Genome-Wide Association Study of Daughter Pregnancy Rate in Crossbred Dairy Cows

Abstract

A genome-wide association study (GWAS) of daughter pregnancy rate (DPR) was conducted using 75,133 SNPs and 40,203 first lactation crossbred dairy cows mostly from Jersey–Holstein crosses. The GWAS analysis detected 6528 additive effects, 65 dominance effects, 1638 additive × additive (A × A) effects, 3 additive × dominance effects, and 18 intra-chromosome dominance × dominance (D × D) effects. Of the 1638 A × A effects, 1634 were intra-chromosome and four were inter-chromosome A × A effects. The distance between two SNPs with intra-chromosome epistasis effects was in the range of 3.61 Kb to 2.68 Mb, and many interacting SNP pairs were within the same genes. The additive and A × A effects were distributed on all chromosomes showing genome-wide involvement in DPR heterosis. The dominance and D × D effects all had homozygous advantages and heterozygous disadvantages. The GWAS results identified four genetic mechanisms underlying DPR heterosis in crossbred dairy cows: complementary additive effects from different breeds and new additive effects due to cross breeding, two-locus allelic interactions between loci and between breeds, within-locus allelic interactions between breeds, and genotype × genotype interactions enabled by allelic interactions between breeds. Results in this study provided a novel understanding about the genetic factors and mechanisms underlying DPR heterosis in crossbred dairy cows.

1. Introduction

Daughter pregnancy rate (DPR) is the percentage of cows that become pregnant during each 21-day period and is an important fertility trait in dairy cows [1]. The Holstein breed is the largest dairy breed in the U.S. with high production levels but low DPR, which was in decline since 1960s until about 2000 when a trend reversal occurred, but the declines since the 1960s were not fully recovered [2]. In contrast to purebred dairy cows, crossbred dairy cows typically have improved fertility and survival rate relative to pure breeds and these advantages are most striking relative to Holsteins [3,4,5,6]. The phenotypic advantage of crossbreds over the phenotypic performance of their parental breeds is termed as heterosis [7,8]. The previously reported DPR heterosis was also confirmed in this study (see Section 3). Several data-driven hypotheses about the genetic mechanism of heterosis have been proposed, including more superior alleles in hybrids than in parental lines [9]; heterozygous advantage where heterozygous genotypes outperformed homozygous genotypes; a genetic phenomenon termed as dominance [10] or overdominance [11,12]; the dominance model where the genotypes with one or two dominant alleles per locus were more favorable than the homozygous recessive genotypes [13]; and negative dominance and dominance × dominance epistasis effects [14]. These different hypotheses indicated that the genetic mechanism of heterosis could be different for different traits and species and that more studies on the genetic mechanism of heterosis are needed to build a consensus towards understanding the genetic mechanism of heterosis. The U.S. dairy genomic evaluation database has accumulated a large dairy crossbred population with DPR phenotypic observations and genotypic data of genome-wide single nucleotide polymorphism (SNP) markers. Utilizing these resources, this study aimed at identifying genetic mechanisms of DPR heterosis through a genome-wide association study (GWAS) of additive, dominance, and pairwise epistasis effects.

2. Results and Discussion

The GWAS analysis detected 6528 additive effects and 65 dominance effects with log₁₀(1/p) > 8, and 1659 pairwise epistasis effects with log₁₀(1/p) > 12 for DPR in the 40,203 crossbred dairy cows. Of the 1659 pairwise effects, 1638 were additive × additive (A × A) effects, 3 were additive × dominance (A × D) or dominance × additive (D × A) effects, and 18 were dominance × dominance (D × D) effects. Of the 1638 A × A effects, 1634 were intra-chromosome and four were inter-chromosome A × A effects. The only inter-chromosome pairwise epistasis effect was a D × A effect. Details of these effects are described below.

2.1. Additive Effects

The 6528 additive effects were distributed over all chromosomes (Figure 1a, Table S1). SNP rs108997231 in the SLX4IP gene of Chr13 had the most significant additive effect with log₁₀(1/p) = 30.49, followed by SNPs on chromosomes 1, 17, 20, and 26 (Figure 1a,

Table 1. Top 20 additive effects of daughter pregnancy rate (DPR).

SNP	Chr	Position	Candidate Gene	α	al+	ae+	f_al+	al-	ae-	f_al-	log10(1/p)
rs108997231	13	3906114	SLX4IP	2.116	1	0.966	0.543	2	−1.150	0.457	30.49
rs110789800	1	76375060	LOC614619; GMNC	−2.005	2	0.655	0.673	1	−1.350	0.327	27.00
rs3423093496	26	29435606	MIR2285K-4(d)	−1.897	2	0.847	0.553	1	−1.050	0.447	26.69
rs109778958	17	29595964	LARP1B	1.928	1	0.698	0.638	2	−1.230	0.362	26.49
rs41638404	20	25229625	ARL15; NDUFS4	1.870	1	0.750	0.600	2	−1.120	0.400	26.30
rs110560119	26	22679137	ARMH3	1.835	1	0.735	0.599	2	−1.100	0.401	25.19
rs43292973	2	22416125	CIR1	1.870	1	0.790	0.577	2	−1.080	0.423	25.10
rs111018678	14	3308936	TRAPPC9	−1.913	2	0.553	0.711	1	−1.360	0.289	24.43
rs109798047	10	5830788	LOC100138889; LOC781028	−1.845	2	0.958	0.481	1	−0.887	0.519	24.41
rs110511554	10	36374183	PPP1R14D	1.856	1	0.991	0.466	2	−0.865	0.534	24.33
rs110600324	15	75950150	PEX16	1.869	1	1.070	0.427	2	−0.799	0.573	23.80
rs43438147	5	43898744	FRS2	−1.781	2	0.681	0.617	1	−1.100	0.383	23.65
rs3423141818	2	103153421	ABCA12	−1.819	2	0.845	0.535	1	−0.974	0.465	23.64
rs3423165011	3	118672880	LOC101907483; LOC112446044	1.811	1	0.651	0.640	2	−1.160	0.360	23.63
rs43708865	9	45754622	LOC785087; LOC112448181	−1.969	2	0.589	0.701	1	−1.380	0.299	23.59
rs29019531	1	97643741	LOC112448275; LOC107132196	−1.718	2	0.809	0.529	1	−0.909	0.471	23.45
rs110080847	24	21456850	GALNT1; INO80C	−1.944	2	0.674	0.653	1	−1.270	0.347	23.40
rs43464313	6	48853315	LOC100298058; LOC112447186	1.790	1	0.710	0.603	2	−1.080	0.397	23.37
rs3423092932	14	1854457	ADGRB1	1.757	1	0.597	0.660	2	−1.160	0.340	23.09
rs42438948	17	7548169	LRBA	−1.883	2	0.633	0.663	1	−1.250	0.337	22.87

‘d’ indicates that the SNP is downstream of the gene. ‘α’ is the additive effect of the SNP as the difference between allelic effects of ‘allele 1’ and ‘allele 2’. ‘ae+’ is the allelic effect of the positive allele. ‘ae-’ is the allelic effect of the negative allele. ‘f_al+’ is the frequency of the positive allele. ‘f_al-’ is the frequency of the negative allele.

