Liviu R. Totir

Sampling genotypes in large pedigrees with loops

Markov chain Monte Carlo (MCMC) methods have been proposed to overcome computational problems in linkage and segregation analyses. This approach involves sampling genotypes at the marker and trait loci. Scalar-Gibbs is easy to implement, and it is widely used in genetics. However, the Markov chain that corresponds to scalar-Gibbs may not be irreducible when the marker locus has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. These problems do not arise if the genotypes are sampled jointly from the entire pedigree. This paper proposes a method to jointly sample genotypes. The method combines the Elston-Stewart algorithm and iterative peeling, and is called the ESIP sampler. For a hypothetical pedigree, genotype probabilities are estimated from samples obtained using ESIP and also scalar-Gibbs. Approximate probabilities were also obtained by iterative peeling. Comparisons of these with exact genotypic probabilities obtained by the Elston-Stewart algorithm showed that ESIP and iterative peeling yielded genotypic probabilities that were very close to the exact values. Nevertheless, estimated probabilities from scalar-Gibbs with a chain of length 235 000, including a burn-in of 200 000 steps, were less accurate than probabilities estimated using ESIP with a chain of length 10 000, with a burn-in of 5 000 steps. The effective chain size (ECS) was estimated from the last 25 000 elements of the chain of length 125 000. For one of the ESIP samplers, the ECS ranged from 21 579 to 22 741, while for the scalar-Gibbs sampler, the ECS ranged from 64 to 671. Genotype probabilities were also estimated for a large real pedigree consisting of 3 223 individuals. For this pedigree, it is not feasible to obtain exact genotype probabilities by the Elston-Stewart algorithm. ESIP and iterative peeling yielded very similar results. However, results from scalar-Gibbs were less accurate.

A nonsense mutation in cGMP-dependent type II protein kinase (PRKG2) causes dwarfism in American Angus cattle.

Historically, dwarfism was the major genetic defect in U.S. beef cattle. Aggressive culling and sire testing were used to minimize its prevalence; however, neither of these practices can eliminate a recessive genetic defect. We assembled a 4-generation pedigree to identify the mutation underlying dwarfism in American Angus cattle. An adaptation of the Elston-Steward algorithm was used to overcome small pedigree size and missing genotypes. The dwarfism locus was fine-mapped to BTA6 between markers AFR227 and BM4311. Four candidate genes were sequenced, revealing a nonsense mutation in exon 15 of cGMP-dependant type II protein kinase (PRKG2). This C/T transition introduced a stop codon (R678X) that truncated 85 C-terminal amino acids, including a large portion of the kinase domain. Of the 75 mutations discovered in this region, only this mutation was 100% concordant with the recessive pattern of inheritance in affected and carrier individuals (log of odds score = 6.63). Previous research has shown that PRKG2 regulates SRY (sex-determining region Y) box 9 (SOX9)-mediated transcription of collagen 2 (COL2). We evaluated the ability of wild-type (WT) or R678X PRKG2 to regulate COL2 expression in cell culture. Real-time PCR results confirmed that COL2 is overexpressed in cells that overexpressed R678X PRKG2 as compared with WT PRKG2. Furthermore, COL2 and COL10 mRNA expression was increased in dwarf cattle compared with unaffected cattle. These experiments indicate that the R678X mutation is functional, resulting in a loss of PRKG2 regulation of COL2 and COL10 mRNA expression. Therefore, we present PRKG2 R678X as a causative mutation for dwarfism cattle.

Improved techniques for sampling complex pedigrees with the Gibbs sampler

Markov chain Monte Carlo (MCMC) methods have been widely used to overcome computational problems in linkage and segregation analyses. Many variants of this approach exist and are practiced; among the most popular is the Gibbs sampler. The Gibbs sampler is simple to implement but has (in its simplest form) mixing and reducibility problems; furthermore in order to initiate a Gibbs sampling chain we need a starting genotypic or allelic configuration which is consistent with the marker data in the pedigree and which has suitable weight in the joint distribution. We outline a procedure for finding such a configuration in pedigrees which have too many loci to allow for exact peeling. We also explain how this technique could be used to implement a blocking Gibbs sampler.

A Two-Stage Approximation for Analysis of Mixture Genetic Models in Large Pedigrees

Information from cosegregation of marker and QTL alleles, in addition to linkage disequilibrium (LD), can improve genomic selection. Variance components linear models have been proposed for this purpose, but accommodating dominance and epistasis is not straightforward with them. A full-Bayesian analysis of a mixture genetic model is favorable in this respect, but is computationally infeasible for whole-genome analyses. Thus, we propose an approximate two-step approach that neglects information from trait phenotypes in inferring ordered genotypes and segregation indicators of markers. Quantitative trait loci (QTL) fine-mapping scenarios, using high-density markers and pedigrees of five generations without genotyped females, were simulated to test this strategy against an exact full-Bayesian approach. The latter performed better in estimating QTL genotypes, but precision of QTL location and accuracy of genomic breeding values (GEBVs) did not differ for the two methods at realistically low LD. If, however, LD was higher, the exact approach resulted in a slightly higher accuracy of GEBVs. In conclusion, the two-step approach makes mixture genetic models computationally feasible for high-density markers and large pedigrees. Furthermore, markers need to be sampled only once and results can be used for the analysis of all traits. Further research is needed to evaluate the two-step approach for complex pedigrees and to analyze alternative strategies for modeling LD between QTL and markers.

