Abstract
Given the universality of autopolyploid species in nature, it is crucial to develop genomic selection methods that consider different allele dosages for autopolyploid breeding. However, no method has been developed to deal with autopolyploid data regardless of the ploidy level. In this study, we developed a modified genomic best linear unbiased prediction (GBLUP) model (polyGBLUP) through constructing additive and dominant genomic relationship matrices based on different allele dosages. polyGBLUP could carry out genomic prediction for autopolyploid species regardless of the ploidy level. Through comprehensive simulations and analysis of real data of autotetraploid blueberry and guinea grass and autohexaploid sweet potato, the results showed that polyGBLUP achieved higher prediction accuracy than GBLUP and its superiority was more obvious when the ploidy level of autopolyploids is high. Furthermore, when the dominant effect was added to polyGBLUP (polyGDBLUP), the greater the dominance degree, the more obvious the advantages of polyGDBLUP over the diploid models in terms of prediction accuracy, bias, mean squared error and mean absolute error. For real data, the superiority of polyGBLUP over GBLUP appeared in blueberry and sweet potato populations and a part of the traits in guinea grass population due to the high correlation coefficients between diploid and polyploidy genomic relationship matrices. In addition, polyGDBLUP did not produce higher prediction accuracy than polyGBLUP for most traits of real data as dominant genetic variance was not captured for these traits. Our study will be a significant promising method for genomic prediction of autopolyploid species.
Keywords: autopolyploid species, genomic prediction, genomic best linear unbiased prediction, allele dosages, dominance effects
INTRODUCTION
In nature, numerous economically important species exhibit autopolyploid, with most being autotetraploids, such as potato (Solanum tuberosum) [1], blueberry (Vaccinium spp.) [2] and guinea grass (Panicum maximum Jacq.) [3]. Additionally, there are species with high ploidy levels, including autohexaploid sweet potato (Ipomoea batatas (L.) Lam) [4], autooctaploid white sturgeon (Acipenser transmontanus) [5] and sugarcane (Saccharum spp., Poaceae) with ploidy levels ranging up to 16 [6]. Unlike diploids, which have only three genotypes per marker allele (
,
,
are encoded as 0, 1, 2), marker alleles in autopolyploid include different allele dosages depending on the ploidy level, e.g. autotetraploids include five genotypes, i.e. 
,
,
,
and 
are encoded as 0, 1, 2, 3, 4, where 
and 
are the reference and alternative allele, respectively. It usually takes longer to cultivate autopolyploid species than diploid species, and the development of improved varieties may take 10 to 20 years or more [7]. The possible reason is that autopolyploidy takes longer to eliminate harmful alleles due to the presence of more heterozygotes (e.g. 
,
,
in autotetraploid) [8]. Therefore, molecular breeding techniques (e.g. genomic selection [9]) are urgently needed to shorten the breeding cycle and improve the genetic gain of autopolyploid species.
Genomic selection (GS) [9], which relies on linkage disequilibrium (LD) between single-nucleotide polymorphisms (SNPs) and quantitative trait loci (QTL), has been widely applied in animal [10], plant [11] and aquaculture [12] breeding. In GS, SNP makers covering the whole genome are used to predict genomic estimated breeding values (GEBVs). Currently, many statistical methods have been proposed to predict GEBV. Among them, genomic best linear unbiased prediction (GBLUP) is widely used in the GS of animal and plant breeding in the world [13]. The core of GBLUP method is to construct a genomic relationship matrix (G) by using molecular markers covering the whole genome, and G is subsequently integrated into the framework of mixed model equations (MMEs) to directly estimate the GEBV [13]. In addition to additive effects, many studies have reported that incorporating non-additive effects (e.g. dominant effects) into the model can improve the accuracy of genomic prediction by constructing an additional dominance genomic relationship matrix (D) [14–16]. However, traditional G and D matrices were designed for diploids and could not handle genotypes of multiple allele dosages in autopolyploids, as heterozygote types could not be correctly distinguished. To address this limitation, hidden heterozygotes have been commonly applied in autopolyploid species by encoding homozygotes as 0 and 2 and all heterozygotes as 1 using GBLUP [17, 18]. It is important to note that in autopolyploids, the phenotype can be influenced by multiple copies of the same allele (additive effects) or more complex interactions between alleles (dominant effects), which may pose challenges for achieving accurate genomic predictions [19].