). In general, the difference between the statistical significance of any two additive effects ranked next to each other were relatively small. Therefore, the results of additive effects indicated genome-wide contributions of additive effects to DPR in the crossbred dairy cows in this study. Among the top 1000 significant SNP effects, negative allelic effects on average had larger sizes (average −0.957) than positive allelic effects (average 0.629), but the positive alleles had higher frequencies (average 0.60) than the frequencies of the negative alleles (average 0.40) (Table S1). SNP rs42052893 in the KDSR gene of Chr24 had the largest positive allelic effect (2.62) with # 663 ranking in statistical significance, whereas SNP rs110314053 upstream of LOC107132214 of Chr01 had the largest negative allelic effect (–2.17) with #562 ranking in statistical significance (Table S1, Figure 1b). Four examples of the distribution of allelic effects of the top 1000 additive effects on different chromosomes are shown in Figure 1c–f. Compared to the Holstein DPR studies, this crossbred study had a larger number of additive effects but lacked chromosome regions that were much more significant than other chromosome regions. The 6528 additive effects in this crossbred study with 40,203 cows were more than six times as many as the 1126 additive effects from the 2019 Holstein study using 245,214 cows [15] that were more than nine times as many as the 40,203 cows in this crossbred study, and were only 14% fewer than the 7567 additive effects from the Holstein study using 1,194,736 cows [16] that were 29 times as many as the 40,203 cows in this crossbred study. Based on the comparison between the crossbred and the 2019 Holstein studies, the number of significant additive effects in the crossbred study was at least six times as many as in Holsteins, indicating that additive effects were a contributing factor to DPR heterosis in crossbred dairy cows.

2.2. Dominance Effects

The 65 dominance effects were distributed over 19 autosomes and the X chromosomes with Chr20 having the largest number (17) of dominance effects (Table S2, Figure 2a). SNP rs109688013 in the MC1R gene of Chr18 had the most significant dominance effect with log₁₀(1/p) = 18.19, the most positive dominance value (d₂₂ = 1.92) and the most negative dominance value (d₁₂ = –1.28), and the largest negative dominance effect, δ = d₁₂ − 0.5 (d₁₁ + d₂₂) = –2.55. Other SNPs with significant dominance effects following rs109688013 included SNPs on chromosomes X, 10, 20, and 12 (Figure 2a,

Table 2. Top 20 dominance effects of daughter pregnancy rate (DPR).

SNP	Chr	Position	Candidate Gene	δ	d_12	f_12	d_11	f_11	d_22	f_22	log10(1/p)
rs109688013	18	14705671	MC1R	−2.555	−1.280	0.421	0.629	0.445	1.920	0.134	18.19
rs109588756	X	56705838	VSIG1	−2.136	−0.846	0.529	1.850	0.092	0.730	0.379	11.83
rs134257923	10	79265033	EIF2S1	−1.973	−0.854	0.511	0.667	0.368	1.570	0.121	11.17
rs135634336	20	27116615	-	−1.826	−0.784	0.563	0.953	0.292	1.130	0.145	10.87
rs109658603	20	26870070	-	−1.873	−0.775	0.560	1.340	0.121	0.855	0.319	10.60
rs29018884	20	27087053	-	−1.848	−0.767	0.559	0.841	0.320	1.320	0.121	10.54
rs110147881	12	20815304	SERPINE3	−1.751	−0.783	0.553	0.973	0.183	0.962	0.264	10.37
rs135312762	20	28614378	EMB	−1.988	−0.788	0.511	1.790	0.088	0.610	0.402	10.35
rs111004899	20	35745330	LOC104975274; LOC101906686	−1.742	−0.750	0.567	1.020	0.151	0.964	0.282	10.03
rs109513293	29	47636529	SHANK2	−1.714	−0.740	0.570	0.857	0.201	1.090	0.230	9.92
rs41608172	13	27232535	TRNAE-UUC_63 (d)	−1.694	−0.752	0.554	0.971	0.168	0.912	0.278	9.83
rs109628824	20	27104916	-	−1.791	−0.745	0.559	0.821	0.320	1.270	0.121	9.78
rs109680456	12	42526721	LOC107132995 (d)	−1.693	−0.713	0.571	0.660	0.235	1.300	0.193	9.74
rs41653703	4	76449707	RAMP3	−1.677	−0.729	0.567	0.924	0.165	0.972	0.268	9.63
rs109588503	4	76953836	NUDCD3	−1.675	−0.727	0.568	0.922	0.165	0.973	0.268	9.59
rs133706373	18	56085584	PRMT1	−1.723	−0.761	0.543	0.803	0.308	1.120	0.148	9.58
rs41632642	15	80951917	SELENOH	−1.651	−0.754	0.544	0.950	0.245	0.844	0.211	9.53
rs41889435	18	56040467	PRR12; RRAS	−1.659	−0.747	0.549	0.905	0.261	0.918	0.189	9.46
rs41654346	6	72581115	IGFBP7; TRNAK-UUU_12	−1.681	−0.754	0.536	0.773	0.314	1.080	0.150	9.40
rs109489228	10	78362097	LOC112448513	−1.930	−0.786	0.512	1.670	0.097	0.617	0.391	9.39

‘d’ indicates that the SNP is downstream of the gene. ‘δ’ is the dominance effect of the SNP as the difference between the heterozygous dominance value and the average of the two homozygous dominance values. ‘d_ij’ is the dominance value (deviation) of the ij SNP genotype. ‘f_ij’ is the frequency of the ij genotype, i = 1, 2 and j = 1, 2.

). The number of dominance effects (65) is much smaller than the number of additive effects (6528) in this study but is far more than the two significant dominance effects from the 2019 Holstein study using 245,214 cows [15].

