Introduction

Gibberella ear rot (GER) caused by Fusarium graminearum is a major disease of maize (Zea mays L.) in Europe and Canada. It reduces the yield and contaminates the grain with mycotoxins, in particular with deoxynivalenol (DON). Breeding resistant cultivars is the most effective approach for combatting the disease due to limited effect of agronomic practices and fungicides (Martin et al. 2011). For genetic improvement of GER, Martin et al. (2012) recommended marker-assisted selection, but this presupposes accurate estimates of the chromosomal location and effect of the underlying quantitative trait loci (QTL).

Linkage mapping with individual bi-parental families derived from divergent parent lines has become routine for dissecting the genetic architecture of complex traits in crops (Holland 2007), although it has certain drawbacks. (i) The mapping population originates from two parent lines and, therefore, represents only a small cross section of the breeding germplasm (Xu 1998; Liu and Zeng 2000). (ii) Mapping results from one family are often not transferable to other families (Beavis 1998; Melchinger et al. 1998), because the expression of QTL depends on its presence and in addition can be influenced by the genetic background. To overcome these limitations, multi-family QTL mapping has been proposed to detect QTL jointly from multiple bi-parental families (Jansen et al. 2003; Blanc et al. 2006; Bink et al. 2012). These populations can be either families routinely generated in practical breeding programs (Bardol et al. 2013) or created using special mating designs, e.g., the diallel design (Blanc et al. 2006), nested association mapping population (NAM, Yu et al. 2008) or multi-parent advanced generation intercross (MAGIC, Huang et al. 2015). The main difference between bi-parental and multi-family QTL mapping concerns the set of QTL that segregate in one versus several populations, which enables testing for QTL × genetic background interactions in the latter case (Blanc et al. 2006). For a given sample size, the former approach has generally higher power to detect rare QTL with large effects, segregating in only one or a small number of families, while the latter approach has higher chances to detect common QTL with small effects shared by a large number of families (Li et al. 2011; Ogut et al. 2015).

Four main categories of biometrical models have been developed for multi-family QTL mapping in plants, which differ in their assumptions about the QTL effects: (1) effects are specific to each population (e.g., FULL model in Jannink and Jansen 2001; disconnected model in Blanc et al. 2006), (2) effects of parental alleles are identical over populations (e.g., REDUCED model in Jannink and Jansen 2001; connected model in Blanc et al. 2006), (3) identical by descent (IBD) segments shared by parents have the same alleles, the effects of which are identically expressed in different populations (e.g., HaploMQM− model in Jansen et al. 2003; LDLA model in Bardol et al. 2013 and Giraud et al. 2014), (4) identical by state (IBS) segments among parents harbor identical alleles with effects consistent across genetic backgrounds (Yu et al. 2008; LDLA-1-marker model in Bardol et al. 2013; Model-B in Würschum et al. 2012). From category (1) to (4), the number of alleles at QTL decreases, resulting in a reduced number of parameters to be estimated. Thus, the power of QTL detection may increase and estimation error of QTL effects may decrease (Rebai and Goffinet 1993, 2000), if a common set of QTL can be assumed. In experimental studies, the performance ranking of these models varied among populations of equal size and among traits (Blanc et al. 2006; Steinhoff et al. 2011; Bardol et al. 2013; Giraud et al. 2014). Therefore, further research is warranted to compare these models and provide guidance for their choice.

In recent years, genomic prediction of breeding values of untested genotypes with genome-wide markers has received considerable interest by breeders due to a dramatic reduction in the costs of genotyping (Meuwissen et al. 2001; Jannink et al. 2010). One important question in genomic prediction, and generally in marker-based prediction, is how to design the training set (TS) for achieving a high prediction accuracy. Major factors identified are the sample size and number of families in the TS and their relatedness to the validation set (VS, Riedelsheimer et al. 2013; Lehermeier et al. 2014). Multi-family QTL mapping offers the possibility to unveil the genetic basis of prediction accuracy in genomic prediction with different composition of the TS.

In our study, we compared five models of QTL mapping with multiple crosses for QTL detection and QTL-based performance prediction evaluated with cross-validation. Besides additive effects, we investigated digenic epistasis and QTL × genetic background interactions. Moreover, we examined several scenarios of composition of the TS for QTL-based prediction. Our analyses were based on a total of 639 doubled-haploid (DH) lines derived from five interconnected crosses genotyped with 56 k SNP and 363 SSR markers and phenotyped for relevant GER resistance traits (DON concentration, GER severity) and a phenological trait (days to silking) in maize.

Materials and methods

Plant material and field trials

Four flint maize inbred lines developed by the University of Hohenheim were used as parents. They represent elite breeding materials of Central Europe displaying good combining ability for grain yield in crosses with dent lines. Pedigree-based coefficients of coancestry among them range between 0.05 and 0.23 (Martin et al. 2011). Regarding resistance against Fusarium graminearum, parent line UH006 is highly resistant, UH007 is moderately resistant, and UH009 and D152 are highly susceptible (Bolduan et al. 2009). The four parent lines, herein denoted as R1, R2, S1 and S2, respectively, were crossed in an incomplete half-diallel design (Fig. S1) and the F1 crosses were used for developing five interconnected families of DH lines ranging in size from 43 to 204 (Table 1). The DH lines were developed by applying the in vivo haploid method detailed by Prigge and Melchinger (2012).

