Relationships between individuals and inbreeding coefficients are commonly used for breeding decisions, but may be affected by the type of data used for their estimation. The proportion of variants with low Minor Allele Frequency (MAF) is larger in whole genome sequence (WGS) data compared to Single Nucleotide Polymorphism (SNP) chips. Therefore, WGS data provide true relationships between individuals and may influence breeding decisions and prioritisation for conservation of genetic diversity in livestock. This study identifies differences between relationships and inbreeding coefficients estimated using pedigree, SNP or WGS data for 118 Holstein bulls from the 1000 Bull genomes project. To determine the impact of rare alleles on the estimates we compared three scenarios of MAF restrictions: variants with a MAF higher than 5%, variants with a MAF higher than 1% and variants with a MAF between 1% and 5%.A) and genomic (G) relationship matrices were lower, and variances of the relationships were larger when accounting for allele frequencies than without accounting for allele frequencies.We observed significant differences between estimated relationships and, although less significantly, inbreeding coefficients from pedigree, SNP or WGS data, and between MAF restriction scenarios. Computed correlations between pedigree and genomic relationships, within groups with similar relationships, ranged from negative to moderate for both estimated relationships and inbreeding coefficients, but were high between estimates from SNP and WGS (0.49 to 0.99). Estimated relationships from genomic information exhibited higher variation than from pedigree. Inbreeding coefficients analysis showed that more complete pedigree records lead to higher correlation between inbreeding coefficients from pedigree and genomic data. Finally, estimates and correlations between additive genetic  contains supplementary material, which is available to authorized users. The use of sequence data has increased considerably in the past few years and is expected to further expand due to technological improvements and a reduction in costs for whole genome sequencing ,2. WhileBreeding decisions are partly based on estimated relationships and inbreeding coefficients analysis of the population from which breeding individuals will be selected. Pedigree, SNP chips or WGS data can be used to estimate these coefficients. Traditional pedigree records have been used in selection strategies for about 30 years and SNP data have proven their efficiency in the last decade . NeverthEven though whole genome sequence data are becoming increasingly abundant, an important question is if it is worth investing in such a technique, or whether traditional data, i.e. a limited number of SNP variants and pedigree, are sufficient for long-term selection strategies and prioritisation of animals for genetic diversity conservation . Thus, sThis study was performed on whole genome sequence and pedigree data from 118 Holstein bulls. All data used were already exiting and no animal experiments were involved. Of these 118, 63 originated from Europe , 19 from North-America (12 from the United States of America and seven from Canada) and 36 from Australia. They were selected as being important ancestors of the current Holstein populations in these countries. Pedigree records were available from the 1950s onwards and contained 4,054 individuals, 1,538 males and 2,516 females. The most represented sire had 53 offspring and the most represented dam had six. From the 118 bulls used for this study, 117 had birth date information and were born between 1968 and 2004. All 118 bulls had both parents recorded in the pedigree. From this group, 61 individuals were involved in a parent-offspring relationship (43 parent-offspring pairs). We counted two full sib pairs and 56 individuals were part of half-sib families containing two to five half-sibs. On average, individuals had partial pedigree records (missing dams or sires after generation one) of 13 generations and complete records of three generations . A subgroup of 60 out of the 118 bulls had full pedigree records of at least two ancestral generations (full record on parent and grand-parent generations), of which 44 had full pedigree records at least up to four ancestral generations. These sub-groups were used for further analysis on inbreeding coefficients.Whole genome sequence data for the selected bulls, including 28,336,153 SNPs (95% of the WGS variants) and 1,668,587 insertion-deletion variants (5% of the WGS variant) (hereafter jointly referred to as variants), were accessible through the 1000 bull genomes project (Run 3.0), and were for each individual obtained as described by Daetwyler et al. . SequencP-values for the F-exact test were computed [P-value ≤ 0.05 for the F-exact test, were calculated in each case.Hardy-Weinberg proportions analysis is traditionally performed as part of the editing process when using SNP data. In general, variants showing extreme departure from Hardy-Weinberg proportions are excluded from the analysis, as they are likely to represent genotyping errors. In our case we estimated the fraction of variants departing from Hardy-Weinberg proportions for each type of data and scenario of MAF restriction used in this study. The F-exact test was used to identify departure from Hardy-Weinberg proportions as it is the most suitable for cases of variants with low MAF . For eaccomputed . The fraA) and genomic (G) relationship matrices were computed. Two different methods were used to calculate the G matrix:Additive genetic  and whole genome sequence data (GWGS) in three scenarios: (1) using all variants with a MAF higher than 5% (5+); (2) using all variants with a MAF higher than 1% (1+); (3) using variants with a MAF between 1% and 5% (1_5) in order to infer whether relationships based on variants with rare alleles were different from relationships based on common variants. After MAF restriction 41,225; 44,548 and 3,323 SNPs were kept for relationship estimation from the 50 K SNP chip (SNP), and 11,953,905; 15,871,933 and 3,918,028 from whole genome sequence (WGS) data, in scenario 5+, 1+ and 1_5, respectively , and between SNP (GSNP) and WGS (GWGS) data with different MAF restriction scenarios in order to infer the impact of rare alleles on estimated relationships. All statistical analyses were conducted in R [Estimated relationships using the three types of data  and the different scenarios  were compared against each other. The relationships were split into groups and the cut-off points between these groups were defined according to pedigree estimated relationships as follows: self-relationships , first degree relationships group such as parent-offspring or full sib relationships (relationships ≥0.5 to <1), second degree relationships group such as half sib, grandparents-offspring or cousin relationships (relationships ≥0.25 to <0.5) and less-related individuals (relationships <0.25) . Only thted in R . The tested in R .Aped matrix using the algorithm of Sargolzaei et al. [G matrix diagonal elements (self-relationship) minus 1. It should be noted that these inbreeding coefficients represent the correlation between uniting gametes in an individual [Inbreeding coefficients for pedigree were computed from the i et al. . Genomicdividual . Individdividual .A uniform distribution of MAF was observed for SNP variants, while a L shaped distribution was observed for sequence variants Figure . As expeEstimated relationships for the three groups of different degrees of relationships  ranged from 0.00 to 0.66 for pedigree data, from –0.14 to 0.60 for SNP data and from –0.11 to 0.55 for WGS data Table . Mean vaGSNP and GWGS had a correlation of 0.95 with Aped, while GSNP and GWGS had a correlation of 0.99 .Both 9 Figure . Correla9 Figure . In factAped versus GSNP5+ or GWGS5+ .Inbreeding coefficients were on average close to zero for SNP and WGS, ranging from 0 to 0.16 for pedigree estimates, from –0.24 to 0.11 for SNP and from –0.21 to 0.07 for WGS. Correlations between pedigree and genomic inbreeding increased with pedigree depth, as expected. Significant differences between correlations were observed between depth1 and depth4, for Estimated relationships for scenario 1+ and 1_5 varied from slightly negative −0.13) for the less related group to highly positive (1.06) for first degree relationships group  and scenario 1_5 (on average 0.83); both being lower than the value of 0.98 observed for 5 +.When comparing scenarios including rare alleles, we observed that the correlations between GSNP1+ and GWGS1+ to Aped , and between depth1 and other depths for GSNP1_5 compared to Aped . Similar as for the relationships, scenario 1_5 showed important differences with scenario 1+ as correlations between Aped and GSNP1_5 for depth1 and all between Aped and GWGS1_5 were not significantly different from zero.Inbreeding coefficients ranged from –0.23 to 0.18 for SNP and from –0.27 to 0.04 for WGS across the two scenarios including rare alleles. Correlations between pedigree and genomic inbreeding coefficients increased with pedigree depth. Difference in correlations was significant between depth1 and depth4 when comparing GSNP and GWGS, between the different MAF scenarios  . Regarding inbreeding coefficients, differences between scenarios were only significant when computed from whole genome sequence data . Correlation between scenario 1+ and 5+ for GSNP, in almost all group of degrees of relationships, did not show significant difference from 1, adding variants with low MAF did not affect estimated relationships when using SNP. As scenario 1_5 and 1+ partly used the same variants, they were, for both GWGS and GSNP, better correlated (0.84 to 0.59) than 1_5 and 5+ (0.65 to 0.50). Moreover, the correlations between scenario 1+ and 1_5 for GWGS were higher than for GSNP, indicating that the exclusive use of variants with a MAF between 1% and 5% gave estimates that were closer to the estimated relationships of WGS data, as the latter type of data contains relatively more of these variants.Hereafter we report correlations within 5) Table . ComparaAlongside the Yang method, which weighs the contribution of each locus by its MAF, we also computed relationships based on similarities between genotypes. This yielded estimated relationships that were generally higher and with smaller variances than those yielded by the Yang method. Estimated relationships for genomic data ranged from 0.40 to 1.94; in particular scenario 1_5 showed high genomic estimated relationships ranging from 1.47 to 1.94 (Table Aped and G estimated relationships for scenarios 5+ and 1+ (0.96). The overall correlations between the Aped and G in scenario 1_5 were smaller for similarities, which where 0.43 for GWGS and 0.39 for GSNP . This observation supported our conclusion that changes between scenarios and type of data were due to low MAF variants, and not because the sequence data also included insertion-deletion variants.Differences between pedigree and marker-based estimators have three main causes. Firstly, pedigree estimators rely on the fact that 50% of the genome is transmitted from parents to offspring. Likewise, two non-inbred full sibs theoretically are expected to share 50% of their genome. Marker-based methods, however, give access to the actual shared proportion. In the case of full sibs, for example, the share of genome might vary from the 50% value due to Mendelian sampling . SecondlAnalogous to our similarity-based method, Pérez-Enciso , in a siBy correcting for allele frequencies, the Yang estimator puts relatively more emphasis on