Equilibria in an epistatic viability model under arbitrary strength of selection

A class of viability models that generalize the standard additive model for the case of pairwise additive by additive epistatic interactions is considered. Conditions for existence and stability of steady states in the corresponding two-locus model are analyzed. Using regular perturbation techniques, the case when selection is weaker than recombination and the case when selection is stronger than recombination are investigated. The results derived are used to make conclusions on the dependence of population characteristics on the relation between the strength of selection and the recombination rate.     

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.     

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.     

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     

随机交配群体中连锁对世代平均数和方差的影响及其测定

本文以黄花烟草N.rustica的两个品种V_2和V_(12)为材料,人工创造4个随机交配轮次不同的群体,对开花期(FT)和最后株高(FH)两个性状在理论上探讨了随机交配群体中连锁对世代平均数和方差的影响,并用上述材料进行验证。结果指出,在平均数和方差两种水平上都测定出连锁的存在;无论是加性方差还是显性方差,都随着交配轮次的增加而减少,这是相引连锁的表现。本文还讨论了基因连锁强度、基因联合程度对世代平均数和方差的影响。     

A FORTRAN program for the calculation and analysis of two-locus linkage disequilibrium coefficients

Summary A FORTRAN program was written that calculates composite linkage disequilibrium coefficients from genotypic data. Chi-square tests determine whether coefficients calculated for allele and locus pairs are significantly greater than zero. A subroutine is provided that partitions the variance in linkage disequilibrium into within- and between-subpopulation components. Output obtained from analysis of allozyme data collected from natural subpopulations of the house fly (Musca domestica L.) are included to illustrate features of the program.Journal Paper No. J-11345 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 2411     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

A two-locus model of speciation.

Speciation is considered as the evolution of partial or complete cross-incompatibility between the carriers of genes (at a locus called "object locus") that distinguish the prospective species populations. The mating relations at the object locus are modified by the alleles at a second mating modifier locus. Based on a widely applicable concept of fitness and mating preference, it is shown that heterozygote disadvantage in fitness at the object locus is necessary for speciation, which corroborates Wallace's hypothesis. It is pointed out that the difference between sympatric and parapatric speciation essentially lies in the mechanisms stabilizing the polymorphism required at the object locus as a prerequisite for speciation. In the presence of recombination between the object and mating modifier locus speciation may be prevented by forces maintaining gametic phase imbalance between these loci such as can result from unidirectional gene flow between parapatric populations.     

The effects of linkage and gene flow on local adaptation: a two-locus continent-island model

Population subdivision and migration are generally considered to be important causes of linkage disequilibrium (LD). We explore the combined effects of recombination and gene flow on the amount of LD, the maintenance of polymorphism, and the degree of local adaptation in a subdivided population by analyzing a diploid, deterministic continent–island model with genic selection on two linked loci (i.e., no dominance or epistasis). For this simple model, we characterize explicitly all possible equilibrium configurations. Simple and intuitive approximations for many quantities of interest are obtained in limiting cases, such as weak migration, weak selection, weak or strong recombination. For instance, we derive explicit expressions for the measures and r² (the squared correlation in allelic state) of LD. They depend in qualitatively different ways on the migration rate. Remarkably high values of r² are maintained between weakly linked loci, especially if gene flow is low. We determine how the maximum amount of gene flow that admits preservation of the locally adapted haplotype, hence of polymorphism at both loci, depends on recombination rate and selection coefficients. We also investigate the evolution of differentiation by examining the invasion of beneficial mutants of small effect that are linked to an already present, locally adapted allele. Mutants of much smaller effect can invade successfully than predicted by naive single-locus theory provided they are at least weakly linked. Finally, the influence of linkage on the degree of local adaptation, the migration load, and the effective migration rate at a neutral locus is explored. We discuss possible consequences for the evolution of genetic architecture, in particular, for the emergence of clusters of tightly linked, slightly beneficial mutations and the evolution of recombination and chromosome inversions.     