Table 1 Family size (N), pedigree-based coefficient of coancestry among parent lines (f), family means (\(\bar{X}\)), and estimates of genotypic variance (\(\sigma_{G}^{2}\)), heritability (h 2), and genetic correlations (r g ) of deoxynivalenol concentration (DON), Gibberella ear rot severity (GER), and days to silking (DS) for five families of doubled-haploid (DH) lines
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All 639 DH lines and their parental lines were tested at two locations in Southwest Germany, namely Stuttgart-Hohenheim (48°43′12″ N, 9°10′48″ E) and Eckartsweier (48°31′12″ N, 7°52′12″ E) in 2 years (2008 and 2009). In each environment (year × location combination), four 10 × 20 α designs, each with two replicates, were grown adjacent to each other as detailed by Martin et al. (2011). The experimental units were 3 m single row plots spaced 0.75 m apart with 20 plants.

Artificial inoculation with an aggressive isolate of F. graminearum (IFA66) was conducted as detailed by Bolduan et al. (2009). Briefly, the inoculum (1 ml, 100,000 conidia) was injected 5–6 days after silk emergence into the silk channels of the primary ears of plants at a similar developmental stage. Six and eight plants per plot were inoculated in 2008 and 2009, respectively. At physiological maturity, the inoculated ears were manually dehusked and visually rated for GER severity from 0 to 100 %. After harvest, the ears were dried to a moisture content of approximately 14 %. DON concentration of each plot was measured with near-infrared spectroscopy (NIRS) as described in detail elsewhere (Martin et al. 2011; Miedaner et al. 2015). Moreover, the number of days to silking (DS) was recorded on a plot basis as the number of days from sowing to silk emergence of the primary ears in 50 % of the plants.

Phenotypic data analysis

Data for GER severity and DON concentration were transformed using the arcsine square root function and the natural logarithm function, respectively, to reduce heterogeneity of variances and approximate the assumption of a Gaussian distribution. After calculation of adjusted entry means in each environment, variance components across environments and entry-mean based heritabilities (h 2) were estimated for each family as detailed by Martin et al. (2011). Genotypic correlation coefficients (r g ) between traits in each family and their standard errors were calculated according to Mode and Robinson (1959). Statistical analyses of the phenotypic data were performed with software PLABSTAT (Utz 2005).

Marker screening and consensus map construction

The Illumina MaizeSNP50 array comprising 56,110 SNP markers (Ganal et al. 2011) was applied for genotyping all DH lines and the four parent lines. In each family, polymorphic SNP markers were selected, if (i) their physical map position was known and (ii) their minor-allele frequency and average call frequency exceeded 0.05 and 0.80, respectively. DH lines with more than 5 % heterozygous SNPs or an average call rate smaller than 0.80 were excluded from further analyses. In addition, the DH lines were genotyped by 123 (R1R2), 106 (R1S1), 129 (R1S2), 113 (R2S1) and 121 (R2S2) polymorphic SSRs as described in detail by Martin et al. (2011).

In each family, markers were grouped into 10 linkage groups with software MSTmap (Wu et al. 2008). Afterwards, a consensus map of each linkage group across all families was constructed with software Carthagene (de Givery et al. 2005) in the following steps. Step 1: merge the dataset of all families with the dsmergen command and then combine each pair of strongly correlated markers (2-point LOD score ≥3) into one locus. Step 2: build a framework map with a limited number of markers, but having a reliable order (buildfw 10 10 {} 0). Step 3: incorporate additional markers into the framework map using the command buildfw keepThres AddThres {“marker order of the framework map”} 0, where the values of keepThres and AddThres are high (≥3), but lower than in step 2. Notably, questionable markers were removed, if they caused considerable inconsistency in the marker order between the genetic and physical map or resulted in an excessive expansion of the linkage group. Step 4: repeat step 3 to add as many markers as possible while ensuring a robust marker order (keepThres and AddThres ≥3).

Clustering of parental alleles at each locus

The alleles of the parent lines at each locus of the consensus map were clustered into ancestral classes based on similarity scores between each pair of lines, which were calculated with a sliding window approach implemented in the R package “clusthaplo” (Leroux et al. 2014). Briefly, for a locus centered at a window of certain size, the similarity score between one pair of lines was computed as weighted measure of the number of IBS loci within that window, the length of the longest common genome segment of that window and, if the marker density in that window was low, the estimated genome-wide relatedness between the two lines. In this study, the first two weights were chosen from an exponential and uniform distribution, respectively. Afterwards, a Hidden Markov Model overcoming the threshold setting issue was applied to cluster the parental alleles at each locus. Clusters were firstly generated based on a set of genome-wide 35 k polymorphic SNP markers on the physical map, the positions of which were transformed into a centiMorgan scale by chromosome-wise ratios calculated from the length of the consensus map of each chromosome over its physical map length (Huang et al. 2011), and secondly extracted for the shared markers between the physical and genetic consensus map.

The window size to be used for clustering was determined in two steps. First, we investigated the LD decay along each chromosome. The LD between each pair of markers on each chromosome was calculated as r 2 (Hill and Robertson 1968) on the basis of 41 flint inbred lines including our four parents. The decay of r 2 on every chromosome was estimated according to Hill and Weir (1988). Second, to investigate how sensitive the clustering is with respect to the choice of the window size, five different window sizes ranging from 5 to 25 cM in steps of 5 cM were examined. The clustering results of each chromosome were evaluated with respect to (i) the average number of clustered ancestor alleles, (ii) the number of cluster changes defined as the change of at least one haplotype in the clustering result from locus to locus (Leroux et al. 2014), and (iii) the Pearson correlation coefficient between the modified Rogers’ distance among the four parental lines (Reif et al. 2005) and their clustering-based dissimilarity, calculated as the proportion of loci not sharing identical clusters. Finally, windows of size 20 and 10 cM were applied for chromosome 7 and the other chromosomes, respectively, on the basis of our findings and following the recommendation of Giraud et al. (2014) to choose as window size twice the genetic distance corresponding to r 2 = 0.2.