A striking feature of the dominance effects was the consistent pattern of homozygous advantage and heterozygous disadvantage according to the three dominance values (d_ij) of each significant dominance effect (δ), showing that the homozygous genotypes all had positive dominance values, one of the two homozygous genotypes of each SNP had the highest dominance value, and the heterozygous SNP genotypes of all 65 SNPs had the lowest dominance values that were all negative (Figure 2b,c; Table S2). Since each dominance value is a deviation of the genotypic value from the population mean of all three genotypic values and the additive value of the SNP genotype, the pattern of each dominance value could be affected by the estimate of the additive value of the genotype. Therefore, the phenomenon of homozygous advantage and heterozygous disadvantage detected by dominance values was further validated by the genotypic means of two types of phenotypic values: the original phenotypic values without removing pedigree additive (breeding) values (y_ij) and the corrected phenotypic values after removing pedigree additive (breeding) values (g_ij). The results of the original phenotypic values confirmed the phenomenon of homozygous advantage and heterozygous disadvantage for all 65 dominance effects, i.e., y₁₁ or y₂₂ had the highest genotypic mean whereas y₁₂ had the lowest genotypic mean. The results of the corrected phenotypic values had only four SNPs that did not have the lowest genotypic means for the heterozygous genotypes (g₁₂) but one of the homozygous genotypes of every SNP was still the most positive genotype for all 65 dominance effects. Therefore, the dominance and original phenotypic values confirmed that all dominance effects had homozygous advantage and heterozygous disadvantage whereas the corrected phenotypic values confirmed homozygous advantage for all 65 dominance effects and heterozygous disadvantage for 61 of the 65 dominance effects (Figure 2d, Table S2).

2.3. Additive × Additive (A × A) Epistasis Effects

The 1658 significant A × A effects detected in the 40,203 crossbred cows were far more than just one significant A × A effect detected in the 2021 Holstein study using 245,214 cows [17]. Of the 1658 A × A effects, 1654 were intra-chromosome and four inter-chromosome A × A effects (Table S3). The 1658 A × A epistasis effects were distributed on all chromosomes but lacked chromosome regions with A × A effects far more significant than in other regions (Figure 3). The significance and distribution patterns of the A × A effects indicated genome-wide contributions of A × A effects to DPR heterosis, like the contributions of the additive effects. Although all chromosomes had intra-chromosome A × A epistasis effects, different chromosomes had large variations in the number of intra-chromosome A × A effects, e.g., Ch20 had 110 effects whereas Chr28 only had five effects. The distance between two interacting SNPs on the same chromosome was in the range of 6702 bp to 2.68 Mb, and many interacting SNP pairs were within the same genes. The A × A epistasis effects had an interesting pattern: the most positive and the most negative allelic combinations always involved one allele of one SNP and two alleles at the other interacting SNP (Table S3). The top 20 A × A effects are shown in

Table 3. Top 20 intra-chromosome A × A effects.

SNP-1	Chr	Position-1	Candidate Gene-1	SNP-2	Position-2	Candidate Gene-2	αα	log10(1/p)
rs136164685	22	17954298	LMCD1	rs109133545	17997196	LOC112443504	2.880	24.42
rs3423093320	14	35595531	LOC101903752; TRPA1	rs3423093613	35751572	TRPA1; LOC112449577	3.290	21.27
rs41579785	19	14094861	ENSBTAG00000053331	rs3423464242	14110279	LOC104969035; HEATR6	3.490	21.09
rs3423363450	14	35558190	LOC101903752; TRPA1	rs3423093613	35751572	TRPA1; LOC112449577	3.290	21.07
rs133491240	17	23169353	LOC783956 (u)	rs110348666	25067804	PCDH10	2.600	20.97
rs111002920	3	27938935	LOC782058; NGF	rs110020978	28078792	NGF; LOC112445967	3.410	20.52
rs41656697	10	10474235	HOMER1	rs29019953	10520078	HOMER1	−2.350	20.16
rs42982833	20	45269875	LOC112443070 (u)	rs137619389	45663409	LOC112443070; LOC112442955	3.330	20.04
rs109874240	1	142636491	TFF1; TMPRSS3	rs42141252	142787751	SLC37A1	−2.810	19.93
rs109520109	5	42304422	CPNE8	rs43436966	43506809	LOC104972423; MYRFL	−2.700	19.76
rs134052004	14	40390594	PEX2; LOC101904449	rs41625934	40600797	LOC101904449; LOC782385	−3.830	19.75
rs42445061	3	60310725	TTLL7(d)	rs134212595	60964021	-	2.470	19.70
rs111002920	3	27938935	LOC782058; NGF	rs109896720	28047336	NGF; LOC112445967	3.430	19.61
rs109621977	3	48432235	ALG14	rs109849660	49515221	ABCA4	−2.390	19.21
rs109619154	29	8688612	LOC101905908; PRSS23	rs110088807	9023536	ME3	2.920	19.09
rs135072252	18	40169055	HYDIN	rs110785717	40219814	HYDIN; LOC112442400	2.730	19.09
rs109367522	5	42871428	PTPRR	rs42791351	43371061	LOC104972423; MYRFL	−2.700	19.09
rs110991306	2	94592808	GPR1; ZDBF2	rs110975979	95586874	LOC112442377; LOC100847666	2.360	19.06
rs43183410	20	45022442	LOC112443070 (u)	rs137619389	45663409	LOC112443070; LOC112442955	−3.230	19.02
rs3423093320	14	35595531	LOC101903752; TRPA1	rs3423093914	35815088	LOC112449577; LOC112449516	2.340	18.83

‘u’ indicates the SNP is upstream of the gene. ‘αα’ is the A × A effect of the SNP pair.

.

Within each chromosome, some chromosome regions had large concentrations of intra-chromosome A × A epistasis effects and six examples of such large concentrations are shown in Figure 4. The 1.08–3.71 Mb region of Chr13 had 31 pairs of A × A epistasis effects mostly involving six genes, PLCB1, PLCB4, PAK5, ANKEF1, SNAP25, and SLX4IP (Table S3, Figure 4a). Through interaction with neighboring genes, the 31 pairs of A × A epistasis effects formed an interacting chromosome region about 2.63 Mb in size. It was interesting to note that the two genes at the two ends of this region, PLCB1 at the upstream end and SLX4IP at the downstream end, were reported to be associated with fertility, with PLCB1 affecting embryonic implantation failure in mice and SLX4IP affecting embryogenesis in zebrafish [18]. It was also interesting to note that SLX4IP had the most significant additive effect in this study. The 31–51 Mb region of Chr14 (Figure 4b), the 24–74Mb region of Chr05 (Figure 4c), and the 22–32 Mb region of Chr17 (Figure 4d) each had a large cluster of A × A epistasis effects; Chr01 (Figure 4e) and Chr03 (Figure 4f) each had A × A epistasis effects covering over half of the chromosome.

