A Million-Cow Validation of a Chromosome 14 Region Interacting with All Chromosomes for Fat Percentage in U.S. Holstein Cows

Abstract

A genome-wide association study (GWAS) of fat percentage (FPC) using 1,231,898 first lactation cows and 75,198 SNPs confirmed a previous result that a Chr14 region about 9.38 Mb in size (0.14–9.52 Mb) had significant inter-chromosome additive × additive (A×A) effects with all chromosomes and revealed many new such effects. This study divides this 9.38 Mb region into two sub-regions, Chr14a at 0.14–0.88 Mb (0.74 Mb in size) with 78% and Chr14b at 2.21–9.52 Mb (7.31 Mb in size) with 22% of the 2761 significant A×A effects. These two sub-regions were separated by a 1.3 Mb gap at 0.9–2.2 Mb without significant inter-chromosome A×A effects. The PPP1R16A-FOXH1-CYHR1-TONSL (PFCT) region of Chr14a (29 Kb in size) with four SNPs had the largest number of inter-chromosome A×A effects (1141 pairs) with all chromosomes, including the most significant inter-chromosome A×A effects. The SLC4A4-GC-NPFFR2 (SGN) region of Chr06, known to have highly significant additive effects for some production, fertility and health traits, specifically interacted with the PFCT region and a Chr14a region with CPSF1, ADCK5, SLC52A2, DGAT1, SMPD5 and PARP10 (CASDSP) known to have highly significant additive effects for milk production traits. The most significant effects were between an SNP in SGN and four SNPs in PFCT. The CASDSP region mostly interacted with the SGN region. In the Chr14b region, the 2.28–2.42 Mb region (138.46 Kb in size) lacking coding genes had the largest cluster of A×A effects, interacting with seventeen chromosomes. The results from this study provide high-confidence evidence towards the understanding of the genetic mechanism of FPC in Holstein cows.

1. Introduction

The fat percentage (FPC) in Holstein cattle has the strongest genetic effects among dairy traits, and the gene of diacylglycerol O-acyltransferase 1 (DGAT1) of chromosome 14 (Chr14) has been widely confirmed to contain the most significant effects of FPC [1,2,3,4,5,6,7], including the effect of the K232A variant [3,4,5] and the effect of the single-nucleotide polymorphism (SNP) marker rs109421300 (ARS-BFGL-NGS-4939) in DGAT1 [1,6]. As an example showing how much more significant the FPC additive effects are than the additive effects of other dairy traits, the log₁₀(1/p) value as a measure of statistical significance was 5150 for FPC, and it was 1320, 820, 374 and 371 for the protein percentage, milk yield, fat yield and protein yield, respectively, from a previous large-scale genome-wide association study (GWAS) using 60,671 SNPs and 294,079 Holstein cows (2019 GWAS) [1]. Although the exact reasons for the highly significant effect of DGAT1 on FPC were not completely understood, the antagonism between milk and fat yields of DGAT1 was a likely genetic mechanism [1,8,9], and the antagonism of the most significant SNP (rs109421300) was extreme, with the lowest milk yield and the highest fat yield among all SNPs on the cattle genome [1]. In addition, a large chromosome segment (6.79 Mb in size) containing DGAT1 had highly significant effects on FPC. A conditional analysis showed that SNP effects in this region had strong linkage disequilibrium (LD) because the removal of the rs109421300 genotypic values from the phenotypic values removed 82% of the significant effects in this region. A follow-up GWAS on epistasis effects showed that this Chr14 region was also involved in the genome-wide epistasis effects of FPC.

Using the same dataset of 294,079 Holstein cows and 76,109 SNPs from the 2019 GWAS for additive effects, genome-wide epistasis tests found that the Chr14 region with the most significant additive effects for FPC interacted with all chromosomes for FPC in the form of inter-chromosome additive × additive (A×A) effects [10]. This was a unique discovery among the five production traits with significant inter-chromosome epistasis effects, including FPC, protein percentage and milk, fat and protein yields. The significant epistasis effects in this 10 Mb region were divided into two sub-regions separated by a gap region without significant inter-chromosome A×A effects, a region including DGAT1 upstream of the gap region and a region downstream of the gap region, which had over 40 coding genes but had no significant inter-chromosome A×A effects. The lack of significant inter-chromosome A×A effects in the gap region likely reflected the true status of this gap region: no A×A effects with other chromosomes. Given that both sides of this gap region had highly significant inter-chromosome A×A effects and the gap region had strong LD as shown by a previous conditional analysis [1], the lack of inter-chromosome A×A effects in the gap region was not due to the lack of LD with the two regions with significant inter-chromosome A×A effects on both sides of this gap region. Under the assumption of no A×A effects with other chromosomes, the statistical significance of this gap region provided a reference of the cut-off statistical significance to declare significant inter-chromosome A×A effects. The highest concentrations of the inter-chromosome A×A effects were in and around the PPP1R16A and CYHR1 genes upstream of the gap region and were in a noncoding region immediately downstream of the gap region. Following the 2019 GWAS and the 2021 epistasis study, the Holstein cows that can be used for GWAS surpassed one million by the end of 2022. Such a large sample size should provide high statistical confidence to establish or deny all or any of the previous findings of the 10 Mb Chr14 region interacting with all chromosomes for FPC. For this purpose, we conducted the genome-wide validation of inter-chromosome A×A effects for FPC using over one million first lactation cows in this study.

2. Results and Discussion

The genome-wide tests of inter-chromosome A×A effects for FPC using 1,231,898 first lactation cows and 75,198 SNPs essentially confirmed the previous result that the 10 Mb Chr14 region interacted with all chromosomes for FPC, confirmed the structure of this 10 Mb region and revealed many new results. This study identified 2763 pairs of significant inter-chromosome A×A effects with log₁₀(1/p) > 32 for FPC (Figure 1, Table S1). Of these, 2761 pairs each involved the 0.14–9.52 Mb Chr14 region (9.38 Mb in size) with 107 SNPs and one of the remaining 29 chromosomes with 1050 SNPs, whereas only two pairs did not involve the Chr14 region, one pair between Chr03 and Chr05 and one pair between Chr05 and Chr20 (Figure 1a). Among the 2763 inter-chromosome A×A effects, the four most significant effects were between Chr06 and Chr14 (Figure 1b). The total number of SNPs in the 2763 SNP pairs was 1160. The circular plots of A×A effects between Chr14 and other chromosomes, as well as the Manhattan plots of the statistical significance of the A×A effects of all non-Chr14 chromosomes, are provided in Figure S1. Detailed test results, including statistical significance and estimated A×A effects and values for each of the 2763 SNP pairs, are provided in Table S1.

2.1. Structure of the 9.38 Mb Chr14 Region Interacting with All Chromosomes for FPC

The significant inter-chromosome A×A effects in the 9.38 Mb Chr14 region (Figure 2a) were in two sub-regions that we named Chr14a and Chr14b, where Chr14a was the 0.14–0.88 Mb region (0.74 Mb in size) with 19 SNPs [11], and Chr14b was the 2.21–9.52 Mb region (7.31 Mb in size) with 88 SNPs [12]. These two sub-regions were separated by a 1.3 Mb gap at 0.9–2.2 Mb, without significant inter-chromosome A×A effects with log₁₀(1/p) > 32 (Figure 2b) but with over 40 coding genes [13]. The Chr14a region had the largest number of significant inter-chromosome A×A effects, 2148 out of the 2761 pairs or 78%, interacting with all the remaining chromosomes, and the Chr14b region had 613 of the 2761 pairs or 22%, interacting with all the remaining chromosomes except Chr24 (Table S1). Details of Chr14a and Chr14b are described below.