Detection of QTL with main effects and study of epistatic effects

Five biometric models, differing in the assumption about the number and effects of alleles at a QTL, were utilized to detect QTL with additive effects for every trait (Table 2), three based on linkage analysis (Model 1–3) and two incorporating LD and linkage information (Model 4 and 5). A detailed description of these models is given in supplementary materials.

Table 2 Overview of the five biometric models used for detecting QTL with additive effects in multiple families
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Calculations for all five models were performed using iterative composite interval mapping (iQTLm, Charcosset et al. 2000) implemented in the MCQTL_LD software (Jourjon et al. 2005). A genome scan was performed for every marker and/or every 2 cM position, using flanking markers to infer the genotype at this position as described by Haley and Knott (1992), with multiple regression. Thresholds for declaring a putative QTL were determined by a permutation test using 1000 permutations to limit the genome-wise Type I error to 10 % for the joint analysis and 2 % for the single-family analysis to make the two comparable according to the Bonferroni correction (Blanc et al. 2006). Cofactors were selected by forward selection, restricting the distance between two adjacent cofactors to be greater than 20 cM. Support intervals of QTL positions were determined on the basis of 1-LOD unit drop. The proportion of genotypic variance (p G ) explained by each QTL (all QTL) was calculated as \(p_{G} \; = \;R_{\text{adj}}^{2} /h^{2}\), where \(R_{\text{adj}}^{2} = 1 - \frac{{RSS_{\text{full}} /df_{\text{full}} }}{{RSS_{\text{red}} /df_{\text{red}} }}\) and RSS full and RSS red refer to the residual sum of squares of the full model including the tested QTL (all QTL) and of the reduced model without the tested QTL (all QTL), respectively, and \(df_{\text{full}}\) and \(df_{\text{red}}\) refer to the degrees of freedom of the residual error in the full and reduced model, respectively (Giraud et al. 2014). Note that in the joint analyses, a family effect was included in both the full and reduced models and h 2 was calculated as the average heritability across all families. QTL detected with different models were considered different if their 1-LOD drop support intervals did not overlap.

QTL detected with Model 3 were further tested for digenic epistasis (Model 6) and QTL × genetic background (family) interaction (Model 7), applying the models detailed in Blanc et al. (2006) and our supplementary materials. Calculations for Model 6 were conducted with the “simple” method implemented in MCQTL_LD software (Jourjon et al. 2005). Calculations for Model 7 were conducted with a self-made program within the R environment (R Development Core Team 2008). The Type I error of both models was confined to 10 %. Estimates of \(R_{\text{adj}}^{2}\) and p G were calculated for each significant interaction as described above for additive effects, except that the difference between the full and reduced model refers to the interaction term.

Cross-validation

Two cross-validation schemes, detailed below, were applied to (i) evaluate and compare Model 1–5 with two sample sizes of the TS and (ii) investigate with the connected model (Model 3), how the composition of the TS affects the prediction accuracy for the validation set (VS). Briefly, in each cross-validation run, QTL detection, localization and estimation of genetic effects were conducted in the TS and validation of the QTL results was performed in the VS as detailed by Utz et al. (2000). A Type I error of 10 % was applied to all models. Calculations were replicated 200 times with different random samples to obtain robust estimates using the R package “cvMCQTL” (Foiada et al. 2015) and our own extensions. To reduce the computation time of cross-validation, highly correlated markers from the dense genetic consensus map were removed by retaining only one marker per cM, which was polymorphic in most of the families compared with other markers within that 1 cM bin.

In Scheme 1, the same number of lines was sampled randomly without replacement from three completely connected families (R1R2, R1S1, and R2S1) to form the TS (Fig. S2). The VS was built in the same way from the remaining lines. Two TS sizes with N TS = 81 (27 from each family) and 180 (60 from each family) and corresponding VS sizes with N VS = 48 (16 from each family, except 15 for Model 1 in the case of R2S1 due to its small family size) and 108 (36 from each family) were employed. To enable direct comparison of prediction accuracies of the five models, the within-family prediction accuracy \(r_{\text{VS}}\) was calculated for each family of the VS as the Pearson correlation between observed and predicted performance divided by the square root of h 2. If no QTL was identified in the TS, the prediction accuracy was set to zero. The prediction accuracy \(r_{\text{VS}}\) was averaged over all cross-validation runs and the corresponding standard deviation was determined. Moreover, the frequency of QTL detected in each 20 cM bin along the genome was recorded across cross-validation runs.

For Scheme 2, seven scenarios using TS of equal size composed of DH lines from a maximum of four families (R2S2 was excluded due to its small sample size) were investigated differing in (i) the number of families from which the DH lines were randomly sampled without replacement and (ii) the relatedness between the DH lines in the TS and VS ranging from full sib (F) and half sib (H) to unrelated (U) lines, ignoring relatedness among the four parent lines. The VS always comprised only one family. The scenarios were coded by the number of families in the TS and adding letters (F, H, U) reflecting the relationship of the genotypes in the TS to the genotypes in the VS (Table 3). If the TS included more than one family, the same number of DH lines was sampled from each family. Note that in all scenarios, both parents of the VS were parents of at least one family in the TS so that allele effects could be estimated with Model 3. The size of the TS varied from 60 to 280 in steps of 20. As with Scheme 1, \(r_{\text{VS}}\) was averaged over all cross-validation runs.