Leveraging allele dosage information in GBLUP to improve the accuracy of GS has been reported in autopolyploid species [20–25]. However, these studies primarily focused on extending the G matrix only to autotetraploid species [20–25]. To the best of our knowledge, no method has been developed to deal with autopolyploid data regardless of the ploidy level. Therefore, in this study, we propose a modified GBLUP method (polyGBLUP) by constructing additive and dominant genomic relationship matrices based on different allele dosages. For its general application, the efficiency of the proposed method was evaluated through simulation study and real data of autotetraploid blueberry, guinea grass and autohexaploid sweet potato.
METHODS AND MATERIALS
Classic genomic best linear unbiased prediction model
GBLUP
The classic GBLUP model [13] including additive genetic effects can be written as
 |
(1) |
where 
is the vector of phenotypic values, 
is the vector of fixed effect, 
is the incidence matrix associating 
with 
, 
is the vector of additive genetic effects, following a normal distribution of 
, where 
is the additive genetic variance and 
is the additive genomic relationship matrix [13]. 
is an incidence matrix linking 
to 
, and 
is the vector of random errors, following a normal distribution of 
, where 
is the identity matrix and 
is the residual variance. 
is constructed using all markers as follows:
 |
(2) |
where 
represents the alternative allele frequency at locus 
, 
. 
is the centered marker matrix. The element of 
for an individual 
at the marker 
is
 |
(3) |
We could know that the genotype (
, 
, 
encoded as 0, 1, 2) frequencies were 
, 
and 
; thus, the mean of additive effects
 |
(4) |
and the additive genetic variance is equal to 
, so
 |
(5) |
GDBLUP
The GBLUP including additive and dominance genetic effects is termed GDBLUP [14]. The statistical model of GDBLUP can be expressed as
 |
(6) |
where 
, 
, 
, 
, 
and 
are same as in Equation (1), 
is the vector of dominance genetic effects, following a normal distribution of 
, where 
is the dominance genetic variance and 
is the dominance genomic relationship matrix [14]. 
is created as follows:
 |
(7) |
The element of 
for the 
individual at the 
th marker is calculated as follows:
 |
(8) |
Also, like additive effect values, the mean of dominance deviation is
 |
(9) |
and the dominance genetic variance is equal to 
, so
 |
(10) |
For GBLUP and GDBLUP, the SNP genotypes of autopolyploid were encoded as diploid allele dosage, i.e. homozygotes were encoded as 0 and 2, and heterozygotes encoded as 1, e.g. autotetraploid SNP genotypes were encoded as 0 (
), 1 (
), 1 (
), 1 (
) and 2 (
).
Modified genomic best linear unbiased prediction model for autopolyploids
polyGBLUP
The modified GBLUP (polyGBLUP) model has the same model as GBLUP in Equation (1), except that
is the polyploid additive genomic relationship matrix (
). According to formulas (3–5), for autotetraploids, the element of 
for an individual 
at the marker 
is
 |
(11) |
The genotype frequencies of 
were 
, 
, 
, 
, 
, respectively; thus, the mean of additive effects
 |
(12) |
and the additive genetic variance is equal to 
, so
 |
(13) |
Therefore, we can infer the additive effect values of different genotypes (Table 1) and additive genetic variances (Table 2) under different ploidy levels; according to formula (2), 
can be generalized as
Table 1.
The additive effect and dominance deviation values of different genotypes under different ploidy levels
| Genotype | Ploidy = 2 | Ploidy = 4 | Ploidy = 6 | Ploidy = 8 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Freq.a | Add.b | Dom.c | Freq.a | Add.b | Dom.c | Freq.a | Add.b | Dom.c | Freq.a | Add.b | Dom.c | |
| 8 |
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| 7 |
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| 6 |
|
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| 5 |
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| 4 |
|
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| 3 |
|
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| 2 |
|
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|
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| 1 |
|
|
|
|
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| 0 |
|
|
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|
aGenotype frequency
bCoefficient of additive effect value
cCoefficient of dominance deviation value
Table 2.
