- Short communication
- Open Access

# Quantifying genomic connectedness and prediction accuracy from additive and non-additive gene actions

- Mehdi Momen
^{1}and - Gota Morota
^{2}Email authorView ORCID ID profile

**Received:**8 May 2018**Accepted:**5 September 2018**Published:**17 September 2018

## Abstract

### Background

Genetic connectedness is classically used as an indication of the risk associated with breeding value comparisons across management units because genetic evaluations based on best linear unbiased prediction rely for their success on sufficient linkage among different units. In the whole-genome prediction era, the concept of genetic connectedness can be extended to measure a connectedness level between reference and validation sets. However, little is known regarding (1) the impact of non-additive gene action on genomic connectedness measures and (2) the relationship between the estimated level of connectedness and prediction accuracy in the presence of non-additive genetic variation.

### Results

We evaluated the extent to which non-additive kernel relationship matrices increase measures of connectedness and investigated its relationship with prediction accuracy in the cross-validation framework using best linear unbiased prediction and coefficients of determination. Simulated data assuming additive, dominance, and epistatic gene action scenarios and real swine data were analyzed. We found that the joint use of additive and non-additive genomic kernel relationship matrices or non-parametric relationship matrices led to increased capturing of connectedness, up to 25%, and improved prediction accuracies compared to those of baseline additive relationship counterparts in the presence of non-additive gene action.

### Conclusions

Our findings showed that connectedness metrics can be extended to incorporate non-additive genetic variation of complex traits. Use of kernel relationship matrices designed to capture non-additive gene action increased measures of connectedness and improved whole-genome prediction accuracy, further broadening the scope of genomic connectedness studies.

Genetic connectedness is used to evaluate the extent to which reliable comparisons of estimated breeding values can be safely performed across management units. The strength of genetic links or connectedness relies on the relatedness of individuals across management units [1]. In turn, genetic evaluations of managed populations such as livestock species rely for their success on sufficient connectedness between different units. In such cases, best linear unbiased prediction (BLUP) provides fair ranking of the estimated breeding values of individuals while minimizing the risk of potential uncertainty in estimated breeding value comparisons [2–4]. The majority of previous studies on connectedness were performed with regard to pedigree relatedness; however, Yu et al. [5] rekindled an interest in this area by evaluating the utility of genome-based connectedness. Using real mice and cattle data, they reported that genomic relatedness enables the enhancement of genetic connectedness measures across management units compared to those obtained from pedigree relationships. This is mainly because genomic information captures relatedness between units that appears disconnected according to the pedigree. The utility of genomic connectedness was further investigated by assessing whether the enhanced estimates of connectedness delivered by genomics also led to an increased accuracy of breeding value prediction [6]. It was found that the use of genomic relatedness yields increased measures of connectedness and improved prediction accuracies (PA) compared to those of pedigree-based models under a purely additive gene action mode when a sufficient number of single-nucleotide polymorphisms (SNPs) is present. This parallels the recent recognition of the impact of non-additive genetic variation marked by SNPs e.g. [7, 8]. By properly accounting for non-additive genetic variation, it is potentially possible to enhance (1) the accuracy of total genetic value prediction, (2) the accuracy of breeding value prediction by clearly separating additive from non-additive genetic variation, and 3) the efficiency of mate allocation procedures as well as crossbreeding or purebred selection schemes [9, 10]. However, the relationship between the estimated level of connectedness and PA in the presence of non-additive genetic variation is less well understood. Accordingly, the objective of the current study was to evaluate the interrelationship between the degree of genomic connectedness and genome-enabled PA by calculating connectedness statistics from either the joint use of additive and non-additive genomic relationship matrices or non-parametric relationship matrices using simulated and real data, further broadening the scope of genomic connectedness studies.

## Methods

### Simulated data

A two-step simulation process was carried out using the QMSim software [11]. A historical population with 1000 individuals was created at the initial generation, followed by a sharp reduction in the population size owing to population bottleneck during generation 1 to 100. This resulted in the population size decreasing to 220 individuals in the last historical generation, creating initial linkage disequilibrium along with mutation and drift. The recent population was formed by randomly sampling 200 females and 10 males from the last historical generation. The individuals were mated for the subsequent five generations with equal probability of males and females, producing a total of 2000 individuals with a structured pedigree for analysis.