Table 3 Composition of the training set (TS) and validation set (VS, in bold face letters) under seven scenarios and average coefficient of coancestry (\(\bar{f}\)) between the TS and VS
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Results

Phenotypic data analysis and consensus map construction

R1R2 had the lowest family means (\(\bar{X}\)) of DON and GER, followed by R2S1 (Table 1). The differences in \(\bar{X}\) of DS between families were small. Significant (P < 0.01) genotypic variances were observed for all traits in all families, with estimates being smallest for R1S1 and largest for R2S1. Heritabilities were generally high with an average of 0.78 for DON and GER and 0.90 for DS, and small differences among families. Genetic correlations were extremely tight between DON and GER (r g  ≥ 0.96) and moderately negative (−0.66 ≤ r g  ≤ −0.21) for DS with DON or GER for all families except R2S2.

In total, 17,800 markers passed the quality check and were employed to construct the consensus map. It had a total length of 1854 cM and 2472 loci made up of 14,421 markers, out of which strongly correlated markers were combined into one locus before determining the marker order (Fig. S3). The genetic distance between adjacent loci ranged from 0 cM to a maximum of 9.4 cM across all chromosomes. The order of the markers on the physical map (Schnable et al. 2009), our consensus genetic map, and the SSR-based genetic map reported by Martin et al. (2012) showed collinearity across the entire genome with minor deviations in a few regions (Fig. S3). Map distances differed occasionally between the consensus map and the SSR-based genetic map, which had a total length of 2060 cM. Some regions on the physical map of each chromosome were devoid of markers. For instance, a huge segment of 75 Mb on chromosome 3, comprising the centromere, was completely lacking any markers, but the markers flanking this segment had a genetic map distance less than 5 cM, indicating extreme suppression of recombination.

Choice of window size and clustering of parental alleles

In the set of 41 flint lines used for examining the decay in LD between the markers on a chromosome, the threshold r 2 = 0.2 was reached at 9.6 cM for chromosome 7 and between 3.2 cM and 5.3 cM for the other chromosomes (Fig. S4). The average number of ancestral alleles at each locus obtained from clustering with five different window sizes was similar for all chromosomes (Fig. S5). The only exception was window size 5 cM, which resulted in a higher number of ancestral alleles on chromosome 6 than the other window sizes. Concerning the number of cluster changes on each chromosome, 5 cM deviated notably from the other window sizes. Nevertheless, for all window sizes, the correlation coefficients between the modified Rogers’ distances among the four parent lines and their clustering-based dissimilarities were above 0.97 for all chromosomes. Therefore, we chose windows of size 20 cM for chromosome 7 and 10 cM for the other chromosomes. Clustering of the four parental alleles at each locus to ancestral allele classes varied across loci and chromosomes (Fig. 1a, b). On average, centromeric regions had fewer clustered ancestral alleles than telomeric regions. Generally, parent lines with a higher coefficient of coancestry shared more often the same ancestral allele than others.

Fig. 1
figure 1

a Number of estimated ancestral alleles and b clusters of the four parental alleles at each locus along the genome obtained with software “clusthaplo”. In a the position of the centromeres is indicated by white darts. In b each distinct ancestral allele is given a single color when it is shared by at least two haplotypes. Note that occurrence of the same color at two different loci does not imply anything about the relatedness of the corresponding alleles. Genome-wide values for −log10 (P value of F test) obtained from bi-parental (c) and multi-family (d) QTL mapping analysis of deoxynivalenol concentration (DON). The location of QTL is indicated above the curves with different symbols for the various models. The horizontal lines refer to the significance thresholds of each model

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Detection of QTL with additive effects and epistatic QTL with different models

For DON, one to two QTL with additive effects were detected with Model 1 in each family explaining together p G  = 14.8–43.9 % of the genotypic variance (Table 4; Fig. 1c). The only exception was family R2S2 with the smallest sample size (N = 43), where no QTL was identified. Each QTL was detected in only one family, except one QTL on chromosome 2, which was shared between R1R2 and R2S1 (Table S1). In general, the favorable alleles originated from the resistant parents R1 and R2 except for one QTL on chromosome 1 in family R1S2, where the favorable allele was contributed by the susceptible parent S2. Interestingly, this QTL had only p G  = 14.8 %, even though the sample size in this family was fairly large (N = 161).

Table 4 Number of detected QTL and proportion (\(p_{G}\)) of the genotypic variance explained by all QTL in a simultaneous fit applying different models for deoxynivalenol concentration (DON), Gibberella ear rot severity (GER), and days to silking (DS)
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Compared with Model 1, the joint analyses (Model 2 to Model 5) of DON with all five families detected considerably more QTL (8–13) and had higher p G values in the simultaneous fit, ranging between 34.4 and 52.9 % (Table 4, Fig. 1d). This included all QTL identified with Model 1 in all families and several new QTL, e.g., on chromosomes 4, 5, 6, 8, 9 and 10 (Table S1). Model 4 detected the largest number of QTL and had the highest p G (52.9 %), whereas Model 5 detected besides Model 2 the least number of QTL with smallest p G (34.4 %) and Model 3 was in between. Most of the favorable QTL alleles detected with Model 3, reducing DON, originated from parent line R1 with the highest resistance level (Table S2). Interestingly, the susceptible parent lines S1 and S2 also contributed resistance alleles with sizeable effects at some QTL.

