- Methodology article
- Open Access
A new mapping method for quantitative trait loci of silkworm
- Hai-Ming Xu^{1},
- Chang-Shuai Wei^{1},
- Yun-Ting Tang^{2},
- Zhi-Hong Zhu^{1},
- Yang-Fu Sima^{3} and
- Xiang-Yang Lou^{1, 4}Email author
https://doi.org/10.1186/1471-2156-12-19
© Xu et al; licensee BioMed Central Ltd. 2011
- Received: 5 September 2010
- Accepted: 28 January 2011
- Published: 28 January 2011
Abstract
Background
Silkworm is the basis of sericultural industry and the model organism in insect genetics study. Mapping quantitative trait loci (QTLs) underlying economically important traits of silkworm is of high significance for promoting the silkworm molecular breeding and advancing our knowledge on genetic architecture of the Lepidoptera. Yet, the currently used mapping methods are not well suitable for silkworm, because of ignoring the recombination difference in meiosis between two sexes.
Results
A mixed linear model including QTL main effects, epistatic effects, and QTL × sex interaction effects was proposed for mapping QTLs in an F_{2} population of silkworm. The number and positions of QTLs were determined by F-test and model selection. The Markov chain Monte Carlo (MCMC) algorithm was employed to estimate and test genetic effects of QTLs and QTL × sex interaction effects. The effectiveness of the model and statistical method was validated by a series of simulations. The results indicate that when markers are distributed sparsely on chromosomes, our method will substantially improve estimation accuracy as compared to the normal chiasmate F_{2} model. We also found that a sample size of hundreds was sufficiently large to unbiasedly estimate all the four types of epistases (i.e., additive-additive, additive-dominance, dominance-additive, and dominance-dominance) when the paired QTLs reside on different chromosomes in silkworm.
Conclusion
The proposed method could accurately estimate not only the additive, dominance and digenic epistatic effects but also their interaction effects with sex, correcting the potential bias and precision loss in the current QTL mapping practice of silkworm and thus representing an important addition to the arsenal of QTL mapping tools.
Keywords
- Composite Interval Mapping
- Dominance Effect
- Epistatic Effect
- Marker Interval
- Digenic Epistatic Effect
Background
Silkworm (Bombyx mori) is the basis of sericultural industry. With nearly 5000 years' domestication, silkworm has an undoubted importance in human history and is still of great value in modern economy. In addition, it is also an ideal model organism of the Lepidoptera. Because silkworm is easy to rear and could produce large amount of mutation, it is second to fruit fly as a model organism in insect genetics study. Over these years, the "old" creature is becoming a new hot spot in genetic research.
Many important traits of silkworm are complex quantitative traits, such as whole cocoon weight and cocoon shell weight, etc. The genetic variation of quantitative traits are usually controlled by a number of genes (quantitative trait loci, QTLs) with epistatic and gene × environment interactions. To locate their positions on chromosome and estimate their contribution to the variation of trait is a key step for positional cloning and follow-up utilization of those genes. With the development of modern molecular biology, it has become possible to dissect the genetic mechanism of quantitative trait and to identify the associated genes and their interacting network by co-segregation analysis of molecular markers in a mapping population based on specific genetic model connecting QTL genotype with a phenotype of interest.
Since the draft sequence of silkworm genome [1] was reported, genetic research of silkworm has been greatly spurred and many linkage-maps have been constructed with various molecular makers, such as random amplified polymorphic DNAs (RAPDs), amplified fragment length polymorphisms (AFLPs), selective amplification of DNA fragments (SADFs), microsatellites (also known as simple sequence repeats, SSRs) and expressed sequence tags (ESTs) [2–6]. Meanwhile, many statistical models were developed for mapping QTLs, such as the interval mapping (IM) method [7] and the composite interval mapping (CIM) method [8]. Along with increasing evidence supporting the claim that epistatic and QTL-environment interactions are usually involved in the genetic variation of complex trait [9–11], several complicated mapping models were developed to analyze epistatic effects [12–18]. Kao et al. [12] expanded CIM to multiple interval mapping (MIM) for detecting epistasis. Wang et al. [15] established a mixed linear model based composite interval mapping (MCIM) method to analyze both epistasis and QE interaction in a double haploid (DH) population. A few years later, the MCIM was extended for other designed populations [19] and improved in searching strategy and estimating QTL parameters [16]. Parallel to these frequentist methods, Bayesian approaches have also been proposed for QTL mapping of complex traits [17, 20–24]. Recently, there are several new Bayesian methods developed for mapping QTLs underlying dynamic traits [25], ordinal traits [26], multiple traits [27–30], expression data [31], and gene frequency data [32], and for multiple inbreed lines designs [33].
Although there are a large number of QTLs reported in other species, relatively fewer QTL mapping studies are performed in silkworm [6, 34–37], which in part result from the fact that the current models and corresponding analysis methods are not appropriate for the silkworm. Silkworm has a particular characteristic called female achiasmata where meiosis occurs with no crossover between homologous chromosome pairs. Yet the majority of recently developed mapping tools are based on the assumption of chiasmata without considering the genetic differences between male and female. Therefore, only specific mapping populations such as a backcross (BC) population are suggested for gene mapping in order to satisfy this assumption [3, 5]. By setting female silkworm as the homozygous parental lines, they can avoid the problem of achiasmata. However, a BC population contains fewer segregant types of molecular markers than F_{2} population. As a result, genetic information is not enough to reveal additive and dominance effects, epistatic effects of QTLs and their interaction effects with environment. Therefore, it is necessary to develop a new method of QTL mapping with consideration of female achiasmata for F_{2} population of silkworm.
In the present study, we proposed a new statistical method for QTL mapping in silkworm. The method can analyze the additive, dominance and digenic epistic effects of QTLs, as well as their interaction with sex. The effectiveness of the method is investigated by intensive Monte Carlo simulations.
