Background

Barriers to genetic exchange between diverging populations promote reproductive isolation and, ultimately, speciation. Genetic divergence of barriers between populations involves the evolution of prezygotic and postzygotic isolation [1,2,3]. Individual isolation barriers may be incomplete, while sequentially acting barriers restrict gene flow between diverging lineages. Reproductive isolation components that act early are often thought to contribute more to total isolation than late-acting postzygotic barriers [4, 5]. However, postzygotic isolation is undoubtedly a major factor in population divergence in the majority of systems [6]. Hybrid sterility and inviability are common forms of intrinsic barriers of postzygotic isolation, which lead to genetic differentiation and maintain species integrity [7,8,9].

It has been hypothesized that postzygotic reproductive isolation accumulates via increasing genomic incompatibilities [10,11,12], and that it is the by-product of genomic divergence [13]. Various genetic mechanisms of hybrid male sterility were proposed [14]. Of them, the Bateson–Dobzhansky–Muller (BDM) model predicts that hybrid sterility is caused by the accumulative negative interaction of two or more loci. For example, a single pair of heterospecific loci results in nearly complete hybrid male sterility in Mimulus [15]. Cytoplasmic male sterility (CMS) is another common genic incompatibility mechanism, which results from a mitochondrial–nuclear mismatch. The hybrid CMS mechanism has been widely studied in model species such as rice [16] and Arabidopsis [17]. In rice, a few genes causing hybrid sterility have been cloned [17,18,19] and several hypotheses have been proposed to explain the genetic mechanisms of hybrid sterility, such as the “duplicate gametophytic lethal model” [20] and the “one-locus allelic interaction model” [21]. In contrast to these simple models, many other studies demonstrated that hybrid sterility is a complex phenotype and is controlled by complicated mechanisms involving multiple loci. For example, empirical works in Drosophila and mice have revealed that hybrid sterility is highly polygenic and complex [22,23,24]. It is likely that genetic bases of hybrid sterility are extremely diverged in different systems. With genomic data increasingly accessible for non-model organisms [25], thousands of loci are easily identified for many lineages, potentially providing unprecedented power to study genetic mechanisms of hybrid sterility.

Theoretically, if there is gene flow between diverging species, tightly linked loci have larger selection coefficients than single loci [26, 27]. This was evidenced by many empirical studies, especially on Mimulus [28, 29]. Furthermore, genetic linkage will be increased between loci involved in adaptation and reproductive isolation [30,31,32]. Thus, a pleiotropic locus or tightly physically linked loci for adaptive traits and reproductive isolation would be the most probable way for reproductive isolation to arise. In plants, hybrid male sterility often occurs concomitantly with pleiotropic effects on floral morphology (reviewed in [33]). In Mimulus, hybrid male fertility is significantly correlated to corolla size phenotypically [34, 35]. In addition to pleiotropy, local adapted adjacent populations may also become reproductively isolated through hitchhiking of ecological adaptation genes and hybrid incompatibility loci [36]. Quantitative trait locus (QTL) mapping is an efficient approach for detecting pleiotropic or tightly linked loci underlying phenotypically correlated traits, and it is one of the most important fundamental methods for further detecting the hitchhiking effect.

In this study, we focus on Primulina depressa and Primulina danxiaensis, a pair of sympatric sister species (2n = 2x = 36). P. depressa is characterized as large and blue-purple flowers [37] while P. danxiaensis flowers are small and light-yellow [38], and before recent phylogenetic analysis, they were placed in two different genera [39]. Recent molecular dating revealed these two species diverged approximately 2.1 million years ago [40]. Our previous study [41] of this species pair found that flower morphological traits, as prepollination barriers, are likely subject to divergent selection. However, several lines of evidence suggest that postpollination isolation barriers are likely to play an important role in blocking hybridization between them, because of their sympatric distribution with overlapping flowering time and shared pollinators such as Amegilla spp., Bombus spp., and Nomia spp.. In our crossing experiments, hybrids of these sympatric sister species showed decreased male fertility. This indicats the potential postzygotic isolation barriers between P. depressa and P. danxiaensis, and makes them an ideal system for investigation of the hybrid sterility mechanisms and their evolution.