For GER, with the exception of R2S2, one to two QTL displaying additive effects were detected with Model 1 in each family with p G values from the simultaneous fit between 18.1 and 48.4 % (Table 4; Fig. S6a). Each QTL was detected in only one family except for one QTL on chromosome 2 shared between R1R2 and R1S1 (Table S1). As expected on the basis of the tight genotypic correlations between GER and DON, QTL for GER showed a high degree of co-localization and congruency of effects with QTL for DON and this applied irrespective of the model applied, but the ranking of the Model 3 and 4 in terms of the number of QTL detected and p G differed.

For DS, Model 1 detected two to four QTL in all families except R2S2, with p G values from the simultaneous fit ranging from 23.2 to 61.4 % (Table 4, Fig. S6c). In contrast to DON and GER, a large number of QTL identified for DS were congruent between two or three families (Table S1). For instance, two out of three QTL detected in R1R2 were also found in R1S2. For family R2S1, the QTL on chromosome 10 for DS and GER co-localized and had p G values of 52.0 and 20.1 %, respectively. Model 2 to Model 5 detected all QTL identified with Model 1 in each family and several additional QTL (Table S1, Fig. S6d). Model 2 to Model 5 detected similar numbers of QTL (11 to 13), but Model 4 and Model 5 had smaller p G values than Model 2 and 3.

No significant (P < 0.05) digenic epistasis was found for DON with Model 3. For GER, only one pair of QTL at position 67.3 cM on chromosome 9 and 62.6 cM on chromosome 10 displayed a significant (P < 0.05) interaction with p G  = 5.5 %. For DS, significant digenic epistasis was detected between the QTL at position 62.6 cM on chromosome 2 and the QTL at 71.9 cM on chromosome 8 with p G  = 4.4 %. With Model 3, several significant (P < 0.05) QTL × genetic background (family) interactions were found, one for DON, three for GER and six for DS (Table S2), but the corresponding p G values were consistently below 2.5 %.

Comparison of QTL mapping models via cross-validation

For Scheme 1 and DON, higher prediction accuracies \(r_{\text{VS}}\) were achieved for N TS = 81 with Model 1 than with the joint analysis of Model 2 to 5 in each VS family except R1S1 (Fig. 2). For N TS = 180, estimates of \(r_{\text{VS}}\) from the joint analysis increased substantially in each family and approached or even exceeded those of Model 1 with N TS = 81. For both values of N TS, there existed only minor differences between Model 2–5 in terms of \(r_{\text{VS}}\) within each family. Family R1S1, which had the lowest h 2 among the three families, had generally smaller \(r_{\text{VS}}\) values than the other two families. With Model 1 and N TS = 81, the frequency of QTL detection in cross-validation runs was high (>0.4) for certain QTL, which were mostly specific for either R1R2 or R2S1, but generally low (<0.15) in R1S1 (data not shown). Joint analysis with Model 2–5 consistently identified QTL in those regions, where they were also detected with Model 1, but with low frequency (<0.15). Increasing the sample size to N TS = 180 increased the QTL frequencies (>0.3) in the joint analysis considerably and all QTL identified by Model 1 with N TS = 81 were detected.

Fig. 2
figure 2

Prediction accuracy (\(r_{\text{VS}}\)) in the validation set (VS indicated above each graph) composed of one family for deoxynivalenol concentration (DON) obtained by applying different models for single-family (model 1) and joint family analysis (model 2–5) of three fully connected families (R1R2, R1S1, R2S1). The height of black and gray columns refer to means across 200 cross-validation runs with N TS = 81 and 180 and N VS = 48 and 106, respectively, and the vertical bars show the corresponding standard deviation

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Similar results were observed for GER and DS, except that (1) for both traits and N TS = 81, Model 1 had in family R1S1 also a slightly higher mean \(r_{\text{VS}}\) than the joint analysis models (Fig. S7); (2) for DS and N TS = 81, the detected QTL frequency in cross-validation runs showed higher consistency among families than DON and GER (data not shown). For N TS = 81, Model 3 or 4 reached generally the highest mean for \(r_{\text{VS}}\) among the joint analysis models. Both were in most cases superior to Model 2 and 5 and differences among models were less pronounced for N TS = 180.

Effect of training set composition on prediction accuracy under different scenarios

The ranking of \(r_{\text{VS}}\) values for the different scenarios remained largely unaffected by the sample size of the TS and was almost identical for all traits (Fig. 3; Fig. S8). Prediction accuracies \(r_{\text{VS}}\) were higher under Scenario 1F than under all other scenarios for all VS families except for R1S1, where Scenario 3FH performed either equally well (DS) or better (DON, GER). In general, \(r_{\text{VS}}\) values obtained for scenarios (1F, 3FH, 4FH, 4FHU), which included different proportions of full sib DH lines in the TS, were higher than those without full sibs (2H, 3H, 3HU), irrespective of N TS. Contrasting scenarios 2H with 3HU and 3FH with 4FHU showed that including unrelated lines generally reduced \(r_{\text{VS}}\) values. The increase in \(r_{\text{VS}}\) with increasing N TS was generally highest for scenario 1F up to N TS = 140, but the slope of the curves varied among the VS families.