The additive and dominance genetic variance and the elements of Z and W matrices under different ploidy levels
aThe element of 
in additive genomic relationship matrix
bAdditive genetic variance
cThe element of 
in dominance genomic relationship matrix
dDominance genetic variance
 |
(14) |
where 
, 
is the 
matrix of allele frequency 
, 
is the number of individuals and 
is the number of SNPs, 
is 
genotype matrix corresponds to the dosage of alternative allele, ploidy is the ploidy level value, e.g. ploidy = 8 for octoploid species,
can be expressed as
 |
(15) |
in which case the 
for genotypes 0, 1, 2, 3, 4, 5, 6, 7 and 8 are 
, 
, 
, 
, 
, 
, 
, 
and 
, respectively, and 
.
polyGDBLUP
The polyGDBLUP including additive and dominance genetic effects for autopolyploids has the same expression as Equation (6), except that it used the polyploid dominance genomic relationship matrix (
). According to the dominance deviation values in formula (8) and the dominance deviation values of genotypes 
, 
and 
can be de-coefficiented as follows [26]:
 |
(16) |
Based on this, we can infer the dominant deviation values of different polyploid genotypes, e.g. for tetraploid species, there are three possible combinations of two 
alleles and three possible combinations of one 
allele with one 
allele for genotype 
, and the dominant deviation value of 
equals to 
; therefore, the dominant deviation values of five different genotypes are as follows:
 |
(17) |
so 
where 
is genotype corresponds to the dosage of alternative allele. According to formulas (9 and 10), the mean of dominance deviation is
 |
(18) |
and the dominance genetic variance is equal to 
, so
 |
(19) |
Therefore, the dominant deviation values and dominance genetic variances for different genotypes at different ploidy levels can be inferred in Tables 1 and 2. According to formula (7), 
can be generalized as
 |
(20) |
 |
(21) |
where 
is an 
diagonal matrix and the diagonal element 
=
; 
denotes the 
diagonal matrix whose diagonal element is ploidy level value (ploidy) minus 
; 
denotes the element-wise (Hadamard) product. The other elements are already defined, e.g. ploidy = 8 for octoploid species, 
can be expressed as
 |
(22) |
in which case the 
for genotypes 0, 1, 2, 3, 4, 5, 6, 7 and 8 are 
, 
, 
, 
, 
, 
, 
， 
and 
, respectively.
We implemented GBLUP, GDBLUP, polyGBLUP and polyGDBLUP for genomic prediction of autopolyploid species as a shell script that is available from https://github.com/Songhailiang307/polyGBLUP allowing for easy use.
Simulated data
The simulation of a breeding programme was performed using AlphaSimR R package [27]. The simulated autopolyploid genome consisted of 10 chromosomes, each with 2 × 107 base pairs length, the mutation rate of 1 × 10−6 per base pair per generation and 1000 segregating sites per chromosome (10 000 in total), from which we assigned 250 sites (2500 in total) as QTL and 750 sites (7500 in total) as SNP markers. The effective population size was set to 50. The initial founder population consisting of 600 individuals with an equal proportion of male and female. The sex of founder individuals was randomly assigned. Based on the founder population, a recent population was created by randomly selecting 50 males and 100 females from 300 males and 300 females to be sires and dams of the next generation. Each selected male mated randomly with two females and each female produced 20 offspring (10 males and 10 females). The recent population underwent selection for 10 generations, with a population size of 2000 in each generation. The simulated population structure is shown in Supplementary Figure S1.
In this study, genetic effects of traits including (I) additive effects only and (II) additive and dominant effects were simulated. In the first case (I), in order to evaluate the influence of ploidy level (ploidy) and heritability (
) on different models, the following four different situations were simulated: (1) ploidy = 4 and 
; (2) ploidy = 4 and 
; (3) ploidy = 8 and 
; and (4) ploidy = 8 and 
. In the second case (II), i.e. genetic effects include additive effects and dominant effects, the dominance effect (
) at a locus is the dominance degree (
) at that locus times the absolute value of its additive allele effect (
):
 |
The dominance degrees were sampled from a normal distribution with variance 0.2 and a mean of either 0.3 (low dominance) or 1 (high dominance). The additive and dominance effects were then scaled to achieve a desired genetic variance of 1. Therefore, four different scenarios were simulated to evaluate the impact of ploidy levels and dominance effects on the different models: (1) ploidy = 4 and 
; (2) ploidy = 4 and 
; (3) ploidy = 8 and 
; and (4) ploidy = 8 and 
. In the second case, the heritability was fixed to 0.3. For each scenario, 20 replicated datasets were simulated.
