Mapping Epistatic Quantitative Trait Loci With One-Dimensional Genome Searches

Corresponding author: Jean-Luc Jannink, Agronomy Department, Iowa State University, Ames, IA 50011-1010., jjannink{at}iastate.edu (E-mail)

Communicating editor: J. A. M. VAN ARENDONK

	ABSTRACT

TOP ABSTRACT METHODOLOGY SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

The discovery of epistatically interacting QTL is hampered by the intractability and low power to detect QTL in multidimensional genome searches. We describe a new method that maps epistatic QTL by identifying loci of high QTL by genetic background interaction. This approach allows detection of QTL involved not only in pairwise but also higher-order interaction, and does so with one-dimensional genome searches. The approach requires large populations derived from multiple related inbred-line crosses as is more typically available for plants. Using maximum likelihood, the method contrasts models in which QTL allelic values are either nested within, or fixed over, populations. We apply the method to simulated doubled-haploid populations derived from a diallel among three inbred parents and illustrate the power of the method to detect QTL of different effect size and different levels of QTL by genetic background interaction. Further, we show how the method can be used in conjunction with standard two-locus QTL detection models that use two-dimensional genome searches and find that the method may double the power to detect first-order epistasis.

TRAITS that show continuous variation among individuals (quantitative traits) are affected by the environment and by many genes [quantitative trait loci (QTL)] that act singly and in interaction with each other. Interaction among QTL, or epistasis, is implicated in a number of important processes. Epistatic variance, as its fraction of the total genetic variance increases, may reduce the resemblance of offspring to their parents (LYNCH and WALSH 1998 ), affect the importance of genetic drift and population structure in evolution (WRIGHT 1980 ; WADE 1992 ), increase the genetic variance remaining in a population after a bottleneck (GOODNIGHT 1987 , GOODNIGHT 1988 ), and lead to either heterosis (LYNCH and WALSH 1998 ) or its reverse, outbreeding depression, which can cause speciation events (PARKER 1992 ; ORR 1995 ).

Methods to study epistasis using quantitative genetic methods that are based strictly on individual phenotypes lack power because epistasis contributes little to the resemblance among relatives (CHEVERUD and ROUTMAN 1995 ). The advent of complete DNA-marker linkage maps offers a new possibility for the study of epistasis: QTL mapping allows for the direct evaluation of interaction among identified loci (e.g., simulation model 4 of HALEY and KNOTT 1992 ; CHASE et al. 1997 ). Nevertheless, such an approach confronts two important problems: low power to detect first-order epistasis (between just two QTL) and the intractability of seeking higher-order epistasis (involving multiple QTL).

To evaluate QTL interactions, methods must search for multiple QTL simultaneously. Such a multidimensional search (KAO et al. 1999 ) necessitates many statistical tests, and a high statistical threshold must be adopted to avoid false-positives among those tests. As an indication, LANDER and BOTSTEIN 1989 (Appendix 6) suggest that an m-fold higher threshold be used to declare significance for an m-dimensional search as for a one-dimensional search. This high threshold would drastically reduce power to find significant QTL interactions. To date, researchers have implemented two partial solutions to this quandary. First, they have increased detection power by restricting the search for QTL interactions to portions of the genome (FIJNEMAN et al. 1996 ). This restriction ensures a feasible number of tests but leaves other portions of the genome unexplored. Second, they have sought QTL interactions that affect only those QTL that have detectable main effects (LARK et al. 1995 ). This option also drastically reduces the number of tests but fails to discover interacting QTL that have no strong individual effect (FIJNEMAN et al. 1996 ; HOLLAND et al. 1997 ).

Despite the problem of an appropriate statistical threshold, methods for two-dimensional searches have been developed (HALEY and KNOTT 1992 ; CHASE et al. 1997 ; HOLLAND 1998 ; WANG et al. 1999 ). These methods will allow the detection of first-order interactions but not necessarily of higher-order interactions. The ability to detect higher-order interactions is desirable because such interactions have been documented (ALLARD 1988 , ALLARD 1999 ; DOEBLEY et al. 1995 ; ALONSO-BLANCO et al. 1998 ). Furthermore, the metabolic pathways that presumably underlie quantitative traits involve multiple interacting gene products (enzymes) and regulatory loci that could generate higher-order epistasis (MCMULLEN et al. 1998 ).