Fig. 3
figure 3

Prediction accuracy (\(r_{\text{VS}}\)) of QTL-based prediction across 200 cross-validation runs for deoxynivalenol concentration (DON) under different scenarios. The validation set (VS indicated above each graph) is composed of one family and the composition of the training set (TS) is reflected by the coding of the scenario: the number refers to the number of families included in the TS and the letter(s) refer(s) to their kinship with the VS, where F, H, and U denote full-sib, half sib and unrelated lines, respectively; for details see Table 4. QTL mapping was based on model 1 for Scenario 1F and on Model 3 for the other scenarios

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Discussion

Historically, QTL mapping in maize started with bi-parental populations (Edwards et al. 1987). Following Lander and Botstein (1989), highly diverse parents were generally chosen to increase the chances of segregation of QTL, especially for resistance traits (Schön et al. 1993). The initial euphoria abated after it was recognized that QTL effects reported in the early studies were oftentimes highly inflated (cf. Schön et al. 2004) due to the so-called Beavis (1998) effect (Xu 2003), first described by Utz and Melchinger (1994). To obtain unbiased estimates of QTL effects, Utz et al. (2000) recommended to use cross-validation for separating QTL detection, corresponding to model selection, from estimation of QTL effects. Further, it was found that with small sample size of the mapping population, the power of QTL detection for quantitative traits with polygenic architecture is low (Schön et al. 2004). Different from academia, maize breeders commonly produce DH lines from several crosses, including resistant and susceptible parents, each family being only of moderate size. The five families of DH lines analyzed here are typical for this situation. The questions to be answered by our study were: (1) Should QTL mapping for marker-assisted selection under such a setting be conducted separately for each family or jointly across all families? (2) Which of the models proposed in the literature for joint analysis across families yield highest prediction accuracy of QTL-based prediction evaluated by cross-validation? (3) How does composition of the TS and its pedigree-relationship(s) to the VS influence the prediction accuracy?

Consensus map construction and recombination landscape

Multi-family QTL mapping requires a joint linkage map for all families included in the analysis. Construction of a consensus map can be complicated, if families differ largely in their recombination rate or even in the linear order of markers, but this is very unlikely with the interconnected families produced from related parents. Therefore, we applied dsmergen command in Carthagene to estimate one single recombination rate for all families and obtain consensus distance over families. Since SNP markers have only two alleles, each of these marker loci can segregate only in a subset of families of a connected design. However, owing to the high marker density provided by the MaizeSNP50 array, we found plenty of tightly linked markers segregating in different families, which enabled construction of a consensus map.

The total length of our consensus map (1854 cM) agreed well with the map lengths reported by Bauer et al. (2013) for families R1R2, R1S1 and R1S2, which ranged between 1655 and 1893 cM. Further, the linear order of markers on our consensus map was in excellent harmony with the high-density linkage map presented by Ganal et al. (2011) for the flint cross F2 × F252, but their map length was expanded due to four generations of intermating. Comparison of the consensus map and the physical map revealed strong recombination suppression in the centromeric regions of all chromosomes, most notably on chromosome 3, where a segment of 75 Mb had a map distance of 5 cM compared to a ratio of 0.07 cM per Mb averaged over the maize genome. Suppression of recombination in pericentromeric regions is in agreement with the results reported by Bauer et al. (2013) for European maize germplasm and Rodgers-Melnick et al. (2015) for US and Chinese maize germplasm.

While the consensus map should be constructed with great care, its influence on QTL mapping with multiple families depends primarily on the map density. If a high marker density is available as in our study, the recombination break points in the meiosis of the parental gamete of each DH line can be determined with high accuracy. Hence, in a genome scan with a high-density map, the genotype of the putative QTL employed for QTL mapping in the regression approach can be inferred with high fidelity from the observable genotype at tightly linked markers provided the population size is sufficiently large (Peleman et al. 2005).

Clustering parental alleles at each locus

Choice of the window size is critical for computing the similarity score between pairs of lines in “clusthaplo” (Leroux et al. 2014). Following Giraud et al. (2014), we initially chose a window size of 20 cM for chromosome 7 and 10 cM for all other chromosomes, based on the decay of LD for the same set of markers using representative lines from the same breeding pool as the four parent lines. To be on the safe side, we also varied the window size from 5 to 20 cM and observed for clustering with 5 cM a much larger average number of ancestral alleles and number of cluster changes. This is because under this setting, IBD segments greater than 5 cM are broken into pieces, which can lead to incorrect estimation of similarity score for loci at both ends of the haplotype. Chromosome 7 had the smallest average number of clustered ancestral alleles in agreement with its slow decay of LD. Centromeric regions had on average fewer clustered ancestral alleles than telomeric regions (Fig. 1a, b) in accordance with the different recombination rates along the genome mentioned above. These results differ from those of Giraud et al. (2014) who detected on average more ancestral alleles in the centromeric than in the telomeric regions, most likely because (i) we used a much higher marker density for clustering and (ii) the parent lines in our study were more closely related to each other, which facilitated accurate detection of IBD segments. Altogether, the number of ancestral alleles obtained from “clusthaplo” varied along the chromosomes and this resulted in different numbers of parameters in the LDLA model (Model 4), which caused an erratic pattern in the curves of the –log (P values) in Figs. 1d; S6b and S6d.

Detection of QTL with additive effects and epistatic QTL based on all five families

All families except R2S2 had been separately analyzed for QTL for GER and DON with low-density maps comprising between 106 and 129 SSR markers (Martin et al. 2011, 2012). We identified with Model 1 only a subset of the QTL detected previously, because different from these authors, we applied a more stringent significance level (α = 2 % vs. 15 %) in permutation tests to protect against a high global Type I error rate in multiple tests with several families. The QTL detected by us were always adjacent to the flanking markers reported previously, but their exact position was shifted primarily as a result of the change in the genetic map caused by the higher marker density.