2.2. Inter-Chromosome A×A Effects of Chr14a

The Chr14a region was a gene-dense area with at least 46 coding genes [11], including the 15 genes shown in Figure 2c, which shows multiple locations with large clusters of inter-chromosome A×A effects. An SNP upstream of LOC789384 (rs136939758) was the upstream boundary of Chr14a with 139 inter-chromosome A×A effects. An SNP in PLEC with one of the lowest log₁₀(1/p) values (32.40) was the downstream boundary of Chr14a.

Two regions within Chr14a were particularly interesting: the PPP1R16A-FOXH1-CYHR1-TONSL (PFCT) region and the CPSF1-ADCK5-SLC52A2-DGAT1-SMPD5-PARP10 (CASDSP) region. The PFCT region was only 29 Kb in size with four SNPs, but had the largest number of inter-chromosome A×A effects (1141 pairs) with all chromosomes, including the most significant inter-chromosome A×A effects (Figure 1b and Figure 2c,

Table 1. Inter-chromosome A×A effects between four SNPs in the PFCT region of Chr14a and six SNPs in SLC4A4- NPFFR2 of Chr06.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs109146371	14	465742	PPP1R16A	rs42766480	6	87156735	GC-NPFFR2	0.0062	67.39	4
rs109146371	14	465742	PPP1R16A	rs110352004	6	87213962	GC-NPFFR2	−0.0058	58.53	35
rs110984572	14	468124	PPP1R16A-FOXH1	rs137302420	6	86751807	SLC4A4	−0.0058	55.72	59
rs110984572	14	468124	PPP1R16A-FOXH1	rs109512265	6	86753255	SLC4A4	0.0058	56.42	53
rs110984572	14	468124	PPP1R16A-FOXH1	rs110953922	6	86755896	SLC4A4	0.0058	56.42	54
rs110984572	14	468124	PPP1R16A-FOXH1	rs109901151	6	86762457	SLC4A4	0.0058	57.82	38
rs110984572	14	468124	PPP1R16A-FOXH1	rs42766480	6	87156735	GC-NPFFR2	−0.0064	70.48	1
rs110984572	14	468124	PPP1R16A-FOXH1	rs110352004	6	87213962	GC-NPFFR2	0.0060	62.15	12
rs137727465	14	487527	CYHR1	rs137302420	6	86751807	SLC4A4	0.0057	55.72	61
rs137727465	14	487527	CYHR1	rs109901151	6	86762457	SLC4A4	−0.0057	57.12	46
rs137727465	14	487527	CYHR1	rs42766480	6	87156735	GC-NPFFR2	0.0063	68.92	3
rs137727465	14	487527	CYHR1	rs110352004	6	87213962	GC-NPFFR2	−0.0059	61.42	21
rs137472016	14	494621	CYHR1-TONSL	rs137302420	6	86751807	SLC4A4	−0.0058	55.72	66
rs137472016	14	494621	CYHR1-TONSL	rs109512265	6	86753255	SLC4A4	0.0058	56.42	55
rs137472016	14	494621	CYHR1-TONSL	rs110953922	6	86755896	SLC4A4	0.0058	56.42	56
rs137472016	14	494621	CYHR1-TONSL	rs109901151	6	86762457	SLC4A4	0.0058	57.82	40
rs137472016	14	494621	CYHR1-TONSL	rs42766480	6	87156735	GC-NPFFR2	−0.0064	69.70	2
rs137472016	14	494621	CYHR1-TONSL	rs110352004	6	87213962	GC-NPFFR2	0.0060	61.42	22

Pos is the chromosome position of the SNP.

). The CASDSP region was 280 Kb in size, with six SNPs. Unlike PFCT, which interacted with all chromosomes, CASDPS mainly (42 out of 98 pairs) interacted with the SLC4A4-GC-NPFFR2 (SGN) region of Chr06 (86.75–87.40 Mb). The epistasis effects between the SGN region of Chr06 and the PFCT and CASDSP regions of Chr14 are an important new discovery in this study.

2.3. Inter-Chromosome A×A Effects between Chr06 and Chr14a

The significant inter-chromosome A×A effects of Chr06 (314 effects) were distributed in the 6.11–114.38 Mb region (Figure 3a,b; Table S1). However, Chr06 nearly exclusively interacted with Chr14a, with 312 of the 314 significant inter-chromosome A×A effects between Chr06 and Chr14a. Only two A×A effects involved Chr14b, between an SNP in the ABCG2 gene of Chr06 and two SNPs in Chr14b (Figure 3a,b; Table S1), noting that ABCG2 had significant effect on milk yield [1]. Although the 312 pairs between Chr06 and Chr14a were distributed in the 6.11–114.38 Mb region, the A×A effects between Chr14a and the SGN region of chr06 were most interesting because the Chr14a and SGN regions were two of the most important regions affecting milk production, and the SGN region also had highly significant effects on somatic cell score, daughter pregnancy rate and cow conception rate [1,2,14,15,16,17].

The PFCT region had four SNPs each interacting with the same eleven SNPs in SGN, and the CPSF1-ADCK5-SLC52A2-DGAT1 (CASD) region had three SNPs each interacting with the same nine SNPs in SGN (Figure 3c, Table S1). The SLC4A4 gene had ten significant SNPs and four of these ten SNPs in the 10.65 Kb tail region (86751807–86762457 bp), i.e., 2.5% of the gene interacted with both the PFCT and CASD regions of Chr14a, noting that the size of SLC4A4 was 427.295 Kb. The GC gene had no significant inter-chromosome A×A effects for FPC. The GC-NPFFR2 region had seven significant SNPs and two of these seven SNPs interacted with the PFCT region (Table 1). The top four most significant inter-chromosome A×A effects were between an SNP in GC-NPFFR2 (rs42766480) of Chr06 and four SNPs in PFCT, rs110984572 in PPP1R16A-FOXH1 (#1), rs137472016 in CYHR1-TONSL (#2), rs137727465 in CYHR1 (#3) and rs10914637 in PPP1R16A (#4) (Table 1; Figure 3c,d). These results showed that rs42766480 likely had a major role in the interactions with the PFCT region. However, this SNP was not ranked high (highest ranking #742) among the A×A effects between the SGN and CASD regions. The most significant inter-chromosome A×A effect between CASD and SGN was between rs211309638 in ADCK5-SLC52A2 and rs110352004 in GC-NPFFR2, with a ranking of #37 among all effects. Other than this SNP, the effect rankings of SNPs in CPSF1, DGAT1 and SMPD5 were in the range of #153–#393 (