A possible resolution to both the search dimension problem and the higher-order interaction problem would be to perform a one-dimensional search for QTL that interact with the genetic background. In the simplest case of two loci, the alleles present at one locus form the genetic background for alleles present at the other locus. Thus, a first-order interaction between these two loci would cause each locus to interact with its genetic background. Higher-order epistasis may also cause each locus to interact with its genetic background. Higher-order epistasis may also cause QTL-by-genetic-background interaction (DOEBLEY et al. 1995 ; ALONSO-BLANCO et al. 1998 ). Detecting such interactions has become more important given improvements in methods to transfer alleles from one background to another using marker-assisted selection (HOSPITAL and CHARCOSSET 1997 ). Such efforts could be fruitless if the transferred alleles failed to affect the phenotype in a new genetic background as they had in their background of origin. CHARCOSSET et al. 1994 and REBAI et al. 1994 , REBAI et al. 1997 developed methods to detect QTL-by-genetic-background interaction that employ genotyped progeny resulting from a diallel mating design. These methods use least-squares tests applied directly to DNA marker data.

In this article, we develop a new method that maps within marker intervals the loci of greatest QTL-by-genetic-background interaction by simultaneous analysis of multiple related inbred-line crosses. Using maximum likelihood, the method contrasts models in which QTL allelic values are either nested within populations or are fixed over populations. High likelihood ratios between these models indicate QTL-by-genetic-background interaction. As a further benefit, the method allows statistical control of genetic noise due to other, nonfocal, QTL using multiple regression on marker data [Jansen's multiple-QTL model (MQM) method; JANSEN and STAM 1994]. We apply the method to simulated doubled-haploid populations derived from a diallel among three parents and evaluate its power to detect QTL that interact with either the polygenic background or with each other. Finally, we show how the method can be used in conjunction with standard two-locus models.

	METHODOLOGY

TOP ABSTRACT METHODOLOGY SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

Consider three doubled-haploid parents, denoted A, B, and C, a diallel of the three possible crosses between them, A x B, A x C, and B x C, and the doubled-haploid populations derived from each of these crosses. Using these populations, QTL may be mapped using the MQM procedure (JANSEN and STAM 1994 ) within each population. The linear model for such a procedure is

where y_ij is the phenotypic value of individual j in population i = 1 ... 3, µ_i is the mean for population i, and _ij N(0, ²_i) is a residual error for individual y_ij. The independent variables x^q_ij and x^c_ij depend on the genotype at the QTL analyzed and at marker cofactor c = 1 ... f_i, respectively, where f_i is the number of cofactors used in population i. These independent variables take on the values given in Table 1. The regression coefficient ₁ estimates ¹/₂(g_A1 - g_B1), where g_X1 is the genetic value of the homozygote of the allele derived from parent X at the QTL locus analyzed in the genetic background of population 1. Equivalently, ₁ estimates the substitution effect between the alleles derived from parent A and parent B in the genetic background of population 1 and under the assumption of no dominance. The regression coefficients ₂, ₃, and ß_ic have similar interpretations. Note that in (FULL), the allelic values g_Xi are nested within populations.

For missing QTL or marker information, JANSEN and STAM 1994 have shown that maximum-likelihood estimates for the parameters µ_i, _i, ß_ic, and _ij can be obtained within each population by an expectation-maximization procedure using weighted multiple regression. Given this procedure, the support level for the presence of a QTL at a map location, using information from all populations simultaneously, derives from the likelihood ratio between (FULL) and a no-QTL null model. (FULL) contains three more estimated parameters than the no-QTL model, that is, one QTL effect per population.

A reduction in the number of estimated parameters is possible if we assume the QTL does not interact with genetic background. In that case, we may consider the allelic value of a QTL fixed over populations and represent the value of the homozygote of the allele from parent X at the QTL locus analyzed as g_X, irrespective of genetic background (i.e., population) in which this genotype occurs. The regression coefficients ₁, ₂, and ₃ then respectively estimate ¹/₂(g_A - g_B), ¹/₂(g_A - g_C), and ¹/₂(g_B - g_C), which we denote ^*₁, ^*₂, and ^*₃. Using the identity (g_A - g_C) = (g_A - g_B) + (g_B - g_C), that is, ^*₂ = ^*₁ + ^*₃, we can develop a second model,

(REDUCED) where the variables x^q*_1j and x^q*_3j take values that cause ^*₁ and ^*₃ to be summed to estimate ^*₂ (Table 1). Other parameters are the same as in (FULL). (REDUCED) contains two more estimated parameters than the no-QTL model, that is, one parameter less than FULL.