In agreement with Blanc et al. (2006) and Ogut et al. (2015), the QTL detected by Model 1 in each family generally showed little congruency across families, suggesting that each family comprised a unique set of segregating QTL. The only congruent QTL for DON was found for families R1R2 and R2S1 on chromosome 2, explaining a high percentage of the genotypic variance (p G  = 20.3 and 30.0 %, respectively), and the favorable allele originated in both families from the common parent R2. This is in accordance with the findings of Blanc et al. (2006) that congruent QTL among families often have large effects and originate from a common parent. In contrast, several QTL for DS detected with Model 1 were consistent across two or three families, suggesting that the level of congruency of QTL across families depends strongly on the trait. While most QTL for all three traits were family specific, we detected with Model 3 only few significant QTL × genetic background (family) interactions. Either the family sizes in our study were too small to warrant sufficient power for detecting this type of epistasis, or epistatic effects are small for the investigated traits. Results from studies with the US NAM panel with 200 recombinant inbred lines from each of 25 families on flowering date (Buckler et al. 2009) and a genome-wide association mapping study of Fusarium verticillioides with 1687 lines from the USDA gene bank (Zila et al. 2013) support the latter explanation.

For all traits, Model 2 detected all the QTL identified with Model 1 and additional QTL even though both models assume that allele effects of QTL are nested within families. Thus, joint analysis of several families with Model 2 can benefit from more replicates of QTL genotypes, which leads to a higher power of QTL detection, especially if common QTL are shared between families. This finding is somewhat different from the results by Blanc et al. (2006), where the QTL detected with these two models displayed greater discrepancies. This may be due to different genetic architecture of the traits and/or the higher marker density in our study which increased the power of QTL detection for both models. Model 3 detected more QTL than Model 2 for DON and GER. This is in line with Blanc et al. (2006) and can be explained by (i) a smaller number of parameters to be estimated in Model 3 than in Model 2, which leads to a higher power of QTL detection (Rebai and Goffinet 2000), and (ii) the low importance of epistasis observed for these traits. Contrary to expectation, Model 4 did not outperform the other joint analysis models for GER and DS. This may be attributable to the small number of parents in our study so that the gain in power for Model 4, expected from reducing the number of parameters by clustering the parental alleles, was limited. This is different from the study of Bardol et al. (2013), which involved more parents so that clustering the parental alleles resulted in a substantial reduction of the parameters in the model. Although the number of QTL detected for GER and DS was not smallest for Model 5, it yielded the lowest values of p G for all three traits. This implies that either some of the QTL detected by Model 5 were false positives or estimates of the allele effects at the detected QTL were inaccurate, as expected if the numbers of alleles at QTL exceed those at adjacent markers. The latter explanation is consistent with Lu et al. (2012) and Bardol et al. (2013), who observed that multi-allelic models capture a greater proportion of the genetic variance than bi-allelic models.

Comparison of models via cross-validation

For marker-assisted selection, breeders are interested in the prediction accuracy of genotypes on the basis of the detected QTL. To warrant a fair comparison of the different models for QTL detection, unbiased estimates of the prediction accuracy were determined by cross-validation using Scheme 1 with the following features: (i) Three completely interconnected families (R1R2, R1S1, and R2S1) were analyzed so that every pair of parental alleles could be contrasted with greatest power (Wu and Jannink 2004). (ii) The same number of DH lines was sampled from each family for composition of the TS and VS so that each family contributed equally to QTL detection, estimation of parameters and prediction. (iii) All models were compared with the same sample size for the TS (N TS = 81 or 180) and VS (N VS = 48 or 108). (iv) The same Type I error of 10 % was applied for all models.

In contrast to Ogut et al. (2015), who found that joint analysis generally had higher prediction abilities than single-family analysis, our results showed that prediction accuracies \(r_{\text{VS}}\) for individual families determined with cross-validation were for most traits and families with N TS = 81 lower for the joint analysis models than for Model 1 (Fig. 2; Fig. S7). Obviously, the superiority of Model 1 for small sample sizes depends strongly on the genetic architecture of the trait across all families. If specific QTL with large effects prevail in each family, as applies to DON and GER, the power of detecting these QTL is lower for Model 2–5 than for Model 1, because only a subset of genotypes (one-third under Scheme 1) will segregate in the TS used by these models. In contrast, if a QTL with a small effect segregates in one family, but has a large effect in the other families, joint analysis will most likely detect this QTL and using it for prediction can help to increase \(r_{\text{VS}}\). If the number of DH lines from each family in the TS is larger so that N TS = 180 for the joint analysis, then Model 3 reached generally similar values for \(r_{\text{VS}}\) as Model 1 for N TS = 81. Thus, the superiority of Model 1 over Model 3 seems to be strongly dependent on the number of individuals from the family to be predicted that are included in the TS besides the genetic architecture of the trait and the congruency of QTL across families.