low MAF variants. Rare alleles are either recent mutations or ancient alleles driven to low allele frequencies through time due to drift, or natural and artificial selection. These alleles have a higher risk for disappearing after a few generations; thus in the framework of genetic diversity conservation, it may be desirable to put a higher priority on rare compared to common alleles in order to balance the potential loss of genetic diversity. This suggests that the Yang estimator may also be most appropriate when computed relationships are used for genetic diversity conservation decisions, which aim to conserve variation at low MAF variants as much as possible.GSNP and GWGS , in agreement with a correlation of 0.92 between both scenarios reported by Pérez-Enciso [GSNP and GWGS on one hand and A on the other hand were considerably lower and variances of estimated relationships were generally higher for both GSNP and GWGS than for A, comparable to results found in other studies [In our study, correlations were high only between z-Enciso . Additio studies -34.Grouping individuals according to their pedigree depths showed that longer pedigree records led to closer correlation between pedigree and genomic inbreeding coefficients. Negative inbreeding coefficients, i.e. self-relationships lower than one, were also observed. With ‘inbreeding’ defined as the mating of individuals that are more related than the average of the population , negativPérez-Enciso  argued tRelationships computed from whole genome sequence data are expected to reflect the true relationships between individuals; therefore, sequence data are considered a valuable resource for improving estimated relationships. In this study, estimated relationships and inbreeding coefficients from pedigree and genomic information were hardly correlated; when from SNP and WGS data they were shown to be strongly correlated. Nevertheless, when using the sequence data, neglecting rare alleles, i.e. variants with a MAF below 5%, led to significant changes in the estimated relationships. Such changes may affect selection strategies for long-term selection and genetic diversity conservation. If conservation of genetic diversity is geared towards safeguarding all accessible variation, then relationship estimators that weigh genotypes by their allele frequencies are to be preferred, possibly combined with the use of sequence data. The following question, however, remains un-answered: to what extent will the use of whole genome sequence data and rare allele information affect selection strategies such as Optimal Contribution Selection in optimising long-term genetic improvement and genetic diversity conservation?


Rosa x hybrida). Due to changes in global trade and an increase in energy costs the ornamental industry has seen a shift in the production and sale of flowers from the US and Europe alone to production in Africa and Latin America. As Kenya is a major exporter of roses to Europe we studied the genetic variation and heritability of specific morphological traits in a tetraploid population grown in the Netherlands and in Kenya. The aim was to estimate genotype by environment interaction (G × E) and to investigate the implications of (G × E) for rose breeding.Global trade has ensured that the ornamental horticulture continues to grow worldwide, with rose hybrids being the most economically important genus (A tetraploid rose population (K5) from a cross between two tetraploid parents was field tested over two seasons in the Netherlands (summer and winter) and two locations in Kenya (Nairobi and Njoro). Ten traits were compared per genotype across the four environments. There were differences in trait association across the four environments showing that the traits were partially influenced by the environment.2ge/σ2g also showed a high value for heritability. For the traits number of petals, prickles on petioles, prickles on stems the interaction is minimal. For the traits chlorophyll content, stem width and side shoots we observed a much higher interaction ratio of 0.83, 1.43 and 3.13 respectively. The trait number of petals had the highest heritability of 0.96 and the lowest σ2ge/σ2g ratio (0.08). The trait number of side shoots (SS) with the lowest heritability (0.40) also had the highest σ2ge/σ2g ratio of 3.13.The traits that had a low ratio of σAttained by this experiment showed that we have different magnitudes of non-crossover G × E interactions. For the traits number of petals, prickles on stems and prickles on petioles with a low interaction and high heritability, selection can be done at any of the environments. Thus, these traits can be confirmed at the breeding site. For the traits stem width, side shoots and chlorophyll content that had a higher interaction selection for or against these traits should be done at the production location or at least be verified there.The online version of this article (doi:10.1186/s12863-014-0146-z) contains supplementary material, which is available to authorized users. Rosa the most important genus, economically, of ornamental horticulture. In the period 2000–2008, rose imports grew by 60% from 2.3 billion in 2000 to 3.7 billion in 2008 [Cut roses have an estimated turnover of 768 million Euro in the Netherlands, compared to 294 million of the number two cut ornamental, chrysanthemum . This maBecause of these trends, there are now two types of growers; the European growers found in the country where the breeding centre and market are located and the foreign growers, usually located in tropical countries, who export their flowers to Europe and the USA. Both growers are interested in high yields, vase life, disease resistance and novel colours. European growers aim for novel colours, big heads and longer stems. Due to the high production costs in Europe, novelties in the roses ask for a higher price at the flower auction to make production economically viable. With the increase in production in the tropics, postharvest longevity is becoming increasingly important , as therMost recurrent blooming roses flower within several weeks of germination allowing selection for floral traits to be made relatively early compared to most woody species . This alA good plant breeding program has to take into consideration the influence of the genotype by environment interactions and the correlations of important traits. This is because the growing areas are now located worldwide while the breeding is still done in temperate regions. There is limited information available about the stability and expression of most of the cut rose morphological traits as well as the correlation of the different traits. In order to improve the efficiency of breeding for quality traits in roses, understanding the variation of these traits in different environments is necessary. The main purpose of multi-environment trials is to observe stability of genotypes across the environments, the identification of superior genotypes and of the location that best represents the target environment for production. Rose growers believe that high altitudes and cool climates lead to deeper colours and longer stems. The lower altitudes give faster maturing, more vigorous plants leading to higher production. So far, a few genetic studies have been performed on a limited number of traits in diploid roses -10 and iThe K5 tetraploid rose population used in this study was described by Yan et al.  and Koni, Kenya, Nairobi, and Kenya, Njoro . In the Netherlands observations were made during the summer of 2007 and the winter 2007/2008. In Kenya the observations were made between January and July of 2009.Trials were established at three locations, The Netherlands, WageningenRooted nodal cuttings of each genotype, including the parents, were produced by Terra Nigra B.V., a Dutch company that is active in the breeding, propagation and marketing of roses. In the Netherlands, these cuttings were planted in pairs in pots of coco peat in a greenhouse. The greenhouse was artificially lit to ensure a day length of 18 hrs, the temperature was kept at 20°C (day temperature) and 17°C (night temperature), and the relative humidity (RH) was kept between 80 and 90%. A randomized complete block design was set up with one replicate pot per block. Each pot had two plants of the same genotype. Cuttings were shipped to Kenya in 2009 and rooted at the Terra Nigra site in Naivasha (Kenya) before transplanting at the two sites in Kenya: Nairobi (Winchester farm) and Njoro (Agriflora farm). The plants were grown in soil with a spacing of 15 cm. The growers were instructed to follow accepted production practices  for rose production. The set up was an incomplete block as the process of producing cuttings in the Netherlands, rooting them in Naivasha and then transplanting to Nairobi and Njoro meant that some genotypes of the original 184 present in Wageningen, did not survive. In total, 148 genotypes were fully represented at the two locations in Kenya, as well as in Wageningen. In each location there were 2 plants per genotype. Four stems were selected in each plant and measurements of the various traits were taken. Thus in total per genotype per flush there were 8 measurements taken. These measurements were done twice, with each repetition described as a flush. The traits were measured at three locations; in the Netherlands in Wageningen (WAG) in 2007, and in Kenya in Nairobi (WIN) 2009 and in Njoro (AGR) 2009. In each location we measured two flushes. In Wageningen, the first and the second flush corresponded to summer (WAG-S) and winter (WAG-W) measurements respectively.Preliminary analysis of the data using Genstat 16  was condIn this study a number of horticulturally important traits were assessed. Three phenotypic growth traits were measured before the plants were bent. The bending of juvenile stems is a standard practice carried out before the plants can begin producing commercially viable stems. The growth traits were: plant Height (H), which was a measure of the height (cm) from the rim of the pot to the apical bud before bending; plant Vigour (V) where the plants were ranked on a scale of 1–5 based on their height, number of stems, number of leaves and the branching present at time of scoring; Bending date (BT) where the bending dates were given the following numerical scores: 1 = 29, 2 = 32, 3 = 37, 4 = 39, and 5 = 44 days after planting. These traits were only measured in Wageningen over summer (2007).The traits stem length (SL) which was the length (cm) from the floral tube to the shoot base, prickles on the stem (PS) which were the number of prickles between the 4th and 6th nodes on the main stem  algorithm of Genstat 16 . DescripThe broad-sense heritabilities were calculated, across the environments and in each environment, using the following equation:2g and s2e represent genetic variance and residual variance for each environment respectively and r the number of replicates of each genotype.Here sPearson correlation coefficients between the phenotypic traits were calculated per environment and over the environments to give a measure of the strength of linear association using Genstat 16 .The trait data was graphically analysed across the four environments using principal components analysis (PCA). This was conducted using mean values from REML analysis of each trait using Genstat 16 . PCA bipIn order to quantify the size of the variation due to genotype × environment interaction relative to main genotypic variation, the morphological traits over locations were analysed using a linear mixed model with the residual maximum likelihood (REML) procedure of Genstat 