Having defined (FULL) and (REDUCED), we see that a QTL-by-genetic-background interaction would cause a difference in their likelihoods. Thus, when using the models to fit a QTL at a locus, a large likelihood ratio between the models provides evidence of a QTL at that locus that interacts with genetic background, in other words, a QTL that interacts epistatically with other loci. In the presence of epistasis, a general relationship between the regression coefficients of (FULL) can be expressed

(1)

where d represents a deviation from the identity ^*₁ - ^*₂ + ^*₃ = 0 used to develop (REDUCED). Interaction between a QTL and genetic background would cause a nonzero deviation contrast: in the presence of epistasis between the locus under consideration and other loci, |d| > 0. CHARCOSSET et al. 1994 proposed a similar contrast but did not use it in developing their linear models.

Further analytical exploration of d reveals its importance in mapping epistatic QTL. We address two questions: first, what is the maximal value of d relative to the substitution effect of the QTL in which it occurs, and second, for a first-order interaction between two loci, locus 1 and locus 2, how are the corresponding values of d₁ and d₂ related?

To answer the first question, consider that in a doubled-haploid population, a QTL with allele-substitution effect induces an additive genetic variance of ². Assuming that ₁, ₂, and ₃ are not equal (as is indeed impossible if at least one is nonzero and epistasis is absent), we ask what the maximal value of d may be, given a mean QTL variance over the three populations of ²_QTL, that is, given

(2)

To use the Lagrange multiplier theorem to solve this constrained maximization problem, define the auxiliary function h(₁, ₂, ₃, ) = ₁ - ₂ + ₃ - (²₁ + ²₂ + ²₃ - 3²_QTL), differentiate it with respect to (₁, ₂, ₃, ), and set the partial derivatives to zero. One obtains 1 - 2₁ = 0, -1 - 2₂ = 0, 1 - 2₃ = 0, and ²₁ + ²₂ + ²₃ = 3²_QTL. Solving gives ₁ = -₂ = ₃ so that 3²₁ = 3²_QTL. Therefore the allele substitution effects (₁, ₂, ₃) = (_QTL, -_QTL, _QTL) yield a maximal d of 3_QTL, that is, a maximal ratio |d|/_QTL of three. Note that these are indeed odd substitution effects: fixing ₁ and ₃ to _QTL, ₂ would be 2_QTL in the absence of epistasis; instead it is -_QTL. We refer below to the ratio between d and _QTL as the "deviation ratio."

To analyze the relationship between the deviation contrasts d₁ and d₂ of two interacting loci, consider the vector of genetic values

where a_ij is the genetic value of a double homozygote at loci 1 and 2, subscript i takes the value 1, 2, and 3, when parent A, B, or C, respectively, confer the allele present at locus 1, and subscript j does likewise for locus 2 (the superscript ^T indicates transpose). With these genetic values, the allele substitution effects at locus 1, (₁₁, ₂₁, ₃₁)^T, and at locus 2, (₁₂, ₂₂, ₃₂)^T, are given by

(3)

Rearranging Equation 3 we find ₁₁ - ₂₁ + ₃₁ + ₁₂ - ₂₂ + ₃₂ = 0. In other words d₁ + d₂ = 0. This relationship indicates that if we assume that the deviation contrast found at a locus was caused by an epistatic interaction with only one other locus, then we can expect that at that other locus we may find a deviation contrast of similar magnitude but opposite sign. A single genome scan mapping loci with high deviation contrasts could therefore simultaneously identify loci affected by epistasis and suggest which other loci might be interacting with them.

	SIMULATIONS

TOP ABSTRACT METHODOLOGY SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

Genome generation:
The genome consisted of 10 chromosomes of 200 cM each. Two hundred marker loci were randomly distributed over the genome. Each marker locus was assumed to be triallelic, with allele frequencies, in order of abundance, of 0.5, 0.3, and 0.2. For each of three doubled-haploid parents denoted A, B, and C below, alleles were randomly assigned at each marker locus with the probability of each allele depending on its frequency. Given this procedure, the probability that two parents shared an allele at a given marker locus was (0.5)² + (0.3)² + (0.2)² = 0.38. Thus, for a given population, the expected number of segregating markers was 200*(1 - 0.38) = 124.

In a first set of simulations (set 1), we explored the power to detect QTL interacting with the polygenic background. Doubled-haploid populations from all three crosses (A x B, A x C, and B x C) were generated by doubling simulated gametes from F₁'s of each of the crosses. We assumed an isolated QTL to be present at the center of each of six of the chromosomes. Each parent carried a different QTL allele and QTL substitution effects (₁, ₂, ₃) were picked randomly but were subject to the constraints of Equation 1 and Equation 2 to obtain desired deviation contrasts and average QTL effects. On each of two chromosomes, the fraction of the phenotypic variance caused by additive QTL effects, averaged over the three populations, was h²_QTL = 0.16. On each of four other chromosomes, the fraction of the phenotypic variance, averaged over the populations, caused by additive QTL effects was h²_QTL = 0.07. We did not generate QTL-by-genetic-background interaction by simulating epistasis among the six QTL. Rather, QTL allelic values were assumed to be affected by other background loci specific to each population. Averaged over the populations, the fraction of the phenotypic variance due to these six QTL was h² = 0.60. The genetic value of each individual depended on the simulated allele substitution effects within its population and on which parent contributed each of its QTL alleles. We added a random normal deviate of variance (1 - h²) to each individual.