Only four significant inter-chromosome A × A effects were detected between Chr02 and Chr12, Chr08 and Chr19, Chr07 and Chr18, and Chr20 and Chr26 (Table S3). The A × A values of the four inter-chromosome A × A effects all had the same pattern as that of the intra-chromosome: the most positive and negative A × A values were due to interactions between one allele of one SNP and two alleles of the other interacting SNP.

2.4. Additive × Dominance (A × D and D × A) Epistasis Effects

The GWAS analysis detected two intra-chromosome A × D effects on Chr20 and one inter-chromosome and D × A effect between Chr04 and Chr19 (Table S4). The HECW1 gene with a D × A effect also had two other SNPs with an intra-chromosome A × A effect. Although these results showed the existence of A × D and D × A effects in the crossbred cows, A × D and D × A effects were unlikely to be an important mechanism underlying DPR heterosis in crossbred cows due to the small number of the A × D and D × A effects.

2.5. Dominance × Dominance (D × D) Epistasis Effects

The GWAS analysis detected 18 intra-chromosome D × D effects on nine chromosomes with Chr20 having the largest number (six) of D × D effects (

Table 4. Significant D × D effects.

SNP-1	Chr-1	Position-1	Candidate Gene-1	SNP-2	Chr-2	Position-2	Candidate Gene-2	δδ	log10(1/p)
rs42670220	20	39445232	RAI14	rs41942366	20	41041011	LOC104975283; SUB1	5.010	15.33
rs110573919	25	3828387	DNAAF8	rs134957967	25	3920360	UBN1	4.350	14.32
rs109289243	17	66235451	TPST2	rs109255241	17	66309374	LOC100847159; LOC614881	4.710	14.28
rs132697751	13	25821584	ENKUR	rs109352819	13	25911339	GPR158	4.750	14.25
rs41942122	20	34603769	LOC782462; LOC112443045	rs109633897	20	34759872	LOC782462; LOC112443045	5.230	13.94
rs41874708	18	32330449	LOC112442399; CDH11	rs41637282	18	32771279	CDH11	4.950	13.90
rs132912044	4	6268973	LOC101904266 (d)	rs3423164105	4	6379484	LOC101904266 (d)	4.600	13.87
rs110405028	18	14014113	APRT	rs110180580	18	14050260	CBFA2T3	4.500	13.48
rs41594638	7	75473623	LOC100296952; LOC112447598	rs41667214	7	76297401	LOC112447528 (u)	4.250	13.29
rs42640157	20	58364680	ANKH	rs110840725	20	58566498	OTULINL	3.930	12.93
rs109014148	20	33024411	PLCXD3	rs43001858	20	33323766	C6	5.010	12.77
rs43278453	1	142383844	UMODL1	rs136138690	1	143444786	CRYAA; LOC112448267	4.130	12.72
rs41951335	20	58357512	ANKH	rs110840725	20	58566498	OTULINL	3.900	12.68
rs134010835	22	35456434	MAGI1	rs110776523	22	35535752	MAGI1	4.950	12.53
rs41894282	18	54627066	BICRA; EHD2	rs3423454787	18	54680124	SELENOW; LOC101902766	5.650	12.52
rs135176671	1	144551556	TSPEAR	rs109840008	1	146320854	PRMT2	4.030	12.37
rs137324140	13	2145874	PLCB4	rs134403374	13	2232129	ENSBTAG00000054174	4.470	12.14
rs3423449147	20	67492982	LOC104969150; ICE1	rs41964197	20	68697504	LOC112443018 (d)	4.880	12.03

‘d’ indicates the SNP is downstream of the gene. ‘u’ indicates the SNP is upstream of the gene. ‘δδ’ is the D × D effect of the SNP pair.

). The chromosome distance between two interaction loci was in the range of 3.61 Kb to 1.77 Mb, and only one pair of interacting loci were within the same gene, MAGI1 of Chr22. All 18 D × D effects had a heterozygous disadvantage where any two-locus genotypes with a heterozygous genotype at one locus and a homozygous genotype at the other locus had a negative D × D value. The doubly heterozygous genotype and 68 of the 72 two-locus homozygous genotypes had positive D × D values (Table S5). These D × D results added evidence to the phenomenon of heterozygous disadvantage and homozygous advantage observed for single-locus dominance effects. The doubly heterozygous genotype (AaBb) in fact had a positive value and was an interesting exception of the heterozygous disadvantage. Of the 18 doubly heterozygous genotypes, seven had DD3 values (3rd most positive values), eight had DD4 values, and one had a DD5 value, noting that all 18 DD3 and 18 DD4 values were positive, and DD5 had fourteen positive values and four negative values (Table S5). The positive D × D values of AaBb underwent negative switching twice: the heterozygous genotype at one locus turned a positive value into a negative value and the heterozygous genotype at the other locus turned the negative value back to positive, but the actual reason for the positivity of AaBb is yet to be understood.

2.6. Genetic Mechanism of DPR Heterosis in Dairy Cattle

The GWAS results indicated that the genetic mechanism of DPR heterosis in crossbred dairy cows included a larger collection of favorable alleles than in Holsteins, and allelic and genotypic interactions. These genetic mechanisms can be summarized as the following: complementary additive effects from different breeds, allelic interactions between different breeds, homozygous advantage due to interaction between the same alleles from different breeds and heterozygous disadvantage, and genotype × genotype interactions.

2.6.1. Complementary Additive Effects from Different Breeds and New Additive Effects Due to Cross Breeding

This crossbred study with 40,203 cows had more than six times as many additive effects as the Holstein study on additive and dominance effects using 245,214 cows [15], as described earlier. This result supports the hypothesis that the large number of additive effects were due to complementary additive effects between different breeds, where a chromosome region lacked favorable alleles in one breed, but another breed had favorable alleles in the same region. For the hypothetical example of Figure 5, the chromosome region from the breed represented in green has nine favorable alleles whereas the breed represented in blue has only three favorable alleles in the same region, but the crossbred from these two breeds has 12 favorable alleles, more than those in either breed. The distribution of the additive effects over the entire genome indicated that chromosome regions with complementary additive effects from different breeds widely existed on the cattle genome. Given that Holsteins and Jerseys had the largest genetic contributions (combined 97.11% breed contributions, see Section 3) to the crossbred cows, the additive results indicated the broad existence of complementary additive effects between Holsteins and Jerseys. However, the summation of additive effects in both Holsteins and Jerseys will unlikely reach the number of additive effects observed in this study, noting that the number of additive effects for the DPR in Jerseys was unknown. Therefore, cross breeding likely activated new additive effects that were either undetectable or non-existent in the pure breeds.