Methods
Genetic model for QTL mapping of silkworm
where μ is the population mean; a_{ k }and d_{ k }are the additive effect and dominant effect of the k-th QTL, respectively; x_{ Aik }and x_{ Dik }are the coefficients of QTL effects which can, when QTL genotypes are unobserved, be derived from the conditional probability of the putative QTL given the genotypes of flanking markers (flanking marker M_{+}, M_{-} of QTL Q) and the QTL position (the recombination frequency r_{ M+Q },r_{ M-Q }), respectively; aa_{ i }, ad_{ i }, da_{ i }and dd_{ i }are the additive-additive, additive-dominance, dominance-additive and dominance-dominance epistatic effects of the l-th pair of QTLs, respectively, with their coefficients x_{ AAil }, x_{ ADil }, x_{ ADil }and x_{ DDil }, which are the products of the corresponding x_{ A }and x_{ D }; all the above additive, dominance and epistatic effects are of our interest and thus considered as fixed; S_{ h }is the sex effect of sex h, S_{ h }~ (0, σ^{ 2 }_{ S }); as_{ hk }and ds_{ hk }are additive-sex and dominance-sex interaction effects, as_{ hk }~ (0, σ^{ 2 }_{ AS }) and ds_{ hk }~ (0, σ^{ 2 }_{ DS }), respectively; aas_{ hl }, ads_{ hl }, das_{ hl }and dds_{ hl }are interaction effects between epistasis and sex, aas_{ hl }~ (0, σ^{ 2 }_{ AAS }), ads_{ hl }~ (0, σ^{2}_{ ADS }), das_{ hl }~ (0, σ^{2}_{ DAS }) and dds_{ hl }~ (0, σ^{2}_{ DDS }); ε_{ hi }is the random residual effect, ε_{ hi }~ (0, σ^{ 2 }_{ e }).
where y is the vector of the phenotypic values; b is the vector of fixed effects and X is the coefficient matrix; e_{ u }is the vector of the u-th random effect and U_{ u }is the corresponding coefficient matrix, and R_{ u }is an identity matrix for every u in this model if the components of e_{ u }are independent.
Conditional probabilities of QTL genotypes and coefficients of additive and dominance effects for achiasmate and chiasmate F_{2} populations
Marker genotype | Achiasmate F_{2}population^{2} | Chiasmate F_{2}population^{3} | ||||||
---|---|---|---|---|---|---|---|---|
QQ ^{ 1 } | x _{ A } | x _{ D } | ||||||
M _{+} M _{+} M _{-} M _{-} | s_{1}s_{2}/s | r_{1}r_{2}/s | 0 | s_{1}s_{2}/s | (r_{1}r_{ 2 }- s_{1}s_{2})/2s | s_{ 1 }^{2}s^{2}_{2}/s^{2} | 2s_{1}s_{2}r_{1}r_{2}/s^{2} | r_{ 1 }^{2}r_{ 2 }^{2}/s^{ 2 } |
M _{+} M _{+} M _{-} m _{-} | s_{1}r_{2}/r | r_{1}r_{2}/r | 0 | s_{1}r_{2}/r | (r_{1}s_{ 2 }- s_{1}r_{2})/2r | s_{1}s_{2}s_{1}r_{2}/sr | (s_{1}s_{2}r_{1}s_{2} + r_{1}r_{2}s_{1}r_{2})/sr | r_{1}r_{2}r_{1}s_{2}/sr |
M _{+} M _{+} m _{-} m _{-} | - | - | - | - | - | s_{ 1 }^{2}r^{2}_{2}/r^{2} | 2r_{1}s_{1}r_{2}s_{2/}r^{2} | r_{1}^{2}s_{2}^{2}/r^{2} |
M _{+} m _{+} M _{-} M _{-} | r_{1}s_{2}/r | s_{1}r_{2}/r | 0 | r_{1}s_{2}/r | (s_{1}r_{ 2 }- r_{1}s_{2})/2r | s_{1}s_{2}r_{1}s_{2}/sr | (s_{1}s_{2}s_{1}r_{2} + r_{1}r_{2}r_{1}s_{2})/sr | r_{1}r_{2}s_{1}r_{2}/sr |
M _{+} m _{+} M _{-} m _{-} | r_{1}r_{2}/2s | s_{1}s_{2}/s | r_{1}r_{2}/2s | 0 | (s_{1}s_{ 2 }- r_{1}r_{2})/2s | $\frac{2{s}_{1}{s}_{2}{r}_{1}{r}_{2}}{{s}^{2}+{r}^{2}}$ | $\frac{{s}_{1}^{2}{s}_{2}^{2}+{r}_{1}^{2}{r}_{2}^{2}+{s}_{1}^{2}{r}_{2}^{2}+{r}_{1}^{2}{s}_{2}^{2}}{{s}^{2}+{r}^{2}}$ | $\frac{2{s}_{1}{s}_{2}{r}_{1}{r}_{2}}{{s}^{2}+{r}^{2}}$ |
M _{+} m _{+} m _{-} m _{-} | 0 | s_{1}r_{2}/r | r_{1}s_{2}/r | -r_{1}s_{2}/r | (s_{1}r_{ 2 }- r_{1}s_{2})/2r | r_{1}r_{2}s_{1}r_{2}/sr | (s_{1}s_{2}s_{1}r_{2} + r_{1}r_{2}r_{1}s_{2})/sr | s_{1}s_{2}r_{1}s_{2}/sr |
m _{+} m _{+} M _{-} M _{-} | - | - | - | - | - | r_{ 1 }^{ 2 }s^{2}_{2}/r^{2} | 2r_{1}s_{1}r_{2}s_{2}/r^{2} | s _{ 1 } ^{2} r ^{2} _{2} r ^{2} |
m _{+} m _{+} M _{-} m _{-} | 0 | r_{1}s_{2}/r | s_{1}r_{2}/r | -s_{1}r_{2}/r | (r_{1}s_{2} - s_{1}r_{2})/2r | r_{1}r_{2}r_{1}s_{2}/sr | (s_{1}s_{2}r_{1}s_{2} + r_{1}r_{2}s_{1}r_{2})/sr | s_{1}s_{2}s_{1}r_{2}/sr |
m _{+} m _{+} m _{-} m _{-} | 0 | r_{ 1 }r_{2}/s | s_{1}s_{2}/s | -s_{1}s_{2}/s | (r_{1}r_{2} - s_{1}s_{2})/2s | r_{ 1 }^{2}r_{ 2 }^{2}/s^{2} | 2s_{1}s_{2}r_{1}r_{2}/s^{2} | s _{ 1 } ^{2} s _{ 2 } ^{2/} s ^{2} |
The above QTL full model, assuming the number of QTLs and their positions are known, can be used to detect the significance of QTL effects. Based on the final QTL full model after excluding insignificant QTLs, all genetic effects of QTL and QTL × sex interaction effects will be estimated by the mixed linear model approach.
