Alternative Ways of Computing the Numerator Relationship Matrix

FnR: R package for computing inbreeding and numerator relationship coefficients

BACKGROUND

Inbreeding and relationship coefficients are essential for conservation and breeding programs. Whether dealing with a small conserved population or a large commercial population, monitoring the inbreeding rate and designing mating plans that minimize the inbreeding rate and maximize the effective population size is important. Free, open-source, and efficient software may greatly contribute to conservation and breeding programs and help students and researchers. Efficient methods exist for calculating inbreeding coefficients. Therefore, an efficient way of calculating the numerator relationship coefficients is via the inbreeding coefficients. i.e., the relationship coefficient between parents is twice the inbreeding coefficient of their progeny. A dummy progeny is introduced where no progeny exists for a pair of individuals. Calculating inbreeding coefficients is very fast, and finding whether a pair of individuals has a progeny and picking one from multiple progenies is computationally more demanding. Therefore, the R package introduces a dummy progeny for any pair of individuals whose relationship coefficient is of interest, whether they have a progeny or not.

RESULTS

Runtime and peak memory usage were benchmarked for calculating relationship coefficients between two sets of 250 and 800 animals (200,000 dummy progenies) from a pedigree of 2,721,252 animals. The program performed efficiently (200,000 relationship coefficients, which involved calculating 2,721,252 + 200,000 inbreeding coefficients) within 3:45 (mm:ss). Providing the inbreeding coefficients (for real animals), the runtime was reduced to 1:08. Furthermore, providing the diagonal elements of D in A = TDT ' (d), the runtime was reduced to 54s. All the analyses were performed on a machine with a total memory size of 1 GB.

CONCLUSIONS

The R package FnR is free and open-source software with implications in conservation and breeding programs. It proved to be time and memory efficient for large populations and many dummy progenies. Calculation of inbreeding coefficients can be resumed for new animals in the pedigree. Thus, saving the latest inbreeding coefficient estimates is recommended. Calculation of d coefficients (from scratch) was very fast, and there was limited value in storing those for future use.

Response to hormonal treatment and conception rates of Sahiwal cows subjected to fixed time artificial insemination in pastoral dairy systems

This study aimed at determining factors influencing response of Sahiwal cows/heifers to fixed time artificial insemination protocol in pastoral systems in Kenya. Available cows/heifers were inspected for conformity to Sahiwal breed characteristics, parity, body condition score, and subsequently rectal palpation to determine pregnancy status, ovarian structures, and estimated ovarian diameter. Consequently, these animals were injected with 100 µg of gonadotrophin-releasing hormone. On days 7 and 9, only responsive cows/heifers were injected with 500 µg of cloprostenol and 100 µg of gonadorelin Acetate, respectively. On day 10, animals were inseminated and separated from bulls for 45 days and pregnancy diagnosis done after 90 days. Analysis of variance was performed to determine the effects of production system, parity, and ovarian structures on ovary diameters pre- and post-hormonal treatment. Logistic regression was used fitting a logit function to account for the binomial distribution of conception. Overall, 56.2%, 23.1%, and 20.7% of the animals had follicles (F), corpus luteum (CL), and corpus albicans (CA), respectively, at day 0, and 16.6%, 68.6%, and 14.8%, respectively, at day 7. Human and environmental factors had no influence on conception. Among the animal factors, only the ovarian structures at day 7 had a significant effect on conception. Ovaries with CL at this time were about 6 times significantly more likely to conceive than those with F. For higher conception rates, animals with ovaries with CL should be recruited into the FTAI program as they are significantly more likely to conceive than those with other ovarian structures.

Understanding factors influencing the estimated genetic variance and the distribution of breeding values

This study investigated the main factors influencing the genetic variance and the variance of breeding values (EBV). The first is the variance of genetic values in the base population, and the latter is the variance of genetic values in the population under evaluation. These variances are important as improper variances can lead to systematic bias. The inverse of the genetic relationship matrix (K -1) and the phenotypic variance are the main factors influencing the genetic variance and heritability (h2). These factors and h2 are also the main factors influencing the variance of EBVs. Pedigree- and genomic-based relationship matrices (A and G as K) and phenotypes on 599 wheat lines were used. Also, data were simulated, and a hybrid (genomic-pedigree) relationship matrix (H as K) and phenotypes were used. First, matrix K underwent a transformation (K* = w K + α 11' + β I), and the responses in the mean and variation of diag(K -1) and offdiag(K -1) elements, and genetic variance in the form of h2 were recorded. Then, the original K was inverted, and matrix K -1 underwent the same transformations as K, and the responses in the h2 estimate and the variance of EBVs in the forms of correlation and regression coefficients with the EBVs estimated based on the original K -1 were recorded. In response to weighting K by w, the estimated genetic variance changed by 1/w. We found that μ(diag(K)) - μ(offdiag(K)) influences the genetic variance. As such, α did not change the genetic variance, and increasing β increased the estimated genetic variance. Weighting K -1 by w was equivalent to weighting K by 1/w. Using the weighted K -1 together with its corresponding h2, EBVs remained unchanged, which shows the importance of using variance components that are compatible with the K -1. Increasing β I added to K -1 increased the estimated genetic variance, and the effect of α 11' was minor. We found that larger variation of diag(K -1) and higher concentration of offdiag(K -1) around the mean (0) are responsible for lower h2 estimate and variance of EBVs.