Here, we estimated hybrid male sterility in the species divergence between these sympatric sister species through heterosis analysis. We conducted a QTL mapping analysis for hybrid pollen fertility with an F2 population. We characterized the number, mode of action, digenic interactions, and phenotypic effects of loci that cause male sterility in F2 hybrids. We then detected whether hybrid male sterility QTLs were coincident with other flower and leaf trait QTLs. Our main objectives are to dissect the genetic basis of hybrid fertility, which may play an important role in reproductive isolation in this system.

Results

Analysis of variance and correlations

The proportion of fertile and sterile pollen in all individuals was evaluated by testing pollen viability with 2,3,5-triphenyltetrazolium chloride (TTC). Under normal cultivation conditions (excluding harsh environments such as drought, flood, and extremely high or low temperatures), the pollen of parental lines (P. depressa and P. danxiaensis) was completely fertile (the pollen fertility rate [PFR] was about 95%), and the pollen of F1 hybrids was semi-fertile (53.02%). The PFR of F2 individuals ranged from 0% (completely sterile) to 69.69%, with an average of 41.17% (Fig. 1). In the F2 population, the PFR exhibited continuous variation, with a novel class of individuals (15.4%) that produced no viable pollen grains (Fig. 1).

Fig. 1
figure 1

Histogram of PFR in the F2 population. Black vertical line indicates pollen fertility of F1 hybrids

Full size image

In F2 hybrids, PC1 and PC2, which represent variation in plant overall size and leaf physiology, showed weak positive but significant correlations with PFR (Fig. 2). For individual traits, corolla length, stamen length, and leaf pigment concentration traits were significantly correlated with PFR (Additional file 1: Table S1).

Fig. 2
figure 2

Scatter plots showing the correlation between PFR and PCs derived from morphological and physiological traits. Percentages of total variation for each PC explained and interpretation for each PC represented were indicated in brackets

Full size image

Mid-parent heterosis in 13 traits

Mid-parent heterosis (MPH) was estimated as the percentage of the difference between the mean of F1 or F2 and the mid-parent mean divided by the mid-parent mean. We estimated MPH for PFR and 12 flower and leaf traits that we studied previously [41]. PFR displayed almost the lowest MPH among all of the traits (Table 1). MPH for PFR in F1 was − 44.5%. The performance of 12 flower and leaf traits for F1 hybrids, the F2 population, and their parental lines is listed in our previously published paper [41]. In these traits, the concentration of leaf pigments showed the highest MPH (an average of 31.9%) in F1 (Table 1).

Table 1 Mid-parent heterosis (MPH) for PFR and 12 flower and leaf traits in our previous study. If %MPH > 0, this represents the percentage of individuals showing MPH > 0 in all of the measured F2 individuals
Full size table

Considerable transgressive segregation was observed for PFR and leaf pigments in F2 individuals (Fig. 1; Table 1; data from [41]). As described above, all of the F2 individuals produced less fertile pollen than parental lines. Almost all of the F2 individuals showed heterosis in leaf pigments (Table 1).

QTL analyses

The linkage map and QTL mapping analysis for floral and leaf traits were described in our previous study [41]. We detected QTL for PFR here with MapQTL v.6.0 software using 1 cM increments [42]. Table 2 summarizes QTL mapping results for PFR, and Fig. 3 shows locations of QTLs. A total of eight QTLs were identified for PFR, which jointly explained 55.4% of the phenotypic variation in the F2 population (Table 2). The phenotypic variation explained by each QTL ranged from 4.2% (PF02) to 17.8% (PF01). In particular, PF01 on LG02 is a major-effect locus for PFR, with an LOD value of 12.09. The P. depressa allele at PF01 resulted in 12.44% less pollen viability. Except PF01, all QTLs explained less than 7% of the total phenotypic variation (Fig. 4a and Table 2). These results suggest that the genetic architecture of pollen fertility contains a major-effect QTL plus numerous minor-effect QTLs.