Table 2. Inter-chromosome A×A effects between four SNPs in CASDS region of Chr14a and the SGN region of Chr06.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs134432442	14	550784	CPSF1	rs137302420	6	86751807	SLC4A4	−0.0056	50.30	178
rs134432442	14	550784	CPSF1	rs109512265	6	86753255	SLC4A4	0.0056	50.96	164
rs134432442	14	550784	CPSF1	rs110953922	6	86755896	SLC4A4	0.0056	50.96	165
rs134432442	14	550784	CPSF1	rs109901151	6	86762457	SLC4A4	0.0055	51.63	153
rs134432442	14	550784	CPSF1	rs110352004	6	87213962	GC-NPFFR2	0.0055	51.63	154
rs211309638	14	572120	ADCK5-SLC52A2	rs137302420	6	86751807	SLC4A4	−0.0059	55.72	68
rs211309638	14	572120	ADCK5-SLC52A2	rs109512265	6	86753255	SLC4A4	0.0059	56.42	57
rs211309638	14	572120	ADCK5-SLC52A2	rs110953922	6	86755896	SLC4A4	0.0059	56.42	58
rs211309638	14	572120	ADCK5-SLC52A2	rs109901151	6	86762457	SLC4A4	0.0059	57.12	49
rs211309638	14	572120	ADCK5-SLC52A2	rs110352004	6	87213962	GC-NPFFR2	0.0060	58.53	37
rs109421300	14	609870	DGAT1	rs137302420	6	86751807	SLC4A4	0.0055	49.64	191
rs109421300	14	609870	DGAT1	rs109512265	6	86753255	SLC4A4	−0.0056	50.30	179
rs109421300	14	609870	DGAT1	rs110953922	6	86755896	SLC4A4	−0.0055	50.30	180
rs109421300	14	609870	DGAT1	rs109901151	6	86762457	SLC4A4	−0.0055	50.96	166
rs109421300	14	609870	DGAT1	rs110352004	6	87213962	GC-NPFFR2	−0.0055	50.30	181
rs135549651	14	775260	SMPD5	rs137302420	6	86751807	SLC4A4	−0.0054	48.33	240
rs135549651	14	775260	SMPD5	rs109512265	6	86753255	SLC4A4	0.0054	48.98	211
rs135549651	14	775260	SMPD5	rs110953922	6	86755896	SLC4A4	0.0054	48.98	212
rs135549651	14	775260	SMPD5	rs109901151	6	86762457	SLC4A4	0.0053	48.98	213
rs135549651	14	775260	SMPD5	rs110352004	6	87213962	GC-NPFFR2	0.0051	44.52	393

Pos is the chromosome position of the SNP.

), less significant than those of the PFCT region (ranking #1–#66, Table 1).

The most important feature of PFCT was that this small 29 Kb region of Chr14a interacted with all chromosomes. The most important feature of CASD was that this region mainly interacted with the SGN region of Chr06. The most important feature of the Chr06 inter-chromosome A×A effects was that the SGN region only interacted with Chr14a, including the PCFT and CASD regions. These results of Chr14a and the SGN region of Chr06 were particularly interesting because Chr14a and the SGN region of Chr06 were two of the most significant regions for milk production traits and the SGN region also was highly significant for somatic cell score and two fertility traits (daughter pregnancy rate and cow conception rate) [1,14].

2.4. Other Inter-Chromosome A×A Effects of Chr14a

The Chr14a region had other highly significant inter-chromosome A×A effects, in addition to those with the SGN region of Chr06, including those with Chr02, Chr05, Chr17, Chr29 and ChrX (

Table 3. Top inter-chromosome A×A effects of the Chr14 excluding those with the SGN region of Chr06.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs110984572	14	468124	PPP1R16A-FOXH1	rs109208465	5	5756462	PHLDA1-BBS10	0.006	62.89	6
rs137472016	14	494621	CYHR1-TONSL	rs109208465	5	5756462	PHLDA1-BBS10	0.006	62.89	7
rs137472016	14	494621	CYHR1-TONSL	rs109210391	29	44322319	KLC2	0.013	62.89	8
rs110984572	14	468124	PPP1R16A-FOXH1	rs43304498	2	41451308	KCNJ3-GALNT13	0.009	62.15	9
rs137727465	14	487527	CYHR1	rs43304498	2	41451308	KCNJ3-GALNT13	−0.009	62.15	10
rs137472016	14	494621	CYHR1-TONSL	rs43304498	2	41451308	KCNJ3-GALNT13	0.009	62.15	11
rs137727465	14	487527	CYHR1	rs109208465	5	5756462	PHLDA1-BBS10	−0.006	62.15	12
rs137472016	14	494621	CYHR1-TONSL	rs41596003	6	43758146	PPARGC1A	0.006	62.15	13
rs109146371	14	465742	PPP1R16A	rs109210391	29	44322319	KLC2	−0.013	62.15	15
rs110984572	14	468124	PPP1R16A-FOXH1	rs109210391	29	44322319	KLC2	0.013	62.15	16
rs137727465	14	487527	CYHR1	rs109210391	29	44322319	KLC2	−0.013	62.15	17
rs109208977	14	243959	LOC789384	rs134212233	31	105004567	ENSBTAG00000045867	0.006	62.15	18
rs110984572	14	468124	PPP1R16A-FOXH1	rs41596003	6	43758146	PPARGC1A	0.006	61.42	19
rs137727465	14	487527	CYHR1	rs41596003	6	43758146	PPARGC1A	−0.006	61.42	20
rs137472016	14	494621	CYHR1-TONSL	rs41615143	2	44981402	RBM43(d)	−0.008	60.69	23
rs136580003	14	399818	ARHGAP39	rs137059769	17	63245988	C12orf76 (u)	0.005	60.69	24
rs109146371	14	465742	PPP1R16A	rs43304498	2	41451308	KCNJ3-GALNT13	−0.009	59.97	25
rs110984572	14	468124	PPP1R16A-FOXH1	rs41615143	2	44981402	RBM43(d)	−0.008	59.97	26
rs137727465	14	487527	CYHR1	rs41615143	2	44981402	RBM43(d)	0.008	59.97	27
rs137727465	14	487527	CYHR1	rs109961025	2	54049623	KYNU-5S-rRNA	0.010	59.97	28

Pos is the chromosome position of the SNP.

, Figure 4). An SNP in LOC789384 (rs109208977), an SNP between ZNF250 and ZNF16 (rs110508680) and an SNP in ARHGAP39 each had a large cluster of inter-chromosome A×A effects, with 260, 195 and 231 inter-chromosome A×A effects, respectively (Figure 2c, Table S1). Of the 20 highly significant effects not involving the SGN region of Chr06 (Table 3), 18 involved SNPs in the PCFT region, further showing a major role of the PCFT region in the inter-chromosome A×A effects for FPC.

2.5. Inter-Chromosome A×A Effects of Chr14b

The Chr14b region about 7.31 Mb in size (Figure 2b,d) [12] was nearly ten times as large as Chr14a (0.74 Mb in size) and had multiple locations with large clusters of inter-chromosome A×A effects. This region was divided into two sub-regions for convenience in describing the results: the 2.28–2.42 Mb region (138.46 Kb in size) as ‘Chr14b1’ and the remaining 7.17 Mb region as ‘Chr14b2’, with Chr14b1 accounting for 2% and Chr14b2 for 98% of Chr14b.

The Chr14b1 region had six significant SNPs in noncoding regions with two uncharacterized coding genes (LOC112449593, LOC112449592) and TRNAC-GCA [18]. These six SNPs had the largest number of inter-chromosome A×A effects (119 pairs) in Chr14b, involving seventeen chromosomes (chromosomes 2, 3, 4, 8, 12, 13, 16, 17, 18, 20, 21, 23, 25, 26, 28, 29, X) (Table S1). The most significant inter-chromosome A×A effect of Chr14b (log₁₀(1/p) = 64.37, #5 ranking) was between rs134537992 in Chr14b1 and rs42368654 about 96.6 Kb downstream of the LMX1A gene of Chr03 (Figure 5a,