Tension versus ecological zones in a two-locus system

Previous theories show that tension and ecological zones are indistinguishable in terms of gene frequency clines. Here I analytically show that these two types of zones can be distinguished in terms of genetic statistics other than gene frequency. A two-locus cline model is examined with the assumptions of random mating, weak selection, no drift, no mutation, and multiplicative viabilities. The genetic statistics for distinguishing the two types of zones are the deviations of one- or two-locus genotypic frequencies from Hardy-Weinberg equilibrium (HWE) or from random association of gametes (RAG), and the deviations of additive and dominance variances from the values at HWE. These deviations have a discontinuous distribution in space and different extents of interruptions in the ecological zone with a sharp boundary, but exhibit a continuous distribution in the tension zone. Linkage disequilibrium enhances the difference between the deviations from HWE and from RAG for any two-locus genotypic frequency.     

Genetic basis of pre-harvest sprouting tolerance using single-locus and two-locus QTL analyses in bread wheat

Quantitative trait loci (QTL) analysis for pre-harvest sprouting tolerance (PHST) in bread wheat was conducted following single-locus and two-locus analyses, using data on a set of 110 recombinant inbred lines (RILs) of the International Triticeae Mapping Initiative population grown in four different environments. Single-locus analysis following composite interval mapping (CIM) resolved a total of five QTLs with one to four QTLs in each of the four individual environments. Four of these five QTLs were also detected following two-locus analysis, which resolved a total of 14 QTLs including 8 main effect QTLs (M-QTLs), 8 epistatic QTLs (E-QTLs) and 5 QTLs involved in QTL × environment (QE) or QTL × QTL × environment (QQE) interactions, some of these QTLs being common. The analysis revealed that a major fraction (76.68%) of the total phenotypic variation explained for PHST is due to M-QTLs (47.95%) and E-QTLs (28.73%), and that only a very small fraction of variation (3.24%) is due to QE and QQE interactions. Thus, more than three-quarters of the genetic variation for PHST is fixable and would contribute directly to gains under selection. Two QTLs that were detected in more than one environment and at LOD scores above the threshold values were located on 3BL and 3DL presumably in the vicinity of the dormancy gene TaVp1. Another QTL was found to be located on 3B, perhaps in close proximity to the R gene for red grain colour. However, these associations of QTLs for PHST with genes for dormancy and grain colour are only suggestive. The results obtained in the present study suggest that PHST is a complex trait controlled by large number of QTLs, some of them interacting among themselves or with the environment. These QTLs can be brought together through marker-aided selection, leading to enhanced PHST.     

Sam Karlin and multi-locus population genetics

Between 1967 and 1982, Sam Karlin made fundamental contributions to many areas of deterministic population genetic theory. This remembrance focuses on his work in multi-locus population genetics, primarily on the interaction between genotypic selection and the rate of recombination.     

Effects of epistasis on tests for linkage in self-pollinated species

Summary Tests for linkage based on covariances among relatives in self-pollinated species are usually based upon an assumption that epistasis is not important. This study was conducted to determine the impact of epistasis on, and to investigate the sensitivity of, such tests. Thirty covariances were calculated for each of ten non-epistatic and ten epistatic genetic models with varying probabilities of recombination between two coupling or repulsion loci. Each set of covariances was tested for linkage by comparing covariances calculated for the model with those expected for an additive-dominance model with no linkage. Results showed that the test for linkage is quite insensitive to the effects of linkage due to the disproportionate influence of inbreeding. Repulsion linkages should be easier to detect than coupling linkages for all models. Epistasis was found to mimic or counteract the effects of linkage. Tests for linkage based on covariances within a hierarchical mating design appear to be insensitive to linkage and may confuse the effects of linkage and epistasis.     