Table 2 Summary of QTLs detected for PFR and overlapping loci for two floral morphological traits
Full size table
Fig. 3
figure 3

Genome location of significant PFR QTLs and two co-localized floral morphology QTLs. Confidence intervals of QTLs are illustrated on the right side of linkage groups

Full size image
Fig. 4
figure 4

The effect size, additive effect distribution, and dominance effect distribution of QTLs for hybrid PFR are shown in (a), (b), and (c). The x-axis indicates the name of each QTL. The y-axis in a indicates the phenotypic variation explained by each QTL (PVE). The y-axis in b and c indicates the additive and dominance effects of the P. depressa allele relative to the P. danxiaensis allele at each QTL

Full size image

For five of the eight pollen fertility QTLs (PF01, PF02, PF05, PF06, and PF07), P. danxiaensis alleles were associated with decreases in pollen viability (Table 2 and Figs. 4b and 5). Analysis of the dominance effect showed that three QTLs (PF02, PF03, and PF06) were dominant with P. danxiaensis alleles, and the other five were dominant with P. depressa alleles (Table 2 and Figs. 4c and 5).

Fig. 5
figure 5

Phenotypic trait means of different genotypes of each QTL

Full size image

Comparing with previous QTLs for 12 flower and leaf traits for the same population, we found two QTLs for pollen fertility (PF02 and PF04) overlapped with QTLs for floral morphology (corolla length and stamen length) (Table 2 and Fig. 3). We calculated the significance of correspondence between these overlapping QTLs and found that both pairs showed significantly more overlap than expected by chance (P < 0.05). In addition, we found two additional floral morphology QTLs (upper petal width and corolla width) were tightly linked (about 5 cM) with PF02 (Table 2 and Fig. 3). As we previously found that divergence in flower morphology was driven by natural selection [41], there was probably a hitchhiking effect between hybrid sterility and floral traits. However, we did not find the evidence of QTL overlap between PFR and leaf physiology.

Gene action and locus interaction

According to gene action estimation, two of the eight identified QTLs (PF01 and PF02) exhibited partial dominance, and the other six exhibited overdominance (Table 3). For the six overdominant loci, heterozygous genotypes reduced the pollen fertility at loci of PF03 and PF06 (Table 3 and Fig. 5). The underdominance evidenced the conflict between parental alleles. Conversely, the heterozygous genotype was favorable for pollen fertility at four loci (PF04, PF05, PF07, and PF08; Table 3 and Fig. 5). The overdominance suggests that heterosis occurs at some loci.

Table 3 The degree of dominance was calculated as dominance effect/|additive effect|
Full size table

We characterized pairwise QTL interactions to summarize epistatic interactions at a threshold of P < 0.05 (Additional file 2: Table S2). Digenic interactions were identified between two QTL pairs: PF02/PF08 and PF03/PF08 (Fig. 6; Tables 3; Additional file 2: Table S2). A high level of male sterility occurred in individuals that harbored alleles from different parents at PF03 and PF08 simultaneously (e.g., heterozygous alleles or homozygous P. danxiaensis alleles at PF03 and homozygous P. depressa alleles at PF08). A similar pattern was observed between PF02 and PF08; lower PFR occurred in individuals that were homozygous for P. danxiaensis alleles at PF08 and homozygous for P. depressa alleles or heterozygous for both parental alleles at PF02.

Fig. 6
figure 6

Digenic interactions between two pairs of QTLs. AA, BB, and AB represent homozygous genotypes of male and female parents and a heterozygous genotype

Full size image

The estimation of locus segregation distortion revealed that peak LOD markers and their surrounding markers of PF02 distorted to the male parent (P < 0.05), while PF05 distorted to the female parent (Tables 3; Additional file 3: Table S3). Peak LOD markers and many surrounding markers of PF04, PF07 and PF08 distorted to female and male parents, respectively, even though not significantly at peak LOD markers (Additional file 3: Table S3).