Table 4. Top 20 inter-chromosome A×A effects of the Chr14b1 region.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs3423093141	14	2282659	Chr14b1	rs42368654	3	3924620	LMX1A (d)	0.0191	47.69	264
rs3423094258	14	2330431	Chr14b1	rs42368654	3	3924620	LMX1A (d)	−0.0182	55.03	81
rs3423094258	14	2330431	Chr14b1	rs41981850	21	41462728	SCFD1-COCH	−0.0126	45.14	359
rs3423094258	14	2330431	Chr14b1	rs109787816	23	30347149	ZSCAN12	−0.0272	50.30	182
rs3423357679	14	2350879	Chr14b1	rs42368654	3	3924620	LMX1A (d)	0.0178	57.12	50
rs3423357679	14	2350879	Chr14b1	rs41981850	21	41462728	SCFD1-COCH	0.0132	55.03	82
rs3423357679	14	2350879	Chr14b1	rs109787816	23	30347149	ZSCAN12	0.0248	45.14	360
rs3423357679	14	2350879	Chr14b1	rs135435373	31	6923026	bta-mir-2285bj-1 (u)	0.0080	48.33	241
rs136475864	14	2372575	Chr14b1	rs42368654	3	3924620	LMX1A (d)	0.0161	52.97	126
rs134537992	14	2421119	Chr14b1	rs43319812	2	112888145	DOCK10	−0.0136	45.14	361
rs134537992	14	2421119	Chr14b1	rs42368654	3	3924620	LMX1A (d)	0.0187	64.37	5
rs134537992	14	2421119	Chr14b1	rs109376678	8	56181140	TLE4(d)	−0.0137	45.14	362
rs134537992	14	2421119	Chr14b1	rs109489404	8	56297348	TLE4(d)	−0.0138	45.77	328
rs134537992	14	2421119	Chr14b1	rs133536911	20	30612345	FGF10 (d)	0.0144	50.30	183
rs134537992	14	2421119	Chr14b1	rs41940594	20	35354207	FYB1-RICTOR	0.0140	48.33	242
rs134537992	14	2421119	Chr14b1	rs41981850	21	41462728	SCFD1-COCH	0.0136	59.25	32
rs134537992	14	2421119	Chr14b1	rs109787816	23	30347149	ZSCAN12	0.0250	46.41	304
rs134537992	14	2421119	Chr14b1	rs109277263	29	41499833	LOC522784	−0.0135	45.14	363
rs134537992	14	2421119	Chr14b1	rs135435373	31	6923026	bta-mir-2285bj-1 (u)	0.0077	45.14	364
rs41661929	14	6113669	Chr14b2	rs136387741	31	87757884	CLCN5	−0.0049	45.77	330

Pos is the chromosome position of the SNP. Chr31 is the nonrecombining region of ChrX.

). This SNP of Chr03 interacted with 14 SNPs in Chr14b, two SNPs immediately downstream of this SNP interacted with an SNP in KCNK9 in Chr14b2, and two other SNPs interacted with an SNP downstream of the FAM135B gene in Chr14b2. All the other Chr03 SNPs (96 total) interacted with Chr14a (Figure 5a, Table S1). Although nearly the entire Chr03 interacted with Chr14a, the small region interacting with Chr14b1 had the most significant effect of Chr03. Other than the Chr03 region near LMX1A, the most significant effect of Chr14b1 was with Chr21 and Chr23 (Figure 5b,c; Table 4), noting that Chr23 almost interacted with Chr14a only, except for the inter-chromosome A×A effects with Chr14b1 (Figure 5c).

Based on the limited gene information in Chr14b1, two alternative hypotheses for the inter-chromosome A×A effects of the six SNPs in the Chr14b1 region could be made: (1) the noncoding sequences in the Chr14b1 region had biological functions in the form of interactions with other chromosomes for FPC, and (2) any or all LOC112449593, LOC112449592 and TRNAC-GCA genes were responsible for the interactions between the six SNPs in the Chr14b1 region and the seventeen chromosomes due to LD with the significant SNPs. Hypothesis (1) should be the most likely reason for the interactions involving the six SNPs and implies major biological functions of the noncoding regions in the Chr14b1 region in the form of inter-chromosome A×A effects for FPC. Hypothesis (2) implies linked effects of the six SNPs through LD with any or all LOC112449593, LOC112449592 and TRNAC-GCA genes. However, this hypothesis of linked effects was unlikely because the inter-chromosome A×A effects for FPC were unlikely affected in a significant way by LD with causal genes, as shown by the 1.3 Mb gap region of Chr14 (Figure 2b), which had at least 40 coding genes but no significant inter-chromosome A×A effects with log₁₀(1/p) > 32.

The Chr14b2 region had significant inter-chromosome A×A effects mostly in or near four genes, PTK2, TRAPPC9, KCNK9 and FAM135B (Figure 2d,

Table 5. Top 20 inter-chromosome A×A effects of the Chr14b2 region.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs132788949	14	2867641	PTK2	(no rs number)	31	25156387	blank	−0.0053	46.41	305
rs41624797	14	2929132	PTK2	(no rs number)	31	25156387	blank	0.0054	48.98	214
rs41624797	14	2929132	PTK2	rs135542379	31	24950173	blank	−0.0051	47.04	283
rs41624797	14	2929132	PTK2	rs41626477	31	9512588	TENM1	0.0048	45.14	365
rs41624797	14	2929132	PTK2	rs110945141	5	36089282	TMEM117	0.0062	44.52	394
rs41624797	14	2929132	PTK2	rs110881559	2	133918945	TAS1R2-PAX7	0.0047	43.90	446
rs55617160	14	3439565	TRAPPC9	(no rs number)	31	25156387	blank	0.0052	46.41	306
rs55617160	14	3439565	TRAPPC9	rs135542379	31	24950173	blank	−0.0050	45.14	366
rs55617160	14	3439565	TRAPPC9	rs110945141	5	36089282	TMEM117	0.0062	43.90	447
rs55617160	14	3439565	TRAPPC9	rs41626477	31	9512588	TENM1	0.0048	43.90	448
rs135838690	14	3687442	KCNK9	rs42368654	3	3924620	LMX1A (d)	0.0151	45.77	329
rs110822835	14	3710917	KCNK9	rs110945141	5	36089282	TMEM117	0.0062	44.52	395
rs110143087	14	3738219	KCNK9 (d)	rs110945141	5	36089282	TMEM117	0.0065	49.64	192
rs110143087	14	3738219	KCNK9 (d)	rs133552324	10	35535274	GPR176	−0.0071	46.41	307
rs110281272	14	4021974	KCNK9 (d)	rs42477574	5	34302710	SCAF11	−0.0056	45.14	367
rs110281272	14	4021974	KCNK9 (d)	rs136387741	31	87757884	CLCN5	0.0050	45.14	368
rs110281272	14	4021974	KCNK9 (d)	rs136157041	31	87819894	CLCN5 (d)	0.0049	45.14	369
rs110281272	14	4021974	KCNK9 (d)	rs42477555	5	34282642	SCAF11	0.0056	44.52	396
rs110979942	14	4543775	FAM135B	rs109127443	16	16191164	5S-rRNA (d)	−0.0100	44.52	397
rs42306021	14	4858211	FAM135B (d)	rs135542883	31	114523882	blank	0.0073	44.52	398

Pos is the chromosome position of the SNP. Chr31 is the nonrecombining region of ChrX.

). The PTK2 gene interacted with eight chromosomes (chromosomes 2, 5, 10, 11, 16, 17, 19, 31), TRAPPC9 with eight chromosomes (chromosomes 2, 5, 10, 11, 17, 20, 28, 31) and KCNK9 with eleven chromosomes (chromosomes 2, 3, 4, 5, 8, 10, 19, 20, 21, 28, 31). This Chr14b2 region had many inter-chromosome A×A effects with ChrX (Figure 4d), which also interacted with Chr14a and Chr14b1. Chr10 almost exclusively interacted with Chr14b2 (Figure 5d). Of the 55 inter-chromosome A×A effects of Chr14b, only six effects were between Chr14a and four SNPs of Chr10, including those between an SNP in GNG2 of Chr10 and three SNPs in DGAT1, PARP10 and PLEC of Chr14a, and between an SNP in LOC789384 of Chr14a and three SNPs of Chr10 (Figure S1, Table S1). The remaining 49 inter-chromosome A×A effects of Chr14b2, except one, were in or near PTK2, TRAPPC9, KCNK9 and GPR20 (Table S1).