Real data
Autotetraploid blueberry
The blueberry population consisted of 1804 genotyped individuals originated from 117 biparental-designed crosses of 146 parents [20, 28, 29]. These individuals were collected from the production seasons of 2014 and 2015. Three phenotypes were investigated: (1) total yield (Yield) (1–5 scale), (2) fruit weight (Weight) (g), and (3) fruit firmness (Firm) (g mm−1 of compression force). Fruit weight and fruit firmness measurements were obtained from five randomly sampled fully mature berries. Least square means (LSMeans) analysis was carried out for each trait, and genotype and year were regarded as fixed effects [24]. Subsequently, the corrected phenotypes after correcting the fixed effects were used as the phenotype values of genomic prediction. For genotype data, autotetraploid genotypes were obtained by FreeBayes software [30]. After quality control, a total of 86 930 SNPs were used for genomic prediction analysis.
Autotetraploid guinea grass
The guinea grass population, including 570 plants, was generated through a polycross mating design from 20 male and 19 female parents [21]. Six phenotypes were evaluated in this population: (1) leaf dry matter (LDM, g/plant), (2) regrowth capacity (RC), (3) percentage of leaf blade (PLB, %), (4) organic matter (OM, %), (5) crude protein (CP, %) and (6) in vitro digestibility of organic matter (IVD, %). LDM, RC and PLB were evaluated for eight harvests, during the years of 2013 (four harvests), 2014 (one harvest) and 2015 (three harvests), and OM, CP and IVD were evaluated for four harvests, during the years of 2013 and 2015 (two harvests each). Furthermore, a longitudinal linear mixed model was performed for each trait to obtain the corrected phenotypes later used as phenotype values in genomic prediction as implemented by Lara et al. [21]. For genotype data, genotyping by sequencing (GBS) was conducted in NextSeq 500 platform and the raw data were analyzed using the Tassel-GBS pipeline [31]. After quality control, a total of 41 424 SNPs were used for genomic prediction analysis.
Autohexaploid sweet potato
The sweet potato dataset consisted of phenotypic records on 282 accessions, which are part of 731 accessions randomly selected from the United States Department of Agriculture (USDA) germplasm bank in Griffin, GA, United States [32]. The accessions were planted in field trials and phenotyped in the years 2012, 2013 and 2014. Three phenotypes were investigated: (1) green–red coordinate (GRC), (2) yellow–blue coordinate (YBC) and (3) color saturation (CS). LSMeans analysis was performed for each trait to obtain the corrected phenotypes, and genotype and year were regarded as fixed effects [24]. The corrected phenotypes were subsequently used to fit the genomic prediction models. For genotype data, autohexaploid genotypes were obtained by HaplotypeCaller tool in the GATK software [33]. After quality control, a total of 77 837 SNPs were used for genomic prediction analysis.
Assessing prediction efficiency
For simulated data, we selected 2000 individuals from generation 9 as reference population to predict 500 randomly selected validation population from generation 10. For real data, genomic prediction was carried out through 20 repetitions of 5-fold cross-validation (CV). Prediction accuracy was calculated as the Pearson’s correlation between y and total genetic values (GV) for the validation individuals obtained from different models, that is r(y, GV), where y is true breeding values and corrected phenotypes for simulated data and real data, respectively. GV included only additive genetic effects for GBLUP and polyGBLUP models, while GV included additive genetic effects and dominant genetic effects for GDBLUP and polyGDBLUP models. The regression coefficient of y on GV was used to evaluate the bias of predictions, and the bias was expressed as the absolute value of the regression coefficient minus 1, i.e. abs(1 − b(y, GV)). In addition, mean squared error (Mse) and mean absolute error (Mae) metrics were used to compare model performance. Mse (Mae) represented the average square (absolute) of the difference between y and GV centered on zero.
