Linkage strategies for genetically complex traits. II. The power of affected relative pairs.

The power to detect disease-susceptibility loci through linkage analysis using pairs of affected relatives and affected-unaffected pairs is examined. Allelic identity by descent (ibd) for a completely polymorphic marker for sibling, uncle-nephew, grandparent-grandchild, half-sib, and first-cousin pairs is considered. Affected-unaffected pairs generally represent a poor strategy. For single-locus models, ibd depends on lambda R, the risk ratio for type R relatives compared with population prevalence, and the recombination fraction theta. The ibd for grandparent-grandchild pairs is least affected by recombination, followed by sibs, half-sib, uncle-nephew, and first-cousin pairs. For diseases with large lambda values and for small theta values, distant relatives offer greater power. For larger theta values, grandparent-grandchild pairs are best; for small lambda values, sibs are best. Additive and multiplicative multilocus models are considered. For the multiplicative model, the same formulas as in the single-locus model apply, except that lambda iR (for the ith contributing locus) is substituted for lambda R. For the additive model, the deviation from null expectation for ibd is divided among all contributing loci. Compared with the multiplicative model, for an additive model there is usually greater advantage in distant relationships. Multipoint analysis using linked marker loci for affected relative pairs is described. Simultaneous use of multiple markers diminishes the effect of recombination and allows for localization of the disease-susceptibility locus.     

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.     

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.     

Accounting for epistasis in linkage analysis of general pedigrees

Complex traits are generally believed to be influenced by multiple loci. Identification of loci involved in complex traits is more difficult for interacting than for additive loci. Here we describe an extension of the program lm_twoqtl in the package MORGAN to handle two quantitative trait loci (QTLs) with gene-gene interaction. We investigate whether parametric linkage analysis that accounts for such epistasis improves prospects for linkage detection and accuracy of localization of QTLs. Through use of simulated data we show that analysis that accounts for epistasis provides higher lod scores and better localization than does analysis without epistasis. In addition, we demonstrate that the difference between lod scores in the presence vs. absence of use of an interaction model in analysis is greater in extended than in nuclear pedigrees.     

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.     

Genotype-specific recurrence risks as indicators of the genetic architecture of complex diseases

A statistic is introduced that relates discoveries made in genome-wide association (GWA) studies to patterns of disease risks among relatives. The genotype-specific recurrence risk (GSR) is the genotype-specific risk to relatives of known relationship to affected probands. The GSRs can be used for three purposes. (1) They can provide an independent test of whether an allele identified in a GWA study is associated with the disease. (2) They can provide a test of whether interactions among loci affecting the disease are multiplicative. (3) They can be used by genetic counselors to incorporate information from GWA studies for predicting the risk to relatives of known genotype. Under a multiplicative model of disease causation, the GSRs for a locus are the genotypic risks in probands for that locus multiplied by lambda(R)/lambda(jR), where lambda(R) is Risch's recurrence risk ratio and lambda(jR) is the contribution to lambda(R) from the locus of interest. If there is saturation of risk with increasing numbers of causative alleles, then observed GSRs for individuals with high-risk genotypes will be lower than predicted by the multiplicative 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.     

Familial recurrence-pattern analysis of cleft lip with or without cleft palate.

Cleft lip with or without cleft palate (CL/P) is a common congenital malformation with an incidence in European white populations of about 1/1,000. The familial clustering of CL/P has been extensively characterized, and epidemiological studies have proposed monogenic models (with reduced penetrance), multifactorial/threshold models, and mixed major-gene/multifactorial models to explain its inheritance. The recognition of an association between two RFLPs at the transforming growth factor alpha (TGFA) locus and CL/P supports a major-gene component to the etiology of CL/P. Risch has shown that the recurrence risk ratio lambda R (risk to relatives, vs. population prevalence) is a useful pointer to the mode of inheritance. Here we further develop the use of lambda R to analyze recurrence-risk data for CL/P. Recurrence risks for first-, second-, and third-degree relatives equate well with oligogenic models with as few as four loci. A monogenic/additive model is strongly rejected. The limited available twin data are also consistent with this model. A "major gene" interacting epistatically with an oligogenic background is shown to be a plausible alternative. Power calculations for a linkage study to map the CL/P major-risk locus suggest that a sample of 50 affected sib pairs will be adequate, but linkage to minor-risk loci will require very much larger samples.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.     