In a second set of simulations (set 2) we evaluated how best to combine QTL-by-background interaction information with standard two-locus models to detect first-order epistasis. We simulated two QTL, each at the center of a chromosome. Genetic values for the alleles derived from each parent were generated as given in Table 2, scaled so that the total genetic variance caused by the QTL together was a fraction H²_QTL = 0.10 or H²_QTL = 0.20 of the phenotypic variance. We added a random normal deviate of variance (1 - H²_QTL) to each individual. Generation of the genome and marker information was the same as above. Populations were of 100 doubled-haploid progeny. Note that, in the above, we use h²_QTL to denote the ratio of the additive genetic variance of a single QTL relative to the phenotypic variance and H²_QTL to denote the ratio of the total genetic variance of a pair of QTL relative to the phenotypic variance. Thus, for a pair of QTL generated as in Table 2, if H²_QTL = 0.20, then for each QTL, h²_QTL 0.07.

QTL analysis, set 1:
For each population separately, three markers were chosen per chromosome to be candidate cofactors in the MQM procedure (leading to 3 x 10 = 30 candidate cofactors over the entire genome). Segregating markers were chosen that allowed the most uniform chromosome coverage. Because the same markers did not necessarily segregate in each population, different sets of candidate cofactors resulted per population. Using all candidate cofactors we calculated a bias-adjusted residual variance for each population. These variances are unbiased (JANSEN 1994 ) and we used them for all further estimations on the three populations. Again, for each population separately, we used a backward elimination procedure to retain in the model only those cofactors that explained significant proportions of variance. To determine whether to retain a cofactor, we used a threshold T such that P(F_{1, d.f.} > T) = 0.02, with d.f. equaling the number of residual degrees of freedom in the all-cofactor model, that is, d.f. = [number of individuals in population - (number of candidate cofactors + 1)].

To locate QTL we then scanned the full genome in 5-cM steps. We first calculated the likelihood of the data under (FULL) in the absence of a QTL (L_NoQTL), but using all retained cofactors with the exception of cofactors within 25 cM of the putative QTL. The likelihood of the data under (FULL) was then calculated in the presence of a QTL (L_Full). Finally, we calculated the likelihood of the data under (REDUCED) (L_Reduced). From these likelihoods we calculated three likelihood ratios:

(4)

The first two statistics indicate the support level for the presence of a QTL using either (FULL) or (REDUCED). The third statistic increases as the level of QTL-by-genetic-background increases.

We ran simulations with population sizes of 50, 100, and 200 doubled-haploid individuals per population, and with deviation ratios of zero to three in one-half increments. To determine genome-wide significance thresholds for these three statistics we performed 3000 simulation runs on individuals generated without genetic variance. We chose as threshold the 95th percentile value of the genome-wide maxima of the statistics. The power of (FULL) or (REDUCED) to detect a QTL under specified conditions is the fraction of simulated QTL for which LR_Full or LR_Reduced exceeded their thresholds. Similarly, the power to detect a QTL-by-genetic-background interaction is the fraction of QTL for which LR_Deviation exceeded its threshold.

QTL analysis, set 2:
We used two methods to detect pairs of interacting QTL, one with information from the deviation contrast method and one without it. In the first, we assumed that population A x B was part of a diallel, as would be typical for a population in an applied plant breeding program. That diallel was analyzed as indicated above and results were used to determine which regions of the genome should be paired for analysis using the two-locus model for epistasis,

(BILOC_b) where y_ij, µ_i, _i, ß_ic, x^c_ij, and _ij have the same interpretation as in (FULL). The ¹_ix^q1_ij and ²_ix^q2_ij regressions account for main effects at two QTL loci being analyzed and the ¹²_ix^q1_ijx^q2_ij regression accounts for interaction between the loci. Statistically controlling for possible epistatic interactions between cofactors was not attempted (but see WANG et al. 1999 for a possible approach). The likelihood ratio between (BILOC_a) and (BILOC_b), LR_Epistasis, indicates the level of support for an epistatic interaction between the loci being analyzed. In a (BILOC) analysis, we scanned the two regions of interest on a 5-cM grid using all retained cofactors with the exception of cofactors within 25 cM of either putative QTL.