RESULTS
Simulated data
When the genetic effects of simulated traits only included additive effects, as shown in Figure 1, polyGBLUP achieved 11.4% and 23.3% higher prediction accuracy than GBLUP when ploidy levels were 4 and 8, respectively. Furthermore, polyGBLUP consistently exhibited lower bias, Mse and Mae than GBLUP in all scenarios. These findings were consistent with the results of heritability estimation, where the heritabilities estimated by polyGBLUP closely approximated the given values (0.1 and 0.3) compared to GBLUP (Supplementary Figure S2). In the case where the genetic effects included both additive and dominant effects (Figure 2), GDBLUP outperformed GBLUP in terms of prediction accuracy only when the ploidy level was 4 and the dominance degree was 1. However, polyGDBLUP produced higher accuracy than ployGBLUP in all scenarios, with on average 1.7% and 6.8% improvement when the dominant degrees were 0.3 and 1, respectively. Similarly, polyGDBLUP achieved the highest accuracy, the lowest bias, Mse and Mae in all cases, which was in line with the result that the heritabilities estimated by polyGDBLUP were closest to the given values among all models.
Figure 1.
(A) Accuracy, (B) bias, (C) Mse and (D) Mae of genomic prediction using GBLUP and polyGBLUP methods for simulated data with ploidy levels (ploidy) equal to 4 and 8 and heritabilities (h2) equal to 0.1 and 0.3. Mse: mean squared error; Mae: mean absolute error; GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids.
Figure 2.
(A) Accuracy, (B) bias, (C) Mse and (D) Mae of genomic prediction using GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for simulated data with ploidy levels (ploidy) equal to 4 and 8 and dominance degrees (Vd) equal to 0.3 and 1. Mse: mean squared error; Mae: mean absolute error; GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects.
Furthermore, when genetic effects included additive effects and dominant effects, variance components estimated by different models were investigated as shown in Figure 3. For both GDBLUP and polyGDBLUP models, the higher the simulated dominance degrees, the higher the estimated dominance effect variances. When the dominance degree was 1, the dominance effect variance estimated by polyGDBLUP and GDBLUP was 8.3% and 14.2% higher than that when the dominance degree was 0.3, respectively. In addition, for the GDBLUP model, the estimated dominance variance was the largest when the ploidy level was 4 and the dominance degree was 1, which was consistent with the result that GDBLUP obtained higher prediction accuracy than GBLUP only in this case (Figure 2), while polyGDBLUP has the highest prediction accuracy in all cases (Figure 2), emphasizing the advantages of the polyGDBLUP model in the context of genomic prediction for autopolyploid species.
Figure 3.
The ratio of additive genetic variance, dominant genetic variance and residual variance estimated by GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for simulated data with ploidy levels equal to 4 and 8 and dominance degrees (Vd) equal to 0.3 and 1. GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects.
Real data
We also assessed the genomic prediction efficiency of GBLUP, GDBLUP, polyGBLUP and polyGDBLUP using real autotetraploid species. For autotetraploid blueberry, the prediction performances of the four models are demonstrated in Figure 4. Compared with GBLUP and GDBLUP models, polyGBLUP and polyGDBLUP produced higher prediction accuracy, with an average improvement of 0.4%, 1.0% and 1.0%, for Yield, Weight and Firm traits, respectively. Among the three traits, GBLUP and GDBLUP did not exhibit significant differences in prediction accuracy. However, polyGDBLUP achieved higher accuracy than polyGBLUP for the Yield trait. Regarding prediction bias, only polyGBLUP and polyGDBLUP models produced lower bias than GBLUP and GDBLUP for the Weight trait. However, the difference in prediction bias between models was not obvious for the other two traits, as all models showed low prediction bias. For Mse and Mae, polyGBLUP and polyGDBLUP produced lower Mse and Mae than GBLUP and GDBLUP in all cases. However, there was no significant difference in Mse and Mae between the polyGBLUP and polyGDBLUP models. For the estimated variance components, the average proportion of dominant effect variances estimated by GDBLUP and polyGDBLUP was 9.3% and 7.2%, respectively (Figure 7A). Notably, the value obtained by GDBLUP was the highest for Firm trait, which was consistent with the result that GDBLUP showed a slightly better prediction accuracy compared to GBLUP (Figure 4).
Figure 4.
(A) Accuracy, (B) bias, (C) Mse and (D) Mae of genomic prediction using GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for autotetraploid blueberry. Mse: mean squared error; Mae: mean absolute error; GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects. Yield, Weight and Firm are three analyzed traits of blueberry: total yield, fruit weight and fruit firmness, respectively.
Figure 7.