Scanning for QTLs with main effects
where a^{+}_{th} (a^{-}_{ th }) and d^{+}_{ th }(d^{-}_{ th }) are the additive and dominance effects due to the right (left) marker of the t-th marker interval in the h-th sex, with corresponding coefficients ζ^{+}_{A}(ζ^{-}_{A}) and ζ^{+}_{D}(ζ^{-}_{D}); the other parameters have the same definition as those in model (1). ζ^{+}_{A}(ζ^{-}_{A}) takes value of 1, 0, -1 when the marker genotype is MM, Mm and mm, respectively. ζ^{+}_{D}(ζ^{-}_{D}) takes value of -0.5 for homozygous genotype (MM and mm) and 0.5 for heterozygote (Mm). If the marker genotype information is missing, the transitional-possibility-matrix algorithm will be employed to calculate their expected values. To determine which pair of adjacent markers should be selected, the F-testing is applied and the threshold value is determined by permutations [40]. After all the marker intervals exceeding the F-critical value are included into the model, stepwise model selection method is performed to eliminate all ghost peaks.
where Q denotes QTL genetic effects with coefficient matrix of X_{ Q }, and M denotes maker effects with coefficient matrix of X_{ M }, X_{ QM }is a matrix catenated by the X_{ Q }and X_{ M }; n is the number of observed values, rank (X_{ QM }) and rank (X_{ M }) are the ranks of matrix X_{ QM }and X_{ M }, respectively; SSR(Q|M) is the extra sum of squares due to the genetic effects of the putative QTL given the inclusion of M in the model; SSE is the residual sum of squares. SSR(Q|M) and SSE can be calculated using Henderson III method [41]. The permutation technique is used to determine the critical value of F-test. For all the QTLs detected to be significant at the level of 0.05, the stepwise selection is conducted to eliminate the false positive peaks.
Scanning for paired QTLs with epistatic effects
where aa^{+}_{ ph }(aa^{-}_{ ph }), ad^{+}_{ ph }(ad^{-}_{ ph }), da^{+}_{ ph }(da^{-}_{ ph }) and dd^{+}_{ ph }(dd^{-}_{ ph }) denote the additive-additive, additive-dominance, dominance-additive and dominance-dominance epistatic effects within the h-th sex between two right (left) markers of the p-th marker interval pairs, respectively; the coefficients of epistatic effects can be calculated by the products of coefficients of marker major effects in model (3); other parameters are defined the same as those in model (4). For each paired marker intervals tested, the F-statistic to test its extra effects is calculated using the formula (5), and the critical value to declare significance is specified by calculating F-statistic in a series of randomly shuffling observation vector y s. All paired intervals above the critical value are then picked up as significant candidate interactions.
where mp is the number of selected interval pairs, mi is the number of QTLs identified in one-dimensional searching; all other parameters are defined the same as those in model (1) and model (6). Similar F-test and selection procedure are applied.
Estimation of QTL parameters in the full model
After the number and positions of QTLs are specified, a full model consisting of all genetic effects of QTLs and their interaction effects with environment (sex) is established. The variance components of random effects can be estimated by restricted maximum likelihood (REML), the fixed effects by generalized least squares (GLS) or ordinary least squares (OLS), and the random effects by adjusted unbiased prediction (AUP). These mixed-model estimates of QTL effects are set to be the initial values of MCMC methods [42]. The sample distributions of QTL parameters are obtained by the Gibbs sampling [16, 43]. Finally, each effect is estimated by the distribution mean, while significance of an effect is tested by t statistic.
Numerical calculation of the difference in the coefficients of QTL effects between the achiasmate and the chiasmate models
As there is no genetic material exchange for female silkworm when producing gamete in meiosis, the marker frequency distribution and the conditional probabilities of QTL genotypes in silkworm F_{2} population are substantially different from those in the normal F_{2} population with chiasma. To demonstrate the difference in QTL detection and investigate the inappropriateness of QTL mapping of silkworm traits based on the traditional genetic model, we compared the conditional probabilities of QTL genotypes given the flanking marker genotypes under the achiasmata F_{2} and the traditional chiasmata F_{2} models that were calculated from Table 1.
We used two cases where r equal to 0.09 (10 cM) and 0.16 (20 cM), respectively, to evaluate the difference in the additive and dominance coefficients between two models summarized in the following steps: (1) to set r fixed and r_{1} changed from 0 to r; (2) to calculate each coefficient in two models based on the QTL conditional probability in Table 1 for every given marker genotype; (3) to calculate the absolute difference (D_{ i }= |x_{ ai }- x_{ ci }|) and relative difference (R_{ i }= |x_{ ai }- x_{ ci }|/x_{ ai }) for the i-th flanking marker genotype (7 totally), x_{ a }(x_{ c }) is the coefficient of QTL effect in the achiasmate (chiasmate) model; and (4) to investigate the maximum and the minimum of the set of D_{ i }and R_{ i }(i = 1, 2,..., 7) for every r_{1}.