*a*,

*d*, and

*ad*are the additive, dominance, and epistatic effects, respectively; \({\mathbf {W}}_{\mathbf {a}}\), \({\mathbf {W}}_{\mathbf {d}}\), and \({\mathbf {l}}_{k}{\mathbf {l}}_{k^{\prime }}\) are SNP codes for additive, dominance, and epistasis, respectively;

*k*denotes the

*k*th QTL; and

*nQTL*is the number of QTL (for the epistatic term, this is only summed over the epistatic QTL). The phenotypic value of each individual \(y_i\) was created by adding a normally distributed residual \(\epsilon _i \sim N(0,\sigma ^2_{\epsilon })\) to the sum of genetic values. Additive effects were drawn from a Gamma distribution with shape and scale parameters equal to 0.42 and 8.282, respectively [12]. Their effect signs were sampled to be positive or negative with probability 0.5. The dominance effect for the

*k*th QTL was determined as the product of the absolute value of the additive QTL effect and the degree of dominance \(d_{k}=\delta _k\mid a_{k}\mid\) [13, 14]. Here, \(\delta _k\) is the degree of dominance sampled from a normal distribution with \(\delta _k \sim N(0,1)\). The epistatic effects were drawn from a normal distribution with \(N(0.02,\sigma ^2=0.03)\) [14]. Additive and dominance components were simulated for the AD scenario; additive, dominance, and epistatic components were included for the ADE scenario; and only epistasis was considered for the PE scenario. Two broad-sense heritability levels (\(H^2\)) equal to 0.4 and 0.8 were simulated, with the partitioning of variance components shown in Table 1. We considered phenotypic variance equal to unity and simulated genetic variance according to the proportion of phenotypic variance explained by additive, dominance, and epistatic QTL effects: \(\sigma ^2_a = \sum _k 2p_kq_k\alpha ^2_k\), \(\sigma ^2_d = \sum _k [2p_kq_kd_k]^2\), and \(\sigma ^2_{ad} = 2\sum _k \sum _{k^{\prime }} p^2_k p_{k^{\prime }} q_{k^{\prime }}(\alpha _k d_{k^{\prime }})^2\), where \(\alpha =[a + d(q-p)]^2\) is the allele substitution effect, and

*p*and

*q*are minor and major allele frequencies, respectively [15, 16].

Simulated heritability value for each gene action scenario

\(H^2\) | Gene action | \(h^2_A\) | \(h^2_D\) | \(h^2_E\) |
---|---|---|---|---|

0.4 | AD | 0.3 | 0.1 | - |

ADE | 0.2 | 0.1 | 0.1 | |

PE | – | – | 0.4 | |

0.8 | AD | 0.6 | 0.2 | - |

ADE | 0.4 | 0.2 | 0.2 | |

PE | – | – | 0.8 |

### Real data

For real data analysis, publicly available PIC swine data was used [17]. We analyzed five traits, T1, T2, T3, T4, and T5, with the corresponding number of individuals equal to 2804, 2715, 3141, 3184, and 3184. Their heritability values were 0.03, 0.23, 0.20, 0.32, and 0.36, respectively. It has been shown that this dataset exhibits a small to moderate amount of dominance genomic variation [18, 19]. Therefore, this dataset was considered suitable to test the extent to which the use of a non-additive genomic kernel relationship matrix might increase the capturing of connectedness measures. After removing SNPs with a minor allele frequency lower than 0.05, 52,842 SNPs remained for the analysis.

### Management unit simulation

*K*-means clustering algorithm applied to a numerator relationship matrix computed from pedigree data such that the overall level of relatedness between individuals in different management units is minimized. There was no exchange of individuals between MU1 and MU2 in scenario 1 (S1), which served as a least connected design. An additional five management unit scenarios (S2 to S6) were considered by exchanging 10, 20, 30, 40, and 50% of individuals between MU1 and MU2 as shown in Fig. 1.