16 . In the The genotype and genotype × environment (GGE) biplot was used to explain the variation due to genotypes and genotype × environments (G × E). The GGE biplot analysis was done using Genstat 16 .The highest average relative humidity was observed in WAG-W at 95% followed by WAG-S at 82%. The average relative humidity in Kenya was considerably lower at 58% in WIN and 49% in AGR. The lowest average temperature was recorded in WAG-W at 19°C followed by WAG-S at 21°C. The average temperature in Kenya was recorded at 28°C in WIN and 25°C in AGR. WAG-W also had the lowest range of temperature with difference of 6°C compared to WAG-S that had a difference of 24°C [see Additional file p < 0.001) for stem length (SL) and number of petals (NP). Number of side shoots (SS), stem width (SW) and chlorophyll content (CHL) were only measured in the second flush. For these two reasons we opted to analyse Wageningen flush 1 and flush 2 as two different environments, Wageningen summer (WAG-S) and Wageningen winter (WAG-W). There was no significant genotype × flush interaction for the two locations in Kenya so that for subsequent analyses they were treated as just two environments WIN and AGR. So in total, four different environments were identified and used for further studies.In Wageningen, The Netherlands there was significant genotype × flush interaction , vigour (V) and the length of time before bending (BT). These traits exhibited transgressive segregation and had high heritabilities . We also compared the performance of the genotypes across the three to four environments and we found significant differences in the performance of the genotypes means for the traits SL, SS, SW and CHL (p < 0.001). The mean SL in WAG-W was 94 cm, while in WAG-S, WIN and AGR the means were 74, 67 and 65 cm respectively . Correlations between H and V in WAG-S (0.81), SW in WAG-W (r = 0.33) and SL in AGR, WAG-S, WAG-W and WIN  were statistically significant (p < 0.001). The trait Vigour (V) was statistically significantly (p < 0.001) correlated to the traits CHL in WIN (r = 0.32) and SS in WAG-W (r = 0.31) (Table The trait BT which were the number of days between planting and bending of the branches was negatively correlated to the traits H and V in WAG-S (r = 0.71 and r = 0.55 respectively), PS in AGR (r = 0.37), PS in WAG-S (r = 0.34), PS in WIN (r = 0.34) and SW in WAG-W (r = 0.33). BT was also negatively correlated to SL in all the environments with AGR(r = 0.46), WAG-S (r = 0.39), WAG-W (r = 0.45) and WIN (r = 0.44). All the correlations were statistically significant  with a range of r = 0.88 to r = 0.93. The trait SL had significant correlations with a range or r = 0.55 to r = 0.71. The traits PS and PP also had high correlations across environments with a range of r = 0.70 to r = 0.84 for PS and r = 0.66 to r = 0.84 for PP. SW had low but significant positive correlations across environments with r = 0.29 to r = 0.32. CHL had a moderate correlation between the environments WAG-W and AGR, r = 0.54, but correlations were low for CHL for the environments WAG-W and WIN with r = 0.29 and WIN and AGR with r = 0.33. The correlation for SW for the environment pair WIN and AGR was only significant at p < 0.01 where r = 0.29 . In WAG-S the trait H had a positive correlation to SL (r = 0.39) and V (r = 0.82). The traits PS was positively correlated to SL (r = 0.48). BT was negatively correlated to H (r = 0.69), PS (r = 0.31), SL (r = 0.31) and V (r = 0.58). In WAG-W there was a positive correlation between the traits PS and SL (r = 0.37), PS and SW (r = 0.31), SL and SS (r = 0.48), SL and SW (r = 0.66) and SS and SW (r = 0.65). The trait SL was negatively correlated to NP (r = 0.28). In WIN there was a positive correlation between PS and SL (r = 0.36) and SW (r = 0.41). The trait SL also had a positive correlation to SW (r = 0.61). In AGR the traits PP and PS were positively correlated (r = 0.35). The trait SW was also positively correlated to PS and SL and SS with r = 0.43, r = 0.63 and r = 0.30 respectively. The trait PS was positively correlated to SL (r = 0.43).We were able to observe the correlation of all the traits within their environments [see Additional file Traits measured on the K5 population in the four environments WAG-S, WAG-W, WIN and AGR are shown in a PCA biplot Figures . In WAG-In WAG-S Figure A the traIn WAG-W . The trait number of side shoots (SS) with the lowest heritability (0.40) also had the highest σ2ge/σ2g ratio of 3.13.For the traits chlorophyll content (CHL), number of petals (NP), prickles on petioles (PP), prickles on stem (PS), and stem length (SL) the genetic variance component was larger than the G × E interaction variance component. For the traits stem width (SW) and side shoots (SS) the G × E component was higher than the genetic variance component Table . The traThe G × E interactions were further explored through a GGE biplot analysis. The first two principal components of the GGE biplots explained 97.52% (PC1 = 95.04% and PC2 = 2.47%) of the totals GGE variation for NP  studies.Most breeders sow their seeds in the Netherlands under controlled glass greenhouse followed by selection under these conditions. Seedlings are selected under controlled greenhouse conditions in the first clonal selection. After further observation cuttings are made from the selected genotypes and they are sent to the tropics, to be grown in soil under plastic green houses. It was important for us to determine if the phenotype observed during the juvenile phase in the Netherlands could be used as an indicator of the performance of the genotypes in the tropics.Within the K5 population we observed that there was a significant strong negative correlation between the length of the juvenile period (BT) and the final stem lengths (SL) of the genotypes, i.e. those genotypes that had a shorter juvenile period produced longer stems during production. This is in agreement with  who foun2ge/σ2g ratio (0.08) illustrating that there is only a very small contribution of the interaction. The observed difference in the number of petals across the environment can be found in the ranges where WAG-S had a lower average number of petals compared to WAG-W, WIN and AGR. This was also corroborated by the GGE biplot which showed the environments did not discriminate the genotypes. The presence of petal numbers lower and higher than the parents is an indication of transgressive segregation. This would suggest that the observed variation within the double flowers is controlled by multiple genes. On the other hand, a single dominant locus responsible for the switch from single flower to double flower phenotype has been identified on LG3 in the integrated consensus map (ICM) [Blo QTL for the single to double flowers switch.The high observed broad sense heritabilities for number of petals were an indicator that this trait has a strong genetic component with not much environmental influence. Heritability for number of petals was found to be high in our tetraploid population similar to what was observed in the diploid populations analysed by . The anaap (ICM) ,8-10. Inp < 0.001) in number of petals among the genotypes. This was first observed by [Within the K5 population there were significant differences  in only one environment AGR (r = 0.35). In the other environments the two traits did not have any significant correlation. In the PCA plots in three of the environments PP and PS did not show any strong positive or negative correlation. The low correlation, in combination with the high heritabilities, is a good indicator that the two traits segregate independently. It was observed in other populations that the absence of prickles on the stem and of prickles on the petiole indeed segregate independently [Within the K5 population in all the environments we observed that the prickles on petioles and the prickles on stem exhibited transgressive segregation. This indicates that multiple genes may be responsible for this trait. The transgressive segregation was unidirectional as the parent P867 did not have prickles on stem or on petioles. Both of these traits had high heritabilities and low genotype by environment interaction. The traits PS and PP had low PC2 percentages of 8.58% and 6.77%. Between PS and PP there was a weak but significant positive correlation  correlations of SW across environments are in agreement with the high observed σ\The trait SS did not have high positive or negative correlations across any of the environments. This trait had the lowest heritability of the traits studied and we observed a very high genotype × environment interaction. The GGE plot shows that the genotypes responded differently for this trait in different environments. The G × E variance component for this trait was much higher than the genotypic variance component thus we can conclude the environment strongly affects this trait.The trait Chlorophyll also exhibited transgressive segregation and had high heritabilities ranging from 0.68 to 0.75 across the environments. This trait also had strong positive correlations across all the environments. This trait showed a high interaction when its genotypic component was compared to the G × E component. This was also observed on the GGE biplot which showed that WAG-W discriminated the genotypes differently for this trait from AGR and WIN. This trait was shown to have a non cross-over G × E interaction. As there was no correlation between this trait and stem length, stem width or vigour trait we can conclude that chlorophyll content is genetically independent of this trait and therefore cannot be used to predict the vigour of a plant.In the last decade rose breeding has had to rapidly adapt to the change in growing climate as production locations have shifted from predominantly seasonal European climate to more constantly warm climates in the tropics. This meant that more trials had to be done to identify if the European selected varieties were suitable for the warmer environments. We choose to study the effects of environment on the traits NP, PS, PP, SL, SW, SS and CHL as these are the traits that form the basic plant structure. If a breeder was able to understand how these traits are inherited and how they respond to the different environments they would be able to make a more structured breeding program. Currently an ideal rose plant would have at least 30 petals, no prickles on the stem or petioles, at least 50 cm stem length and very few side shoots. The stem width is relative to the stem length.The traits had a high heritability thus enabling the breeder to actively breed for or against these traits. It was also important to understand how the environment would affect these traits. We can now conclude that in colder climates you get longer stems but the traits prickles on stem and petals are not affected by the environment. For the number of petals we saw that to increase the number of petals we need to reduce the difference in day and night temperatures. The lack of strong positive correlations to chlorophyll is an indicator that the darker green leaves is aesthetically more pleasing but that does not translate to a more vigorous plant.Results attained by this experiment showed that we have different magnitudes of non-crossover G × E interactions. For the traits NP, PS and PP with a low interaction and high heritability, selection can be done at any of the environments. Thus these traits can be confirmed at the breeding site. For the traits SL, SW, SS and CHL that had a higher interaction selection for or against these traits should be done at or at least verified at the production location.