Simulation scenarios
Estimation of QTL positions and main effects with Models I, II and III ^{a}
QTL^{b} | Chr. | Pos. | A | D | Power | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | |||||||||||
I | II | III | I | II | III | I | II | III | I | II | III | |||||
Q1 | 1 | 44 | 44.02 (3.14) | 44.07 (3.43) | 44.02 (3.14) | 3.88 | 3.88 (0.38) | 3.92 (0.52) | 3.91 (0.43) | -2.3 | -2.21 (0.53) | -2.27 (0.73) | -2.22 (0.60) | 96.7 | 93.3 | 96.7 |
Q2 | 2 | 45 | 45.20 (4.09) | 45.36 (5.40) | 45.20 (4.09) | -2.4 | -2.43 (0.42) | -2.48 (0.54) | -2.48 (0.48) | 1.9 | 1.80 (0.51) | 1.83 (0.76) | 1.84 (0.61) | 91 | 84.3 | 91 |
Q3 | 3 | 50 | 50.18 (2.13) | 50.21 (3.27) | 50.18 (2.13) | 3.2 | 3.25 (0.38) | 3.29 (0.48) | 3.33 (0.44) | 2.1 | 2.05 (0.52) | 2.13 (0.74) | 2.09 (0.61) | 98.3 | 94.7 | 98.3 |
Q4 | 4 | 73 | 73.24 (4.23) | 73.29 (5.77) | 73.24 (4.23) | -2.8 | -2.67 (0.41) | -2.74 (0.50) | -2.71 (0.44) | -1.9 | -1.84 (0.53) | -1.87 (0.73) | -1.85 (0.60) | 92 | 87.7 | 92 |
Q5 | 5 | 15 | 15.28 (6.40) | 16.52 (7.94) | 15.28 (6.40) | 1.9 | 1.98 (0.36) | 2.17 (0.52) | 2.13 (0.42) | 0 | -0.01 (0.55) | -0.09 (0.85) | -0.08 (0.67) | 62.7 | 49.7 | 62.7 |
Q6 | 2 | 75 | - | - | - | 0 | - | - | - | 0 | - | - | - | - | - | - |
Q7 | 4 | 24 | - | - | - | 0 | - | - | - | 0 | - | - | - | - | - | - |
Estimation of the positions and epistatic effects of the paired QTLs with Models I and II ^{a}
Epi. | Pos. i | Pos. j | AA | AD | DA | DD | Power | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | I | II | |||||||
I | II | I | II | I | II | I | II | I | II | I | II | |||||||||
E1 (Q1-Q7) | 44 | 44.22 (3.61) | 44.49 (3.82) | 24 | 24.13 (4.66) | 22.02 (5.55) | 3.09 | 2.92 (0.70) | 3.12 (0.89) | 0 | -0.00 (0.79) | -0.14 (1.13) | 2.34 | 2.21 (0.95) | 2.12 (1.70) | 0 | 0.11 (1.14) | 0.21 (2.87) | 44.7 | 27.3 |
E2 (Q3-Q5) | 50 | 50.04 (2.47) | 50.27 (3.35) | 15 | 15.41 (5.56) | 15.05 (6.13) | 2.6 | 2.34 (0.72) | 2.65 (0.78) | -2.1 | -1.84 (1.31) | -2.17 (1.09) | 0 | 0.04 (2.12) | -0.01 (1.16) | 3.2 | 2.93 (1.60) | 3.21 (1.57) | 81 | 49 |
E3 (Q6-Q7) | 75 | 75.03 (4.62) | 75.44 (5.03) | 24 | 24.07 (4.64) | 23.83 (4.94) | -3.7 | -3.33 (0.69) | -3.70 (0.82) | 0 | -0.05 (0.80) | -0.29 (1.24) | 0 | -0.06 (0.85) | 0.16 (1.27) | -1.9 | -1.74 (1.28) | -2.20 (1.63) | 58 | 32.7 |
In simulations, three different strategies were employed to conduct a genome-wide search for QTLs. The first one used the proposed model (1) (the silkworm F_{2} model), called Model I hereafter, the second used the traditional chiasmate F_{2} model (i.e., all coefficients in model (1) are replaced with those determined according to the genetic structure of a normal chiasmate F_{2} population), called Model II hereafter, and the third used the reduced version of model (1) where all epistatic effects and interaction effects of epistasis with sex were excluded, called Model III hereafter. The above three strategies were used to analyze the same simulated data sets generated by the silkworm F_{2} model with QTL effects, QTL × sex interaction effects.
We first examined the performance of the newly proposed strategy (Model I) and the traditional strategy (Model II) in mapping for silkworm traits and demonstrated the potential bias and loss of power caused by Model II. These simulation results were summarized in Tables 2 and 3. As the role of epistasis in the genetic control of complex traits has been well recognized, the comparison between Model I and Model III could offer us insight into epistasis detection. These results are listed in Tables 2 and 5.
Results
Comparison of coefficients in models for the achiasmate and the normal chiasmate F_{2} populations
As shown by the formula of conditional probability in Table 1, we could see that each probability value in achiasmate model is approximately equal to a first order function of r_{1}, r_{2} or r, while, each one in chiasmate model approximates to a second order function of recombination rate, suggesting that there should be considerable difference between the two models, which can potentially result in estimation bias and loss in accuracy.
According to the above comparison, it could be concluded that the traditional chiasmate F_{2} model would lead to a biased estimation when it was applied to mapping QTLs underlying silkworm traits and our proposed method would improve QTL mapping accuracy.
Comparison between the models for the achiasmate silkworm and the normal chiasmate F_{2} populations
Model I and the traditional Model II were used to analyze the simulated data, and were compared for their abilities in estimating the position and genetic effects of QTLs. The results suggested that Model I had better estimation accuracy (relatively smaller bias and standard deviation) in QTL position and effects than Model II (Table 2). All bias of genetic main effects from Model I were less than 5% of the true values, whereas Model II sometimes gave a larger bias, e.g., the bias of the Q5 additive effect > 10% of the true value (Table 2). Model I had a considerably larger power to detect the five QTLs (ranging from 62.7 ~ 98.3) than Model II (ranging from 49.7 ~ 94.7) (Table 2), regardless of whether the QTL has interaction effects with sex (Q2, Q3, Q4) or not (Q1, Q5) (Table 5). For all QTLs detected in the 300 simulations, the false discovery rate of Model I is 6.17%, prominently smaller than that of Model II (9.56%) with a z-statistic value of -2.063 and a probability of 0.020 by Wilcoxon two-sample test. Model I also provided an estimation of QTL position closer to the true value and had smaller standard deviation (SD) than the model II did (Table 2).
We also compared the estimation accuracies of epistatic effects from Model I with those from Model II (Table 3). Although both the Model I and the Model II could estimate all epistatic effects reasonably well, Model I had relative smaller SD and greater power than Model II in detecting the three pairs of QTLs involved in epistatic interactions, wherein E1 stands for the interaction between one QTL with main effects and another without main effects, E2 stands for the interaction between both QTLs with main effects and E3 stands for the interaction between two QTLs without main effects. As for estimation of QTL position, Model I also outperformed Model II in accuracy.
Comparison of the silkworm F_{2} models with and without epistasis
For the estimation of QTL positions, it could be found there was a slight difference in the estimated values between two models when the detected QTL has additive or dominance effects, or their interaction effects with sex, as well as in the corresponding SD s (Table 2). Two models had the same power in detecting QTL (Table 2), although Model I included more parameters of QTL than Model III. As for the paired QTLs with purely epistatic effects, they couldn't be identified by Model III, but could be detected by Model I. Such a result is expected, considering that (1) Model I and Model III used the same marker genotype information and quantitative trait values, (2) one dimensional scanning procedure did not include epistasis in Model III, and (3) the position of QTL was determined by the result of one dimensional scanning.