Two-loci interaction confirms arthritis-regulating quantitative trait locus on rat chromosome 6

A form of genetic interaction, or epistasis, occurs when one gene interferes with the phenotypic effect of another nonallelic gene. In pristane-induced arthritis (PIA) in rats we have previously identified Pia3, on chromosome 6, to be a locus that regulates onset of disease. In a single congenic strain containing Pia3 on the arthritis-susceptible DA background, DA.Pia3, no difference in onset of disease or early disease severity could be detected. After a two-loci interaction analysis of (E3 x DA)F2 intercross data, Pia3 was found to interact with Pia4 (chromosome 12). Subsequently, the DA.Pia3 congenic strain was combined with the DA.Pia4 congenic strain so that an effect of Pia3 could be observed. The effect of heterozygosity in Pia4 results in lower severity and thus in combination with Pia3 made it possible to observe that Pia3 alleles from the arthritis-resistant E3 strain rendered more severe arthritis into the otherwise 100% susceptible DA strain. As the introduction of Pia4 heterozygosity results in a lower level of arthritis severity we regard this as an additive interaction with a severity threshold-lowering effect.     

On the evolution of epistasis II: a generalized Wright-Kimura framework

The evolution of fitness interactions between genes at two major loci is studied where the alleles at a third locus modify the epistatic interaction between the two major loci. The epistasis is defined by a parameter epsilon and a matrix structure that specifies the nature of the interactions. When epsilon=0 the two major loci have additive fitnesses, and when these are symmetric the interaction matrices studied here produce symmetric viabilities of the Wright [1952. The genetics of quantitative variability. In: Reeve, E.C.R., Waddington, C.H. (Eds.), Quantitative Inheritance. Her Majesty's Stationary Office, London]-Kimura [1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278-281] form. Two such interaction matrices are studied, for one of which epistasis as measured by |epsilon| always increases, and for the other it increases when the linkage between the major loci is tight enough and there is initial linkage disequilibrium. Increase of epistasis does not necessarily coincide with increase in equilibrium mean fitness.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others. Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.      

Mapping genes conditioning in vitro androgenesis in maize using RFLP analysis

Summary This research was designed to map the genes in maize which condition a high response to anther culture using RFLP analysis. A set of 98 S₁ families were developed from the cross of B73 × 139/39-05. In vitro-cultured anthers of 139/39-05 produce numerous embryolike structures while anthers cultured from B73 produce none. Plants from each of the families were grown in the greenhouse. Tassels were harvested from ten individual plants within each family and pretreated prior to culture. Up to three Petri dishes with 60 anthers each were cultured from each tassel. Response was measured as the number of embryo-like structures per 100 anthers cultured. In excess of 105 RFLP clones were screened to detect polymorphism among the parents. A subset of 75 widely distributed clones were scored in the 98 families. Based on the analysis of the resulting genetic data set, the high anther culture response observed in 139/39-05 is conditioned by two major recessive genes, which are epistatic, and two minor genes. One of the two major loci resides in the proximal region of the long arm of chromosome 3 near the indeterminate gametophyte (ig1) gene. The second major locus maps to the centromeric region of chromosome 9. The minor genes reside on chromosomes 1 and 10. Fifty seven percent of the variability among the 98 family means is explained by a genetic model which includes these four chromosomal regions. Moreover, segregation at these loci explains much of the variability observed within the families.     

A simple model of co-evolutionary dynamics caused by epistatic selection

Epistasis is the dependency of the effect of a mutation on the genetic background in which it occurs. Epistasis has been widely documented and implicated in the evolution of species barriers and the evolution of genetic architecture. Here we propose a simple model to formalize the idea that epistasis can also lead to co-evolutionary patterns in molecular evolution of interacting genes. This model epistasis is represented by the influence of one gene substitution on the fitness rank of the resident allele at another locus. We assume that increasing or decreasing fitness rank occur equally likely. In simulations we show that this form of epistasis leads to co-evolution in the sense that the length of an adaptive walk between interacting loci is highly correlated. This effect is caused by episodes of elevated rate of evolution in both loci simultaneously. We find that the influence of epistasis on these measures of co-evolutionary dynamics is relatively robust to the details of the model. The main factor influencing the correlation in evolutionary rates is the probability that a substitution will have an epistatic effect, but the strength of epistasis or the asymmetry of the initial fitness ranks of the alleles have only a minor effect. We suggest that covariance in rates of evolution among loci could be used to detect epistasis among loci.