The overall analysis proceeded as follows. The maximal LR_Deviation (Equation 4) for each chromosome determined the position of maximal support for the presence of an epistatic QTL. If the sum of LR_Deviation for two chromosomes exceeded a threshold T, and the signs of the deviation contrast d calculated from Equation 1 were opposite, 90-cM regions surrounding the points of maximal support were analyzed in each population separately using (BILOC). As T increases, the number of locus pairs analyzed using (BILOC) declines and therefore the likelihood ratio necessary to ensure a 5% type I error rate also declines. We obtained thresholds for BILOC at different T from 1000 simulation runs on individuals generated without genetic variance.

In the second method, we assumed that the population A x B was analyzed for epistatic QTL without the benefit of information from a diallel analysis. (BILOC) was therefore applied over the whole genome leading to a much greater number of tests. We report the power to detect both QTL of an interacting pair, mapped to within 25 cM of their simulated positions. For both methods, we used significance thresholds for a 5% type I error rate on a population-wise basis. If several populations were analyzed, a further Bonferroni correction would be applied.

	RESULTS

TOP ABSTRACT METHODOLOGY SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

QTL-by-background interaction:
Analysis results from a single simulation run in Fig 1 illustrate the type of output from the method. The likelihood ratios between (FULL) and a no-QTL model show support for the presence of QTL at their simulated locations, irrespective of whether the QTL is epistatic to others. In contrast, the likelihood ratios between (FULL) and (REDUCED) specifically identify QTL that are epistatic to others segregating in the background of the populations. Using the regression coefficients estimated from (FULL), a deviation contrast can be calculated using Equation 1. Consistent with theory, the estimated deviation contrasts for two QTL involved in first-order epistatic interaction are of similar magnitude but opposite sign. At the loci of maximal LR_Deviation, the deviation ratios estimated for the two QTL were 2.15 and -2.09, values that exceed in magnitude the ratio of 1.73 = expected given the genetic values of Table 2.

View larger version (20K): In this window In a new window Download PPT slide   Figure 1. Diallel analysis results. The top shows the likelihood ratios between (FULL) and a no-QTL model and between (FULL) and (REDUCED) (as described in the text). The bottom shows estimates of the deviation contrast. QTL were simulated at the center of the first three chromosomes (triangles), and none were simulated on the fourth chromosome. The first two QTL were simulated to interact using genetic values given in Table 2, and the third QTL did not interact. Values were scaled to obtain a phenotypic variance of 100 with h2QTL = 0.07 for all QTL. These results derive from an analysis of a diallel of three doubled-haploid populations of 100 individuals each.

Empirical powers to detect QTL with nonnull deviation contrasts are graphed in Fig 2. The magnitude of the deviation contrast increases with both the average variation caused by the QTL and with the deviation ratio. For a given h²_QTL, the power to detect a deviation contrast increases with the deviation ratio; conversely, for a given deviation ratio, the power to detect a deviation contrast increases with h²_QTL. Even though the QTL simulated accounted for a fairly large fraction of the phenotypic variance, large population sizes were necessary to obtain adequate power to detect them, unless their deviation ratio was high (>2). What deviation ratio values might occur in real circumstances is an empirical problem that has not been addressed. For the simple configuration of epistatic effects given in Table 2, the deviation ratio is = 1.73. To get a feel for what deviation ratios might arise from interacting loci under other conditions, consider all double-homozygote genetic values a_ij (Equation 3) as independently and identically distributed with a_ij N(0, ²). In that case, manipulation of Equation 3 shows that the deviation contrast d N(0, [³/₈]²) and _i from (FULL) is distributed N(0, ²) so that E(²_QTL) = var(_i) = ². Analytically we find is 0.98. The expected deviation ratio, E(), is more difficult to derive analytically; by simulation we found a mean deviation ratio of 1.16. Given the desire to detect QTL with deviation ratios between 1 and 2, population sizes of 200 individuals per cross in the diallel would seem necessary.

View larger version (22K): In this window In a new window Download PPT slide   Figure 2. Power to detect nonzero deviation contrasts for QTL of different effects with diallels formed of populations of three sizes, as affected by the deviation ratio (from QTL analysis set 1, see text).