Although there was a small difference in position estimation, the estimates of additive and dominance effects were apparently different between two models (Table 2). Most of the estimated values in Model I were closer to the true values as compared with those in Model III. We could also see that, in Model I every estimate of additive or dominance effects had smaller SD than that in Model III, suggesting that Model I could provide more stable and more unbiased estimation (Table 2). As for QTL × sex interaction, although Model III could also give relatively accurate estimation, it was clear that the results of Model I were better than those of Model III, in terms of biasedness or SD (see Table 5).
Prediction of QTL × sex interaction effects
Estimation of epistasis-sex interaction effects with Model I ^{a}
Epi. | AAS1 | AAS2 | ADS1 | ADS2 | DAS1 | DAS2 | DDS1 | DDS2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | |
E1 | 0 | -0.05(0.39) | 0 | 0.05(0.39) | 1.7 | 1.25(0.95) | -2 | -1.25(0.95) | 0 | 0.05(0.44) | 0 | -0.05(0.44) | 0 | -0.00(0.66) | 0 | 0.01(0.66) |
E2 | -1 | -1.02(0.75) | 1.4 | 1.04(0.75) | 1.6 | 1.05(0.99) | -2 | -1.05(0.99) | 0 | -0.02(0.43) | 0 | 0.02(0.43) | 0 | -0.03(0.62) | 0 | 0.03(0.62) |
E3 | 0 | -0.00(0.40) | 0 | 0.00(0.40) | 0 | 0.03(0.54) | 0 | -0.03(0.54) | 1.7 | 1.39(1.00) | -2 | -1.38(1.00) | 1.8 | 1.12(1.27) | -2 | -1.12(1.28) |
Estimation of QTL-sex interaction effects with Model I and Model III ^{a}
QTL | AS1 | AS2 | DS1 | DS2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | Par. | Est.( SD ) | |||||
I | III | I | III | I | III | I | III | |||||
Q1 | 0 | 0.02(0.28) | -0.01(0.29) | 0 | -0.02(0.28) | 0.01(0.29) | 0 | -0.01(0.33) | -0.01(0.36) | 0 | 0.00(0.33) | 0.01(0.36) |
Q2 | -1.7 | -1.53(0.42) | -1.52(0.48) | 1.7 | 1.54(0.42) | 1.53(0.48) | 0 | 0.02(0.40) | 0.04(0.38) | 0 | -0.01(0.39) | -0.03(0.38) |
Q3 | -1.6 | -1.53(0.41) | -1.54(0.47) | 1.6 | 1.54(0.41) | 1.54(0.47) | -1.5 | -1.28(0.60) | -1.23(0.74) | 1.4 | 1.30(0.60) | 1.19(0.71) |
Q4 | 0 | -0.02(0.22) | -0.03(0.26) | 0 | 0.02(0.22) | 0.03(0.26) | 1.6 | 1.36(0.65) | 1.30(0.75) | -1.6 | -1.36(0.65) | -1.30(0.75) |
Q5 | 0 | 0.02(0.22) | -0.01(0.27) | 0 | -0.02(0.22) | 0.01(0.27) | 0 | 0.01(0.29) | 0.01(0.38) | 0 | -0.01(0.29) | -0.01(0.38) |
Discussion
Crossing-over is an important issue in organisms with meiosis, which can not only enrich phenotypic variation among individuals but also speed up the process of evolution. However, there are many exceptions such as Drosophila and silkworm, which are not only of great value as model organism for biology study but also important for agriculture. Drosophila and silkworm have a common characteristic in reproduction that crossing-over occurs only in one sex, while there is a difference that such a phenomenon occurs in female parent for silkworm and in male parent for Drosophila. The particular action in meiosis, also called achiasmata, needs to be considered in the process of gene mapping. In order to avoid the problem on the achiasmate and use available genetic model and software for a normal chiasmate mapping population, investigators have proposed to conduct QTL analysis based on BC population of silkworm. However, there are obvious disadvantages in this solution: recombination information and diversity of genotype in the BC population are not so rich as the F_{2} population for unraveling genetic architecture of complex traits where complex epistasis and gene × sex or environment interaction are involved. The existing studies on constructing linkage map or genetic mapping with F_{2} populations for silkworm [5, 44] chose to simply neglect such a recombination difference between two sexes because of lack of appropriate analytical method. This potentially leads to bias and loss in precision as, in silkworm F_{2} population, every individual receives one gamete from female parent without crossing-over and another one generated by potential sister-chromatid exchange from male parent, resulting in a different population structure from the normal chiasmate F_{2} population. Thus, our proposed method that can accommodate achiasmata phenomenon and also effectively handle epistatic effects and the interaction effects of QTL and environment, represents a necessary addition to the current toolkit of QTL mapping.
Many of the widely used statistical methods and software, such as IM and CIM, do not include sex effects in the models because they are mainly designed for plants such as Arabidopsis thaliana and rice. But in animals, many quantitative traits are sex dependent and behave much differently between males and females, such as the cocoon traits of silkworm. Sex specific traits can be categorized into three types of inheritance: sex-limited, sex-influenced (also known as sex-controlled), and sex-linked; the former two of which are controlled by autosomal gene(s) and sex, in our term, in which there are sex and/or gene × sex interaction effects, and the latter one is caused by gene(s) carried on the sex chromosomes. It has been well documented that there are sex differences in terms of the presence/absence and locations of QTLs [45], as well as the interaction of QTL with sex [46–48]. Thus, for the purpose of improving analysis power, the sex effect and the QTL × sex interaction effect should be generally included into the analysis model as a covariate to eliminate the influence from sex. In our study, as in some literature [49], the sex effect is considered in the QTL model as a random effect for the purpose of background control. Simulation results showed the proposed method could improve both statistical power to detect a QTL and estimation accuracy for genetic effects of QTL and QTL × sex interaction. We like to point out that the sex and the sex related interaction effects can also be treated as fixed ones in our model when necessary.