Depending on the trait, Ogut et al. (2015) reported generally poor consistency of the QTL detected by Model 1 and 2. We found that QTL detected by Model 1 with high frequency were all identified by the joint analysis models with low (N TS = 81) or high (N TS = 180) frequencies (data not shown). Furthermore, for Scheme 1 and N TS  = 81, Model 2 generally had lower \(r_{\text{VS}}\) than Model 3 and Model 4 (Fig. 2; Fig. S7). Although the difference was not very big, this finding still suggests that Model 2 is most likely not the best choice in the experiments involving families of small size (N ≤ 27). Moreover, since Model 2 assumes QTL effects are specific to families (Table 2), full sibs must be included in the TS to estimate the QTL effects in the TS and predict other full sibs in the VS. This feature imposes considerable restrictions on possible composition of the TS, and, therefore, makes Model 2 less attractive than Model 3 and 4. For Scheme 1 and N TS = 180, no substantial differences in \(r_{\text{VS}}\) among the four joint analysis models were observed for all traits, contrary to the findings on p G based on the full data set (Table 4). This could be explained by the composition of the TS with different sampling of genotypes from the individual families and different assumptions about the number of QTL alleles. For all traits, Model 5 generally reached for N TS = 81 slightly lower \(r_{\text{VS}}\) values than Model 3 and Model 4. This finding is consistent with that of p G for the full data set. Thus, Model 5 most likely explains a smaller proportion of the genotypic variance than the other models allowing for multi-allelic QTL, even though it had a similar power of QTL detection, as reflected by the number detected QTL (Table 4). In conclusion, multi-family QTL analysis is superior to single-family analysis only if each family is represented by an adequate sample size (generally >60) in the TS and common QTL do exist. Model 3 or 4 exceeded Model 2 and 5, when evaluated with cross-validation, but differences among these models were generally small.

It must be noted that the p G values presented in Table 4 are mostly likely inflated, because they were determined without cross-validation. To obtain an idea about the upward bias and its dependency on the model, we determined for Scheme 1 in addition to the p G values for the VS, which correspond to the square of the \(r_{\text{VS}}\) values, also the p G values for the TS, using the same method as described for the full data set. Compared to the TS, the p G values in the VS averaged for Model 2–5 only about 45 % for N TS = 81 and about 70 % for N TS = 180 (data not shown). Moreover, the relative size of the upward bias in the p G values of the TS [(p G in TS/p G in VS) – 1] was almost the same for Model 2–5 so that choice of the best model could be determined from analyses of the full data set without cross-validation.

Design of the training set for QTL-based prediction

Riedelsheimer et al. (2013) examined the prediction accuracy \(r_{\text{VS}}\) for genomic prediction with GBLUP using the same five families and the phenotypic and genotypic data (excluding SSR markers) as used in this study. Regarding the composition of the TS, they found that \(r_{\text{VS}}\) was highest for scenario 1F and much lower for scenarios 2H and 3H with a further minor reduction for 3HU. We found in most cases the same ranking for these scenarios (Fig. 3; Fig. S8). Scenarios (1F, 3FH, 4FH, 4FHU), where the TS included various proportions of full sibs to the genotypes in the VS, had generally higher \(r_{\text{VS}}\) in our study than scenarios (2H, 3H, 3HU) with only half sibs or with half sibs and unrelated lines. This is in agreement with experimental result on genomic prediction with full-sib and half-sib families by Lehermeier et al. (2014) and Foiada et al. (2015). Moreover, \(r_{\text{VS}}\) increased in most cases linearly with increasing \(N_{\text{TS}}\) and the slope was generally steeper for the scenarios with full sibs in the TS. Thus, the average kinship \(\bar{f}\) between the TS and VS was the main factor determining \(r_{\text{VS}}\) for a given sample size. The only exception was family R1S1, where \(r_{\text{VS}}\) for DON and GER was higher for scenario 3FH with \(\bar{f}\) = 0.33 than for 1F with \(\bar{f}\) = 0.50. Possible explanations, why prediction including full sibs in the TS generally achieved higher accuracy than all other scenarios, could be that related families share more QTL than less related ones. These QTL could be rare QTL, which segregate and have significant effects in only one or a limited number of families as observed in our study for DON and GER. The number of families included in the TS hardly affected \(r_{\text{VS}}\), once both parents of the VP were parents of at least one population in the TS. However, the kinship between the TS and VS seems less influential, when both related and less related families comprise a large number of common QTL, as demonstrated for R1R2 for DS, where \(r_{\text{VS}}\) was similar for all scenarios (Fig. S8b). In addition, we found that the number of QTL and the size of QTL effects for all traits depended strongly on the family. The deviation observed for R1S1 can be explained by the observation that R1S1 comprised both rare and common QTL for all the three traits, and the average kinship \(\bar{f}\) between the TS and VS did not accurately reflect the resemblance at the QTL for scenarios 3FH and 1F. Altogether, our results suggest a major influence of \(\bar{f}\) on \(r_{\text{VS}}\), but the association seems to depend on the trait. While kinship measurements between genotypes, either based on pedigree or genome-wide markers, basically estimate the genome-wide resemblance, they may fail to reflect the specific resemblance with respect to the QTL influencing the trait of interest (Würschum and Kraft 2015). Nevertheless, identical linkage phases between marker and QTL may be more persistent between related materials than less related ones, as discussed by Riedelsheimer et al. (2013) and supported by our analysis of ancestral alleles with “clusthaplo”.

In conclusion, our results strongly emphasize to apply QTL-based prediction only, if the TS includes at least 60 genotypes being full sibs to the VS. If no full sibs are available, both parents of the VS should be included as parents in half-sib families in the TS. Inclusion of unrelated DH lines seems of questionable value. Without a sufficient number of full sibs in the TS, the risk of having a very low prediction accuracy is high even with the most advanced methods of multi-family QTL mapping. Thus, breeders are advised to ascertain a high degree of connectedness among the families, which they want to use for QTL-based prediction in marker-assisted selection or for genomic selection.

Author contribution statement

AEM and TM designed the experiments; TM and AEM conducted the experiments with the help of others mentioned in Acknowledgements; EB and CCS generated the SNP data used in this study; MS conducted the phenotypic data analysis, conducted the marker quality check and compiled the linkage maps for each family; SH constructed the consensus map and performed all QTL analyses with the help of HFU and WL; SH and AEM drafted the manuscript which was edited by all authors.