The ratio of additive genetic variance, dominant genetic variance and residual variance estimated by GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for autotetraploid (A) blueberry, (B) guinea grass and (C) autohexaploid sweet potato. GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects. Yield, Weight and Firm are three analyzed traits of blueberry: total yield, fruit weight and fruit firmness, respectively; OM, CP, IVD, LDM, RC and PLB are six analyzed traits of guinea grass: organic matter, crude protein, in vitro digestibility of organic matter, leaf dry matter, regrowth capacity and percentage of leaf blade, respectively. GRC, YBC and CS are three analyzed traits of sweet potato: green–red coordinate, yellow–blue coordinate and color saturation, respectively.
For autotetraploid guinea grass, genomic prediction was performed for six traits, as shown in Figure 5. Among the six traits, polyGBLUP and polyGDBLUP models achieved higher accuracy than GBLUP and GDBLUP only for OM, CP and IVD, with an average improvement of 1.2%, 2.3% and 0.7%, respectively. In addition, considering dominant effects in the models did not improve the accuracy of genomic prediction compared to only considering additive effects in the models, and even the prediction accuracy was slightly lower when dominance effect was considered, e.g. polyGDBLUP and GDBLUP produced slightly lower prediction accuracy than ployGBLUP and GBLUP for the IVD traits. For prediction bias, Mse and Mae, in most cases, these values obtained by different methods were not significantly different, except that GDBLUP produced higher prediction bias (for IVD trait) and higher Mse and Mae (for OM and IVD traits) compared to other methods. When examining the estimated variance components, it was found that the dominance effect variances estimated by GDBLUP and polyGDBLUP were almost zero in most cases, except that GDBLUP obtained 23.7%, 3.8% and 35.0% dominance effect variance ratios for the OM, CP and IVD traits (Figure 7B), respectively, which was consistent with the results that the accuracy obtained with and without considering dominance effects in the model was similar (Figure 5).
Figure 5.
(A) Accuracy, (B) bias, (C) Mse and (D) Mae of genomic prediction using GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for autotetraploid guinea grass. Mse: mean squared error; Mae: mean absolute error; GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects. OM, CP, IVD, LDM, RC and PLB are six analyzed traits of guinea grass: organic matter, crude protein, in vitro digestibility of organic matter, leaf dry matter, regrowth capacity and percentage of leaf blade, respectively.
We also performed genomic prediction in autohexaploid sweet potato population. As shown in Figure 6, polyGBLUP and polyGDBLUP models yielded a 0.6%, 2.0% and 1.6% higher accuracy than GBLUP and GDBLUP for GRC, YBC and CS, respectively, while the accuracy was similar between polyGBLUP and polyGDBLUP, as well as between GBLUP and GDBLUP. In terms of prediction bias, polyGBLUP and polyGDBLUP models produced lower bias than GBLUP and GDBLUP models for these three traits. For Mse and Mae, these values obtained by different methods were not significantly different. For the estimated variance components (Figure 7C), GDBLUP and polyGDBLUP displayed minimal capture of dominance effects, with the estimated dominance effect variance nearly zero for the three traits. This is consistent with the results that the accuracy obtained with and without considering dominance effects in the model was similar (Figure 6).
Figure 6.
(A) Accuracy, (B) bias, (C) Mse and (D) Mae of genomic prediction using GBLUP, GDBLUP, polyGBLUP and polyGDBLUP methods for autohexaploid sweet potato. Mse: mean squared error; Mae: mean absolute error; GBLUP: genomic best linear unbiased prediction; polyGBLUP: modified GBLUP for autopolyploids; GDBLUP: GBLUP including additive and dominance genetic effects; polyGDBLUP: polyGBLUP including additive and dominance genetic effects. GRC, YBC and CS are three analyzed traits of sweet potato: green–red coordinate, yellow–blue coordinate and color saturation, respectively.
DISCUSSION
In GS, the GBLUP method, which estimates the genetic merit of an individual by constructing a genomic relationship matrix, has been widely used in animal, plant and aquaculture breeding programs [10–13]. Since the development of GBLUP is based on the genetic evaluation of diploid animals such as dairy cattle, the classical GBLUP method is not suitable for autopolyploid species, and each SNP allele in its genome includes different allele dosages, which widely exists in nature [34]. In this study, we proposed polyGBLUP, a modified GBLUP for genomic breeding value estimation of autopolyploid species. The core of polyGBLUP is to construct additive and dominant genomic relationship matrices based on different allele dosages. The comprehensive simulation study and real data of autotetraploid blueberry, guinea grass and autohexaploid sweet potato have demonstrated the superiority of the proposed method.