The method presented here is mainly to detect QTLs on non-sex chromosomes. Sex chromosomes usually play a unique role in many biological processes and phenomena, including sex determination, epigenetic chromosome-wide regulation of gene expression [50]. Sex chromosomes have many different genetic features compared with autosomes and there is extra complexity in mapping of sex-linked genetically inherited traits. First, there are two categories of sex determination systems: heterogametic male (XY) and heterogametic female (ZW), and the heterogametic sex is hemizygous in which gene dosage effect or dosage compensation mechanism may occur. Second, sex chromosomes can show sex-biased transmission. Third, there may also exist random inactivation of the sex chromosome. Broman et al. [51] addressed that if the sex chromosome is treated like an autosome, a sex difference in the phenotype can lead to spurious linkage on the sex chromosome. Further, the number of degrees of freedom for the linkage test may be different for the sex chromosome than for autosomes, and so sex chromosome-specific significance threshold is required. Given the complexity of sex-linked inheritance, tailored mapping methods are needed to effectively hunt sex-linked genes. Thus, Broman et al. [51] proposed a method for mapping QTL on X chromosome in experimental crosses population. Zhang et al. [52] developed a family-based association test to detect QTLs on X-chromosome under consideration of the dosage effect due to female X chromosome inactivation. It is possible to extend the proposed method to mapping QTLs on the sex chromosomes.
The genetic variation in continuous traits is usually governed by a polygenic network system, composed of many genes with a small effect, and sometimes one or a few genes of large effect. Recently, intensive studies on quantitative variation have pointed to that epistasis is usually involved in genetic variation of quantitative traits. Strong interactions between QTLs have been observed in maize [53], soybean [54] and Drosophila[55]. In addition, QTLs with minor or no individual effect can also be involved in epistatic interaction [56]. More and more attention has been paid to molecular dissection of epistasis. Our proposed model includes not only the digenic epistatic effects, but also their interaction effects with sex. Therefore, this model can well tackle the complexity of quantitative trait in silkworm. Simulations revealed that the proposed method could present better estimation of QTL parameters no matter whether or not the epistasis and/or their interaction with sex exists.
Lastly but not least, it should be pointed out that only seven different genotypes of two QTL loci on the same chromosome can be generated in F_{2} population because of the female achiasmate of silkworm [2], while, in the full model, eight genetic effects of a pair of QTLs (two additive effects, two dominance effects and four epistatic effects) need to be estimated. Therefore, under this situation, the proposed method could not produce unbiased estimate of all eight fixed effects. An elaborately planned design is required to effectively detect epistases between interacting loci located on the same chromosome due to the insufficient number of segregating genotypes in an achiasmate F_{2} population. One alternative choice is excluding the higher-order genetic effects of QTL (additive-dominance, dominance-additive, dominance-dominance epistatic effects) from the model. However, for the case in which two QTLs are residing on two different chromosomes, there are still nine QTL genotypes segregated in F_{2} population of silkworm since the chromosome of female parent could be passed independently to its progeny. It explained why Model I could well estimate all epistatic effects in our simulation study. Furthermore, fortunately, the position of QTL could be estimated unbiasedly no matter whether the QTLs are distributed on the same chromosome or not, since the QTL position is distinguished based on the F-statistic measuring the total extra effects due to tested variables in the model which is not affected by the correlation between these variables.
Conclusion
We have developed a genetic model for mapping QTL in silkworm F2 population which could analyze the additive effect, dominance effect, digenic epistatic effect and their interaction effects with sex, and correct the potential bias and precision loss in the current QTL mapping practice of silkworm, thus representing an important addition to the arsenal of QTL mapping tools.
Declarations
Acknowledgements
This work was supported in part by the National Basic Research Program of China (973 Program) 2007CB109007, the National Special Program for Breeding New Transgenic Variety 2009ZX08009-004B, the National Natural Science Foundation 30571405, and the National Institutes of Health Grant DA025095.
Authors’ Affiliations
References
- Xia QY, Zhou ZY, Lu C, Cheng DJ, Dai FY, Li B, Zhao P, Zha XF, Cheng TC, Chai CL: A draft sequence for the genome of the domesticated silkworm (Bombyx mori). Science. 2004, 306 (5703): 1937-1940. 10.1126/science.1102210.View ArticlePubMedGoogle Scholar
- Nagaraju J, Goldsmith MR: Silkworm genomics - progress and prospects. Curr Sci India. 2002, 83 (4): 415-425.Google Scholar
- Tan YD, Wan CL, Zhu YF, Lu C, Xiang ZH, Deng HW: An amplified fragment length polymorphism map of the silkworm. Genetics. 2001, 157 (3): 1277-1284.PubMed CentralPubMedGoogle Scholar
- Yamamoto K, Nohata J, Kadono-Okuda K, Narukawa J, Sasanuma M, Sasanuma S, Minami H, Shimomura M, Suetsugu Y, Banno Y: A BAC-based integrated linkage map of the silkworm Bombyx mori. Genome Biol. 2008, 9 (1): 10.1186/gb-2008-9-1-r21.Google Scholar
