Mathematical frameworks for phenotypical selection and epistasis

A mathematical approach to interactions between genotypes and phenotypes in a multilocus multiallele population is developed. No a priori information on a fitness function is required. In particular, some structural definitions of epistasis and the position effect are given in terms of a decomposition of phenotypical structures. On this base a distance to the additive non-epistasis is introduced and an explicit formula for it is obtained. A class of phenotypical structures including multilocus dominance is described in terms of directed graphs. The evolutionary equations are adjusted to a fitness function compatible with a phenotypical structure. Some results on the finiteness of the equilibria set are presented.     

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 2 ⁿ 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.Supported in part by NIH grant GM 28016, and by grant 81-5458 from the Danish Natural Science Research Council     

The Effect of an Experimental Bottleneck upon Quantitative Genetic Variation in the Housefly

Effects of a population bottleneck (founder-flush cycle) upon quantitative genetic variation of morphometric traits were examined in replicated experimental lines of the housefly founded with one, four or 16 pairs of flies. Heritability and additive genetic variances for eight morphometric traits generally increased as a result of the bottleneck, but the pattern of increase among bottleneck sizes differed among traits. Principal axes of the additive genetic correlation matrix for the control line yielded two suites of traits, one associated with general body size and another set largely independent of body size. In the former set containing five of the traits, additive genetic variance was greatest in the bottleneck size of four pairs, whereas in the latter set of two traits the largest additive genetic variance occurred in the smallest bottleneck size of one pair. One trait exhibited changes in additive genetic variance intermediate between these two major responses. These results were inconsistent with models of additive effects of alleles within loci or of additive effects among loci. An observed decline in viability measures and body size in the bottleneck lines also indicated that there was nonadditivity of allelic effects for these traits. Several possible nonadditive models were explored that increased additive genetic variance as a result of a bottleneck. These included a model with complete dominance, a model with overdominance and a model incorporating multiplicative epistasis.     

The estimation of epistasis in components of fitness in experimental populations of Drosophila melanogaster I. A two-stage maximum likelihood model

Laboratory populations of Drosophila melanogaster bearing the Curly and Plum marked second chromosome inversions were observed in selection experiments for ten discrete generations. Maximum likelihood estimates of the relative fitnesses of Curly, Plum, Curly-Plum, and wild phenotypes were obtained from selection trajectories. Using these estimates, measures of multiplicative and additive epistasis were calculated. These were partitioned into pre-sampling and post-sampling components, and both were found to be significant. In several cases the sign of the epistasis of the two components was reversed, and the direction of net epistasis depended on the particular inversion. The significance of partitioning epistasis into components is discussed in the light of two locus population genetic theory.     

Linkage strategies for genetically complex traits. I. Multilocus models

In order to investigate linkage detection strategies for genetically complex traits, multilocus models of inheritance need to be specified. Here, two types of multilocus model are described: (1) a multiplicative model, representing epistasis (interaction) among loci, and (2) an additive model, which is shown to closely approximate genetic heterogeneity, which is characterized by no interlocus interaction. A ratio lambda R of risk for type R relatives that is compared with population prevalence is defined. For a single-locus model, lambda R - 1 decreases by a factor of two with each degree of relationship. The same holds true for an additive multilocus model. For a multiplicative (epistasis) model, lambda R - 1 decreases more rapidly than by a factor of two with degree of relationship. Examination of lambda R values for various classes of relatives can potentially suggest the presence of multiple loci and epistasis. For example, data for schizophrenia suggest multiple loci in interaction. It is shown in the second paper of this series that lambda R is the critical parameter in determining power to detect linkage by using affected relative pairs.     

Statistical Methods for Linkage Analysis of Complex Traits from High-Resolution Maps of Identity by Descent

A multilocus model for complex traits is described that generalizes the additive and multiplicative models and hence allows simultaneously for both heterogeneity and gene interaction (epistasis). Statistical methods of linkage analysis are discussed under the assumption that identity by descent data from a dense set of polymorphic markers are available. Three methods, single locus search, simultaneous search and conditional search, are described and compared.     

Morphometric Differentiation among Experimental Lines of the Housefly in Relation to a Bottleneck

Differentiation in morphometric traits among experimental populations of the housefly subjected to an experimental bottleneck was examined for replicate lines founded with one, four or 16 pairs of flies. Differentiation among lines within a bottleneck size was significantly greater than predicted by drift in relation to the additive genetic variation for these traits within the founding population. Two models of nonadditive genetic variance were investigated to interpret these results, one involving dominance of allelic effects within loci and another incorporating multiplicative epistasis. Both models generated more variation among lines as a direct result of sampling during the bottleneck than predicted by a model with additive gene action. The pattern of differentiation among our experimental lines in relation to these models conformed more to the model incorporating epistasis. Nevertheless, it may be difficult to distinguish differentiation among lines occurring during a bottleneck as a result of nonadditive gene action from that caused by diversifying selection among lines after the bottleneck.     

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.     