### Genomic relationship kernel matrix

Three types of genomic relationship kernel matrices (\({\mathbf {K}}\)) were used in the present study.

The additive genomic relationship matrix (\({\mathbf {K}} ={\mathbf {G}}\)) was used to capture the pattern of additive inheritance \({\mathbf {G}} = {\mathbf {W}}_{\mathbf {a}}{\mathbf {W}}_{\mathbf {a}}^{\prime } / 2 \sum _{k=1}^{m} p_k(1-p_k)\), where \({\mathbf {W}}_{\mathbf {a}}\) is the centered marker incidence matrix taking values of \(0-2p_k\) for zero copies of the reference allele, \(1-2p_k\) for one copy of the reference allele, and \(2-2p_k\) for two copies of the reference allele [20]. Here, \(p_k\) is the allele frequency at SNP \(k = 1, \ldots , m\). The dominance genomic relationship matrix (\({\mathbf {K}} = {\mathbf {D}}\)) aimed at capturing dominance gene action \({\mathbf {D}}={\mathbf {W}}_{\mathbf {d}}{\mathbf {W}}_{\mathbf {d}}^{\prime } / \sum _{k=1}^{m}(2p_k(1-p_k))^2\), where \({\mathbf {W}}_{\mathbf {d}}\) is the dominance marker incidence matrix defined according to Vitezica et al. [21]. The additive by dominance genomic relationship matrix was constructed as \({\mathbf {G}}\#{\mathbf {D}}\), where \(\#\) denotes the Hadamard product [22].

#### Gaussian kernel

*i*and

*j*with their genotype vectors \({\mathbf {w}}_i \in (0,1,2)\) and \({\mathbf {w}}_j \in (0,1,2)\) is given by:

**GK**entries closer to 0 (i.e., local kernel) and smaller \(\theta\) produces entries closer to 1 (i.e., global kernel). Therefore, \(\theta\) controls the extent of genomic similarity between individuals.

### Coefficient of determination

**b**and

**u**are the vectors of systematic effects and genetic values, and \(\varvec{\epsilon }\) is the vector of residuals. By defining \(\text {var}({\mathbf {u}}) = {\mathbf {K}}\sigma ^2_u\) we have:

*l*and \(l^{\prime }\) consisting of \(n_l\) and \(n_{l^{\prime }}\) individuals is given by [26, 27]:

*l*th, \(l^{\prime }\)th, and the remaining units. Here, the sum of contrast vector elements is zero. The greater the CD of contrast, the greater the connectedness. A large CD is expected when prediction error covariance in the numerator is large, reflecting errors that are in the same direction between units. Alternatively, the measure of CD decreases when the relationship between individuals across units is large in the denominator. Therefore, the CD of contrast combines the prediction error variance of the difference (PEVD) [2] and genetic variability. This metric was chosen because it was found to represent the most stable connectedness metric in a recent study [5].

### Connectedness measures and prediction accuracy

Measures of CD between MU1 and MU2 were inferred from estimated variance components followed by assessing genomic PA by two-fold cross-validation using a BLUP type model. In the first fold, MU1 was treated as a training set and MU2 was treated as a testing set. This was reversed in the second fold such that MU2 was used to train the model and MU1 was used to test prediction performance. The multi-kernel \({\mathbf {G}}\) and \({\mathbf {D}}\) approach in the AD scenario, the multi-kernel \({\mathbf {G}}\), \({\mathbf {D}}\), and \({\mathbf {G}}\#{\mathbf {D}}\) approach in the ADE scenario, and the \({\mathbf {GK}}\) matrix in the PE scenario were benchmarked against the baseline \({\mathbf {G}}\) matrix (i.e., genomic BLUP). Note that the use of \({\mathbf {GK}}\) corresponds to fitting a reproducing kernel Hilbert spaces regression (e.g. [28]). For a multi-kernel approach, we weighted each kernel by its relative contribution to the marked total genetic variation, also known as kernel averaging or multiple kernel learning [29], to measure connectedness and assess PA. For instance, the kernel matrix \({\mathbf {K}} = \frac{\sigma ^2_g}{\sigma ^2_g + \sigma ^2_d} {\mathbf {G}} + \frac{\sigma ^2_d}{\sigma ^2_g + \sigma ^2_d} {\mathbf {D}}\) was used when \({\mathbf {G}}\) and \({\mathbf {D}}\) were fitted together, where \(\sigma ^2_g\) and \(\sigma ^2_d\) were additive and dominance genomic variances, respectively. PA was obtained as the correlation between true and predicted genetic values for the simulated data averaged across 10 replicates (cor(\({\mathbf {g}}, \hat{{\mathbf {g}}}\))) and the correlation between phenotypes and predicted genetic values for the real data (cor(\({\mathbf {y}}, \hat{{\mathbf {g}}}\))).