Discussion

Hybrid inviability and sterility

Studying mechanisms of reproductive isolation barriers is important for improving our understanding of the process of speciation [3, 5]. We identified multiple potential prezygotic reproductive isolation barriers that could contribute to the divergence between sympatric P. depressa and P. danxiaensis [41]. However, the incomplete pollinator divergence and the overlapping flowering time indicate that the prezygotic reproductive isolation might be insufficient for blocking the hybridization between P. depressa and P. danxiaensis. In this study, approximately one-sixth of the F2 hybrids showed complete male sterility (Fig. 1), which indicates hybrid male sterility has contributed, at least partially, to reproductive isolation between these two species.

Hybrid inviability and sterility are two of the most common forms of postzygotic isolation in plants, because they prevent gene flow between lineages. In our case, we studied MPH for hybrid fertility and 12 flower and leaf traits to unravel the mechanism of postzygotic isolation between P. depressa and P. danxiaensis in sympatry. In F1 hybrids, pollen fertility showed a greater underdominance than all of the flower and leaf traits investigated (Table 1), suggesting that male sterility plays an important role in postzygotic reproductive isolation in this case. F1 hybrids showed high concentrations of chlorophylls which represent potentially better photosynthesis capacities, but they did not show significant heterosis or depression for most of the traits for floral morphology and leaf size (abstract values of MPH were less than 10%; Table 1). Considerable transgressive segregation of leaf pigments in F2 population indicated that the photosynthetic capacity of most F2 hybrids was better than the parental lines (Table 1; data from [41]). This heterosis for photosynthesis was consistent with the action of loci in leaf pigments, 27 out of 30 loci were dominant or overdominant [41]. MPH value of all F2 individuals for PFR were negative, and this was possibly caused by genetic conflict (Table 1). Collectively, these results indicate hybrid sterility might play an important role in postzygotic reproductive isolation, while we did not observe apparent hybrid inviability, at least in our greenhouse. Further study of the relative importance of hybrid male sterility and hybrid inviability will be needed in the future.

Artificial crossing between P. depressa and P. danxiaensis produced vigorous but semi-sterile hybrids. Most Primulina species are narrow endemics with small population sizes [43]. Small-size populations usually have lower genetic diversity and are prone to accumulating deleterious recessive alleles [44,45,46,47]. If chlorophyll content and photosynthetic capacity are maladaptive in this system, it will make sense that genetic incompatibilities decrease fertility but increase vigor in interspecific hybrids here. Similar conclusions have also been drawn in many other plant species, such as Luffa [48] and rice [49,50,51,52].

Genetic architecture of hybrid male sterility

We found a polygenic basis for PFR in P. depressa × P. danxiaensis F2 hybrids. We identified one major-effect QTL, which explained 17.8% of the hybrid population phenotypic variation. Seven minor-effect individual QTLs explained 4.2–6.9% of the phenotypic variance (Table 2). These QTLs jointly accounted for 55.4% of the total phenotypic variation in hybrid pollen fertility. Even though a low number of QTLs may have remained hidden due to the modest size of the mapping population (N = 195), the genetic architecture of postzygotic isolation in our system was polygenic, as we found a major-effect QTL plus numerous minor-effect QTLs.

The polygenic genetic basis of hybrid male fertility is consistent with that observed in other plant species. For example, eight-QTL hybrid sterility was identified between two closely related species of Solanum [53], and a number of different QTLs for male sterility were mapped in two subsequent mapping experiments [54, 55]. Likewise, studies in Drosophila indicated that interspecific male and female sterility were highly polygenic (e.g., [56, 57]). Tao et al. [22] identified a total of approximately 60 QTLs contributing to hybrid male sterility when Drosophila mauritiana is introgressed into a Drosophila simulans background, indicating that a polygenic genetic basis of hybrid male sterility is prevalent in both plants and animals. However, two-locus genetic incompatibility was reported to cause male sterility between species or subspecies in rice [58, 59] and Mimulus [15]. Lowry et al. [60] detected three sterility loci in hybrids of two Panicum hallii ecotypes. In general, fewer loci contribute to hybrid sterility in plants than in Drosophila, perhaps because plants largely lack sex chromosomes [56, 61].