2.6.2. Two-Locus Allelic Interactions Between Loci and Between Breeds

This crossbred study with 40,203 cows detected 1658 significant additive A × A effects, compared to only one significant A × A effect from the Holstein study on epistasis effects using 245,214 cows [17], and the only significant Holstein A × A effect was not detected in the crossbred cows. As the genetic effect type with the second largest number of significant effects, A × A effects were the second most common effects in the crossbred cows after additive effects. Two-locus allelic interactions are the genetic interpretation of pairwise A × A effects and such allelic interactions in this study were allelic interactions between breeds.

A significant A × A effect requires at least one pair of alleles, one allele from each locus interacting with each other as described in Figure 6. For two biallelic loci with each locus having one allele from Jersey and one allele from Holstein (Figure 6a), four allelic combinations are possible, AB from Jersey, ab from Holstein, and the Ab and aB allelic combinations each have one allele from Jersey and one allele from Holstein (Figure 6b). The A × A effect of the two loci is a contrast of the mean genotypic values of the four allelic combinations in the population, and allelic interactions must be present for at least one of the four allelic combinations for the A × A effect to be statistically significant (Figure 6c). Although the method in this study does not identify which alleles are interacting with each other, the interacting alleles could be deduced. Since none of the A × A effects in crossbred cows was detected in Holstein cows, the ab allelic combination from Holstein could be assumed to have no interaction between the two alleles. Since Jersey epistasis study was unavailable and Jersey also had lower DPR levels than in crossbred cows, it should be reasonable to assume most of the A × A effects in the crossbred cows were insignificant in Jersey cows. Under this assumption, the AB allelic combination from Jersey also had no interaction between the two alleles, leaving only Ab and aB allelic combinations to be the candidates with allelic interactions responsible for the significant A × A effect, noting that each of these two allelic combinations has one allele from Jersy for one locus and one allele from Holstein for the other locus. Therefore, the mostly likely reason for the significant A × A effects in the crossbred cows was allelic interactions between loci and between breeds.

2.6.3. Within-Locus Allelic Interaction Between Different Breeds Underlying Homozygous Advantage and Heterozygous Disadvantage

Within-locus allelic interactions of a biallelic SNP locus are reflected in the dominance values of the three genotypes of the locus including the two homozygous genotypes and the heterozygous genotype. Assuming a Jersey–Holstein cross, each SNP genotype of a crossbred cow consists of one Jersey allele and one Holstein allele (Figure 7). The dominance value (d_ij) of a genotype is the deviation of the genotypic value (g_ij) from the additive value (a_ij) of the genotype and the population average of all three genotypic values (µ), where the additive value of the genotype is the sum of the two allelic effects. A large dominance value of a genotype, positive or negative, indicates the presence of allelic interactions within the genotype. The example of Figure 7 is the dominance values of rs109688013 in the MC1R gene of Chr18 with the most significant dominance effect (Table 2 and Table S2). The two homozygous genotypes (A_jA_h or a_ja_h) had positive dominance values (d₂₂ = 1.92 and d₁₁ = 0.629) and the heterozygous genotype (a_jA_h or A_ja_h, not distinguishable in this study) had a negative dominance value (d₁₂ = –1.28). Since the two alleles of every genotype originated from different breeds, the within-locus allelic interactions were allelic interactions between different breeds. These dominance values indicated that every genotype had within-locus allelic interaction between breeds that was responsible for the homozygous advantage and heterozygous disadvantage. All the advantages and disadvantages revealed by the dominance values were further confirmed by the original and corrected phenotypic values, as shown in Figure 2d. The interaction between two same alleles from different breeds within a homozygous genotype (A_jA_h or a_ja_h) was interesting because the concept of dominance is typically associated with the heterozygous genotype. For this reason, the terminology ‘within-locus interaction’ is a more general description than ‘dominance’ for interpreting the dominance values, particularly those of the homozygous genotypes. The quantitative genetics approach of dominance values in this study led to the finding of allelic interactions between breeds within both homozygous and heterozygous genotypes but had no indication as to why two same alleles from different breeds interacted with each other or why homozygous genotypes had advantages and heterozygous genotypes had disadvantages. Compared to the additive and A × A effects, the number of significant dominance effects is relatively small and should have limited contributions to the DPR heterosis.

2.6.4. Genotype × Genotype Interactions Enabled by Allelic Interactions Between Breeds

Genotype × genotype (G × G) interactions are the reasons for the 18 significant D × D effects which also had homozygous advantage and heterozygous advantage. Figure 8 is the example of the most significant D × D effect between rs42670220 and rs41942366 of Chr20 showing G × G interactions reflected in the nine D × D values of DD1-DD9 ranked from the most positive to the most negative. Any two-locus genotype with a heterozygous genotype at one locus and a homozygous genotype at the other locus was negative (highlighted in yellow) and all homozygous genotypes were positive. The D × D values clearly demonstrated the presence of G × G interactions in crossbred cows. The AAbb genotype had the most positive D × D value of DD1 = 1.22, but AA and bb individually were also associated with very negative D × D values: the Aabb genotype had the most negative D × D value of DD9 = –1.36 and the AABb genotype had the second most negative D × D value of DD8 = –1.22. Since parents do not transmit any genotype to the next generations and each breed transmits only one allele per locus to the offspring, the G × G interactions were not direct interactions between breeds but were enabled by the allelic interactions between the parental breeds. Moreover, no G × G interaction was detected in the 2021 Holstein study using 245,214 cows [17]. Therefore, the G × G interaction in crossbred cows enabled by the allelic interactions between the parental breeds was a genetic mechanism underlying DPR heterosis in crossbred cows.

