Further analysis of complex allozyme polymorphisms in a barley population

New theory has recently been developed for two-locus models. In the light of this theory, an earlier analysis of esterase allozyme data from an experimental barley population has been modified to take proper account of initial gametic phase (linkage) disequilibria. The results show that the directions in which two-locus genotypic frequencies deviated from products of one-locus frequencies in this population followed those predicted by neutral descent theory. The observed departures were, however, much larger in size than predicted by the new descent measure theory, indicating that selection is operating in the population.     

The effect on neutral gene flow of selection at a linked locus

The effect on gene flow at a neutral locus of a selective cline at a linked locus is investigated. A diffusion approximation for a two-locus island model is derived in which only one locus is subject to selection. The moments of the stationary distribution are obtained and compared to the corresponding moments from a one-locus, neutral island model. This comparison yields an effective migration rate. The effective migration rate is always less than the actual migration rate, but this effect is seen to be small for weak selection and loose linkage in the case of adult migration. The importance of selection at linked loci to the question of genetic differentiation in a subdivided population is discussed.     

Reinforcement and the genetics of hybrid incompatibilities

Recent empirical studies suggest that genes involved in speciation are often sex-linked. We derive a general analytic model of reinforcement to study the effects of sex linkage on reinforcement under three forms of selection against hybrids: one-locus, two-locus, and ecological incompatibilities. We show that the pattern of sex linkage can have a large effect on the amount of reinforcement due to hybrid incompatibility. Sex linkage of genes involved in postzygotic isolation generally increases the strength of reinforcement, but only if genes involved in prezygotic isolation are also sex linked. We use exact simulations to test the accuracy of the approximation and find that qualitative predictions made assuming weak selection can hold when selection is strong. Our simulations also show that incompatibilities that evolve in allopatry by drift or weak selection are likely to be lost by swamping during secondary contact, even when selection against hybrids is strong.     

Natural selection opposing artificial selection: a two-locus deterministic simulation

A two-locus, two-allele metric trait was submitted to artificial truncation selection and to three types of opposing natural selection (two-locus extensions of directional selection, overdominance and underdominance) by numerical simulation in a large random-mating population. Limits to selection were generally reached by generation 100. Intermediate selection plateaus were found, with minor genes, for all three modes of opposing natural selection, but they were least frequent with underdominance. Multiple outcomes were common. In particular, fixation of the genotype favored by artificial selection was often associated with fixation of another genotype and/or with a central equilibrium; the end point actually reached depended on the genetic starting point of the simulation. In general, when the alleles favored by truncation selection were combined (positive linkage disequilibrium) in the base population, or when the trait was determined by major genes, artificial selection would prevail. Limitations inherent to this type of work are discussed, and possible avenues for further work on the antagonism between artificial and natural selection are proposed.     

Two-locus inbreeding measures for recurrent selection

Summary For a population undergoing recurrent selection, a method is presented for determining the average inbreeding coefficients at the end of each breeding cycle. The coefficients are derived in terms of probability measures that genes are identical by descent. For the one-locus case, two digametic measures are defined and employed in the derivation of a recurrence formula for the inbreeding coefficient. Two further classes of measures, trigametic and quadrigametic, are required for transition from one cycle to the previous one to allow the calculation of the inbreeding function for the two-locus case. Numerical values of the average probability of double identity by descent for populations with various imposed assumptions are listed to illustrate the effects of linkage and population size on the accrual of inbreeding and hence of homozygosity.Paper number 5018 of the Journal Series of the North Carolina Agricultural Experiment Station, Raleigh, North Carolina. This investigation was supported in part by NIH research grant number GM 11546 from the National Institute of General Medical Sciences.     

Secondary theorem of natural selection in biocultural populations.

The "Secondary Theorem of Natural Selection," an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.     