Linear mixed model (LMM) analysis has been recently used extensively for estimating additive genetic variances and narrow-sense heritability in many genomic studies. While the LMM analysis is computationally less intensive than the Bayesian algorithms, it remains infeasible for large-scale genomic data sets. In this paper, we advocate the use of a statistical procedure known as symmetric differences squared (SDS) as it may serve as a viable alternative when the LMM methods have difficulty or fail to work with large datasets. The SDS procedure is a general and computationally simple method based only on the least squares regression analysis. We carry out computer simulations and empirical analyses to compare the SDS procedure with two commonly used LMM-based procedures. Our results show that the SDS method is not as good as the LMM methods for small data sets, but it becomes progressively better and can match well with the precision of estimation by the LMM methods for data sets with large sample sizes. Its major advantage is that with larger and larger samples, it continues to work with the increasing precision of estimation while the commonly used LMM methods are no longer able to work under our current typical computing capacity. Thus, these results suggest that the SDS method can serve as a viable alternative particularly when analyzing ‘big’ genomic data sets. Recent surge of genome-wide association studies (GWAS) based largely on the use of single nucleotide polymorphisms (SNPs) has enabled plant/animal breeders and human geneticists to identify hundreds of SNPs responsible for the genetic variation of quantitative traits or complex diseases. While such identification contributes to an in-depth understanding of the genetic architecture of complex traits, the numerous SNPs identified have collectively accounted for only a small amount of genetic variation in many studies. Several possible explanations have been put forward to explain this phenomenon known as the “missing heritability” Since Meuwissen et al. R2 of ∼ 0.02%. However, the LMM analysis of such large data sets creates a much heavier computational burden because the computing time required for constructing the genetic relationship matrix and solving LMM equations increases with the cube of the number of individuals fit as a random effect. The computing time is further increased because iteration is needed to estimate population parameters, such as variance components that are due to the effects of individual tested markers. Despite recent proposed improvements , the computational burden with the LMM analysis will remain to be a major issue in anticipation of larger GWAS or GS studies in the future.In order to achieve sufficient statistical power of detecting numerous variants with small-sized effects that collectively contribute to the genetics of most complex traits, large sample sizes are often used It has been well known since the founding days of quantitative genetics r >0.98) with the high marker density (m = 20000) regardless of sample sizes. However, the situation is different for lower marker densities: while there is little change in the correlations for smaller samples, the correlations for larger samples can be substantially reduced. For example, at the marker density of m = 200, the correlation for the population of size n = 200 is r = 0.95, but the correlation for the sample of size n = 10000 is nearly halved at r = 0.48.The effects of sample size and marker density on the correlation between the actual and theoretical genetic relatedness are depicted in t-tests show that with a few exceptions (seven in the REML estimates and one in the SDS estimates), the means of the estimates are not significantly different from their theoretical values. However, while the ranges of the estimated heritability by the REML approach are relatively consistent over different sample sizes and marker densities, the ranges by the SDS approach are much larger for n = 500 than for n = 5000 under different h2 and m values. For example, for the parameter combination of h2 = 0.8 and m = 200, the ranges of REML estimates are 0.075 ( = 0.828–0.753) for n = 500 and 0.014 (0.806–0.792) for n = 5000, but the ranges of SDS estimates are 0.558 (1.096–0.538) for n = 500 and 0.268 (0.955–0.687) for n = 5000. Unlike the sample size, the marker density has a minor effect on the estimates of heritability though there are larger standard deviations and ranges of heritability estimates when too many markers (m = 20000) are used.Presented in n = 500 to n = 40000. These records of time and memory requirements are taken from the runs of the analyses on Dell Precision T1650 with Intel Xeon E3-1280 (3.8 GHz) and 32GB of RAM under the Redhat Linux 6.4 operating system. In anticipation of insufficient RAMs required for the REML-based analyses of large data sets, we decide that any analysis is terminated if its RAM usage exceeds 30GB. Thus, the recorded memory for a given analysis is the maximum memory usage over the entire process of the analysis if the memory usage at any given time of the analysis is within 30GB RAMs, or the maximum memory usage during the time period from the beginning to the termination of the analysis if the memory usage is beyond 30GB RAMs. In this study, the time period for terminating an analysis is set to be one hour after the 30GB RAM usage criterion is reached. For GCTA, only one CPU core is used in each analysis run to measure the computational time and memory usage for a fair comparison with the computational efficiencies for rrBLUP and SDS. For rrBLUP, the memory usage is measured in a clean R environment with only the The computational time and memory requirements for the two REML-based analyses (rrBLUP and GCTA) and the SDS analysis are recorded in n = 20000, the rrBLUP analysis is terminated because its memory requirement (65.6GB) is far beyond our criterion of 30GB RAMs. With this same sample size, the GCTA analysis is successfully completed in ∼27 minutes but its memory requirement (28.9GB) is near the RAM capacity of our computer. Obviously, both rrBLUP and GCTA are terminated for n = 40000 because the memory requirement certainly exceeds our criterion of 30GB RAMs.It is evident from ilth element of W matrix for the genotype of the ith individual at lth locus is coded as, ilz is 1 for the reference homozygote and 0 for the alternative homozygote, and lth locus. Both rrBLUP and GCTA do not allow for inputting haploid data for constructing W matrix and thus the GRM, but GCTA does have an option to allow for inputting the GRM. Thus the comparison will be between the SDS and GCTA analyses for this empirical data.We will analyse two empirical examples. The first empirical data set is taken from Crossa et al. The estimates of heritability for the GY in E1-E4 are presented in Enhancer of split Complex [(E(spl)-C] with sternopleural bristle number (SBN) and abdominal bristle number (ABN) for 2000 Drosophila melanogaster individuals  sampled from a location in Napa Valley, California. The 203 genotyped polymorphisms actually consisted of 191 SNPs and 12 insertion/deletion events, but for simplicity, they were all referred to as SNPs. There was little evidence of associations between individual SNPs and SBN or ABN in the female or male population. In fact, Macdonald et al. p<0.01) departure from Hardy-Weinberg proportions of three genotypes at each locus. Thus, 950 individuals with 162 SNPs in the E(spl)-C are retained for the analysis. To be consistent with the analysis of the first data set, only the GCTA is used for comparison with the SDS.The second empirical data set is taken from Macdonald et al. As shown in n = 50000 and marker density of m = 2000 and heritability of h2 = 0.8. We analyze this data set using the SDS only because the other methods have already stopped to work with a sample of smaller size n = 40000 and the same marker density (m = 2000) as shown in n = 5000. On average, the per-sample time requirement for constructing the GRM and the SDS analysis is 6355.3 seconds or about 1 hour 46 minutes.The computational burdens of many LMM applications including heritability estimation in large-scale genomic studies have recently stimulated a huge amount of research interest in the development of faster and memory-efficient algorithms for feasible and successful analyses of large-scale genomic data. Despite these efforts, the computational challenges remain. In this study, we investigate the statistical properties and computational efficiency of the least-squares-based SDS method in comparison to the two LMM methods (rrBLUP and GCTA) to see the feasibility of the SDS method as a viable alternative to LMM-based methods. Our results  show than = 1000 with m = 2000, comparing to only 17.856 seconds by rrBLUP, 0.746 seconds by GCTA and 0.061 seconds by the SDS as shown in n and m. It would have been hardly feasible to run the data sets from a number of replicated simulations. Furthermore, de los Campos et al. It is evident from actual computational time and memory requirements under different sample sizes  have provided excellent computing resources to researchers with larger computing needs. However, access to these servers is a complicated process. It involves (i) submission of separate proposals for those systems with considerably large computing capacity in the supercomputing network; and (ii) a long waiting time after submitting batch jobs. In addition, some servers only serve certain countries or regions. Another possibility is the use of GPU (graphics processing unit)-accelerated computer for more efficient and even faster solutions to the large-scale GRM and its inverse essential in GWAS and genomic prediction. There are recent studies on the use of GPU-based parallel computing in GWAS In our study, any analysis would be terminated if its RAM usage exceeds 30GB, the maximum allowable RAMs after accounting for the RAM requirements by the operating system and other essential utility programs. It may be argued that we can increase the memory for the current computer or tap into supercomputing resources to address the issue of insufficient memory. However, neither solution is very feasible in our situation and perhaps in many other situations. Our workstation has a motherboard that can only support the maximum RAMs of 32GB and thus there is no room to add more memory. Many supercomputing servers such as the Westgrid in Canada  and low marker density (m = 1279). At the first glance, this result is somewhat surprising because the SDS estimates would have been more fluctuating than the REML at this level of sample size and marker density as can be inferred from the simulation results of A matrix obtained from the pedigree information only) and the M-RKHS model . The heritability estimates are subsequently calculated. Our GRM-based estimates of heritability, by either SDS or GCTA, are close to or slightly higher than those based on the P model, but they are all smaller than those based on the M-RKHS model. The elements of the K matrix derived from the M-RKHS model are a nonlinear  function of squared-Euclidean distance between markers in pairs of individuals, thereby probably capturing more genetic variation. It remains to be investigated how the use of nonlinear functions of marker distances generally affects the estimation of the genetic relationship between individuals, thereby influencing the estimation of variance components or heritability. The analysis of the drosophila data taken from Macdonald et al. The analysis of the wheat yield data supplied by Crossa et al. A matrix) for two major reasons. First, the GRM would capture much of the Mendelian sampling variation that is missing in the A matrix. Second, the use of marker SNP data (rather than pedigree information) allows for estimation of relatedness of distantly related individuals, thereby controlling confounding effects from the environmental correlation between relatives due to their shared (common) environment. However, the unbiased estimate of GRM is achieved only if all the markers used for the estimation are the causal variants m = 