2.7. Candidate Genes with Fertility and Reproductive Functions

The significant genetic effects detected in this study were associated with many candidate genes known to have fertility and reproductive functions. We searched for such genes among the top 20 additive effects, the top 20 dominance effects, the top 20 intra-chromosome A × A effects, the three inter-chromosome A × A effects, the three A × D and D × A effects’ top 20 dominance effects, and the 18 D × D effects. In addition, we noted that some genes with highly significant DPR effects in Holstein cows [15,16] were also significant in the crossbred cows, although the effects of those genes in crossbred cows were not highly ranked. Together, 82 candidate genes known to affect many aspects of fertility and reproduction were identified (Appendix A), indicating that fertility and reproductive genes likely were broadly involved in DPR heterosis in crossbred dairy cows. Important fertility and reproductive functions of the candidate genes included embryo development (KALRN, DAZL, IFNG, SERPINE3, BMP7, PCDH10, FOXC2, SIPA1L3, NDUFS4, GHR, DROSHA, EOMES, KDSR, and RARB), embryonic implantation failure (PCLB1), embryogenesis (SLX4IP), maintenance of pregnancy (IGFBP7 and IGFBP3), placentation and pregnancy maintenance (EIF2S1), uterine contraction and pregnancy maintenance (OXTR), progesterone synthesis (FGF12), early pregnancy (SLC4A4), delayed puberty (DLG2), oogenesis (HECW1 and LARP1B), oocyte maturation and female fertility (CNOT2), sex hormone production (DIAPH1), gonadotropin-releasing hormone (HOMER1), growth hormone (GHR), and thyroid hormone (THRB). Some genes with highly significant effects on the Holstein DPR were also significant but not ranked high in the crossbred cows, including SLC4A4 and GC of Chr06, KALRN of Chr01, and PEPD, CHST8, and SIPA1L3 of Chr18. The KALRN, PEPD, and CHST8 genes had extremely negative recessive alleles for the DPR in Holstein cows [16] but those extremely negative effects were not observed in the crossbred cows.

3. Materials and Methods

3.1. Dairy Crossbred Population and SNP Data

The dairy crossbred population in this study had 40,203 cows with first lactation phenotypic observations on DPR resulting from crosses among five dairy breeds: Ayrshire, Brown Swiss, Guernsey, Holstein, and Jersey. Based on breed base representation (BBR), a measure for identifying breed origin, a cow or bull is considered a purebred animal if the animal’s BBR values is 94–100% [19]. The breed contributions to the 40,203 crossbred cows measured by average BBR per breed as the percentage of all BBR values of the five breeds were 51.80% from Jersey, 45.31% Holstein, 1.60% Ayrshire, 0.99% Brow Swiss, and 0.30% from Guernsey. Therefore, the crossbreeds were primarily due to the Jersey × Holstein cross given that Jersey and Holstein had a combined 97.11% breed contribution to the crossbred cows. Ignoring the ‘0’ BBR values that every breed had, the range of the BBR values was 2.5–89.5% for Jersey, Holstein, Ayrshire, and Brown Swiss; and 2.5–89.2% for Guernsey, indicating varying degrees of purebred genetic contributions among different crossbred cows that should provide necessary contrasts between favorable and unfavorable genetic factors underlying DPR in the crossbred cows. The phenotypic values used in the GWAS analysis were the phenotypic residuals after removing fixed non-genetic effects available from the December 2023 U.S. Holstein genomic evaluation by the Council on Dairy Cattle Breeding (CDCB). Among Holstein, Jersey, and crossbred cows, crossbreds had the highest average DPR values (54.97), followed by Jersey (53.43) and Holstein (49.06) (

Table 5. Daughter pregnancy rate (DPR) of crossbred, Holstein, and Jersey cows based on the CDCB first lactation DPR data of 2015–2023.

	Crossbred	Jersey	Holstein
Mean	54.9703 ± 33.178	53.434 ± 33.199	49.06 ± 32.01
n	31,338	722,949	4,956,093

). Since 1% DPR is equivalent to four fewer days open, crossbred cows on average had about 24 fewer days open than Holstein cows and six fewer days open than Jersey cows, showing that crossbred cows had DPR heterosis over Holstein and Jersey cows. The SNP genotypes were from 32 SNP chips with various densities and were imputed into 78,964 SNPs via the FindHap algorithm [20] as a routine procedure for genomic evaluation by CDCB [21]. The SNP genotyping quality control by CDCB had checks and requirements at the individual and SNP levels, including call rate, parent–progeny conflicts, sex verification using X-specific SNPs, and Hardy-Weinberg equilibrium [22,23]. In addition, we applied minor allele frequency (MAF) of 5% for SNP filtering in this study. With the requirement of 5% MAF, the number of SNPs for the GWAS analysis was 75,133. The threshold p-value for declaring significant effects for the Bonferroni correction with 0.05 genome-wide false positives was 0.05/[(2)(75,133)] = 10⁻⁸ or log₁₀(1/p) = 8 for additive and dominance effects and was 0.05/[(4)(75,133)(75,132)/2] = 10⁻¹² or log₁₀(1/p) = 12 for pairwise epistasis effects. The SNP and gene positions were those from the ARS-UCD1.3 cattle genome assembly [24]. Genes containing or in proximity to highly significant SNP effects were identified as candidate genes for DPR in the crossbred dairy cows.

3.2. GWAS Analysis

The GWAS analysis used an approximate generalized least squares (AGLS) method. The AGLS method combines the least squares (LS) tests implemented by EPISNP1mpi [25,26] with the phenotypic correction using the estimated breeding values from routine genetic evaluation using the entire U.S. Holstein population. For each SNP pair, the statistical model was

y = µ1 + X_gg + Za + e = Xb + Za + e

(1)

where y = column vector of phenotypic deviation after removing fixed nongenetic effects such as heard–year–season (termed as ‘yield deviation’ for any trait) using a standard procedure for the CDCB genetic and genomic evaluation; µ = common mean; 1 = column vector of 1’s; g = column vector of genotypic values of the nine genotypes of the SNP pair; X_g = model matrix of g; $\mathbf{b}=(\mathsf{\mu },\mathbf{g}\prime )\prime$; X = (1, X_g); a = column vector of pedigree additive values; Z = model matrix of a; and e = column vector of random residuals. The first and second moments of Equation (1) are E(y) = Xb and $\mathrm{v}\mathrm{a}\mathrm{r}\left(\mathbf{y}\right)=\mathbf{V}=\mathbf{Z}\mathbf{G}{\mathbf{Z}}^{\prime }+\mathbf{R}={\mathsf{\sigma }}_{\mathrm{a}}^2\mathbf{Z}\mathbf{A}{\mathbf{Z}}^{\prime }+{\mathsf{\sigma }}_{\mathrm{e}}^2\mathbf{I}$, where ${\mathsf{\sigma }}_{\mathrm{a}}^2$ = additive variance, A = additive relationship matrix, ${\mathsf{\sigma }}_{\mathrm{e}}^2$= residual variance, and I = identity matrix. The problem of estimating the b vector that includes SNP genotypic values in Equation (1) is the requirement of inverting the V if the generalized least squares (GLS) method is used or solving the mixed model equations (MMEs) [27]. Either the GLS or MME method for each of the genome-wide SNPs is computationally challenging. To avoid these computing difficulties, the GWAS used the AGLS method that replaces the pedigree additive values (a) with the best linear unbiased prediction based on pedigree relationships [15] that were readily available from CDCB’s routine genetic and genomic evaluations. The AGLS method is based on the following results:

$$\widehat{\mathbf{b}}={(\mathbf{X}\prime {\mathbf{V}}^{-1}\mathbf{X})}^{-}\mathbf{X}\prime {\mathbf{V}}^{-1}\mathbf{y}$$