Statistical modeling of interlocus interactions in a complex disease: rejection of the multiplicative model of epistasis in type 1 diabetes

In general, common diseases do not follow a Mendelian inheritance pattern. To identify disease mechanisms and etiology, their genetic dissection may be assisted by evaluation of linkage in mouse models of human disease. Statistical modeling of multiple-locus linkage data from the nonobese diabetic (NOD) mouse model of type 1 diabetes has previously provided evidence for epistasis between alleles of several Idd (insulin-dependent diabetes) loci. The construction of NOD congenic strains containing selected segments of the diabetes-resistant strain genome allows analysis of the joint effects of alleles of different loci in isolation, without the complication of other segregating Idd loci. In this article, we analyze data from congenic strains carrying two chromosome intervals (a double congenic strain) for two pairs of loci: Idd3 and Idd10 and Idd3 and Idd5. The joint action of both pairs is consistent with models of additivity on either the log odds of the penetrance, or the liability scale, rather than with the previously proposed multiplicative model of epistasis. For Idd3 and Idd5 we would also not reject a model of additivity on the penetrance scale, which might indicate a disease model mediated by more than one pathway leading to beta-cell destruction and development of diabetes. However, there has been confusion between different definitions of interaction or epistasis as used in the biological, statistical, epidemiological, and quantitative and human genetics fields. The degree to which statistical analyses can elucidate underlying biologic mechanisms may be limited and may require prior knowledge of the underlying etiology.     

Evidence and implications for multiplicative interactions among loci predisposing to human common disease

OBJECTIVE: The aim of this study was to utilize information on monozygotic twin concordance rates and linkage studies results for common diseases to predict the likely mode of interaction between susceptibility loci. METHODS: We calculated combinations of allele frequency and genotypic relative risk (GRR) that would generate linkage results typically observed in common human diseases. Given these single locus effects, we calculated the expected monozygotic twin concordance assuming different numbers of loci under different interaction models. RESULTS: We demonstrate that, for disorders like schizophrenia, a purely additive model of interaction among loci is not consistent with the available evidence. Instead there are likely significant multiplicative or stronger interactions. Given these interactions, we show that in a diagnostic test based on a subset of predisposing loci, the marginal increase of predictive value rises with each additional locus that is discovered. Our model was consistent with susceptibility alleles being common or rare. CONCLUSIONS: Evidence from monozygotic twin concordance rates and linkage results point to a significant degree of multiplicative interaction among loci.     

Marker-Based Inferences about Epistasis for Genes Influencing Inbreeding Depression

We describe a multilocus, marker-based regression method for inferring interactions between genes controlling inbreeding depression in self-fertile organisms. It is based upon selfing a parent heterozygous for several unlinked codominant markers, then analyzing the fitness of progeny marker genotypes. If loci causing inbreeding depression are linked to marker loci, then viability selection is manifested by distorted segregation of markers, and fecundity selection by dependence of the fecundity character upon the marker genotype. To characterize this selection, fitness is regressed on the proportion of loci homozygous for markers linked to deleterious alleles, and epistasis is detected by nonlinearity of the regression. Alternatively, fitness can be regressed on the proportion of heterozygous loci. Other modes of selection can be incorporated with a bivariate regression involving both homozygote and heterozygote marker genotypes. The advantage of this marker-based approach is that ``purging' is minimized and specific chromosomal segments are identified; its disadvantage lies in low statistical power when linkage is not strong and/or the linkage phase between marker and selected loci is uncertain. Using this method, in the wildflower Mimulus guttatus, we found predominant multiplicative gene interaction determining fecundity and some negative synergistic (nonmultiplicative) interaction for viability.     