Note that two forms of type I error may occur in QTL-by-genetic-background mapping: a significant deviation contrast may be declared either in the absence of true genetic variation or in the presence of a QTL that does not interact with genetic background. We set significance thresholds using simulations of genomes without genetic variance and therefore obtained = 5% for the first form of type I error. Fig 2 shows the second form of type I error rate on a per QTL basis as the "detection power" when the deviation ratio is zero. This rate depended on the variation caused by the QTL and on the population size (Fig 2). Because the rate is given on a per QTL basis, it can be <5%, for example, when h²_QTL = 0.07 and population size = 100. The genome-wide error rate would depend on the number of segregating nonepistatic QTL. Even on a per QTL basis, however, for QTL of large effect, we found second form type I error rates >5%. CHARCOSSET et al. 1994 have pointed out that in the presence of a QTL with simple additive effects a nonnull deviation contrast can be generated by heterogeneous recombination frequencies among the populations between markers and the QTL. In this study, we did not simulate heterogeneous recombination frequencies, but sampling variation in the number of recombinants between a marker and a QTL, irrespective of the expected number of recombinants, could cause a similar nonnull deviation contrast. With more progeny per population such sampling variation effects should decrease, but, in contrast to this prediction, we found the highest type I error rate with the largest population size (Fig 2). As an alternate explanation, consider that because the same markers do not segregate in all populations, marker information content at the simulated QTL position will not be equal over the populations. Low information content at that position in population i will cause a downward bias in the estimation of _i. If such a bias does not occur in the other populations a nonnull deviation contrast will also result. This mechanism therefore seems better able to explain the increased level of type I error observed in the presence of QTL with large additive effects.

High values for either LR_Full or LR_Reduced should indicate the presence of a QTL segregating within the diallel. In effect the difference between (FULL) and (REDUCED) is that QTL allele values are nested within populations in (FULL) while they are considered fixed in (REDUCED). Because (REDUCED) estimates one parameter less than (FULL) it may be more powerful to detect a segregating QTL if its assumption of fixed allele effects is correct. Based on our simulations, (REDUCED) is indeed more powerful than (FULL) for QTL with a deviation ratio lower than one (Fig 3). The gain in power, however, appeared relatively small, and for QTL with larger deviation ratios, a possibly large loss in power occurred. In contrast, (FULL) was impervious to changes in the deviation ratio. Methods have been described in the literature that model nested QTL effects over multiple populations (XIE et al. 1998 ; XU 1998 ). Our simulation results indicate that for methods that seek to gain power by assuming fixed allele effects over multiple populations, a consequence of epistasis might precisely be loss of power. Methods applied to complex human or animal pedigrees that combine segregation and linkage analysis to map QTL (e.g., LIN 1999 ) often assume fixed allele effects because the sizes of the families they analyze are small. This assumption may also be made for plant populations. In either case, epistasis may reduce the QTL detection power of these methods.

View larger version (17K): In this window In a new window Download PPT slide   Figure 3. Power to detect segregating QTL per se. Diamonds and squares are for QTL with h2QTL = 0.16 and 0.07, respectively.

First-order QTL interaction:
If a population is embedded within a diallel, information from a prior analysis using (FULL) and (REDUCED) will provide evidence as to which regions of the genome carry epistatic QTL. That information, in turn, may be used to decrease the number of pairwise tests that must be carried out to detect first-order epistasis using (BILOC). Under a null model containing no QTL, reducing the number of tests performed should also reduce the level of highest excursion of (BILOC)'s likelihood ratio. In consequence, a lower threshold could be used to declare significant first-order epistasis and higher power should result. We investigated the effectiveness of such a two-step procedure by simulation.

In a two-step procedure, for an interacting pair of QTL to be detected, they must both pass both steps. Statistical thresholds for both steps must be chosen to attain the desired overall type I error rate (here, = 5%). High stringency in step 1 reduces the power to pass step 1 but also drastically reduces the number of tests performed in step 2, thereby increasing the power to pass step 2. A trade-off between the powers to pass each step results (Fig 4A). Under the conditions simulated, the stringency for step 1 that is optimal for obtaining the greatest overall power occurs for a minimal sum of the deviation likelihood ratios ( LR_Deviation) at two loci under consideration between 8 and 13 (Fig 4B). The rationale for using LR_Deviation in step 1 derives from the result that even for two loci that are simulated to be interacting, the test statistics associated with each locus were independent. By simulation we found correlations between LR_Deviation statistics for two loci of r = 0.01 and r = 0.03 for H²_QTL = 0.20 and 0.10, respectively. That is, for first-order interacting QTL, LR_Deviation is distributed as a noncentral ² with the same noncentral parameter but with independent draws for each locus. Presumably for QTL present on different chromosomes, test statistics are independently affected by errors associated with sampling of recombination events and with microenvironment. Because of the test statistic independence, for a given type I error rate in passing step 1, the power to pass that step is higher using LR_Deviation than requiring each LR_Deviation to exceed a minimal threshold.