(2)

$$\begin{array}{ll}\widehat{\mathbf{b}}& ={\left({\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{X}\right)}^{-}\left({\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{y}-{\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{Z}\widehat{\mathbf{a}}\right)={\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-}{\mathbf{X}}^{\prime }(\mathbf{y}-\mathbf{Z}\widehat{\mathbf{a}})\\ & ={\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-}\mathbf{X}\prime {\mathbf{y}}^{\mathbf{\ast }}\end{array}$$

(3)

where ${\mathbf{y}}^{\mathbf{\ast }}=\mathbf{y}-\mathbf{Z}\widehat{\mathbf{a}}$ and $\widehat{\mathbf{a}}$ is the best linear unbiased prediction (BLUP) of a. Equation (2) is the GLS solution and Equation (3) is the MME solution of b. These two equations yield identical results and $\widehat{\mathbf{b}}$ from either equation is termed the best linear unbiased estimator (BLUE) [27]. If $\widehat{\mathbf{a}}$ is known, the LS version of BLUE given by Equation (3) is computationally efficient relative to the GLS of Equation (2) that requires the V inverse or the joint MME solutions of $\widehat{\mathbf{b}}$ and $\widehat{\mathbf{a}}$. The AGLS method uses two approximations. The first approximation is to use $\tilde{\mathbf{a}}$ from routine genetic evaluation as an approximation of $\widehat{\mathbf{a}}$ in Equation (3):

$$\widehat{\mathbf{b}}={\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-}\mathbf{X}\prime \left(\mathbf{y}-\mathbf{Z}\tilde{\mathbf{a}}\right)={\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-}\mathbf{X}\prime {\mathbf{y}}^{\mathbf{\ast }}$$

(4)

where ${\mathbf{y}}^{\mathbf{\ast }}=\mathbf{y}-\mathbf{Z}\tilde{\mathbf{a}}$ and $\tilde{\mathbf{a}}$ is the column vector of 2(PTA) with PTA being the predicted transmitting ability from the routine genetic evaluation. Equation (4) achieves the benefit of sample stratification correction from mixed models using pedigree relationships without the computing difficulty of inverting V or the joint MME solutions of $\widehat{\mathbf{b}}$ and $\widehat{\mathbf{a}}$ for every SNP. The second approximation of the AGLS approach is the t-test using the LS rather than the GLS formula of the t-statistic to avoid using the V inverse in the GLS formula. The significance tests were used for the t-tests of the contrasts of the estimated SNP genotypic values of the nine genotypes of each SNP pair where locus 1 had AA, Aa, and aa genotypes and locus 2 had BB, Bb, and bb genotypes [25,28]. The t-statistic of the AGLS was calculated as

$${\mathrm{t}}_{\mathrm{j}}={\displaystyle \frac{\left|{\mathrm{L}}_{\mathrm{j}}\right|}{\sqrt{\mathrm{v}\mathrm{a}\mathrm{r}\left({\mathrm{L}}_{\mathrm{j}}\right)}}}={\displaystyle \frac{\left|{\mathbf{s}}_{\mathrm{j}}\widehat{\mathbf{g}}\right|}{\mathrm{v}\sqrt{{\mathbf{s}}_{\mathrm{j}}{(\mathbf{X}\prime \mathbf{X})}_{\mathrm{g}\mathrm{g}}^{-}{\mathbf{s}}_{\mathrm{j}}^{\prime }}}},\hspace{1em}\mathrm{j}=\mathrm{A},\mathrm{D},\mathrm{A}\times \mathrm{A},\mathrm{A}\times \mathrm{D},\mathrm{D}\times \mathrm{A},\mathrm{D}\times \mathrm{D}$$

(5)

where L_j = effect contrast; $\sqrt{\mathrm{v}\mathrm{a}\mathrm{r}\left({\mathrm{L}}_{\mathrm{j}}\right)}$= standard deviation of the effect contrast; ${\mathbf{s}}_{\mathrm{j}}$ = row vector of the contrast coefficients for the ${\mathrm{j}}^{\mathrm{t}\mathrm{h}}$ effect type; $\widehat{\mathbf{g}}$ = column vector of the AGLS estimates of the nine SNP genotypic values for two loci from Equation (4); A = additive; D = dominance; A × A = additive × additive; A × D = additive × dominance; D × A = dominance × additive; D × D = dominance × dominance; ${\mathrm{v}}^2=(\mathbf{y}-\mathbf{X}\widehat{\mathbf{b}})\prime (\mathbf{y}-\mathbf{X}\widehat{\mathbf{b}})/(\mathrm{n}-\mathrm{k})$= estimated residual variance; ${(\mathbf{X}\prime \mathbf{X})}_{\mathbf{g}\mathbf{g}}^{-}$ = submatrix of ${\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-}$ corresponding to $\widehat{\mathbf{g}}$; n = number of observations; and k = rank of X. The formula of ${\mathbf{s}}_{\mathrm{a}}$ defined above allows Hardy-Weinberg disequilibrium [28]. To understand the genetic effects, the genetic values for each effect contrast (L_j) were calculated based on Kempthorne’s definitions [29,30] with an extension to allow for Hardy–Weinberg and linkage disequilibria [28] implemented by the EPISNP1mpi computing tool [25,26]:

$${\mathrm{a}}_{\mathrm{i}}={\mathrm{\mu }}_{\mathrm{i}}-\mu ,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}=A, a$$

(6)

$${\mathrm{a}}_{\mathrm{k}}={\mathrm{\mu }}_{\mathrm{k}}-\mu ,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{k}=B, b$$

(7)

$${\mathrm{d}}_{\mathrm{i}\mathrm{j}}={\mathrm{\mu }}_{\mathrm{i}\mathrm{j}}-\mathsf{\mu }-{\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{j}},\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{ij}=\mathit{AA}, \mathit{Aa}, \mathit{aa}$$