In simulated data, our results indicated polyGBLUP was generally superior to GBLUP in terms of accuracy, bias, Mse and Mae. The superiority is due to the fact that in polyGBLUP, the polyG matrix can distinguish different heterozygote genotype classes, whereas GBLUP based on a diploid model cannot capture such heterozygote complexity in autopolyploids; similar results have been reported in autotetraploids [20, 21, 25]. This advantage becomes more pronounced as the ploidy level increases, e.g. the prediction accuracy of polyGBLUP is 23.3% and 11.4% higher than that of GBLUP when the ploidy level is 8 and 4, respectively (Figure 1). The reason may be that the higher the ploidy level, the higher the category of heterozygotes, so GBLUP ignores more information by treating all heterozygotes equally. In autopolyploids, the dominance effect is more complex than additive effect, because there are more combinations among alleles [26], especially in species with high ploidy levels [4–6]. To evaluate the influence of dominance effects on genomic prediction in autopolyploids, data with high (1) and low (0.3) dominance degrees were simulated. The results showed that the genomic prediction performance was optimal in all cases when the dominant effect was included in polyGBLUP (polyGDBLUP). Furthermore, the advantage of polyGDBLUP becomes more pronounced with higher dominance degrees (Figure 2). However, in most cases, there was no obvious difference between GDBLUP and GBLUP, indicating that the diploid model (GDBLUP) could not effectively distinguish and utilize the dominance effect present in the autopolyploid data.
In addition to simulated data, the advantages of polyGBLUP are also verified in real data. However, unlike the simulated data, the superiority of polyGBLUP over GBLUP only appeared in blueberry and sweet potato populations and OM, CP and IVD traits in guinea grass population, and the improvement (<3%) in accuracy of polyGBLUP over GBLUP was smaller than that of simulation data. To explain this discrepancy, the correlation coefficients of upper off-diagonal elements of different relationship matrices, which represent the similarity between the two relationship matrices, were calculated (Supplementary Figures S3 and S4). The results showed that in simulated data, as the ploidy level increased, the correlation coefficient between G and polyG decreased (Supplementary Figure S3). This lower similarity between G and polyG is consistent with the higher prediction accuracy of polyGBLUP compared to GBLUP in autooctoploid (Figure 1). However, in real data from blueberry, guinea grass and sweet potato populations, the correlation coefficients between G and polyG exceeded 0.97 (Supplementary Figure S4). This high similarity between G and polyG resulted in no significant advantage of polyGBLUP over GBLUP in real data. Additionally, the proportion of heterozygous genotypes in real data (15.3–39.5%) is smaller than in simulated data (46.7–64.2%) (Supplementary Figures S5 and S6). Therefore, the use of the polyGBLUP method did not leverage much information from different heterozygotes, making the advantage of polyGBLUP over GLBUP smaller in real data compared to simulated data. Furthermore, polyGDBLUP did not produce higher prediction accuracy than polyGBLUP for most traits, especially in guinea grass and sweet potato populations (Figures 5 and 6). This is because the analyzed traits did not capture sufficient dominant genetic variance (Figure 7). Although GDBLUP can capture more dominant genetic variance than polyGDBLUP in both real and simulated data (Figures 3 and 7), GDBLUP did not show improved predictive accuracy over GBLUP. One possible explanation is that in the diploid model, the number of heterozygotes in the diploid genotype is increased by encoding heterozygotes into the same genotype (Supplementary Figures S5 and S6), which leads to more dominant effects [14, 35, 36]. Obviously, this dominant effect is not true, which can be explained by the fact that Mse and Mae produced by GDBLUP are higher than other methods (Figures 2, 4 and 5). Therefore, for traits analyzed in autopolyploids, in the absence of dominant genetic variation, we recommend using polyGBLUP. In the presence of dominant genetic variation, we suggest using polyGDBLUP instead of polyGBLUP, and if the diploid method is used in this scenario, we suggest using GBLUP instead of GDBLUP, as the dominance effect estimated by GDBLUP is incorrect.