2.6. Inter-Chromosome A×A Effects of Chr20 and Chr05 Interacting with Chr14

Chr20 and Chr05, along with Chr14 and Chr06, also had highly significant additive effects for milk production traits. Therefore, it was of interest to determine whether the Chr20 and Chr05 regions affecting milk production traits also interacted with the Chr14 region for FPC.

The inter-chromosome A×A effects of Chr20 covered a large distance of 63.52 Mb (6.58–70.10 Mb). Chr14a interacted with the 6.58–28.8 Mb region (mostly the 20–28 Mb region) and Chr14b with the 30.61–42.14 Mb region, whereas both Chr14a and Chr14b interacted with the remaining regions of Chr20 (Figure 6a). The most significant inter-chromosome A×A effect of Chr20 (log₁₀(1/p) = 57.12) was that between rs136653182 about 332 Kb downstream of the ITGA1 gene of Chr20 and rs109208977 in LOC789384 of Chr14a (Figure 6a,

Table 6. Top inter-chromosome A×A effects of the Chr05 and Chr20.

SNP-1	Chr-1	Pos-1	Gene-1	SNP-2	Chr-2	Pos-2	Gene-2	Effect	log10(1/p)	Rank
rs110984572	14	468124	PPP1R16A-FOXH1	rs109208465	5	5756462	PHLDA1-BBS10	0.0061	62.89	6
rs137472016	14	494621	CYHR1-TONSL	rs109208465	5	5756462	PHLDA1-BBS10	0.0061	62.89	7
rs137727465	14	487527	CYHR1	rs109208465	5	5756462	PHLDA1-BBS10	−0.0060	62.15	12
rs109146371	14	465742	PPP1R16A	rs109208465	5	5756462	PHLDA1-BBS10	−0.0059	59.25	31
rs137727465	14	487527	CYHR1	rs137444512	5	1183045	LGR5	0.0057	54.34	89
rs137472016	14	494621	CYHR1-TONSL	rs137444512	5	1183045	LGR5	−0.0057	54.34	90
rs109146371	14	465742	PPP1R16A	rs137444512	5	1183045	LGR5	0.0056	53.65	102
rs110984572	14	468124	PPP1R16A-FOXH1	rs137444512	5	1183045	LGR5	−0.0056	53.65	103
rs110143087	14	3738219	KCNK9 (d)	rs110945141	5	36089282	TMEM117	0.0065	49.64	188
rs110984572	14	468124	PPP1R16A-FOXH1	rs109706757	5	4507251	KCNC2	−0.0054	48.33	219
rs109208977	14	243959	ZNF250	rs136653182	20	26615565	ITGA1 (d)	0.0059	57.12	50
rs134537992	14	2421119	blank	rs133536911	20	30612345	FGF10 (d)	0.0144	50.30	181
rs136939758	14	146715	OR10AG83 (u)	rs136653182	20	26615565	ITGA1 (d)	0.0055	48.98	209
rs109208977	14	243959	ZNF250	rs133862450	20	26701720	ITGA1 (d)	−0.0050	48.98	210
rs109208977	14	243959	ZNF250	rs135333478	20	27147364	blank	0.0052	48.98	211
rs109208977	14	243959	ZNF250	rs136075841	20	28123462	U6-PARP8	−0.0052	48.98	212
rs109968515	14	490055	CYHR1	rs135236809	20	23802929	MTREX	0.0051	48.33	238
rs136939758	14	146715	U6-OR10AG83	rs29024419	20	28231492	U6-PARP8	−0.0054	48.33	239
rs134537992	14	2421119	blank	rs41940594	20	35354207	FYB1-RICTOR	0.0140	48.33	240
rs136939758	14	146715	OR10AG83 (u)	rs132937608	20	28111718	PARP8 (u)	−0.0055	47.69	258

Pos is the chromosome position of the SNP.

). The 20–28 Mb region of Chr20 interacting with Chr14a was near the location with the most significant effects of Chr20 at 31–33 Mb. In contrast, the 30.61–42.14 Mb region of Chr20 interacting with Chr14b had the most significant effects for milk yield among Chr20 SNPs. In particular, the NNT gene had highly significant effects for milk yield and had an SNP interacting with two SNPs in the Chr14b1 region that had the largest cluster of inter-chromosome A×A effects of Chr14b (Figure 4b, Table 3). The most significant A×A effect in the 30.61–42.14 Mb region of Chr20 was that between rs133536911 about 10.59 Kb downstream of the FGF10 gene of Chr20 and rs134537992 in Chr14b1, and rs133536911 also interacted with four other SNPs of Chr14b (Table S1). These results showed that the inter-chromosome A×A effects between Chr20 and the Chr14 region involved the Chr20 region with highly significant effects for milk yield.

The inter-chromosome A×A effects of Chr05 were mostly in the 0.5–10.8 Mb region interacting with Chr14a (Figure 4c, Table 3). The most significant SNP of chr05 was an intergenic SNP (rs109208465) about 71.13 Kb upstream of the BBS10 gene that interacted with six SNPs in PFCT (log₁₀(1/p) = 59.25–62.89) and four SNPs in CASD and SMPD5 (log₁₀(1/p) = 32.40–34.56) of Chr14a. However, the 0.5–10.8 Mb Chr05 region did not have highly significant effects for milk and fat yields or FPC [1]. The MGST1-SLC15A5 region (93.51–93.63 Mb) of Chr05 had highly significant additive effects on fat yield and FPC but this region did not interact with Chr14a or Chr14b for FPC. The SNP closest to the MGST1-SLC15A5 region was rs134855280 at 92.59 Mb, which had a significant inter-chromosome A×A effect with an SNP in LOC789384 of Chr14a. It was interesting that an SNP in EPS8 about 701 Kb downstream of the MGST1-SLC15A5 region had inter-chromosome A×A effects with an SNP in Chr03 and an SNP in Chr20, and these two inter-chromosome A×A effects were the only ones not involving Chr14a or Chr14b (Table S1), noting that EPS8 had highly significant additive effects for FPC. The 23–44 Mb region had significant SNP additive effects for milk and fat yields, and this large region interacted with both Chr14a and Chr14b for FPC (Table S1).

2.7. Patterns of A×A Epistasis Effects

The A×A values of the four allelic combinations (AB, Ab, aB, ab) of each pair of loci typically had large absolute values for the most positive and negative allelic combinations. Let AC1, AC2, AC3 and AC4 represent the four allelic combinations from the most positive combination to most negative combination and let aa1-aa4 represent the A×A values of AC1-AC4, where ‘AC’ stands for ‘allelic combination’. Then, AC1 and AC4 had the largest absolute A×A values, whereas AC2 and AC3 had considerably smaller absolute A×A values than those of AC1 and AC4 (Table S1,

Table 7. Patterns of A×A epistasis effects between Chr14a and the SGN region of Chr06.