2000) for most simulation trials. With the high marker density (m = 20000), there are more irrelevant markers, thereby costing some accuracy in the estimation of heritability. Similar results were observed in other simulation studies .A key part of both LMM and SDS analyses is to estimate the GRM based on a set of SNPs or other genetic markers. The use of GRM is more advantageous over the use of the traditional pedigree-based relationship matrix . In WLS, these near-zero values would be overemphasized , thereby having a much stronger influence on the slope of the regression line (estimated additive genetic variance or narrow-sense heritability) than the values over the rest of the genetic similarity range. This issue certainly needs to be further investigated.Another possible question with our binning procedure is that the number of observations may vary from bin to bin and thus the residual variance may also vary from bin to bin. At the first glimpse, the use of weighted least square (WLS) analysis would be a natural solution to this problem. However, our preliminary analysis shows that the WLS results are similar to those from the regression analysis of un-binned data . We are not exactly sure why WLS does not work well as it should. We suspect the following reason. For the GRM constructed for a large sample, the distribution of genetic similarity between pairs of individuals is often highly skewed towards zero, that is, the majority of genetic similarity values are clustered around zero (see D) based on marker genotypes is needed to estimate the dominance genetic variance; similarly different epistatic relationship matrices based on D matrices  are needed to estimate epistatic genetic variances. While it is quite straightforward for the SDS method to estimate all non-additive variances by extending the simple regression analysis to the multiple regression analysis, it is computationally even more challenging for the LMM-REML methods to estimate non-additive genetic variances even for a moderate-sized data set. Furthermore, far fewer software packages are available for the estimation of non-additive genetic variances. There is an ongoing debate on how much non-additive genetic variances really contribute to the ‘missing heritability’ Our study focuses on a comparison of LMM-REML and SDS methods for estimating additive genetic variance and thus narrow-sense heritability. Such comparison can be easily extended to include non-additive genetic variances. As shown in Su et al. Our study uses all individuals, related and unrelated, in the simulated and empirical populations for the SDS and the LMM-REML analyses. This is somewhat in contrast to some recent studies, more specifically by Yang et al. n >100000) and m >500000) genomic studies  and the use of LMM-REML analysis for the aggregate effect of all markers across the genome in our study and other studies. It should not be forgotten that a key motivation of using LMM-REML analysis is to help recover the portion of ‘missing’ heritability encountered in the single-marker analysis n <10000) and m <100000) genomic data under our current typical computing capacity and suggested the SDS approach as a viable alternative when the data becomes larger and larger. With our binning approach to the SDS analysis, it is possible to handle genomic data sets of any size because the regression analysis is based on a fixed number of bins for phenotypic and genetic similarity indexes regardless of sample size. However, it is presently not feasible yet for the LMM-REML analysis to handle a very large data set. For example, Yang et al. m = 500000 and n = 4000, it would have taken GCTA more than 400 ( = 100×4) CPU hours or more than 16 days to complete this simulation alone. Thus, in most recent LMM-REML analyses , a moderate-sized data set is used.A distinction needs to be made between the use of simple regression analysis or its refinements for genome-wide scan of individual marker effects in a large-scale  was originally proposed by Koch n individuals is partitioned into genetic and residual components, V = G+R = ijθ} is the n×n genetic relationship matrix (GRM) with ijθ being the coefficient of genetic relationship between ith and jth individuals, In is an identity matrix order n, i and j (ijv) in V in terms of the phenotypic correlation between the two individuals , θ = Vech(p = Vech(P) with Vech(X) being the “vector-half” function X. Thus, for an n×n matrix X, there are n(n-1)/2 elements in Vech(X). The residual term in In) is a vector of zeros (0). The results in Multiplying both sides of ith and jth individuals can be written as,ijθ) measures the genetic distance between the ijth pair of individuals for i<j. The usual assumptions are: (i) the additive genetic effects are independent of the residual deviations and (ii) there is no correlation between the residual deviations of the ijth pair of individuals. Collecting the partitioning results for all pairs of individuals including the trivial result of zero SDS for individuals with themselves, we have in matrix form,Jn is the n×n matrix of ones, In is the identity matrix of order n, and d is a diagonal matrix with the diagonal elements being the same as in matrix Above we show that the linear relationship between phenotypic and additive genetic covariances or correlations between pairs of individuals can be used for predicting additive genetic variance or heritability. Similarly, the expected value of difference squared (DS) between the phenotypic values of a pair of individuals can be partitioned into the two components due to additive genetic effect and residual deviation as done in Grimes and Harvey ijth pair of individuals is derived as,The SDS1 and SDS2 (covariance) versions are obviously related to each other. This relationship between the phenotypic values of the ijd) and the covariance (ijv) with the intercept being the phenotypic variance (ijd) values are known, we can immediately obtain the corresponding covariances for all pairs of individuals by reversing the relationship in ijv = ijd.It is evident from Similar relationships can be found for additive genetic SDS and residual SDS with ijd/It is evident from the above theoretical analysis that the estimation of variance components or heritability can be carried out using either the SDS1 (difference) model or SDS2 (covariance) model and both models would lead to the same estimates of variance components or heritability. Thus we will focus on our estimation procedure under the SDS2 model as it is more directly connected to the LMM analysis.n individuals, there are n(n-1)/2 pairs of individuals and phenotypic values of each pair are used to calculate the phenotypic similarity. Thus, the sample phenotypic similarity between the ith and jth individuals for i<j is given by,n phenotypic values. Thus, the additive genetic variance can be estimated from the linear regression of the sample phenotypic similarity on the genetic relatedness as,β is the regression coefficient which estimates the additive genetic variance  within the bins, thereby leading to the estimate of additive genetic variance with minimal bias. A similar argument can be made for binning the standardized phenotypic similarity index . With a nearly perfect starting genetic correlation, AR1 model guarantees a full range of genetic relatedness for each and every simulated sample of sufficient size. Such high genetic correlation may be artificial and unrealistic for many human and livestock populations to which the LMM-REML analysis is often applied. However, it does approximate some situations in crop breeding where modern cultivars are genetically highly similar because (i) strong directional selection has been practiced and (ii) they all trace back to one or a few founders, for example, Canadian wheat ith and jth individuals which are t = |i – j| individuals apart is n individuals under the AR1 model is,We assume that the population consists of n×mZc matrix consisting of m random observations from an n-variate standard normal distribution with the mean vector of zeros and the covariance matrix being given in We further assume that n. For this reason, a more efficient sampling scheme based on the definition of the AR1 model is employed and implemented in the following three steps. First, a ith vector, i-1)th vector nz vectors generated in such way are collected to form the matrix Zc = However, while this direct sampling scheme can be easily implemented using existing software packages such as the mvrnorm function in MASS/R Zc matrix is generated by directly sampling from a multivariate normal distribution or by the recursive relationship, it needs to be converted into an indicator genotype matrix Z = zi containing only three values of 0, 1 and 2 to indicate three possible genotypes at each of m independent loci for the ith individual. For simplicity, we consider the proportions of the three genotypes in each individual to be 1∶ 2 : 1 so that the ranges for converting a normally distributed variate into the three genotypes coded as 0, 1 and 2 are: (−Regardless of whether the y) for µ is the population mean, W is an n×m standardized genotype matrix with the ilth element being lth simulated marker, and e is a vector of n residual effects taken from a multivariate normal distribution, u is a vector of m random additive genetic effects that are taken from a standard multivariate normal distribution a = Wu are taken from a multivariate normal distribution, To simulate random additive genetic effects and phenotypic values, we assume that a quantitative trait is controlled by the quantitative trait loci (QTL) which are randomly located at one tenth of the total simulated markers . In other words, 10% of the total simulated markers are causal (relevant) variants and the remaining 90% are irrelevant markers. Thus a vector of phenotypic values . For each simulated population, the theoretical GRM between n individuals is obtained using the AR1 model as in We consider three simulation scenarios. In the first scenario, we investigate the degree of agreement between the actual and theoretical genetic relatedness. This scenario simulates five populations of sizes n) and marker density (m) on the estimation of narrow-sense heritability. This simulation scenario consists of all combinations of the following parameter values: three levels of the heritability ; and three levels of marker density . These simulation trials are replicated 100 times. The variance components and heritability are estimated using (i) the LMM approach as implemented by two software packages, rrBLUP http://statgen.ualberta.ca/index.html?open=software.html). In each of 100 replicated simulations, the LMM approach as in rrBLUP and GCTA estimates the additive genetic variance and residual variance and the heritability is subsequently calculated as the ratio of the estimated additive genetic variance to the sum of the estimated two variance components. On the other hand, the SDS approach directly estimates the heritability through the regression of the phenotypic correlation between pairs of individuals on the corresponding values of genetic relatedness.In the second scenario, we want to examine the effects of sample size  and one heritability  required by the different estimation methods for the analysis of simulated data for each of the five replicates are recorded using proc.time, a R core function In the third scenario, we want to compare and contrast the computational efficiency of the SDS method of heritability estimation to the REML-based methods as implemented in the two software packages, rrBLUP and GCTA over a wide range of sample sizes. We choose seven sample sizes for this scenario: Figure S1The genetic similarity and phenotypic similarity in a simulated population under an AR1 model with aθ = 0.95 and h2 = 0.5, n = 4000 and m = 2000.(TIF)Click here for additional data file.Text S1Overview of linear mixed models for genomic data.(DOC)Click here for additional data file.Program S1R code for step-by-step description of simulating positions and effects of causal markers.(R)Click here for additional data file.