In our Primulina study, the major-effect QTL contributes nearly 18% of the hybrid male sterility. This large effect makes it possible that divergence of this QTL is sufficient to impede gene flow and contribute to species integrity in the presence of vigorous hybrids. According to mean phenotype values at this major-effect locus (PF01), homologous P. danxiaensis alleles increase pollen fertility two-fold compared with homologous P. depressa alleles. Because mitochondrial genomes of hybrids were inherited maternally from P. danxiaensis, we suspect that CMS is a possible mechanism underlying hybrid male sterility here. Further elucidation of the molecular basis of PF01 will provide insight into its normal function within pure species.

The QTL overlap between PFR and flower morphology (corolla length and stamen length) suggests that hybrid male sterility is genetically correlated with floral morphological traits (Fig. 3 and data from [41]). In addition, a PFR QTL (PF02) was tightly linked to upper petal width and corolla width QTLs (Fig. 3 and data from [41]). This genotypic correlation pattern, including pleiotropic and tightly linked loci, was consistent with our phenotypic correlation analysis (Fig. 2; Additional file 1: Table S1). When a favorable mutation arises, the allele frequency of closely linked surrounding loci will also increase (i.e., genetic hitchhiking) [62]. Inversely, deleterious mutations will also eliminate the variation at their surrounding loci (i.e., background selection) [63]. In our case, flower morphology was divergently selected [41], and it may affect selection on hybrid male sterility genes when hitchhiking occurs. In consideration of both QTL overlap and close linkage of male sterility and morphological traits, we suspect that hybrid sterility evolved as an incidental by-product of hitchhiking. Groups of tightly linked minor-effect loci have larger selection coefficients than single loci with small phenotypic effects [25]. Therefore, a single locus that affects both adaptive and reproductive isolation traits would be the most probable way for selection to maintain differentially adapted species, even in the face of gene flow. These results are consistent with recent studies on Mimulus (e.g. [28, 29, 64]).

The coincidental (phenotypic and genotypic) correlation between PFR and flower morphology suggests developmental integration. In accordance with our results, a recent study in Mimulus found a genetic correlation between flower size and hybrid male fertility [35]. The inconsistent directionality of QTL effects suggests that reduced flower size in hybrids increases male fertility. In consideration of the association between self-fertilization and reduced flower size [64], small flowers theoretically suppressed gene exchange between P. depressa and P. danxiaensis yet promote hybrid male fertility. However, further crossing experiments for estimating outcrossing rates are necessary to evidence the correlation between gene exchanging rate, flower size, and hybrid fertility.

Leaf pigments play important roles in responses to biotic and abiotic stresses and thus are sensitive to environmental changes [65,66,67]. Pollen fertility is easily affected by environmental factors as well [18, 19, 68]. Drought stress, a common instance of environmental stress, was reported to decrease both pollen fertility [69] and leaf chlorophyll content [70, 71] in wheat. This is one of the probable reason why, in our study, pollen fertility was correlated with leaf pigments phenotypically but not genotypically. Further work is needed to test the relationships among stresses, leaf pigments and pollen fertility.

Complexity of hybrid sterility

Significant negative epistatic interactions were observed between two pairs of QTLs: PF02/PF08 and PF03/PF08 (Fig. 6; Table 3; Additional file 2: Table S2). This indicates that the BDM model involving three loci contributes to hybrid male sterility. According to the BDM model, sterility is more polymorphic within plant species than animal species (reviewed in [61]). The polygenic BDM model in our case implies the sterility may be under complex genetic control.