- Yasukochi Y: A dense genetic map of the silkworm, Bombyx mori, covering all chromosomes based on 1018 molecular markers. Genetics. 1998, 150 (4): 1513-1525.PubMed CentralPubMedGoogle Scholar
- Zhan SA, Huang JH, Guo QH, Zhao YP, Li WH, Miao XX, Goldsmith MR, Li MW, Huang YP: An integrated genetic linkage map for silkworms with three parental combinations and its application to the mapping of single genes and QTL. Bmc Genomics. 2009, 10: 10.1186/1471-2164-10-389.Google Scholar
- Lander ES, Botstein D: Mapping Mendelian Factors Underlying Quantitative Traits Using Rflp Linkage Maps. Genetics. 1989, 121 (1): 185-199.PubMed CentralPubMedGoogle Scholar
- Zeng ZB: Precision Mapping of Quantitative Trait Loci. Genetics. 1994, 136 (4): 1457-1468.PubMed CentralPubMedGoogle Scholar
- Mao YC, London NR, Ma L, Dvorkin D, Da Y: Detection of SNP epistasis effects of quantitative traits using an extended Kempthorne model. Physiol Genomics. 2006, 28 (1): 46-52. 10.1152/physiolgenomics.00096.2006.View ArticlePubMedGoogle Scholar
- Symington LS: Role of RAD52 epistasis group genes in homologous recombination and double-strand break repair. Microbiol Mol Biol R. 2002, 66 (4): 630-+. 10.1128/MMBR.66.4.630-670.2002.View ArticleGoogle Scholar
- Tabanao DA, Yu J, Bernardo R: Multilocus epistasis, linkage, and genetic variance in breeding populations with few parents. Theor Appl Genet. 2007, 115 (3): 335-342. 10.1007/s00122-007-0565-6.View ArticlePubMedGoogle Scholar
- Kao CH, Zeng ZB, Teasdale RD: Multiple interval mapping for quantitative trait loci. Genetics. 1999, 152 (3): 1203-1216.PubMed CentralPubMedGoogle Scholar
- Ljungberg K, Holmgren S, Carlborg O: Simultaneous search for multiple QTL using the global optimization algorithm DIRECT. Bioinformatics. 2004, 20 (12): 1887-1895. 10.1093/bioinformatics/bth175.View ArticlePubMedGoogle Scholar
- Piepho HP: A mixed-model approach to mapping quantitative trait loci in barley on the basis of multiple environment data. Genetics. 2000, 156 (4): 2043-2050.PubMed CentralPubMedGoogle Scholar
- Wang DL, Zhu J, Li ZK, Paterson AH: Mapping QTLs with epistatic effects and QTLxenvironment interactions by mixed linear model approaches. Theor Appl Genet. 1999, 99 (7-8): 1255-1264. 10.1007/s001220051331.View ArticleGoogle Scholar
- Yang J, Zhu J, Williams RW: Mapping the genetic architecture of complex traits in experimental populations. Bioinformatics. 2007, 23 (12): 1527-1536. 10.1093/bioinformatics/btm143.View ArticlePubMedGoogle Scholar
- Yi NJ, Shriner D, Banerjee S, Mehta T, Pomp D, Yandell BS: An efficient Bayesian model selection approach for interacting quantitative trait loci models with many effects. Genetics. 2007, 176 (3): 1865-1877. 10.1534/genetics.107.071365.PubMed CentralView ArticlePubMedGoogle Scholar
- Lou XY, Chen GB, Yan L, Ma JZ, Zhu J, Elston RC, Li MD: A generalized combinatorial approach for detecting gene-by-gene and gene-by-environment interactions with application to nicotine dependence. Am J Hum Genet. 2007, 80 (6): 1125-1137. 10.1086/518312.PubMed CentralView ArticlePubMedGoogle Scholar
- Gao YM, Zhu J: Mapping QTLs with digenic epistasis under multiple environments and predicting heterosis based on QTL effects. Theor Appl Genet. 2007, 115 (3): 325-333. 10.1007/s00122-007-0564-7.View ArticlePubMedGoogle Scholar
- Satagopan JM, Yandell YS, Newton MA, Osborn TC: A Bayesian approach to detect quantitative trait loci using Markov chain Monte Carlo. Genetics. 1996, 144 (2): 805-816.PubMed CentralPubMedGoogle Scholar
- Yi NJ, Yandell BS, Churchill GA, Allison DB, Eisen EJ, Pomp D: Bayesian model selection for genome-wide epistatic quantitative trait loci analysis. Genetics. 2005, 170 (3): 1333-1344. 10.1534/genetics.104.040386.PubMed CentralView ArticlePubMedGoogle Scholar
- Fang M, Jiang D, Gao HJ, Sun DX, Yang RQ, Zhang Q: A new Bayesian automatic model selection approach for mapping quantitative trait loci under variance component model. Genetica. 2009, 135 (3): 429-437. 10.1007/s10709-008-9291-5.View ArticlePubMedGoogle Scholar
- Durrant C, Mott R: Bayesian Quantitative Trait Locus Mapping Using Inferred Haplotypes. Genetics. 2010, 184 (3): 839-U375. 10.1534/genetics.109.113183.PubMed CentralView ArticlePubMedGoogle Scholar
- Fang M: Bayesian shrinkage mapping of quantitative trait loci in variance component models. Bmc Genet. 2010, 11: 10.1186/1471-2156-11-30.Google Scholar
- Yang RQ, Xu SZ: Bayesian shrinkage analysis of quantitative trait loci for dynamic traits. Genetics. 2007, 176 (2): 1169-1185. 10.1534/genetics.106.064279.PubMed CentralView ArticlePubMedGoogle Scholar
- Yi NJ, Banerjee S, Pomp D, Yandell BS: Bayesian mapping of genomewide interacting quantitative trait loci for ordinal traits. Genetics. 2007, 176 (3): 1855-1864. 10.1534/genetics.107.071142.PubMed CentralView ArticlePubMedGoogle Scholar
- Liu JF, Liu YJ, Liu XG, Deng HW: Bayesian mapping of quantitative trait loci for multiple complex traits with the use of variance components. Am J Hum Genet. 2007, 81 (2): 304-320. 10.1086/519495.PubMed CentralView ArticlePubMedGoogle Scholar
- Xu CW, Wang XF, Li ZK, Xu SZ: Mapping QTL for multiple traits using Bayesian statistics. Genetics Research. 2009, 91 (1): 23-37. 10.1017/S0016672308009956.View ArticlePubMedGoogle Scholar
- Yi NJ, Banerjee S: Hierarchical Generalized Linear Models for Multiple Quantitative Trait Locus Mapping. Genetics. 2009, 181 (3): 1101-1113. 10.1534/genetics.108.099556.PubMed CentralView ArticlePubMedGoogle Scholar
- Zou F, Huang HW, Lee S, Hoeschele I: Nonparametric Bayesian Variable Selection With Applications to Multiple Quantitative Trait Loci Mapping With Epistasis and Gene-Environment Interaction. Genetics. 2010, 186 (1): 385-U600. 10.1534/genetics.109.113688.PubMed CentralView ArticlePubMedGoogle Scholar