View larger version (21K): In this window In a new window Download PPT slide   Figure 4. (a) Power to pass each step in a two-step procedure to detect first-order epistasis, as affected by the threshold used to pass step 1, the sum of LRDeviation for the two loci. Diamonds and squares are for QTL with H2QTL = 0.20 and 0.10, respectively. (b) Overall power to detect first-order epistasis, as affected by the threshold used to pass step 1.

With this two-step procedure the powers obtained to pass both steps and map both QTL to within 25 cM of their simulated positions were 46 and 11% for QTL pairs with H²_QTL = 0.20 and 0.10, respectively (Fig 4B). Those powers compare favorably to the powers to detect the epistatic pairs in the absence of a prior QTL-by-background analysis, which were 24 and 5% for QTL pairs with H²_QTL = 0.20 and 0.10, respectively.

	DISCUSSION

TOP ABSTRACT METHODOLOGY SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

Combining across- and within-population information:
The two-step procedure that we have explored constitutes a method to combine information obtained across populations (QTL-by-genetic-background interaction) and within populations (QTL-by-QTL interaction). A possible drawback of this combination method is that interacting QTL will not be found unless test statistics for both steps ( LR_Deviation and LR_Epistasis) exceed minimal thresholds. Low correlations found by simulation between LR_Deviation and LR_Epistasis statistics (r = 0.21 and r = 0.10 for H²_QTL = 0.20 and 0.10, respectively) indeed indicate that passing one step is a poor predictor of passing the next step. A rationale we invoked above to justify the use of a minimum for LR_Deviation as the criterion for step 1 rather than separate minima for each LR_Deviation may therefore apply to these two statistics. That is, using a compound statistic ( LR_Deviation + LR_Epistasis) may result in higher power using each separately. In simulations using the compound statistic, we found powers to detect interacting QTL pairs of 61 and 13% for H²_QTL = 0.20 and 0.10, respectively. These powers were obtained despite the fact that full-genome two-dimensional searches were performed. In general the two-step and the compound-statistic procedures can be contrasted as alternate methods of combining information from different sources. In one method, across-population information focuses the within-population search, in the other method both information sources contribute to a joint test statistic.

Thus far, we have applied across- and within-population models in separate analyses. As a further refinement, however, it would be possible to combine the linear models used in each analysis. A combined-models approach would test a two-locus extension of (REDUCED) against a two-locus extension of (FULL) combined with (BILOC) interaction regression parameters within each population [call these models (REDUCED)² and (FULL)², respectively]. In the case of a three-population diallel, (FULL)² would differ from (REDUCED)² by five parameters: one parameter per locus for genetic background interaction and one parameter per population in the diallel for locus-by-locus interaction. Relative to performing separate analyses, the combined-model analysis would gain power by estimating nuisance parameters (the mean and cofactor parameters) only once. It would also more powerfully detect epistatic interactions when they were manifest in all three populations rather than in only a single population as we simulated (Table 2). Finally, we note that while the compound statistic appears to increase the power to detect epistatic QTL, it also creates interpretation problems. If a QTL pair was found to be significant, further analysis would be required to determine what sources contributed to that significance.

We also note that we have only discussed combining information in the ideal situation where two loci interact with each other but with no further loci. When a locus interacts with more than one other locus, the simple equation linking deviation ratios among two interacting loci, d₁ + d₂ = 0, will no longer hold. For three loci, d₁ + d₂ + d₃ = 0 will hold if no three-way interaction occurs among loci. This zero sum, however, does not imply any simple pairwise relationship between, say, d₁ and d₂. As the complexity of interactive sets of QTL increases, therefore, QTL-by-genetic-background interaction information will likely become less useful for the detection of first-order interactions.

Generalization to other population structures:
When QTL affecting the same trait are mapped in several populations they are generally not found in the same locations over the populations (e.g., BRUMMER et al. 1997 ; ORF et al. 1999 ). BEAVIS 1994 pointed out a number of sources of difference between different QTL experiments that reduce reproducibility. First, when the power to detect a QTL is less than one, it may be detected in some experiments but not others. A QTL's detection is enhanced or reduced by colinearity with error residuals caused by random measurement variation and by colinearity with other QTL caused by random sampling of particular genotypes among segregating progeny. With proper randomization, these sampling variations will not be repeated from experiment to experiment. Second, a QTL will be detected only in populations where it segregates: even in the absence of known coancestry, some parents may carry identical-in-state alleles. Third, QTL-by-environment interaction may affect QTL substitution effects in experiments conducted in different environments. Finally, QTL-by-genetic-background interaction may affect QTL substitution effects in experiments conducted with different populations. This plethora of possible causes leading to discrepancies between QTL mapping results makes the design of experiments to support/reject the QTL-by-genetic-background interaction hypothesis challenging. Clearly, to attribute observed differences in QTL effects to that hypothesis requires mating designs that bring single alleles of known origin into different genetic backgrounds. The diallel mating design achieves this task, but other designs are possible.