Strategies for conditional two-locus nonparametric linkage analysis

In this article we deal with two-locus nonparametric linkage (NPL) analysis, mainly in the context of conditional analysis. This means that one incorporates single-locus analysis information through conditioning when performing a two-locus analysis. Here we describe different strategies for using this approach. Cox et al. [Nat Genet 1999;21:213-215] implemented this as follows: (i) Calculate the one-locus NPL process over the included genome region(s). (ii) Weight the individual pedigree NPL scores using a weighting function depending on the NPL scores for the corresponding pedigrees at speci fi c conditioning loci. We generalize this by conditioning with respect to the inheritance vector rather than the NPL score and by separating between the case of known (prede fi ned) and unknown (estimated) conditioning loci. In the latter case we choose conditioning locus, or loci, according to prede fi ned criteria. The most general approach results in a random number of selected loci, depending on the results from the previous one-locus analysis. Major topics in this article include discussions on optimal score functions with respect to the noncentrality parameter (NCP), and how to calculate adequate p values and perform power calculations. We also discuss issues related to multiple tests which arise from the two-step procedure with several conditioning loci as well as from the genome-wide tests.     

Schizophrenia: the testing of genetic models by pedigree analysis.

Simulated pedigrees of schizophrenia generally show a clear peak in their likelihood surface corresponding to analysis by the genetic models, which served as the basis for the simulation. The likelihood surface obtained with real data permits determination of the allelic frequency and the selection of an optimal one-locus, two-locus, and four-locus model. These three models have certain features in common, notably, a relatively high frequency of the allele predisposing to schizophrenia (about 20%) and a relatively low index of genetic determination (23%--34%). However, direct likelihood comparisons do not permit distinctions between the one-locus, two-locus, and four-locus models. The most likely interpretation of this finding is that the etiology of schizophrenia is heterogeneous or even nongenetic. However, a simple model with a single completely recessive locus and incomplete penetrance in the homozygote also produces a flat likelihood surface closely resembling that obtained with the real data. With reservation, this single-locus model may be put forward as a potentially useful working hypothesis.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.Supported in part by NSF Grant DMS-8704369Supported in part by NSF Grant BSR-8512844     

Evidence for two unlinked loci regulating total serum IgE levels.

Studies investigating the genetic control of total serum IgE levels are of major importance in understanding basic pathophysiologic mechanisms in atopy and asthma, since IgE levels predict onset and correlate with the clinical expression of these disorders. Previous analysis of data from 92 families, ascertained through a parent with asthma, showed evidence for recessive inheritance of high IgE levels with linkage to chromosome 5q. Since there was significant residual familial correlation in the one-locus segregation analysis, two-locus segregation and linkage analyses were performed. Segregation analyses provided evidence for a second major locus unlinked to the locus on 5q. Utilization of this two-locus model corroborates the previous evidence for linkage between this trait and markers on 5q31-q33. The LODs for the most informative marker D5S436 increased from 3.00 at 10% recombination to 4.67 at 9% recombination, when the two-locus model was used. Additional linkage studies are needed to map this second locus. These results demonstrate the importance of performing multilocus segregation and linkage analyses for quantitative traits that are related to the phenotype of a complex disorder. This approach has given further insight into the genetics of allergy and asthma by providing evidence for a two-locus model.     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

The two-locus model of Gaussian stabilizing selection

We study the equilibrium structure of a well-known two-locus model in which two diallelic loci contribute additively to a quantitative trait that is under Gaussian stabilizing selection. The population is assumed to be infinitely large, randomly mating, and having discrete generations. The two loci may have arbitrary effects on the trait, the strength of selection and the recombination rate may also be arbitrary. We find that 16 different equilibrium patterns exist, having up to 11 equilibria; up to seven interior equilibria may coexist, and up to four interior equilibria, three in negative and one in positive linkage disequilibrium, may be simultaneously stable. Also, two monomorphic and two fully polymorphic equilibria may be simultaneously stable. Therefore, the result of evolution may be highly sensitive to perturbations in the initial conditions or in the underlying genetic parameters. For the special case of equal effects, global stability results are proved. In the general case, we rely in part on numerical computations. The results are compared with previous analyses of the special case of extremely strong selection, of an approximate model that assumes linkage equilibrium, and of the much simpler quadratic optimum model.     

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.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.     