A comparison of alternative methods to compute conditional genotype probabilities for genetic evaluation with finite locus models

An increased availability of genotypes at marker loci has prompted the development of models that include the effect of individual genes. Selection based on these models is known as marker-assisted selection (MAS). MAS is known to be efficient especially for traits that have low heritability and non-additive gene action. BLUP methodology under non-additive gene action is not feasible for large inbred or crossbred pedigrees. It is easy to incorporate non-additive gene action in a finite locus model. Under such a model, the unobservable genotypic values can be predicted using the conditional mean of the genotypic values given the data. To compute this conditional mean, conditional genotype probabilities must be computed. In this study these probabilities were computed using iterative peeling, and three Markov chain Monte Carlo (MCMC) methods – scalar Gibbs, blocking Gibbs, and a sampler that combines the Elston Stewart algorithm with iterative peeling (ESIP). The performance of these four methods was assessed using simulated data. For pedigrees with loops, iterative peeling fails to provide accurate genotype probability estimates for some pedigree members. Also, computing time is exponentially related to the number of loci in the model. For MCMC methods, a linear relationship can be maintained by sampling genotypes one locus at a time. Out of the three MCMC methods considered, ESIP, performed the best while scalar Gibbs performed the worst.

The effect of using approximate gametic variance covariance matrices on marker assisted selection by BLUP

Under additive inheritance, the Henderson mixed model equations (HMME) provide an efficient approach to obtaining genetic evaluations by marker assisted best linear unbiased prediction (MABLUP) given pedigree relationships, trait and marker data. For large pedigrees with many missing markers, however, it is not feasible to calculate the exact gametic variance covariance matrix required to construct HMME. The objective of this study was to investigate the consequences of using approximate gametic variance covariance matrices on response to selection by MABLUP. Two methods were used to generate approximate variance covariance matrices. The first method (Method A) completely discards the marker information for individuals with an unknown linkage phase between two flanking markers. The second method (Method B) makes use of the marker information at only the most polymorphic marker locus for individuals with an unknown linkage phase. Data sets were simulated with and without missing marker data for flanking markers with 2, 4, 6, 8 or 12 alleles. Several missing marker data patterns were considered. The genetic variability explained by marked quantitative trait loci (MQTL) was modeled with one or two MQTL of equal effect. Response to selection by MABLUP using Method A or Method B were compared with that obtained by MABLUP using the exact genetic variance covariance matrix, which was estimated using 15 000 samples from the conditional distribution of genotypic values given the observed marker data. For the simulated conditions, the superiority of MABLUP over BLUP based only on pedigree relationships and trait data varied between 0.1% and 13.5% for Method A, between 1.7% and 23.8% for Method B, and between 7.6% and 28.9% for the exact method. The relative performance of the methods under investigation was not affected by the number of MQTL in the model.

A study on the minimum number of loci required for genetic evaluation using a finite locus model

For a finite locus model, Markov chain Monte Carlo (MCMC) methods can be used to estimate the conditional mean of genotypic values given phenotypes, which is also known as the best predictor (BP). When computationally feasible, this type of genetic prediction provides an elegant solution to the problem of genetic evaluation under non-additive inheritance, especially for crossbred data. Successful application of MCMC methods for genetic evaluation using finite locus models depends, among other factors, on the number of loci assumed in the model. The effect of the assumed number of loci on evaluations obtained by BP was investigated using data simulated with about 100 loci. For several small pedigrees, genetic evaluations obtained by best linear prediction (BLP) were compared to genetic evaluations obtained by BP. For BLP evaluation, used here as the standard of comparison, only the first and second moments of the joint distribution of the genotypic and phenotypic values must be known. These moments were calculated from the gene frequencies and genotypic effects used in the simulation model. BP evaluation requires the complete distribution to be known. For each model used for BP evaluation, the gene frequencies and genotypic effects, which completely specify the required distribution, were derived such that the genotypic mean, the additive variance, and the dominance variance were the same as in the simulation model. For lowly heritable traits, evaluations obtained by BP under models with up to three loci closely matched the evaluations obtained by BLP for both purebred and crossbred data. For highly heritable traits, models with up to six loci were needed to match the evaluations obtained by BLP.