SNP-1	Gene-1	SNP-2	Gene-2	AC1	aa1	AC2	aa2	AC3	aa3	AC4	aa4
rs109146371	PPP1R16A	rs42766480	GC-NPFFR2	1_1	0.0045	2_1	0.0006	2_2	−0.0007	1_2	−0.0030
rs109146371	PPP1R16A	rs110352004	GC-NPFFR2	1_2	0.0034	2_2	0.0003	2_1	−0.0005	1_1	−0.0031
rs110984572	PPP1R16A-FOXH1	rs42766480	GC-NPFFR2	2_1	0.0046	1_1	0.0006	1_2	−0.0007	2_2	−0.0031
rs110984572	PPP1R16A-FOXH1	rs110352004	GC-NPFFR2	2_2	0.0035	1_2	0.0003	1_1	−0.0005	2_1	−0.0032
rs110984572	PPP1R16A-FOXH1	rs109901151	SLC4A4	2_2	0.0034	1_2	0.0003	1_1	−0.0005	2_1	−0.0031
rs137727465	CYHR1	rs42766480	GC-NPFFR2	1_1	0.0046	2_1	0.0006	2_2	−0.0007	1_2	−0.0031
rs137727465	CYHR1	rs110352004	GC-NPFFR2	1_2	0.0035	2_2	0.0003	2_1	−0.0005	1_1	−0.0032
rs137727465	CYHR1	rs109901151	SLC4A4	1_2	0.0034	2_2	0.0003	2_1	−0.0005	1_1	−0.0031
rs137472016	CYHR1-TONSL	rs42766480	GC-NPFFR2	2_1	0.0046	1_1	0.0006	1_2	−0.0007	2_2	−0.0031
rs137472016	CYHR1-TONSL	rs110352004	GC-NPFFR2	2_2	0.0035	1_2	0.0003	1_1	−0.0005	2_1	−0.0032
rs137472016	CYHR1-TONSL	rs109901151	SLC4A4	2_2	0.0034	1_2	0.0003	1_1	−0.0005	2_1	−0.0031
rs211309638	ADCK5-SLC52A2	rs110352004	GC-NPFFR2	2_2	0.0036	1_2	0.0004	1_1	−0.0006	2_1	−0.0034
rs109421300	DGAT1	rs109901151	SLC4A4	1_2	0.0036	2_2	0.0005	2_1	−0.0008	1_1	−0.0032
rs109421300	DGAT1	rs109512265	SLC4A4	1_2	0.0036	2_2	0.0005	2_1	−0.0008	1_1	−0.0033
rs109421300	DGAT1	rs110953922	SLC4A4	1_2	0.0036	2_2	0.0005	2_1	−0.0008	1_1	−0.0033
rs109421300	DGAT1	rs110352004	GC-NPFFR2	1_2	0.0036	2_2	0.0006	2_1	−0.0009	1_1	−0.0033
rs109421300	DGAT1	rs137302420	SLC4A4	1_1	0.0036	2_1	0.0005	2_2	−0.0008	1_2	−0.0033
rs109421300	DGAT1	rs110434046	GC-NPFFR2	1_2	0.0032	2_2	0.0006	2_1	−0.0015	1_1	−0.0039
rs109421300	DGAT1	rs137844449	NPFFR2	1_2	0.0038	2_2	0.0004	2_1	−0.0003	1_1	−0.0018
rs109421300	DGAT1	rs109034709	NPFFR2	1_2	0.0032	2_2	0.0006	2_1	−0.0015	1_1	−0.0038

AC1–AC4 are the four allelic combinations of the two loci, and aa1–aa4 are the A×A epistasis values of AC1–AC4.

and

Table 8. Patterns of A×A epistasis effects of Chr14b.

SNP-1	Gene-1	SNP-2	Gene-2	AC1	aa1	AC2	aa2	AC3	aa3	AC4	aa4
rs134537992	Chr14b1	rs42368654	LMX1A (d)	1_1	0.0124	2_2	0.0001	1_2	−0.0007	2_1	−0.0055
rs3423357679	Chr14b1	rs42368654	LMX1A (d)	1_1	0.0120	2_2	0.0001	1_2	−0.0007	2_1	−0.0050
rs3423094258	Chr14b1	rs42368654	LMX1A (d)	2_1	0.0114	1_2	0.0001	2_2	−0.0008	1_1	−0.0059
rs136475864	Chr14b1	rs42368654	LMX1A (d)	1_1	0.0116	2_2	0.0000	1_2	−0.0006	2_1	−0.0038
rs3423093141	Chr14b1	rs42368654	LMX1A (d)	1_1	0.0150	2_2	0.0001	1_2	−0.0011	2_1	−0.0030
rs135838690	KCNK9	rs42368654	LMX1A (d)	1_1	0.0103	2_2	0.0001	1_2	−0.0005	2_1	−0.0043
rs134537992	Chr14b1	rs41981850	SCFD1-COCH	1_1	0.0094	2_2	0.0001	1_2	−0.0009	2_1	−0.0032
rs3423357679	Chr14b1	rs41981850	SCFD1-COCH	1_1	0.0094	2_2	0.0001	1_2	−0.0009	2_1	−0.0028
rs134537992	Chr14b1	rs133536911	FGF10_U6	1_1	0.0100	2_2	0.0001	1_2	−0.0007	2_1	−0.0037
rs3423094258	Chr14b1	rs109787816	ZSCAN12	2_1	0.0172	1_2	0.0001	2_2	−0.0006	1_1	−0.0094
rs134537992	Chr14b1	rs109787816	ZSCAN12	1_1	0.0162	2_2	0.0001	1_2	−0.0005	2_1	−0.0083
rs110143087	KCNK9 (d)	rs110945141	TMEM117	2_2	0.0037	1_1	0.0004	2_1	−0.0007	1_2	−0.0018
rs134537992	Chr14b1	rs41940594	FYB1_RICTOR	1_1	0.0099	2_2	0.0001	1_2	−0.0007	2_1	−0.0033
rs3423357679	Chr14b1	rs135435373	bta-mir-2285bj-1 (u)	1_1	0.0051	2_2	0.0002	2_1	−0.0010	1_2	−0.0017
rs41624797	PTK2	rs135542379	blank	2_1	0.0016	1_2	0.0007	1_1	−0.0003	2_2	−0.0025
rs110143087	KCNK9 (d)	rs133552324	GPR176	2_1	0.0048	1_2	0.0002	2_2	−0.0006	1_1	−0.0016
rs134537992	Chr14b1	rs109489404	blank	1_2	0.0094	2_1	0.0001	1_1	−0.0008	2_2	−0.0035
rs134539615	ZFAT (d)	rs29016827	STXBP6	1_2	0.0016	2_1	0.0008	1_1	−0.0009	2_2	−0.0015
rs134539615	ZFAT (d)	rs109853041	STXBP6	1_2	0.0016	2_1	0.0009	1_1	−0.0009	2_2	−0.0015
rs41661929	blank	rs136387741	CLCN5	1_2	0.0012	2_1	0.0012	2_2	−0.0008	1_1	−0.0018

AC1–AC4 are the four allelic combinations of the two loci, and aa1–aa4 are the A×A epistasis values of AC1–AC4.

). Consequently, the size of the A×A effect of two loci as a contrast of aa1-aa4 (Equations (1) and (2) in Materials and Methods) was mostly determined by the A×A values of AC1 and AC4. Therefore, the discussion of A×A patterns focused on the A×A values of AC1 and AC4, which had two patterns: (1) the two A×A values involved the same chr14 allele and two non-Chr14 alleles such as the 1_1 allelic combination for AC1 and 1_2 allelic combination for AC4, and (2) the A×A values involved two chr14 alleles and the same non-Chr14 allele, such as 1_1 for AC1 and 2_1 for AC4. Most Chr14a A×A values (1554 out of 2148, or 72%) had pattern (1), whereas most Chr14b A×A values (346 out of 613, or 56%) had pattern (2). It was interesting that no AC1 and AC4 of any SNP pair involved completely different alleles, such as 1_1 for AC1 and 2_2 for AC4, or 1_2 for AC1 and 2_1 for AC4.