- Zhang W, Zhu J, Schadt EE, Liu JS: A Bayesian Partition Method for Detecting Pleiotropic and Epistatic eQTL Modules. Plos Comput Biol. 2010, 6 (1): 10.1371/journal.pcbi.1000642.Google Scholar
- He W, Fernando RL, Dekkers JCM, Gilbert H: A gene frequency model for QTL mapping using Bayesian inference. Genet Sel Evol. 2010, 42: 10.1186/1297-9686-42-21.Google Scholar
- Fang M, Jiang D, Chen X, Pu LJ, Liu SC: Bayesian analysis of genetic architecture of quantitative trait using data of crosses of multiple inbred lines. Genetica. 2008, 134 (3): 367-375. 10.1007/s10709-008-9244-z.View ArticlePubMedGoogle Scholar
- Lu C, Li B, Zhao AC, Xiang ZH: QTL mapping of economically important traits in silkworm (Bombyx mori). Sci China Ser C. 2004, 47 (5): 477-484. 10.1360/03yc0260.View ArticleGoogle Scholar
- Mirhoseini SZ, Rabiei B, Potki P, Dalirsefat SB: Amplified fragment length polymorphism mapping of quantitative trait loci for economically important traits in the silkworm, Bombyx mori. J Insect Sci. 2010, 10: 10.1673/031.010.14113.Google Scholar
- Lie Z, Cheng L, Fang-yin D, Shou-min F: Mapping of major quantitative trait loci for economic traits of silkworm cocoon. Genet Mol Res. 2010, 9 (1): 78-88. 10.4238/vol9-1gmr676.View ArticlePubMedGoogle Scholar
- Mirhosseini SZ, Bizhannia AR, Rabiei B, Taeb M, Siedavi AR: Identification of AFLP markers linked with cocoon weight genes in silkworm (Bombyx mori L.). Afr J Biotechnol. 2010, 9 (10): 1427-1433.Google Scholar
- Jiang CJ, Zeng ZB: Mapping quantitative trait loci with dominant and missing markers in various crosses from two inbred lines. Genetica. 1997, 101 (1): 47-58. 10.1023/A:1018394410659.View ArticlePubMedGoogle Scholar
- Piepho HP, Gauch HG: Marker pair selection for mapping quantitative trait loci. Genetics. 2001, 157 (1): 433-444.PubMed CentralPubMedGoogle Scholar
- Doerge RW, Churchill GA: Permutation tests for multiple loci affecting a quantitative character. Genetics. 1996, 142 (1): 285-294.PubMed CentralPubMedGoogle Scholar
- Searle SR, Casella G, Mcculloch C: Variance components. 1992, New York: A Wiley-Interscience Publication, Hohn Wiley & Sons, INCView ArticleGoogle Scholar
- Macedo FWM, Gianola D: Bayesian analysis of univariate mixed models with informative priors. European Association for Animal Production, 38th Annual Meeting, Lisbon, Portugal. 1987, 35-Google Scholar
- Wang CS, Rutledge JJ, Gianola D: Bayesian-Analysis of Mixed Linear-Models Via Gibbs Sampling with an Application to Litter Size in Iberian Pigs. Genet Sel Evol. 1994, 26 (2): 91-115. 10.1186/1297-9686-26-2-91.PubMed CentralView ArticleGoogle Scholar
- Promboon A, Shimada T, Fujiwara H, Kobayashi M: Linkage Map of Random Amplified Polymorphic Dnas (Rapds) in the Silkworm, Bombyx-Mori. Genet Res. 1995, 66 (1): 1-7. 10.1017/S0016672300034339.View ArticleGoogle Scholar
- Yu HR, Edderkaoui B, Cortez A, Davidson H, Wergedal J, Baylink D, Mohan S: Mapping of the chromosome 17 BMD QTL in the F-2 male mice of MRL/MpJ × SJL/J. Genetica. 2009, 135 (1): 59-66. 10.1007/s10709-008-9258-6.View ArticlePubMedGoogle Scholar
- Abasht B, Pittel F, Lagarrigue S, Le Bihan-Duval E, Le Roy P, Demeure O, Vignoles F, Simon J, Cogburn L, Aggrey S: Fatness QTL on chicken chromosome 5 and interaction with sex. Genet Sel Evol. 2006, 38 (3): 297-311. 10.1186/1297-9686-38-3-297.PubMed CentralView ArticlePubMedGoogle Scholar
- Farber CR, Medrano JF: Fine mapping reveals sex bias in quantitative trait loci affecting growth, skeletal size and obesity-related traits on mouse chromosomes 2 and 11. Genetics. 2007, 175 (1): 349-360. 10.1534/genetics.106.063693.PubMed CentralView ArticlePubMedGoogle Scholar
- Saless N, Franco GEL, Litscher S, Kattappuram RS, Houlihan MJ, Vanderby R, Demant P, Blank RD: Linkage mapping of femoral material properties in a reciprocal intercross of HcB-8 and HcB-23 recombinant mouse strains. Bone. 2010, 46 (5): 1251-1259. 10.1016/j.bone.2010.01.375.PubMed CentralView ArticlePubMedGoogle Scholar
- Zhu J, Weir BS: Diallel analysis for sex-linked and maternal effects. Theor Appl Genet. 1996, 92 (1): 1-9. 10.1007/BF00222944.View ArticlePubMedGoogle Scholar
- Kaiser VB, Bachtrog D: Evolution of sex chromosomes in insects. Annu Rev Genet. 2010, 44: 91-112. 10.1146/annurev-genet-102209-163600.PubMed CentralView ArticlePubMedGoogle Scholar
- Broman KW, Sen S, Owens SE, Manichaikul A, Southard-Smith EM, Churchill GA: The X chromosome in quantitative trait locus mapping. Genetics. 2006, 174 (4): 2151-2158. 10.1534/genetics.106.061176.PubMed CentralView ArticlePubMedGoogle Scholar
- Zhang L, Martin ER, Morris RW, Li YJ: Association Test for X-Linked QTL in Family-Based Designs. Am J Hum Genet. 2009, 84 (4): 431-444. 10.1016/j.ajhg.2009.02.010.PubMed CentralView ArticlePubMedGoogle Scholar
- Lukens LN, Doebley J: Epistatic and environmental interactions for quantitative trait loci involved in maize evolution. Genet Res. 1999, 74 (3): 291-302. 10.1017/S0016672399004073.View ArticleGoogle Scholar
- Lark KG, Chase K, Adler F, Mansur LM, Orf JH: Interactions between Quantitative Trait Loci in Soybean in Which Trait Variation at One Locus Is Conditional Upon a Specific Allele at Another. P Natl Acad Sci USA. 1995, 92 (10): 4656-4660. 10.1073/pnas.92.10.4656.View ArticleGoogle Scholar
- Gurganus MC, Nuzhdin SV, Leips JW, Mackay TFC: High-resolution mapping of quantitative trait loci for sternopleural bristle number in Drosophila melanogaster. Genetics. 1999, 152 (4): 1585-1604.PubMed CentralPubMedGoogle Scholar
- Montooth KL, Marden JH, Clark AG: Mapping determinants of variation in energy metabolism, respiration and flight in Drosophila. Genetics. 2003, 165 (2): 623-635.PubMed CentralPubMedGoogle Scholar
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