Besides the BDM model, segregation distortion is very common at loci surrounding hybrid male fertility QTLs in our study. Two QTLs (PF02 and PF05) showed significant segregation distortion at their peak LOD and surrounding loci (Additional file 3: Table S3). PF04, PF07, and PF08 did not show significant segregation distortion at peak LOD loci, but their surrounding loci did (Additional file 3: Table S3). The segregation distortion may resulted from gametes abortion (prezygotic selection) or the selective fertilization of particular genotypes (postzygotic selection). Unfortunately, we couldn’t distinguish them from a single F2 linkage map (reviewed by [72, 73]). Correlations between locus segregation distortion and male sterility has been reported in a few systems, including Drosophila (e.g., [74,75,76]). In rice, segregation distortion sometimes has significant effects on hybrid male sterility [77,78,79]. To our knowledge, however, this is the first study on correlation between locus segregation distortion and hybrid male fertility in non-model plants.

In contrast to Mimulus species [15], we here show a complex genetic basis for hybrid male sterility between P. depressa and P. danxiaensis. Hybrid male sterility between Mimulus guttatus and Mimulus nasutus were almost results from a simple genetic incompatibility between a single pair of heterospecific loci. In contrast, our results show a complex genetic basis of hybrid male sterility, including a BDM model involving three loci and genetic correlations with flower morphology. Further genetic dissection may reveal that more complicated mechanisms underling each locus. A more complex genetic architecture of a reproductive isolation barrier will reduce the rate of gene flow across the entire genome to a greater extent than a barrier governed by only a few major loci [6]. Therefore, the identified complex genetic architecture of hybrid male sterility might make a critical contribution to the maintainance of species identity between P. depressa and P. danxiaensis.

Conclusions

In this study, we found that hybrid male sterility in Primulina is controlled by a polygenic genetic basis. The genetic incompatibility involves a three-locus BDM model, even though the single largest-effect locus is not involved in. Hybrid male sterility is genetically correlated with floral morphology and divergence hitchhiking may have occurred between them. This complex pattern is in contrast to the simple genetic incompatibility model found in Mimulus [15]. The identified complex genetic architecture of hybrid male sterility might play an important role in maintaining species identity between P. depressa and P. danxiaensis. Continued studies of reciprocal backcross can provide detailed incompatibility patterns and further our understanding of evolutionary forces driving postzygotic reproductive isolation in this system.

Methods

Plant materials and growth conditions

We collected parental lines (P. depressa and P. danxiaensis) at the sympatric site of Danxia Mountain, Guangdong, South China. With the permission of Danxia Mountain National Park Commission, all samples were originally collected following methods that met the guidelines of Regulations on Wild Plants Protection (People’s Republic of China) for the use of plants in research. Voucher specimens of P. depressa (DXS02) and P. danxiaensis (DXS04) were deposited in the South China Botanical Garden Herbarium (IBSC). We created a F2 mapping population, including 201 individuals, by self-fertilizing one F1 plant derived from a cross between an individual of P. depressa (♂) and an individual of P. danxiaensis (♀). F1 hybrids, male and female parents, and 201 F2 individuals (the mapping population) were genotyped by restriction site-associated DNA sequencing (RAD-seq). A high-density linkage map was constructed with 2484 markers that were spaced across 18 linkage groups. For further details, see [41].

To determine male fertility, we planted parental populations, F1 hybrids, and the F2 population in a greenhouse in South China Botanical Garden under fluorescent light to provide a 14-h day length, with a temperature of ~ 26 °C. Plants were watered by sub-irrigation as needed and fertilized weekly.

Male fertility assessment and pollen staining

Pollen viability was assessed by staining pollens with TTC [80]. For pollen staining, we collected the first three flowers per individual on the day the flower opened. These flowers contained mature pollen, and there was no bias in the analysis of each individual. We halved anthers with a pair of forceps, and then pollen was squeezed into 50 μl of TTC (1% w/v in 50% sucrose) solution on a glass slide and incubated in the dark at room temperature for 15 min. TTC white compound can be enzymatically converted to red 1,3,5-triphenylformazan (TPF) by various dehydrogenases in metabolically active pollen. Therefore, red stained pollen grains were considered viable, while faintly stained or empty pollen were non-viable. More than 100 pollen grains were observed for each flower under the microscope. The percentage of viable pollens in the total examined pollens was calculated as the pollen fertility rate (PFR).