For populations produced from matings among N inbred lines a maximum of N alleles may segregate at a given locus. For a (REDUCED) model assuming QTL allelic values g_i (i = 1 ... N) that are fixed over populations, N - 1 values may be estimated with the constraint ^N_i=1g_i = 0 (REBAI et al. 1997 ). If crosses are made to produce P populations, a (FULL) model assuming QTL allelic values nested within populations will estimate P QTL allele substitution effects. For P > N - 1, the likelihood ratio between (FULL) and (REDUCED) will be distributed as ² with P - N + 1 d.f., if there is no QTL-by-genetic-background interaction. We have presented the simple case where N = 3 and P = 3, which derives from a three-parent diallel, but we briefly describe other possibilities. In a North Carolina design II with two inbred dams (A and B) and sires (C and D), N = 4 and P = 4, with populations (1) A x C, (2) A x D, (3) B x C, and (4) B x D. Then (FULL) estimates four substitution effects (g_A1–g_C1), (g_A2–g_D2), (g_B3–g_C3), and (g_B4–g_D4); (REDUCED) arises from the algebraic identity (g_A–g_D) = (g_A–g_C) + (g_B–g_D) - (g_B–g_C), which is obtained under the assumption of no QTL-by-genetic-background interaction.

As a final example consider two inbred parents (A and B) and two recombinant-inbred-line or doubled-haploid populations, one produced from the F₂ generation and the other produced from a BC₁ generation. Here N = 2 and P = 2, and the populations have different genetic backgrounds because in the first, segregating alleles come from each parent equally, while in the second, segregating alleles come from one parent 75% of the time. BEAVIS et al. 1994 noted such a background difference as a possible factor explaining differences between their results and those of STUBER et al. 1992 for QTL affecting yield in maize (Zea mays L.). In this case, (FULL) estimates two substitution effects (g_A1–g_B1) and (g_A2–g_B2) while the algebraic identity leading to (REDUCED) is particularly simple: (g_A1–g_B1) = (g_A2–g_B2) = (g_A–g_B). This example is a special case of the mechanism discussed by GOODNIGHT 1988 whereby founder events convert epistasis into additivity: partial fixation at loci "will tend to incorporate their epistatic interaction into additive genetic variance." Thus, (g_A1–g_B1) and (g_A2–g_B2) may differ if an interaction between loci in the F₂-derived population is converted to a main effect in the BC₁-derived population.

As illustrated in Table 2, a similar mechanism can lead to QTL-by-genetic-background interaction when more than two parents are used but the epistatically interacting QTL they carry are only biallelic. When several populations are derived from such parents, one may expect that in some populations some but not all of the interacting QTL will be segregating. In such cases, the epistatic variance will be converted to additive variance at the segregating loci, leading to detectable deviation contrasts. For crop species that have gone through major bottlenecks in the course of their domestication, such as soybean [Glycine max (L.) Merrill.] in the United States (GIZLICE et al. 1993 ), this mechanism would appear as a simple and possibly frequent explanation for QTL-by-genetic-background interaction.

Implications for marker-assisted selection:
Research reports and theory surrounding marker-assisted selection often side-step the issue of QTL-by-genetic-background interaction. For example, HOSPITAL and CHARCOSSET 1997 discuss optimal marker-assisted QTL introgression approaches "provided the expression of the gene is not reduced in the recipient genomic background." However, within a marker-assisted selection program in which progeny from different crosses are routinely genotyped, the methods described would make it possible to systematically uncover QTL that interact with genetic background. Within such a context, the information obtained could refine genotype-based selection indices, either by avoiding QTL that interact with genetic background or by improving the prediction of the genetic value conferred by specific QTL combinations.

	ACKNOWLEDGMENTS

We thank two anonymous reviewers for suggestions and improvements. A National Science Foundation North Atlantic Treaty Organization Postdoctoral Fellowship in Science and Engineering, DGE-9902466, supported J.-L.J.'s work on this research.

Manuscript received March 26, 2000; Accepted for publication October 9, 2000.

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