On the evolution of epistasis III: the haploid case with mutation

Whether interaction between genes is better represented by synergistic or antagonistic epistasis has been a focus of experimental research in bacterial population genetics. Our previous research on evolution of modifiers of epistasis in diploid systems has indicated that the strength of positive or negative epistasis should increase provided linkage disequilibrium is maintained. Here we study a modifier of epistasis in fitness between two loci in a haploid system. Epistasis is modified in the neighborhood of a mutation-selection balance. We show that when linkage in the three-locus system is tight, an increase in the frequency of a modifier allele that induces either more negative or more positive epistasis is possible. Epistasis here can be measured on either an additive or multiplicative scale.     

The additive genetic variance after bottlenecks is affected by the number of loci involved in epistatic interactions

Abstract We investigated the role of the number of loci coding for a neutral trait on the release of additive variance for this trait after population bottlenecks. Different bottleneck sizes and durations were tested for various matrices of genotypic values, with initial conditions covering the allele frequency space. We used three different types of matrices. First, we extended Cheverud and Routman's model by defining matrices of "pure" epistasis for three and four independent loci; second, we used genotypic values drawn randomly from uniform, normal, and exponential distributions; and third we used two models of simple metabolic pathways leading to physiological epistasis. For all these matrices of genotypic values except the dominant metabolic pathway, we find that, as the number of loci increases from two to three and four, an increase in the release of additive variance is occurring. The amount of additive variance released for a given set of genotypic values is a function of the inbreeding coefficient, independently of the size and duration of the bottleneck. The level of inbreeding necessary to achieve maximum release in additive variance increases with the number of loci. We find that additive-by-additive epistasis is the type of epistasis most easily converted into additive variance. For a wide range of models, our results show that epistasis, rather than dominance, plays a significant role in the increase of additive variance following bottlenecks.     

A complete classification of epistatic two-locus models

Background

The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results

We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion

The circuits provide information on epistasis beyond that contained in the additive × additive, additive × dominance, and dominance × dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.     

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.     

Capturing pair-wise epistatic effects associated with three agronomic traits in barley

Genetic association mapping has been widely applied to determine genetic markers favorably associated with a trait of interest and provide information for marker-assisted selection. Many association mapping studies commonly focus on main effects due to intolerable computing intensity. This study aims to select several sets of DNA markers with potential epistasis to maximize genetic variations of some key agronomic traits in barley. By doing so, we integrated a MDR (multifactor dimensionality reduction) method with a forward variable selection approach. This integrated approach was used to determine single nucleotide polymorphism pairs with epistasis effects associated with three agronomic traits: heading date, plant height, and grain yield in barley from the barley Coordinated Agricultural Project. Our results showed that four, seven, and five SNP pairs accounted for 51.06, 45.66 and 40.42% for heading date, plant height, and grain yield, respectively with epistasis being considered, while corresponding contributions to these three traits were 45.32, 31.39, 31.31%, respectively without epistasis being included. The results suggested that epistasis model was more effective than non-epistasis model in this study and can be more preferred for other applications.     

WILL POPULATION BOTTLENECKS AND MULTILOCUS EPISTASIS INCREASE ADDITIVE GENETIC VARIANCE?

We apply new analytical methods to understand the consequences of population bottlenecks for expected additive genetic variance. We analyze essentially all models for multilocus epistasis that have been numerically simulated to demonstrate increased additive variance. We conclude that for biologically plausible models, large increases in expected additive variance--attributable to epistasis rather than dominance--are unlikely. Naciri-Graven and Goudet (2003) found that as the number of epistatically interacting loci increases, additive variance tends to be inflated more after a bottleneck. We argue that this result reflects biologically unrealistic aspects of their models. Specifically, as the number of loci increases, higher-order epistatic interactions become increasingly important in these models, with an increasing fraction of the genetic variance becoming nonadditive, contrary to empirical observations. As shown by Barton and Turelli (2004), without dominance, conversion of nonadditive to additive variance depends only on the variance components and not on the number of loci per se. Numerical results indicating that more inbreeding is needed to produce maximal release of additive variance with more loci follow directly from our analytical results, which show that high levels of inbreeding (F > 0.5) are needed for significant conversion of higher-order components. We discuss alternative approaches to modeling multilocus epistasis and understanding its consequences.     

META-X: Generic Software for Metapopulation Viability Analysis

The major tools used to make population viability analyses (PVA) quantitative are stochastic models of population dynamics. Since a specially tailored model cannot be developed for every threatened population, generic models have been designed which can be parameterised and analysed by non-modellers. These generic models compromise on detail so that they can be used for a wide range of species. However, generic models have been criticised because they can be employed without the user being fully aware of the concepts, methods, potentials, and limitations of PVA. Here, we present the conception of a new generic software package for metapopulation viability analysis, META-X. This conception is based on three elements, which take into account the criticism of earlier generic PVA models: (1) comparative simulation experiments; (2) an occupancy-type model structure which ignores details of local population dynamics (these details are integrated in external submodels); and (3) a unifying currency to quantify persistence and viability, the intrinsic mean time to extinction. The rationale behind these three elements is explained and demonstrated by exemplary applications of META-X in the three fields for which META-X has been designed: teaching, risk assessment in the field, and planning. The conception of META-X is based on the notion that PVA is a tool to deal with rather than to overcome uncertainty. The purpose of PVA is to produce relative, not absolute, assessments of extinction risk which support, but do not supplant, management decisions.     