Table 7 shows examples of the Chr14a A×A values where the AC1 and AC4 of each SNP pair had the same Chr14a allele and two different non-Chr14 alleles. SNP rs109421300 of DGAT1 should be a highly recognizable SNP because this SNP had the most significant effects for all five production traits: milk, fat and protein yields and fat and protein percentages. Allele 1 of rs109421300 had an extreme antagonism between fat yield and milk and protein yields, with the most positive effect for fat yield and FPC and most negative effects for milk and protein yields [1]. In this study, allele 1 of rs109421300 was the common Chr14a allele of AC1 and AC4 for all nine pairs of A×A values, and each of the nine SNPs in the SGN region of Chr06 had both alleles in AC1 and AC4. The AC1 and AC4 for seven of the nine SNP pairs had similar absolute values, indicating that these AC1 and AC4 were approximately symmetric. The combination of allele 1 of rs109421300 with a Chr06 allele was positive (aa1, Table 7), whereas the combination of allele 1 of rs109421300 with the alternative Chr06 allele of each Chr06 SNP was negative (aa4, Table 7). The other A×A values of Chr14a had similar patterns. The results of Table 7 indicate that one allele of a Chr14a SNP interacted with both alleles of a non-Chr14 SNP for most of the significant SNP pairs involving Chr14a.

Table 8 shows examples of the Chr14b A×A values where the AC1 and AC4 of each SNP pair had the same non-Chr14 allele and two different Chr14b alleles. Allele 1 of SNP rs42368654 downstream of LMX1A of Chr03 was the common allele of AC1 and AC4 with five SNPs of Chr14b1.The size of AC1 was 2–5 times as large as that of AC4 for the five A×A values, indicating that the interaction between allele 1 of rs42368654 of Chr03 and one Chr14b1 was the main contributor of the five A×A values. Of the twenty SNP pairs in Table 8, the size of AC1 was larger than that of AC4 for eighteen pairs, AC4 was larger than AC1 for two pairs, and AC1 and AC2 had approximately the same sizes for two pairs.

3. Materials and Methods

3.1. Holstein Population and SNP Data

The Holstein population in this study had 1,231,898 first lactation cows with phenotypic observations of fat percentage (FPC) and genotypes of 78,964 original and imputed SNPs. The SNP genotypes were from 32 SNP chips with various densities and were imputed to 78,964 SNPs via the FindHap algorithm [19] as a routine procedure for genomic evaluation by the Council on Dairy Cattle Breeding (CDCB) [20]. The phenotypic values used in the GWAS analysis were the phenotypic residuals after removing fixed non-genetic effects available from the December 2022 U.S. Holstein genomic evaluation by the CDCB. Basic statistics of the cows and phenotypic data of FPC are given in Table S2. With the requirement of a 0.05 minor allele frequency, the number of SNPs for the GWAS analysis was 75,198 SNPs. A strict criterion of log₁₀(1/p) > 32 was used to declare the statistical significance of any inter-chromosome epistasis effect. This requirement ensured that any significant effect had better statistical significance than that of the highest statistical significance of log₁₀(1/p) = 32 shown in Figure 2a,b. The log₁₀(1/p) > 32 requirement was stricter than the requirement of log₁₀(1/p) > 12 for the Bonferroni correction with 0.05 genome-wide false positives. The SNP and gene positions were those from the ARS-UCD1.3 cattle genome assembly [21]. Genes containing or in the proximity of highly significant effects were identified as candidate genes affecting FPC.

3.2. GWAS Analysis

The A×A value of each of the four allelic combinations (AB, Ab, aB, ab) of two loci was calculated as the deviation of the mean of the allelic combination from the population mean and the additive values of the two alleles in the allelic combination [22,23]:

$${(\mathrm{aa})}_{\mathrm{ik}}={\mathsf{\mu }}_{\mathrm{ik}}-\mathsf{\mu }-{\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{k}}$$

(1)

where ${(\mathrm{aa})}_{\mathrm{ik}}$ = A×A value of allelic combination of the ${\mathrm{i}}^{\mathrm{th}}$ allele of locus 1 (A or a allele) and the ${\mathrm{k}}^{\mathrm{th}}$ allele of locus 2 (B or b allele), ${\mathsf{\mu }}_{\mathrm{ik}}$ = the mean of the genotypic values with allelic combination of the ${\mathrm{i}}^{\mathrm{th}}$ allele of locus 1 (A or a allele) and the ${\mathrm{k}}^{\mathrm{th}}$ allele of locus 2 (B or b allele), $\mathsf{\mu }$ = the population mean of genotypic values, ${\mathrm{a}}_{\mathrm{i}}={\mathsf{\mu }}_{\mathrm{i}}-\mathsf{\mu }$ (i = A or a) = additive value of ${\mathrm{i}}^{\mathrm{th}}$ allele of locus 1 (A or a allele), ${\mathrm{a}}_{\mathrm{k}}={\mathsf{\mu }}_{\mathrm{k}}-\mathsf{\mu }$ (k = B or b) = additive value of the ${\mathrm{k}}^{\mathrm{th}}$ allele of locus 2 (B or b allele), ${\mathsf{\mu }}_{\mathrm{i}}$ = the mean of genotypic values with the ${\mathrm{i}}^{\mathrm{th}}$ allele of locus 1 (A or a allele), and ${\mathsf{\mu }}_{\mathrm{k}}$ = the mean of genotypic values with the ${\mathrm{k}}^{\mathrm{th}}$ allele of locus 2 (B or b allele).

The A×A effect of two loci was calculated as a contrast of the four A×A values and this contrast was further expressed as the A×A contrast of the nine genotypic values for epistasis testing [24]:

$$\begin{array}{l}\mathrm{\alpha }\mathrm{\alpha }=[{\left(\mathrm{aa}\right)}_{AB}-{\left(\mathrm{aa}\right)}_{Ab}]-[{\left(\mathrm{aa}\right)}_{aB}-{\left(\mathrm{aa}\right)}_{ab}]\\ =[{\left(\mathrm{aa}\right)}_{AB}-{\left(\mathrm{aa}\right)}_{aB}]-[{\left(\mathrm{aa}\right)}_{Ab}-{\left(\mathrm{aa}\right)}_{ab}]\\ =({\mathsf{\mu }}_{AB}-{\mathsf{\mu }}_{Ab})-({\mathsf{\mu }}_{aB}-{\mathsf{\mu }}_{ab})\\ =({\mathsf{\mu }}_{AB}-{\mathsf{\mu }}_{aB})-({\mathsf{\mu }}_{Ab}-{\mathsf{\mu }}_{ab})\\ = {\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}=s_{\mathrm{a}\times \mathrm{a}}^{}g\end{array}$$