Overdominant epistatic loci are the primary genetic basis of inbreeding depression and heterosis in rice. II. Grain yield components

The genetic basis underlying inbreeding depression and heterosis for three grain yield components of rice was investigated in five interrelated mapping populations using a complete RFLP linkage map, replicated phenotyping, and the mixed model approach. The populations included 254 F(10) recombinant inbred lines (RILs) derived from a cross between Lemont (japonica) and Teqing (indica), two backcross (BC) and two testcross populations derived from crosses between the RILs and the parents plus two testers (Zhong413 and IR64). For the yield components, the RILs showed significant inbreeding depression and hybrid breakdown, and the BC and testcross populations showed high levels of heterosis. The average performance of the BC or testcross hybrids was largely determined by heterosis. The inbreeding depression values of individual RILs were negatively associated with the heterosis measurements of the BC or testcross hybrids. We identified many epistatic QTL pairs and a few main-effect QTL responsible for >65% of the phenotypic variation of the yield components in each of the populations. Most epistasis occurred between complementary loci, suggesting that grain yield components were associated more with multilocus genotypes than with specific alleles at individual loci. Overdominance was also an important property of most loci associated with heterosis, particularly for panicles per plant and grains per panicle. Two independent groups of genes appeared to affect grain weight: one showing primarily nonadditive gene action explained 62.1% of the heterotic variation of the trait, and the other exhibiting only additive gene action accounted for 28.1% of the total trait variation of the F(1) mean values. We found no evidence suggesting that pseudo-overdominance from the repulsive linkage of completely or partially dominant QTL for yield components resulted in the overdominant QTL for grain yield. Pronounced overdominance resulting from epistasis expressed by multilocus genotypes appeared to explain the long-standing dilemma of how inbreeding depression could arise from overdominant genes.     

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.     

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.     

Linkage strategies for genetically complex traits. III. The effect of marker polymorphism on analysis of affected relative pairs.

The results from the second paper of this series are reexamined for markers that are not completely polymorphic. A maximum lod score (MLS) criterion is defined for affected relative pairs. The expected MLS (EMLS) is calculated as a function of the marker polymorphic information content (PIC) for various values of lambda R (relative risk ratio) and different relative types by using simulations. An m-allele model with equal allele frequencies is employed. The EMLS is calculated for two sampling strategies: scheme 1, which uses pairs only, and scheme 2, which also includes additional informative relatives. For scheme 2, the percent of the maximum achievable EMLS (i.e., for a marker with a PIC of 1.0) is approximately equal to the marker PIC value for all relative types. For scheme 1, the EMLS is greatly diminished unless PIC is high, especially for distant relatives. For example, scheme 1 is not cost-effective for sibs unless PIC greater than .7; for second- and third-degree relatives, PIC must be greater than .85. Therefore, in general, it will be worthwhile to type additional relatives in linkage studies using affected pairs. The comparative value of sibs versus distant relatives depends on lambda R, recombination theta, and PIC. For large lambda R and PIC values, distant relatives are preferred. Alternatively, for smaller lambda R and PIC values, sibs are best.     

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.     

The consequences of gene flow for local adaptation and differentiation: a two-locus two-deme model

We consider a population subdivided into two demes connected by migration in which selection acts in opposite direction. We explore the effects of recombination and migration on the maintenance of multilocus polymorphism, on local adaptation, and on differentiation by employing a deterministic model with genic selection on two linked diallelic loci (i.e., no dominance or epistasis). For the following cases, we characterize explicitly the possible equilibrium configurations: weak, strong, highly asymmetric, and super-symmetric migration, no or weak recombination, and independent or strongly recombining loci. For independent loci (linkage equilibrium) and for completely linked loci, we derive the possible bifurcation patterns as functions of the total migration rate, assuming all other parameters are fixed but arbitrary. For these and other cases, we determine analytically the maximum migration rate below which a stable fully polymorphic equilibrium exists. In this case, differentiation and local adaptation are maintained. Their degree is quantified by a new multilocus version of $F_\mathrm{ST}$ and by the migration load, respectively. In addition, we investigate the invasion conditions of locally beneficial mutants and show that linkage to a locus that is already in migration-selection balance facilitates invasion. Hence, loci of much smaller effect can invade than predicted by one-locus theory if linkage is sufficiently tight. We study how this minimum amount of linkage admitting invasion depends on the migration pattern. This suggests the emergence of clusters of locally beneficial mutations, which may form ‘genomic islands of divergence’. Finally, the influence of linkage and two-way migration on the effective migration rate at a linked neutral locus is explored. Numerical work complements our analytical results.