Genotypic Variation for a Quantitative Character Maintained under Stabilizing Selection without Mutations: Epistasis

A model of the gene action on a quantitative character is suggested. The model takes into account epistasis by combining multiplicative with the traditional additive approximation of the action of loci. It is demonstrated on the basis of this model that a high level of genotypic variation can be maintained in a population for a quantitative character under stabilizing selection in the absence of mutations, if there is epistasis. It is also shown that a large amount of additive variation as well as high heritability can be "hidden" in such a population and "released" if stabilizing selection is relaxed.     

Stickbreaking: a novel fitness landscape model that harbors epistasis and is consistent with commonly observed patterns of adaptive evolution

In relating genotypes to fitness, models of adaptation need to both be computationally tractable and qualitatively match observed data. One reason that tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fitness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as fitness trajectories, the distribution of fitness achieved, and epistasis. Stickbreaking is compared to the additive and multiplicative models. We conclude that the stickbreaking model is qualitatively consistent with several commonly observed patterns of adaptive evolution.     

The contrasting genetic architecture of wing size, viability, and development time in a rainforest species and its more widely distributed relative

Divergence among populations can occur via additive genetic effects and/or because of epistatic interactions among genes. Here we use line-cross analysis to compare the importance of epistasis in divergence among two sympatric Drosophila species from eastern Australia, one (D. serrata) distributed continuously and the other (D. birchii) confined to rainforest habitats that are often disjunct. For D. serrata, crosses indicated that development time and wing size differences were due to additive genetic effects, while for viability there were digenic epistatic effects. Crosses comparing geographically close populations as well as those involving the most geographically distant populations (including the southern species border) revealed epistatic interactions, whereas crosses at an intermediate distance showed no epistasis. In D. birchii, there was no evidence of epistasis for viability, although for development time and wing size there was epistasis in the cross between the most geographically diverged populations. Strong epistasis has not developed among the D. birchii populations, and this habitat specialist does not show stronger epistasis than D. serrata. Given that epistasis has been detected in crosses with other species from eastern Australia, including the recently introduced D. melanogaster, the results point to epistasis not being directly linked to divergence times among populations.     

Mapping QTLs with epistatic effects and QTL×environment interactions by mixed linear model approaches

A new methodology based on mixed linear models was developed for mapping QTLs with digenic epistasis and QTL×environment (QE) interactions. Reliable estimates of QTL main effects (additive and epistasis effects) can be obtained by the maximum-likelihood estimation method, while QE interaction effects (additive×environment interaction and epistasis×environment interaction) can be predicted by the-best-linear-unbiased-prediction (BLUP) method. Likelihood ratio and t statistics were combined for testing hypotheses about QTL effects and QE interactions. Monte Carlo simulations were conducted for evaluating the unbiasedness, accuracy, and power for parameter estimation in QTL mapping. The results indicated that the mixed-model approaches could provide unbiased estimates for both positions and effects of QTLs, as well as unbiased predicted values for QE interactions. Additionally, the mixed-model approaches also showed high accuracy and power in mapping QTLs with epistatic effects and QE interactions. Based on the models and the methodology, a computer software program (QTLMapper version 1.0) was developed, which is suitable for interval mapping of QTLs with additive, additive×additive epistasis, and their environment interactions. Received: 23 October 1998 / Accepted: 11 May 1999     

Statistical methods for estimating doubling time in in vitro cell growth

Summary Doubling time has been widely used to represent the growth pattern of cells. A traditional method for finding the doubling time is to apply gray-scaled cells, where the logarithmic transformed scale is used. As an alternative statistical method, the log-linear model was recently proposed, for which actual cell numbers are used instead of the transformed gray-scaled cells. In this paper, I extend the log-linear model and propose the extended log-linear model. This model is designed for extra-Poisson variation, where the log-linear model produces the less appropriate estimate of the doubling time. Moreover, I compare statistical properties of the gray-scaled method, the log-linear model, and the extended log-linear model. For this purpose, I perform a Monte Carlo simulation study with three data-generating models: the additive error model, the multiplicative error model, and the overdispersed Poisson model. From the simulation study, I found that the gray-scaled method highly depends on the normality assumption of the gray-scaled cells; hence, this method is appropriate when the error model is multiplicative with the log-normally distributed errors. However, it is less efficient for other types of error distributions, especially when the error model is additive or the errors follow the Poisson distribution. The estimated standard error for the doubling time is not accurate in this case. The log-linear model was found to be efficient when the errors follow the Poisson distribution or nearly Poisson distribution. The efficiency of the log-linear model was decreased accordingly as the overdispersion increased, compared to the extended log-linear model. When the error model is additive or multiplicative with Gamma-distributed errors, the log-linear model is more efficient than the gray-scaled method. The extended log-linear model performs well overall for all three data-generating models. The loss of efficiency of the extended log-linear model is observed only when the error model is multiplicative with log-normally distributed errors, where the gray-scaled method is appropriate. However, the extended log-linear model is more efficient than the log-linear model in this case.