In GS, non-additive effects include epistatic effects in addition to dominant effects [15]. In this study, we deduced the additive (polyG) and dominant (polyD) genetic relationship matrix based on autopolyploid characteristics; thus, epistatic genomic relationship matrix of autopolyploids (polyE) can be computed using Hadamard products 
and traces (tr) as 
, 
and 
for additive-by-additive, additive-by-dominance and dominance-by-dominance epistatic effects, respectively. However, epistatic effects have not been considered in this study, and the influence of epistatic effects on genomic prediction of autopolyploids is still worth exploring in the future.
In conclusion, we first developed the polyGBLUP in genomic prediction through constructing additive and dominant genomic relationship matrices based on different allele dosages. The advantage of polyGBLUP is that it can process autopolyploid genotypes regardless of the ploidy level, so it is universal to all autopolyploid species existing in nature. We demonstrated the superiority of polyGBLUP over GBLUP in prediction accuracy, bias, Mse and Mae using both simulated data and real data. We also confirmed that polyGDBLUP (polyGBLUP including dominant effects) produced better predictive efficiency than polyGBLUP when dominant effects existed in the data. Therefore, we recommend using polyGBLUP in genomic prediction of autopolyploids because of its superior prediction performance, especially when the autopolyploid has a high ploidy level. In the future, polyGBLUP will play a significant role in genomic prediction of autopolyploid species.
Key Points
We first developed polyGBLUP for genomic prediction of autopolyploid species regardless of ploidy level.
polyGBLUP exhibits superiority in genomic prediction of autopolyploids, especially when the ploidy level of autopolyploid is high.
The prediction performance of polyGDBLUP is better than that of polyGBLUP when the dominance effect appears.
Supplementary Material
Author Biographies
Hailiang Song mainly engaged in statistical genetics, focusing on gene mapping, genomic prediction. His work was published in Reviews in Aquaculture, Evolutionary Applications, Genetics Selection Evolution and Journal of Animal Science and Biotechnology. He conceived the method, designed the experiment and wrote the paper.
Qin Zhang mainly focuses on statistical genetics and its application in breeding, focusing on gene mapping, genomic prediction, genome analysis in animal. His work published in Nature Ecology & Evolution, Genetics, Heredity and Genetics Selection Evolution. In this paper, he designed the experiment and revised the paper.
Hongxia Hu mainly focuses on the theory and application of genetic breeding. She conceived the method, designed the experiment and revised the paper.
Contributor Information
Hailiang Song, Fisheries Science Institute, Beijing Academy of Agriculture and Forestry Sciences & Beijing Key Laboratory of Fisheries Biotechnology, Beijing 100068, China; Key Laboratory of Sturgeon Genetics and Breeding, Ministry of Agriculture and Rural Affairs, Hangzhou, 311799, China.
Qin Zhang, Shandong Provincial Key Laboratory of Animal Biotechnology and Disease Control and Prevention, Shandong Agricultural University, Taian 271001, China.
Hongxia Hu, Fisheries Science Institute, Beijing Academy of Agriculture and Forestry Sciences & Beijing Key Laboratory of Fisheries Biotechnology, Beijing 100068, China; Key Laboratory of Sturgeon Genetics and Breeding, Ministry of Agriculture and Rural Affairs, Hangzhou, 311799, China.
FUNDING
This work was supported by National Natural Science Foundation of China (32202915, 32341059), Beijing Natural Science Foundation (6222014), Beijing Joint Research Program for Germplasm Innovation and New Variety Breeding (G20220628008) and the Youth Foundation of Beijing Academy of Agriculture and Forestry Sciences (QNJJ202105).
DATA AVAILABILITY
Blueberry dataset: https://datadryad.org/stash/dataset/doi:10.5061/dryad.8pk0p2nk9. Guinea grass dataset: https://github.com/leticia-lara/PM_data. Sweet potato dataset: https://github.com/Songhailiang307/polyGBLUP/blob/main/data/Sweetpotato/SweetPotato_data.RData. The shell code for simulated data and the programme of polyGBLUP are available at https://github.com/Songhailiang307/polyGBLUP.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
Blueberry dataset: https://datadryad.org/stash/dataset/doi:10.5061/dryad.8pk0p2nk9. Guinea grass dataset: https://github.com/leticia-lara/PM_data. Sweet potato dataset: https://github.com/Songhailiang307/polyGBLUP/blob/main/data/Sweetpotato/SweetPotato_data.RData. The shell code for simulated data and the programme of polyGBLUP are available at https://github.com/Songhailiang307/polyGBLUP.