations are derived for the concentration of individual solutes. Application of these equations to a two solute system shows that a central core system can concentrate with all transport being down a concentration gradient. This analysis applied to the renal medulla shows that mixing of urea from the collecting duct (CD) and salt from the loop of Henle in the central core of the inner medulla contributes to the concentration of urine during antidiuresis. It also sets criteria for completely passive function of the loop in the inner medulla, but whether these are satisfied cannot be determined from present experimental data.     

Properties of Equilibria in Multi-Locus Genetic Systems

The classical mathematical theory of population genetics considered, for simplicity, almost exclusively one-locus systems. In the last two decades much work has been done on two-locus and, less frequently, multi-locus systems. This research has usually involved investigating properties of systems with given, and usually rather special, fitness parameters. Real genetic fitness systems are undoubtedly multi-locus and seldom will possess simplifying characteristics. One aim of this paper is to study generalized systems where no special assumptions are made about fitness structure, the number of alleles at each locus, the number of loci involved or the recombination structure between loci. A second aim is to consider marginal properties (often one-locus properties) of complex systems: the fact that many observations involve data from only on locus makes this second aim relevant.     

Multilocus Behavior in Random Environments. II. Linkage Disequilibrium in an Additive Model

The effect of a stochastic environment on an additive, two-locus model of a diploid population is examined. The appropriate diffusion equation is derived and its asymptotic properties are approximated by an Orstein-Uhlenbeck process. The first and second order moments of this approximating process are given. The mean linkage disequilibrium will be nonzero if the alleles at the different loci are correlated. The sign of the mean disequilibrium is determined by the sign of the correlation.     

Dynamics of Correlated Genetic Systems. IV. Multilocus Effects of Ethanol Stress Environments

Four replicate populations of Drosophila melanogaster, two reared on medium supplemented with ethanol and two reared on standard medium, were electrophoretically monitored for 28 generations. During the first 12 generations, allelic, genotypic and gametic frequencies were determined for eight polymorphic enzymes: GOT, alpha-GPDH, MDH, ADH, TO, E6, Ec and ODH. Samples from generation 18 and 28 were electrophoretically typed for ADH and alpha-GPDH. In addition, samples from generation 27 were analyzed for the presence of inversion heterozygotes. The experimental results showed rapid gene-frequency divergence between control and treatment populations at the Adh locus in a direction consistent with the activity hierarchy of Adh genotypes. Gene-frequency divergence between control and treatment populations also occurred at the alpha-Gpdh locus, although the agreement among replicates appeared to have broken down by generation 28. No differential gene-frequency change occurred at any of the six remaining marker loci. Furthermore, values of linkage disequilibria among all linked pairs of genes were initially small and remained small throughout the course of the experiment. Taking these facts into account, it is argued that the gene-frequency response observed at ADH is most probably caused by selection at the Adh locus. The gene frequency response at alpha-Gpdh can also be be accounted for in terms of the effect of ethanol on energy metabolism, although other explanations cannot be excluded.     

Multilocus Population Genetics with Weak Epistasis. I. Equilibrium Properties of Two-Locus Two-Allele Models

Using perturbation techniques, I determine the equilibrium of two-locus two-allele models with overdominance and weak epistasis. To lowest order, the allele frequencies, the mean fitness and the covariance between heterokaryotic and homokaryotic flies arising in the Sturtevant experimental design are independent of the recombination rate, r. The disequilibrium varies as one divided by the recombination rate, in contrast to neutral models. Although the disequilibrium generated by weak epistasis is small, too small to be experimentally detected, it can be large enough to have biological importance.     

Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes

The generalized nonepistatic selection regime encompasses combinations of multiplicative and neutral viability effects distributed across a set of loci. These subsume, in particular, mixtures of the classical modes of multiplicative and additive fitness evaluations for multilocus traits. Exact analytic conditions for existence and stability of a multilocus Hardy-Weinberg (H-W) polymorphic equilibrium configuration are ascertained. It is established that the central H-W polymorphism is stable only if the component loci are "over-dominant" and sufficient recombination is in force. The H-W central equilibrium is never stable for tight linkage whenever some multiplicative selection effects are contributed by at least two of the loci involved. In the case of additive selection expression and individual overdominant loci, the H-W polymorphism is stable independently of the level of recombination. In the context of "natural" recombination schemes, "more recombination" enhances the stability of the H-W polymorphic equilibrium.     

Solubility Equilibria in Chiral Systems and Their Importance for Enantioseparation

Ternary solubility equilibria are studied for three chiral systems in various aqueous and nonaqueous solvents. The chosen systems were a pharmaceutical intermediate, threonine and mandelic acid. Measured solubility data are presented and the nature of the ternary solubility phase diagrams is described. On this basis possible procedures for a crystallization based enantioseparation are derived. Also, the impact of solubility equilibria on the resolution of racemates by liquid chromatography is analyzed and discussed for the systems under investigation. Finally, a hybrid approach coupling both separation techniques for an efficient chiral resolution is demonstrated by means of the fundamental solubility phase diagrams.     

Inference on Collapsibility in Generalized Linear Models

GREENLAND and MICKEY (1988) derived a closed-form collapsibility test and confidence interval for IxJxK contingency tables with qualitative factors, and presented a small simulation study of its performance. We show how their method can be extended to regression models linear in the natural parameter of a one-parameter exponential family, in which the parameter of interest is the difference of “crude” and “adjusted” regression coefficients. A simplification of the method yields a generalization of the test for omitted covariates given by HAUSMAN (1978) for ordinary linear regression. We present an application to a study of coffee use and myocardial infarction, and a simulation study which indicates that the simplified test performs adequately in typical epidemiologic settings.     

Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

Models of parent-offspring conflict. II. Promiscuity

The population genetics of Trivers (1974) concept of parent-offspring is examined for species in which the effects of the conflict are felt by future half-sibs, as in promiscuous mating systems in which the male shows no parental care. Whether or not a rare conflictor gene will spread in a non-conflictor population depends on f(m) greater than (m + 1)/(0.5m + 1.5) for a dominant gene, and on f(m) greater than 1/4(7 + 3) for a recessive gene; f(m) is the fitness gained by a conflictor relative to a non-conflictor offspring [f(m) greater than 1], and m is the amount of parental investment taken by a conflictor relative to m = 1 for a non-conflictor. The ESS value for conflict (mo) in promiscuous species with zero male parental care has mo = f(mo)/4[df(mo)/dmo]. However, where the male maintains the same harem for several breeding seasons, or where there is promiscuity but both sexes contribute equally to parental care, conditions for conflict are equivalent to monogamy.