## Results

### AD scenario

The relationships between CD and PA across the six management unit simulation scenarios (S1 to S6) are shown in Fig. 2. The joint fit of \({\mathbf {G}}\) and \({\mathbf {D}}\) kernel relationship matrices was benchmarked using the \({\mathbf {G}}\) matrix alone. A sharp increase in PA was observed with the increasing proportion of exchanged individuals from S1 to S3, which reached a plateau after 30% exchange rate between MU1 and MU2 in S4. Overall, PA improved as more individuals between MU1 and MU2 were shared. Higher PA values were achieved by accounting for dominance \({\mathbf {G}}+{\mathbf {D}}\) compared to \({\mathbf {G}}\) alone for the two heritability levels considered (0.4 and 0.80). The lowest PA (0.368) was obtained in S1 with \({\mathbf {G}}\) and the highest PA (0.632) was obtained in S4 with \({\mathbf {G}}+{\mathbf {D}}\).

### ADE scenario

The results of PA and CD from the ADE scenario are shown in Fig. 3. We found that the overall pattern resembled that of the AD scenario. That is, with increasing degree of similarity among management units, PA increased and then reached a plateau after S4. The highest PA (0.731) was obtained with \({\mathbf {G}}+{\mathbf {D}}+{\mathbf {G}}\#{\mathbf {D}}\) kernel matrices in S4 and the smallest PA (0.245) with \({\mathbf {G}}\) in S1. The PA results suggested that increasing the number of linking individuals improves PA and the use of non-additive genomic relationship matrices simultaneously further increased PA.

### PE scenario

### Real data

## Discussion

The assessment of genetic connectedness originated from testing the estimability of linear functions of fixed effects in *n*-way cross classifications to determine the absence or presence of connectedness [30, 31]. It was subsequently extended to the random effects framework [1] to quantify the uncertainty associated with the accuracy of breeding value comparisons involving different management units. In this sense, connectedness is a measure germane to the capability to have estimable comparisons [3]. In the genomics era, the concept of genetic connectedness offers insights on two aspects of the prediction of genetic values. The first is relevant to improving the quality of genomic breeding value comparisons [5, 32] whereas the other is related to improving the accuracy of genomic prediction [33]. Notably, it is possible to reconcile these two items by quantifying a genomic connectedness level between reference and validation sets in the whole-genome prediction paradigm. Toward this end, Yu et al. [6] investigated the relationship between connectedness measures and PA using pedigree and genomic information under an additive model.

Concurrently, it has been shown that whole-genome prediction models designed to capture non-additivity yield slightly to moderately higher PA than additive counterparts when the underlying genetic architecture is governed by dominance or epistasis e.g. [28, 34]. Although the extent of non-additive genetic variance may not be big in general, this type of variance is particularly important for fitness-related traits [35]. These recent findings served as the impetus for the present study, extending the scope of connectedness applications by further considering non-additive genetic variation.