Phenotypic data analysis

We previously performed principal component analysis (PCA) on seven floral traits (upper petal width, lower petal width, corolla width, corolla height, corolla length, pistil length and stamen length), three leaf physiological traits (concentrations of chlorophyll a, chlorophyll b, and carotenoids), and two morphological traits (leaf length and leaf width). The first two principal components (PCs) accounted for 48.4 and 20.5% of the total measured phenotypic variation, with the first component correlating with overall size (in particular the size of floral traits) and the second with leaf physiology traits. For detail information, see [41]. Here, we calculated Pearson’s correlation coefficients between PFR and the 12 previously studied traits and PCs derived from them.

In order to evaluate the hybrid phenotypic performances, a previously developed mid-parent heterosis (MPH) statistical test [81, 82] was used. MPH was calculated using the following formula:

$$ \mathrm{MPH}=\frac{\mathrm{Hybrid}\ \mathrm{mean}-\mathrm{Mid}-\mathrm{parent}\ \mathrm{mean}}{\mathrm{Mid}-\mathrm{parent}\ \mathrm{mean}}\times 100\%, $$

Where mid-parent mean represents the average of both parents with respect to a trait of interest. In this study, we performed MPH analysis on 13 phenotypic traits mentioned above.

QTL mapping

The linkage map used in this study was a highly resolved RAD-seq-based SNP map with 2484 markers distributed across the 18 linkage groups with an average distance of 0.96 cM [41]. The QTL intervals of PFR were defined on the linkage map with MapQTL v.6.0 software [42]. Before QTL mapping, a genome-wide and chromosome-level LOD score threshold (P < 0.05) for declaring the presence of QTLs was determined using the permutation test (10,000 replications). Based on the permutation results, an LOD score threshold of 3.0 was set for the trait to declare the presence of a significant QTL. The potential QTLs were initially detected employing the Interval Mapping (IM) algorithm. Then, markers with the highest LOD values were selected as cofactors, and the final set of markers detected at P < 0.05 after automatic cofactor selection were further included in the Multiple QTL Model (MQM). Cofactor selection and MQM analysis were repeated until the best possible set of QTLs was found, and then each QTL was characterized by its maximal LOD score. The 1.5-LOD support intervals were estimated in centimorgans (cM) all of the significant QTLs detected. Final QTL maps were drawn with the help of the graphical package MapChart v2.2 [83].

To evaluate the significance of correspondence between QTL and PFR or other traits that we studied previously [41], we calculated the probability of obtaining the observed number of matching QTLs by chance alone [84]. This probability was estimated using a hypergeometric probability distribution function [85]. The equation used is shown in [56].

Gene action and locus interaction analysis

Gene action was proposed to estimate the relative importance of the dominance effect and the additive effect, and it was estimated by calculating |d/a| = |dominance effects/additive effects| [86]. Results were defined as follows: A, additive (|d/a| ≤ 0.20); PD, partial dominance (0.20 < |d/a| ≤ 0.80); D, dominance (0.80 < |d/a| ≤ 1.20); and OD, overdominance (|d/a| > 1.20). Thresholds were firstly proposed by [86], and commonly used in many following studies (e.g., [87,88,89]).

Digenic epistatic interactions were analyzed with markers closer to QTL peaks using R/qtl [90]. The proportion of variance explained by epistasis was tested by comparing the residual of the full model containing all of the single-locus effects and two-locus interaction effects with that of the reduced model containing all of the single-locus effects but excluding two-locus interaction effects.

In addition to gene action and digenic interactions, segregation distortion of mapped QTLs and their closely linked markers was estimated. The degree of marker segregation distortion in the F2 generation was determined by marker data comparison against the expected 1:2:1 ratio using the Chi square test, where significant distortion was declared at P < 0.05.