A novel reparametrization-based INLA approach as a fast alternative to MCMC for the Bayesian estimation of genetic parameters in multivariate animal model is presented. Multi-trait genetic parameter estimation is a relevant topic in animal and plant breeding programs because multi-trait analysis can take into account the genetic correlation between different traits and that significantly improves the accuracy of the genetic parameter estimates. Generally, multi-trait analysis is computationally demanding and requires initial estimates of genetic and residual correlations among the traits, while those are difficult to obtain. In this study, we illustrate how to reparametrize covariance matrices of a multivariate animal model/animal models using modified Cholesky decompositions. This reparametrization-based approach is used in the Integrated Nested Laplace Approximation (INLA) methodology to estimate genetic parameters of multivariate animal model. Immediate benefits are: (1) to avoid difficulties of finding good starting values for analysis which can be a problem, for example in Restricted Maximum Likelihood (REML); (2) Bayesian estimation of (co)variance components using INLA is faster to execute than using Markov Chain Monte Carlo (MCMC) especially when realized relationship matrices are dense. The slight drawback is that priors for covariance matrices are assigned for elements of the Cholesky factor but not directly to the covariance matrix elements as in MCMC. Additionally, we illustrate the concordance of the INLA results with the traditional methods like MCMC and REML approaches. We also present results obtained from simulated data sets with replicates and field data in rice. Estimation of variance components and associated breeding values is an important topic in classic  methods are a popular choice for Bayesian inference of animal models  can be done by modifying the terms of Eq. . Additive genetic relationship matrix  dataset variance components were estimated using MCMCglmm, R-INLA and ASReml-R packages. For MCMC analysis using MCMCglmm package, we considered a total chain length of 50,000 iterations with a burning period of 10,000 iterations. The MCMCglmm package assign inverse-Wishart prior distribution for the random and residual covariance matrices. In our MCMC analysis, we used identity matrix as the scaling matrix of the prior distribution (ones for the variances and zeros for the covariances) assigned for the genetic covariance matrix  using 50 simulation replicates for the (co)variance components and then we plotted the box plots for the estimation errors to visualize the estimation accuracy of different methods. We show those box plots for the estimation errors for the variance Fig.  and covaAdditionally, instead of covariances we report the estimated genetic and residual correlation coefficients as well as the 95 % empirical confidence intervals (in brackets) for each trait in Table Heritability and breeding values are of great interest to breeders in order to plan an efficient breeding program. In our study, we also calculated the correlation coefficients between the estimated and true breeding values using different estimation methods. We used average over 50 simulation replicates for both datasets to calculate the correlation coefficients. For the high heritability dataset, the correlation coefficients were 0.85, 0.84 and 0.83 for REML, MCMC and INLA methods, respectively. However, for the low heritability dataset the correlation coefficients were relatively low being 0.64, 0.65 and 0.58 for REML, MCMC and INLA, respectively.We chose the same starting values for the simulated data and real data in our REML analysis. For MCMC analysis we used empirical phenotypic variance of each trait as the variances and zeros as the covariance as the scale matrix of the prior distribution, whereas, starting values for other parameters were set randomly. For the REML analysis we chose the same starting values that we used for the simulated dataset. Both REML and Bayesian methods gave similar results in our analysis using real dataset Table . Due to Multi-trait analysis of mixed models tend to be powerful and provide more accurate estimates than the single-trait analysis because the former method can take into account the underlying correlation structure found in a multi-trait dataset. However, Bayesian and non-Bayesian inference of multi-trait mixed model analysis are complex and computationally demanding. In this study, we explained how to do Bayesian inference of a multivariate animal model using recently developed INLA and the counter part MCMC, while comparing the results with the commonly used REML estimates. Our results show that reparametrization-based INLA approach can be used as a fast alternative to MCMC methods for the Bayesian inference of multivariate animal model. The reparametrization approach, that was here applied for INLA analysis, can be used also more generally together with other tools to speed up the multi-trait animal model computations.Drawback of the reparametrization-based approach is that priors are assigned for elements in the Cholesky factor instead of the original covariance matrix. Thus, here it is not possible to make a direct comparison between the MCMC and INLA results due to the differences in the prior distributions, however, it is possible to compare both approaches if we choose the same prior distributions. For the MCMC analysis we used inverse-Wishart distributions for the covariance matrices; whereas, for INLA we used Gaussian prior distribution for the elements in the Cholesky factor  (Nowadays, molecular markers are widely used in animal and plant breeding programs as a valuable tool for genetic improvement. Therefore, we also showed how to estimate genetic parameters in a multivariate animal model using molecular marker information with the reparametrization-based INLA approach and frequentist framework. Finally, our results imply that the reparametrization-based INLA approach can be used as a fast alternative to MCMC methods in order to estimate genetic parameters with a multivariate animal model using pedigree information as well as with molecular marker information.BM, AH, PK, JL and MS were involved in the conception and design of the study. BM performed the simulations and preprocessing of the data. BM and AH implemented the method, and performed the statistical analyses. BM drafted the manuscript. BM, AH, JL and MS participated in the interpretation of results. All the authors critically revised the manuscript.