We observed that the inclusion of non-additive genetic relationship kernel matrices or non-parametric relationship matrices in a BLUP type model increased PA as more individuals were exchanged between MU1 and MU2, and that this was associated with stronger measures of connectedness up to S3 or S4. This reinforced the view that the commonly observed higher prediction performance in non-additive or non-parametric models in the presence of non-linear gene action is due to improved capturing of connectedness between units. We also found that the choice of smoothness parameter \(\theta\) not only influences PA but also the extent of CD. This indicates the importance of the smoothness parameter in evaluating PA and CD, especially when a complex trait is controlled by non-additive gene actions. In general, our results showed that when the optimum \(\theta\) is selected, PA and CD of **GK** will be better than those of **G**, and that even **GK** constructed from additive coding of SNPs only captures additive by additive epistasis theoretically [23]. We note that many studies have shown that PA decreases when the reference population has a lower relatedness to the validation population e.g. [36, 37]. This is equivalent to when two units exhibit weak connectedness. Use of connectedness thereby opens up the possibility for an alternative way to measure the strength of relationship between these two populations instead of using an average relationship.

Moreover, once the rate of exchange reached S3 or S4, the estimated level of CD gradually leveled off in all management unit simulation scenarios, contrary to PA. This is because when there are sufficient numbers of individuals linking MU1 and MU2, the denominator of CD becomes smaller thus increasing the second term, which in turn renders the CD of contrast to become small. This agrees with the findings in other studies dealing with only additive genetic variation [5, 6]. Together, these findings suggest that the use of CD holds great potential to identify an optimal breeding program design in terms of genetic diversity while maximizing PA, whereas other connectedness metrics such as PEVD aim at increasing PA regardless of how closely individuals between units become related [5]. Note that PA is one of the criteria to determine the most appropriate model to fit (for example, \({\mathbf {G}}\) vs. \({\mathbf {G}} + {\mathbf {D}}\)). Once the model is chosen, CD can be used to identify an appropriate level of relatedness or diversity between two units while maintaining high PA.

Although we applied *K*-means clustering of a numerator relationship matrix, the choice of \({\mathbf {K}}\) for clustering may impact our results. Thus, we further constructed management units based on clustering of \({\mathbf {G}}\) or \({\mathbf {G}}\) + \({\mathbf {D}}\) under the AD scenario. As shown in Figures S1 and S2 (see Additional file 1: Figures S1 and S2), *K*-means clustering of \({\mathbf {G}}\) or \({\mathbf {G}}\) + \({\mathbf {D}}\) produced patterns of PA and CD that are similar to those generated using the numerator relationship. We also repeated our analyses using forward validation rather than *K*-means clustering. We treated 1200 individuals in generations 1 to 3 as the training set (MU1) and 800 individuals in generations 4 to 5 as the testing set (MU2) under the AD scenario. We found that using \({\mathbf {G}} + {\mathbf {D}}\) yielded higher PA and greater amount of CD compared to using \({\mathbf {G}}\) (Additional file 1: Figure S3).

The utility of genomic connectedness does not preclude its application in management units. For instance, connectedness measured by CD is currently gaining recognition for training population formation in plant breeding [38]. We contend that the use of CD holds promise to tackle a multitude of challenges related to increasing genomic prediction while maintaining genetic diversity.

## Conclusion

Here, the genetic connectedness metric, CD, was used to assess genomic connectedness measures between reference and validation sets in a whole-genome prediction framework using simulated and real data in the presence of non-additive gene action. Joint fitting of additive and non-additive genomic kernel relationship matrices or non-parametric relationship matrices could yield enhanced capture of connectedness and improved PA compared to those obtained through baseline additive models. Our approach shows promise to measure connectedness levels and investigate their relationship with genomic PA when the linear assumption of genotype-phenotype mapping may not hold.

## Declarations

### Authors' contributions

MM performed analyses, interpreted the results, and drafted the manuscript. GM conceived the study, interpreted the results, and revised the manuscript. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

All information supporting the results is included in the text, figures and tables of this article. The publicly available swine dataset can be downloaded from Cleveland et al. [17].

### Ethics approval and consent to participate

Not applicable.

### Funding

GM acknowledges funding from the University of Nebraska and Virginia Polytechnic Institute and State University startup funds.

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## Authors’ Affiliations

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