Population trajectories for single locus additive fecundity selection and related selection models

A selection model which comprises models of additive fecundities as well as models of viability, fecundity, or differential mating selection acting only in one sex, is investigated for an autosomal gene locus in a population reproducing in nonoverlapping generations. The recurrence equations and basic properties of the genotypic population trajectories and equilibrium points are formulated for the multiallelic case. For the diallelic case, the trajectory development is discussed in more detail, and it is proven that every population trajectory converges to a Hardy-Weinberg equilibrium point.     

Preferential mating in symmetric multilocus systems: limits for multiallelism and for many Loci

Models in which general forms of preferential mating have been superimposed on the framework of the symmetric heterozygosity selection regime have been examined previously with respect to the existence and local stability of a central polymorphic equilibrium. The results are now extended to produce the limiting form of the stability conditions in two cases: First, where the number of alleles per locus is assumed to be very large; second, where the number of loci affecting the character is very large. It is argued that some type of frequency dependence in the mating pattern must be included, and a particular case is examined in detail. It is shown that multiallelism is ambiguous in its effect on stability, while an increasing number of loci, at least under zero linkage, leads to a simple stability condition which is analogous to the one-locus heterosis principle. Assortative mating appears to be more likely to produce a stable central polymorphism under high levels of allelism than is sexual selection, but is relatively very much weaker than sexual or viability selection if the number of loci involved is large.     

Selection component analysis of natural polymorphisms using population samples including mother-offspring combinations, II

Population samples including mother-offspring combinations provide information on the selection components: zygotic selection, sexual selection, gametic seletion and fecundity selection, on the mating pattern, and on the deviation from linkage equilibrium among the loci studied. The theory for the analysis of a sample consisting of adult individuals including combinations of a mother and one randomly chosen offspring in the case of n polymorphic autosomal loci with m_a alleles at the ath locus, a = 1, 2,…, n, is developed for an organism with discrete non-overlapping generations. In this general analysis complete determination of the genotypes is assumed. The modifications in the analysis for the case n = 2 and m₁ = m₂ = 2 when the two double heterozygotes cannot be separated are discussed, and the analysis is illustrated by data from a population of Zoarces viviparus (a marine teleost).     

Constraints on the origin and maintenance of genetic kin recognition

Kin-recognition mechanisms allow helping behaviors to be directed preferentially toward related individuals, and could be expected to evolve in many cases. However, genetic kin recognition requires a genetic polymorphism on which recognition is based, and kin discriminating behaviors will affect the evolution of such polymorphism. It is unclear whether genetic polymorphisms used in kin recognition should be maintained by extrinsic selection pressures or not, as opposite conclusions have been reached by analytical one-locus models and simulations exploring different population structures. We analyze a two-locus model in a spatially subdivided population following the island model of dispersal between demes of finite size. We find that in the absence of mutation, selection eliminates polymorphism in most cases, except with extreme spatial structure and low recombination. With mutation, the population may reach a stable limit cycle over which both loci are polymorphic; however, the average frequency of conditional helping can be high only under strong structure and low recombination. Finally, we review evidence for extrinsic selection maintaining polymorphism on which kin recognition is based.     

Selection intensity against deleterious mutations in RNA secondary structures and rate of compensatory nucleotide substitutions

A two-locus model of reversible mutations with compensatory fitness interactions is presented; single mutations are assumed to be deleterious but neutral in appropriate combinations. The expectation of the time of compensatory nucleotide substitutions is calculated analytically for the case of tight linkage between sites. It is shown that selection increases the substitution time dramatically when selection intensity Ns > 1, where N is the diploid population size and s the selection coefficient. Computer simulations demonstrate that recombination increases the substitution time, but the effect of recombination is small when selection is weak. The amount of linkage disequilibrium generated in the process of compensatory substitution is also investigated. It is shown that significant linkage disequilibrium is expected to be rare in natural populations. The model is applied to the mRNA secondary structure of the bicoid 3' untranslated region of Drosophila. It is concluded that average selection intensity Ns against single deleterious mutations is not likely to be much larger than 1.