Ovis aries) to investigate the genetic architecture of individual autosomal recombination rates. Individual rates were high relative to other mammal systems and were higher in males than in females . The heritability of autosomal recombination rate was low but significant in both sexes . In females, 46.7% of the heritable variation was explained by a subtelomeric region on chromosome 6; a genome-wide association study showed the strongest associations at locus RNF212, with further associations observed at a nearby ∼374-kb region of complete LD containing three additional candidate loci, CPLX1, GAK, and PCGF3. A second region on chromosome 7 containing REC8 and RNF212B explained 26.2% of the heritable variation in recombination rate in both sexes. Comparative analyses with 40 other sheep breeds showed that haplotypes associated with recombination rates are both old and globally distributed. Both regions have been implicated in rate variation in mice, cattle, and humans, suggesting a common genetic architecture of recombination rate variation in mammals.Meiotic recombination breaks down linkage disequilibrium (LD) and forms new haplotypes, meaning that it is an important driver of diversity in eukaryotic genomes. Understanding the causes of variation in recombination rate is important in interpreting and predicting evolutionary phenomena and in understanding the potential of a population to respond to selection. However, despite attention in model systems, there remains little data on how recombination rate varies at the individual level in natural populations. Here we used extensive pedigree and high-density SNP information in a wild population of Soay sheep ( MEIOTIC recombination is a fundamental feature of sexual reproduction in nearly all multicellular organisms. In many species, it ensures the proper segregation of homologous chromosomes during meiosis, avoiding deleterious outcomes such as aneuploidy  and can ring finger protein 212 (RNF212), complexin 1 (CPLX1), meiotic recombination protein REC8 (REC8), and PR domain zinc finger protein 9 (PRDM9), among others     is a Neolithic breed of domestic sheep that has lived unmanaged on the St. Kilda archipelago in Scotland since the Bronze Age , (2) to determine whether individual recombination rates were heritable, (3) to identify specific genetic variants associated with recombination rate variation, and (4) to determine whether the genetic architecture of recombination rate variation is similar to that observed in other mammal species.Here we examine the genetic architecture of recombination rate variation in a wild mammal population. The Soay sheep  have been studied on an individual basis since 1985  .To allow unbiased phasing of the SNP data, a standardized pedigree approach was used to identify crossovers that had occurred within the gametes transferred from a focal individual to its offspring; hereafter, focal individual (FID) refers to the sheep in which meiosis took place. For each FID-offspring combination in the Soay sheep pedigree, a subpedigree was constructed to include both parents of the FID (“Father” and “Mother”) and the other parent of the “Offspring” (“Mate”), where all five individuals had been genotyped . This suprepare function; incompatible genotypes were removed from all affected individuals, and subpedigrees containing parent-offspring relationships with more than 0.1% mismatching loci were discarded. Second, sex-specific and sex-averaged linkage map positions (in Kosambi centimorgans) were obtained using the map function, where SNPs were ordered relative to their estimated positions on the sheep genome assembly Oar_v3.1 (GenBank assembly ID GCA_000298735.1)  were assumed to be incorrectly mapped and were removed from the analysis, with the map function rerun until all map distances were below this threshold; in total, 76 SNPs were assumed to be incorrectly mapped . Third, the chrompic function was used to identify informative SNPs  on chromosomes transmitted from the FID to its offspring; crossovers were deemed to have occurred where there was a switch in the grandparental origin of a SNP allele  and, in turn, an overestimation of recombination rate. To reduce the likelihood of calling false crossover events, runs of grandparental origin consisting of a single allele  were recoded as missing  . In the remaining cases of double crossovers, the base pair distances between immediately adjacent SNPs spanning a double crossover were calculated (hereafter span distance) . Informative SNPs that occurred within double-crossover segments with a log10 span distance <2.5 SD from the mean log10 span distance  also were recoded as missing  . The autosomal crossover count (ACC), the number of informative SNPs, and the informative length of the genome  then were calculated for each FID. A simulation study was conducted to ensure that our approach accurately characterized ACC and reduced phasing errors. Autosomal meiotic crossovers were simulated given an identical pedigree structure and population allele frequencies (Nsimulations = 100) (see File S1 for detailed methods and results). Our approach was highly accurate in identifying the true ACC per simulation across all individuals and per individual across all simulations (adjusted R2 > 0.99) but indicated that accuracy was compromised in individuals with high values of R2 ≤ 0.95 between simulated and detected crossovers in the simulation analysis were removed from the study (N = 8) (File S1).Errors in determining the grandparental origin of alleles can lead to false calling of double crossovers (x) regression in R v3.1.1.Relationships between chromosome length and linkage map length and male and female linkage map lengths were analyzed using linear regressions in R v3.1.1. The relationship between chromosome length and chromosomal recombination rate (defined as cM length/Mb length) was modeled using a multiplicative inverse . Briefly, the probability of crossing over within a bin was the sum of all recombination fractions x), multiplicative inverse (1/x), quadratic (x2 + x), cubic (x3 + x2 + x), or log term (log10x); sex, fitted as a main effect and as an interaction term with distance to the nearest telomere; number of SNPs within the bin; and GC content of the bin  (4). Initial models also included a term indicating whether a centromere was present or absent on the 60-Mb region, but this term was not significant in either model.Variation in crossover probability relative to proximity to telomeric regions on each chromosome arm was examined using general linear models with a Gaussian error structure. The response variable was crossover probability per bin; the fitted covariates were as follows: distance to the nearest telomere, defined as the best fit of either a linear (ar_v3.1) . An addii.e., narrow-sense heritability h2, hereafter heritability), a genomic relatedness matrix (GRM) at all autosomal markers was constructed for all genotyped individuals using GCTA v1.24.3 , and variance components estimated from models with and without adjustment were highly similar, suggesting that adjusting for sampling error in this way did not introduce bias. Matrices were not pruned to remove related individuals  because there is substantial relatedness within this population, and models included common environment and parental effects, controlling for some consequences of shared environment among relatives (see below). Trait variance was analyzed first with the following univariate model:y is a vector of the ACC per transferred gamete; X is an incidence matrix relating individual measures to a vector of fixed effects; β, Z1, and Zr are incidence matrices relating individual measures with additive genetic effects and random effects, respectively; a and ur are vectors of additive genetic effects from the GRM and additional random effects, respectively; and e is a vector of residual effects. The heritability h2 was calculated as the ratio of the additive genetic variance to the sum of the variance estimated for all random effects. Model structures were initially tested with a number of fixed effects, including sex, permanent environment effect), maternal and paternal identity, and common environment effects of FID birth year and offspring birth year. The significance of fixed effects was determined using a Wald test, whereas the significance of random effects was calculated using likelihood-ratio tests (LRTs) between models with and without the focal random effect. Only sex and additive genetic effects were significant in any model, but ACC was modeled as a trait of the FID. Phenotypic variance in ACC was partitioned using a restricted maximum-likelihood (REML) animal model  implemenrA was determined using the CORGH error-structure function in ASReml-R (correlation with heterogeneous variances) with rA set to be unconstrained; models fitted sex-specific inbreeding coefficients and individual identity effects. To test whether the genetic correlation was significantly different from 0 and 1, the unconstrained model was compared to models with rA fixed at a value of 0 or 0.999. Differences in additive genetic variance in males and females were tested by constraining both to be equal values using the CORGV error-structure function in ASReml-R. Models then were compared using LRTs with 1 degree of freedom.To investigate whether the additive genetic variation underlying male and female ACC was associated with sex-specific variation in ACC, bivariate models were run. The additive genetic correlation P after multiple testing at α = 0.05 was 2.245 × 10−6. Although sex chromosome recombination rate was not included in the analysis, all GWASs included the X chromosome and SNP markers of unknown position (N = 314). The proportion of phenotypic variance attributed to a given SNP was calculated using the following equation  .Genome-wide association studies (GWASs) of autosomal recombination rates under different scenarios were conducted using ASReml-R  and fitting them in models with an additional GRM of all autosomal markers not present in region i . GRMs were constructed in the software GCTA v1.24.3 and were adjusted using the −grm-adj 0 argument (see earlier) , significant model comparison tests were disregarded. A Bonferroni approach was used to account for multiple testing across the genome by taking the number of tests and dividing by 2 to account for overlap of the sliding windows (because each genomic region was modeled twice).Although a powerful tool to detect regions of the genome underlying heritable traits, the single-locus approach of GWAS has reduced power to detect rare variants and variants with small effect sizes  from those due to trans effects . By using the total ACC within a gamete, we incorporated both cis and trans effects into a single measure. To examine trans effects only, we determined associations between each SNP and ACC minus crossovers that had occurred on the chromosome on which the SNP occurred; e.g., for a SNP on chromosome 1, association was examined with ACC summed across chromosomes 2–26. We found that in this case, examining trans variation  obtained similar results to cis and trans variation (ACC) for both regional heritability and genome-wide association analyses, leading to the same biological conclusions.In the preceding analyses, we wished to separate potential associations with ACC due to p is a vector of the proportion of genetic variation in the population captured by m selected animals, Am is the corresponding subset of a pedigree relationship matrix, and c is a vector of the mean relationship of the m selected animals  (Table S1). In both the Soay and non-Soay data sets of the Ovine SNP50 BeadChip, we extracted 58 SNPs corresponding to ∼4 Mb of the subtelomeric region on chromosome 6 and phased them using Beagle v4.0  approximately 150 years ago  contains genomic data after quality control measures, pedigree information and subpedigree structures, autosomal GRMs, population-wide crossover probabilities, and individual recombination rate results. All scripts used for the analysis are provided at https://github.com/susjoh/GENETICS_2015_185553.The Supplemental Material contains information on additional analyses conducted and is