(8)

$${\mathrm{d}}_{\mathrm{k}\mathrm{l}}={\mathrm{\mu }}_{\mathrm{k}\mathrm{l}}-\mathsf{\mu }-{\mathrm{a}}_{\mathrm{k}}-{\mathrm{a}}_{\mathrm{l}},\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{kl}=\mathit{BB}, \mathit{Bb}, \mathit{bb}$$

(9)

$${\mathrm{a}}_{\mathrm{i}\mathrm{k}}={\mathrm{\mu }}_{\mathrm{i}\mathrm{k}}-\mathsf{\mu }-{\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{k}},\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{ik}=\mathit{AB}, \mathit{Ab}, \mathit{aB}, \mathit{ab}$$

(10)

$$\begin{gathered} {\left(\mathrm{a}\mathrm{d}\right)}_{\mathrm{i}\mathrm{k}\mathrm{l}}={\mathrm{\mu }}_{\mathrm{i}\mathrm{k}\mathrm{l}}-\mathsf{\mu } - {\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{k}}-{\mathrm{a}}_{\mathrm{l}}-{\mathrm{d}}_{\mathrm{k}\mathrm{l}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{k}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{l}} \\ \mathrm{i}=A, a; \mathrm{kl}=\mathit{BB}, \mathit{Bb}, \mathit{bb} \end{gathered}$$

(11)

$$\begin{gathered} {\left(\mathrm{d}\mathrm{a}\right)}_{\mathrm{i}\mathrm{j}\mathrm{k}}={\mathrm{\mu }}_{\mathrm{i}\mathrm{j}\mathrm{k}} - \mathsf{\mu } - {\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{j}}-{\mathrm{a}}_{\mathrm{k}}-{\mathrm{d}}_{\mathrm{i}\mathrm{j}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{k}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{j}\mathrm{k}}, \\ \mathrm{ij}=\mathit{AA}, \mathit{Aa}, \mathit{aa}; \mathrm{k}=B, b \end{gathered}$$

(12)

$$\begin{gathered} {\left(\mathrm{d}\mathrm{d}\right)}_{\mathrm{i}\mathrm{j}\mathrm{k}\mathrm{l}}={\mathrm{g}}_{\mathrm{i}\mathrm{j}\mathrm{k}\mathrm{l}}-\mathsf{\mu }-{\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{j}}- {\mathrm{a}}_{\mathrm{k}}-{\mathrm{a}}_{\mathrm{l}}-{\mathrm{d}}_{\mathrm{i}\mathrm{j}}-{\mathrm{d}}_{\mathrm{k}\mathrm{l}} \\ - {\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{k}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{l}}- {\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{j}\mathrm{k}}-{\left(\mathrm{a}\mathrm{a}\right)}_{\mathrm{i}\mathrm{l}}-{\left(\mathrm{a}\mathrm{d}\right)}_{\mathrm{i}\mathrm{k}\mathrm{l}}-{\left(\mathrm{a}\mathrm{d}\right)}_{\mathrm{j}\mathrm{k}\mathrm{l}}-{\left(\mathrm{d}\mathrm{a}\right)}_{\mathrm{i}\mathrm{j}\mathrm{k}}-{\left(\mathrm{d}\mathrm{a}\right)}_{\mathrm{i}\mathrm{j}\mathrm{l}}, \\ \mathrm{ij}=\mathit{AA}, \mathit{Aa}, \mathit{aa}; \mathrm{kl}=\mathit{BB}, \mathit{Bb}, \mathit{bb} \end{gathered}$$

(13)

where ${\mathrm{a}}_{\mathrm{i}}$ = allelic effect of allele i at locus 1 (i = A, a), ${\mathrm{a}}_{\mathrm{k}}$ = allelic effect of allele k at locus 2 (k = B, b), ${\mathrm{d}}_{\mathrm{i}\mathrm{j}}$ = dominance value of genotype ij at locus 1, ${\mathrm{d}}_{\mathrm{k}\mathrm{l}}$ = dominance value of genotype kl at locus 2, (aa)_ik = additive × additive epistasis value of genotypes with alleles ik, ${\left(\mathrm{a}\mathrm{d}\right)}_{\mathrm{i}\mathrm{k}\mathrm{l}}$ = additive × dominance epistasis value of genotypes with alleles ikl, ${\left(\mathrm{d}\mathrm{a}\right)}_{\mathrm{i}\mathrm{j}\mathrm{k}}$ = dominance × additive epistasis value of genotypes with alleles ijk, ${\left(\mathrm{d}\mathrm{d}\right)}_{\mathrm{i}\mathrm{j}\mathrm{k}\mathrm{l}}$ = dominance × dominance epistasis value of ijkl genotype, μ = the mean genotypic value in the population, ${\mathrm{\mu }}_{\mathrm{i}}$ = the marginal mean of genotypic values for individuals with allele i (i = A, a), ${\mathrm{\mu }}_{\mathrm{k}}$ = the marginal mean of genotypic values for individuals with allele k (k = B, b), ${\mathrm{\mu }}_{\mathrm{i}\mathrm{j}}$ = the marginal mean of genotypic values for individuals with genotype ij at locus 1 (ij = AA, Aa, and aa), ${\mathrm{\mu }}_{\mathrm{i}\mathrm{k}}$ = the marginal mean of genotypic values for individuals with genotype kl at locus 2 (kl = BB, Bb, and bb), ${\mathrm{\mu }}_{\mathrm{i}\mathrm{k}}$ = the marginal mean of genotypic values for individuals with allele i at locus 1 and allele k at locus 2 (i = A, a; k = B, b), ${\mathrm{\mu }}_{\mathrm{i}\mathrm{k}\mathrm{l}}$ = the marginal mean of genotypic values for individuals with allele i at locus 1 and genotype kl at locus 2 (i = A, a; kl = BB, Bb, and bb), and ${\mathrm{\mu }}_{\mathrm{i}\mathrm{j}\mathrm{k}}$ = the marginal mean of genotypic values for individuals with genotype ij at locus 1 and allele k at locus 2 (ij = AA, Aa, aa; k = B, b).

4. Conclusions

The GWAS using 40,203 crossbred dairy cows showed that genome-wide complementary additive effects between breeds and A × A epistasis effects due to between-breed and between-locus allelic interactions were the primary genetic mechanism of DPR heterosis. Interesting and complex genetic mechanisms of DPR heterosis were observed, including dominance effects due to between-breed and within-locus allelic interactions with homozygous advantage and heterozygous disadvantage and D × D effects enabled by between-breed and between-locus allelic interactions also with homozygous advantage and heterozygous disadvantage.