(2)

where $\mathrm{\alpha }\mathrm{\alpha }$ = A×A effect of the two loci as a contrast of the four A×A values of the four allelic combinations of the two loci; ${(\mathrm{aa})}_{AB}$, ${\left(\mathrm{aa}\right)}_{Ab}$, ${\left(\mathrm{aa}\right)}_{aB}$ and ${\left(\mathrm{aa}\right)}_{ab}$ are the four A×A values of the four allelic combinations of $AB$, $Ab$, $aB$ and $ab$, respectively, defined by Equation (1); ${\mathsf{\mu }}_{AB}$= the mean of genotypic values with the $AB$ allelic combination, ${\mathsf{\mu }}_{Ab}$= the mean of genotypic values with the $Ab$ allelic combination, ${\mathsf{\mu }}_{aB}$ = the mean of genotypic values with the $aB$ allelic combination, ${\mathsf{\mu }}_{ab}$ = the mean of genotypic values with the $ab$ allelic combination; $g$ = column vector of the nine SNP genotypic values of the two loci: ${\mathrm{g}}_{AABB}$, ${\mathrm{g}}_{AABb}$, ${\mathrm{g}}_{AAbb}$, ${\mathrm{g}}_{AaBB}$, ${\mathrm{g}}_{AaBb}$, ${\mathrm{g}}_{Aabb}$, ${\mathrm{g}}_{aaBB}$, ${\mathrm{g}}_{aaBb}$, ${\mathrm{g}}_{aabb}$; $s_{\mathrm{a}\times \mathrm{a}}^{}$ = row vector of the A×A contrast coefficients of nine SNP genotypic values; and ${\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}$ = A×A effect of the two loci as a contrast of the nine SNP genotypic values. In the absence of allelic interactions between the two loci, the A×A effect of Equation (2) is expected to be null because each A×A value is expected to be null. In the presence of an allele × allele interaction between the two loci, the $[{(\mathrm{aa})}_{AB}-{\left(\mathrm{aa}\right)}_{Ab}]-[{\left(\mathrm{aa}\right)}_{aB}-{\left(\mathrm{aa}\right)}_{ab}]$ definition of the A×A effect indicates that the allelic difference of locus 2 changes in the presence of the two different alleles of locus 1, whereas the $[{\left(\mathrm{aa}\right)}_{AB}-{\left(\mathrm{aa}\right)}_{aB}]-[{\left(\mathrm{aa}\right)}_{Ab}-{\left(\mathrm{aa}\right)}_{ab}]$ definition of A×A effect indicates that the allelic difference of locus 1 changes in the presence of the two different alleles of locus 2. Therefore, a significant A×A effect expressed as ${\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}=s_{\mathrm{a}\times \mathrm{a}}^{}g$ indicates the presence of an allele × allele interaction between the two loci due to the equivalence between this expression and any of the other four expressions in Equation (2).

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

$$y=\mathsf{\mu }I+X_{\mathrm{g}}g+Za+e=Xb+Za+e$$

(3)

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/USDA genetic and genomic evaluation; µ = common mean; I = identity matrix; g = column vector of genotypic values; $X_{\mathrm{g}}$ = model matrix of g; $b=(\mathsf{\mu }, g\prime )\prime$, $X=(I, X_{\mathrm{g}})$; a = column vector of additive polygenic values; Z = model matrix of a; and e = column vector of random residuals. The first and second moments of Equation (3) are $\mathrm{E}(y)=Xb$ and $\mathrm{var}(y)=V=ZGZ\prime +R$ = ${\mathsf{\sigma }}_{\mathrm{a}}^{2}ZAZ\prime +{\mathsf{\sigma }}_{\mathrm{e}}^{2}I$, where ${\mathsf{\sigma }}_{\mathrm{a}}^2$ = additive variance, A = additive relationship matrix and ${\mathsf{\sigma }}_{\mathrm{e}}^2$ = residual variance. The problem of estimating the b vector that includes SNP genotypic values in Equation (3) is the requirement of inverting the V if the generalized least squares (GLS) method is used, or inverting the A matrix and the coefficient matrix of the mixed model equations (MME) if the MME method is used [27]. However, both V and MME could not be inverted for our sample size. To avoid inverting these large matrices, the GWAS used the method of approximate GLS (AGLS), which replaces the polygenic additive values (a) with the best linear unbiased prediction based on pedigree relationships [1]. The AGLS method is based on the following results:

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

(4)

$$\begin{array}{l}\widehat{b}={(X\prime R^{-1}X)}^{-}(X\prime R^{-1}y-X\prime R^{-1}Z\widehat{a})\\ ={(X\prime X)}^{-}X\prime (y-Z\widehat{a})={(X\prime X)}^{-}X\prime y_{*}\end{array}$$

(5)

where $y_{*}=y-Z\widehat{a}$ and $\widehat{a}$ is the best linear unbiased prediction (BLUP) of a. Equation (4) is the GLS solution, and Equation (5) is the MME solution of b. These two equations yield identical results, and $\widehat{b}$ from either equation is termed the best linear unbiased estimator (BLUE) [27]. If $\widehat{a}$ is known, the LS version of BLUE given by Equation (5) is computationally efficient relative to the GLS of Equation (4), requiring the V inverse, or the joint MME solutions of $\widehat{b}$ and $\widehat{a}$, requiring the inverse of the coefficient matrix of the MME. The AGLS method uses two approximations. The first approximation is to use $\tilde{a}$ from routine genetic evaluation as an approximation of $\widehat{a}$ in Equation (5):

$$\widehat{b}={(X\prime X)}^{-}X\prime (y-Z\tilde{a})={(X\prime X)}^{-}X\prime y^{*}$$

(6)

where $y^{*}=y-Z\tilde{a}$, and $\tilde{a}$ is the column vector of 2(PTA), with PTA being the predicted transmission ability from the routine genetic evaluation. Equation (6) achieves the benefit of sample stratification correction from mixed models using pedigree relationships without the computing difficulty of inverting V or A. 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 for A×A SNP effects used the t-tests of the A×A contrast of the estimated two-locus SNP genotypic values [24,25]:

$${\mathrm{t}}_{\mathrm{a}\times \mathrm{a}}=\frac{{|\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}|}{\sqrt{{\mathrm{var}(\mathrm{L}}_{\mathrm{a}\times \mathrm{a}})}}=\frac{\left|s_{\mathrm{a}\times \mathrm{a}}^{}\widehat{g}\right|}{\mathrm{v}\sqrt{s_{\mathrm{a}\times \mathrm{a}}^{}{(X_{}^{\prime }{X}_{})}_{\mathrm{gg}}^{-}{s}_{\mathrm{a}\times \mathrm{a}}^{\prime }}}$$

(7)

where ${\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}$ = A×A contrast of the nine genotypic values defined by Equation (1); $\sqrt{{\mathrm{var}(\mathrm{L}}_{\mathrm{a}\times \mathrm{a}})}$ = standard deviation of ${\mathrm{L}}_{\mathrm{a}\times \mathrm{a}}$; $s_{\mathrm{a}\times \mathrm{a}}^{}$ = row vector of A×A contrast coefficients; ${\mathrm{v}}^2=(y-X\widehat{b}{)}^{\prime }(y-X\widehat{b})/(\mathrm{n}-\mathrm{k})$= estimated residual variance; $\widehat{g}$ = column vector of the AGLS estimates of the nine SNP genotypic values of the two loci; and ${(X_{}^{\prime }{X}_{})}_{\mathrm{gg}}^{-}$ = submatrix of ${(X_{}^{\prime }X)}_{}^{-}$ corresponding to $\widehat{g}$.

4. Conclusions

This GWAS using over 1.2 million Holstein cows confirmed that a Chr14 region about 9.38 Mb region in size had significant inter-chromosome additive × additive (A×A) effects with all chromosomes for FPC in two sub-regions separated by a gap region without significant inter-chromosome A×A effects. Inside this 9.38 Mb region, a 0.75 Mb region known to have highly significant additive effects of FPC had most of the inter-chromosome A×A effects, including those with a Chr06 region that was known to have highly significant additive effects for some production, reproduction and health traits. This GWAS using an unprecedentedly large sample provides high-confidence evidence that FPC is affected by genome-wide allele × allele interactions centered in the 9.38 Mb Chr14 region.