referenced within the text. Figure S2 and Table S2). There was a linear relationship between the length of autosomes in megabases and linkage map lengths  was consistent across all autosomes . Increased crossover probabilities were associated with higher GC content . Investigation of the relative distances between crossovers (in cases where two or more crossovers were observed on a single chromatid) indicated that there may be crossover interference within this population, with a median distance between double crossovers of 48 Mb .Finer-scale probabilities of crossing over were calculated for 1-Mb windows across the genome for each sex using recombination fractions from their respective linkage maps. Crossover probability varied relative to proximity to telomeric regions, with a significant interaction between sex and distance to the nearest telomere fitted as a cubic polynomial function . Males hh2 = 0.145, SE = 0.027), with the remainder of the phenotypic variance being explained by the residual error term . However, females had marginally higher additive genetic variance (PLRT = 0.040) and higher residual variance (PLRT = 1.11 × 10−3) in ACC than males (P > 0.05). A bivariate model of male and female ACC showed that the cross-sex additive genetic correlation (rA) was 0.826 (SE = 0.260); this correlation was significantly different from 0 (PLRT = 1.14 × 10−3) but not different from 1 (PLRT = 0.551).Individual ACC was heritable  but was not associated with male ACC  in an otherwise gene-poor region of the genome . A second SNP on chromosome 3, OAR3_87207249.1, was associated with ACC in both sexes  and 371 kb from the 5′ end of an ortholog of ribosomal protein L10 (RPL10). Full results of GWASs are provided in Table S4.Two further regions on chromosome Table S5). There was a strong sex-specific association of ACC in females within a subtelomeric region on chromosome 6 (20-SNP sliding window) (RNF212 and CPLX1. The region explained 8.02% of the phenotypic variance (SE = 3.55%) and 46.7% of the additive genetic variance in females (PLRT = 9.78 × 10−14) but did not contribute to phenotypic variation in males . There was an additional significant association between ACC in both sexes and a region on chromosome 7 corresponding to a 1.09-Mb segment containing ∼50 protein-coding regions, including RNF212B  and REC8 ; this region had not shown any significant associations using the GWAS approach alone. The region explained 4.12% of phenotypic variance (SE = 2.3%) and 26.2% of the additive genetic variance in both sexes combined; however, in sex-specific models, the significant association with ACC did not remain after correction for multiple testing (Table S5). No association was observed in the regional heritability analysis with the two regions on chromosome 3 identified using the GWAS approach. Full results for the regional heritability analysis are provided in Table S5.The contribution of specific genomic regions to ACC was determined by partitioning the additive genetic variance in sliding windows  ( window) . This cocis and trans effects on recombination rate. When repeated with trans effects only, all variants associated with ACC in both the GWAS and regional heritability analyses remained significant , meaning that they are likely to affect recombination rate globally  rather than being in LD with polymorphic recombination hotspots.The results presented earlier were ACC incorporating both 6 showed that this region has elevated levels of LD, with the two most significant SNPs from the 50K chip tagging a haplotype block of ∼374 kB . This block contained three candidate genes, CPLX1, cyclin-G-associated kinase (GAK), and polycomb group ring finger 3 (PCGF3), and was 177 kb away from the candidate locus RNF212 . A bivariate animal model including an interaction term between ACC in each sex and the genotype at oar3_OAR6_116402578 confirmed that this locus had an effect on female ACC only; this effect was additive, with a difference of 4.91 (SE = 0.203) autosomal crossovers per gamete between homozygotes (Table S6). There was no difference in ACC between the three male genotypes. Full results for univariate models at imputed SNPs are given in Table S6.Genotyping of 187 sheep at a further 122 loci in the subtelomeric region of chromosome 6. Two were perfectly associated with the A allele at oar3_OAR6_116402578 (conferring reduced ACC), and five were perfectly associated with the G allele (conferring increased ACC) (Table S8). The extent of HS between Soays and non-Soays sheep was low, and there was no evidence of long-range HS between Soays and Boreray sheep compared with other domesticated breeds . This test is not definitive owing to the relatively small size of the Boreray sample (N = 20), meaning that it is possible that either allele occurs in Boreray sheep but has not been sampled. Nevertheless, low levels of HS with other breeds throughout the sample suggest that the alleles at oar3_OAR6_116402578 have not been introduced recently to the Soay sheep population. For example, HS of core haplotypes with Boreray sheep for coat color, coat pattern , suggesting that both high and low ACC haplotypes are ancient across sheep breeds.Seven core haplotypes of six SNPs in length tagged different alleles at oar3_OAR6_116402578 at the subtelomeric region of chromosome RNF212 and CPLX1, loci that have been implicated previously in recombination rate variation in other species. The narrow-sense heritability (h2) was 0.15 across both sexes and was lower than estimates in some mammal species  (h2 = 0.16 compared to h2 = 0.12). Here we discuss the genetic architecture of the trait in more detail, the observation of sexual dimorphism and male-biased recombination rates, and how our findings inform the broader topic of understanding the genetic architecture of recombination rates in mammals.In this study, we have shown that ACC is heritable in Soay sheep and that variation in female ACC is strongly influenced by a genomic region containing RNF212, occurring 88.4 kb from an ∼374-kb block of high LD (r2 > 0.8) containing three further candidate loci, CPLX1, GAK, and PCGF3  with effects on males only and both sexes, respectively, and a 1.09-Mb region of chromosome 7 affecting rates in both sexes . Although the chromosome 7 region was large and specific loci cannot be pinpointed, it contained REC8, which codes a protein required for the separation of sister chromatids and homologous chromosomes during meiosis .Additional genomic regions associated with recombination rate included two loci at 48.1 and 82.4 Mb on chromosome PRDM9 on chromosomes 1, 5, 18, and X, it is not currently known whether any of these copies are functional in sheep. Here we did not identify any association between recombination rate and any of these regions using either GWAS or regional heritability approaches. This may not be surprising because this locus is primarily associated with recombination hotspot usage. Nevertheless, PRDM9 has been associated with recombination rate in cattle and male humans . There were some differences in sample sizes between the sexes, with twice as many meioses characterized in females than in males, so it could be argued that low sample sizes in males may have had less power to identify specific loci. While possible, it is unlikely that the absence of associations between the RNF212/CPLX1 region and male recombination rate is due to low power to detect the effect because (1) models including both sexes showed reduced, rather than increased, significance in this region, (2) bivariate models accounting for variation in RNF212 as a fixed effect supported a sexually dimorphic genetic effect with a lower degree of error than the bivariate approach .This study identified sexual dimorphism in the genetic architecture of recombination rate in Soay sheep. Using a classical quantitative genetics approach, the between-sex genetic correlation was not significantly different from 1, indicating that male and female recombination rate variation had a shared genetic basis—albeit with a relatively large error around this estimate. However, females had significantly higher additive genetic and residual variance in the trait in comparison with males, and GWAS and regional heritability analyses showed that the approach , and (3)i.e., “missing heritability”) , leaving the remaining additive genetic and phenotypic variance unexplained. Our sample size is small relative to such studies in model systems, and there may have been reduced power to detect genetic variants, particularly in males, which were under-represented in the data set. However, our findings of low heritability and unexplained additive genetic variance are consistent with recent results from Icelandic humans, where despite a larger sample size  and greater marker density (N = 30.3 × 106), the fraction of phenotypic variance explained by specific loci remained small; identified variants including RNF212 and CPLX1 explained just 2.52 and 3.15% of male and female phenotypic variance, respectively, accounting for 29 and 24.8% of the additive genetic variance  but, unlike in cattle, was still significantly higher than that in females . Outside the subtelomeric region, recombination rates were more similar between the sexes, with females showing slightly higher recombination rates between 18.1 and 40 Mb from the telomere .Males had considerably higher recombination rates than females, which was driven mainly by large differences in crossover frequencies in the subtelomeric regions between 0 and 18.11 Mb ; recombitelomere . This obIn placental mammals, females usually exhibit higher recombination rates than males , and it O. canadensis) have not undergone domestication and have female-biased recombination rates  (N = 232 microsatellites) in the bighorn sheep study may have failed to resolve crossovers in the subtelomeric regions; furthermore, a recent study of chiasma count in wild progenitor vs. domestic mammal species found that recombination rates had not increased with domestication in sheep , and only the two relatively gene-poor regions identified on chromosome 3 are novel. Furthermore, examination of haplotypes around RNF212 suggests that variation in the RNF212/CPLX1 region affecting recombination rate has been segregating for a long time in sheep. Examining variation in the wild also allowed us to quantify the effects of the individual and the environment on recombination rate; however, we found no effect of common environment , individual age, or inbreeding on recombination rate; rather, most of the variation was attributed to residual effects. This contrasts with the observation in humans that recombination rates increase with age in females (A principal motivation for this study was to determine how recombination rate and its genetic architecture may vary relative to model species that have undergone strong selection in their recent history. We found that the heritability of recombination rate in Soay sheep was much lower than in cattle and mice and was comparable with recent estimates in humans, which also can be considered a wild population (In this study we have shown that recombination rates in Soay sheep are heritable and have a sexually dimorphic genetic architecture. The variants identified have been implicated in recombination rates in other mammal species, indicating a conserved genetic basis across distantly related taxa. However, the proportion of phenotypic variation explained by identified variants was low; this was consistent with studies in humans and cattle, in which although genetic variants were identified, and most of both additive genetic and phenotypic variance has remained unexplained. Similar studies in both mammalian and nonmammalian wild systems may provide a broader insight into the genetic and nongenetic drivers of recombination rate variation, as well its evolution in contemporary populations. However, the question remains as to whether variation in recombination rate is adaptive or merely a by-product of other biological processes. Overall, the approaches and findings presented here provide an important foundation for studies examining the evolution of recombination rates in contemporary natural populations.