Computations with Markers

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1 Computations with Markers Paulino Pérez 1 José Crossa 1 1 ColPos-México 2 CIMMyT-México September, SLU, Sweden Computations with Markers 1/20

2 Contents 1 Genomic relationship matrix 2 Examples 3 Big Data! SLU, Sweden Computations with Markers 2/20

3 Genomic relationship matrix Genomic relationship matrix The genomic relationship matrix (G) appears naturally in several models used routinely in Genomic selection. VanRaden (2008) studied efficient methods to compute genomic predictions using this matrix. There are several ways of computing the G matrix, SLU, Sweden Computations with Markers 3/20

4 Genomic relationship matrix 1 2 G = XX, where X is the matrix of marker genotypes of dimensions n p. For SNPs x ij {0, 1, 2}. G = (X E)(X E) 2 p j=1 p j(1 p j ), where p j is the minor allele frequency of SNP j = 1,..., p, and E is a matrix of expected frequencies of x ij under Hardy-Weiberg equilibrium from estimates of allelic frequencies. 3 G = ZZ p, where Z is the matrix of centered and standardized SNPs codes and p is the number of SNPs, that is z ij = (x ij 2p j )/ 2p j (1 p j ). SLU, Sweden Computations with Markers 4/20

5 Continue... Genomic relationship matrix G = XX appears naturally when we assume that we can predict the phenotypes using the linear model: y = 1µ + Xβ + e, where e N(0, σ 2 ei) and β N(0, σ 2 β I). Let u = Xβ, by using the multivariate normal distribution, it can be shown that u N(0, XX ), and the model is equivalente to y = 1µ + u + e, which is usually known as G-BLUP. We will talk about this model later on. SLU, Sweden Computations with Markers 5/20

6 Examples Examples Figure 1: Toy example for markers. SLU, Sweden Computations with Markers 6/20

7 SNP coding Examples 1 Additive effects 1 if the SNP is homozygous for the major allele x = 0 if the SNP is heterozygous 1 if the SNP is homozygous for the other allele 2 Dominant effects x = { 1 if the SNP is heterozygous 0 if the SNP is homozygous SLU, Sweden Computations with Markers 7/20

8 Continue... Examples #Clear workspace rm(list=ls()) #Set working directory setwd("~/2. Gmatrix/examples") source("recode.r") source("impute.r") Genotype_info=read.csv(file="TC-10-Genotypes-ACGT.csv", header=true,na.strings="?_?",stringsasfactors=false) entry_genotype_info=genotype_info$entry Genotype_info=Genotype_info[,-c(1,2)] X=recode(Genotype_info)$X #Impute missing genotypes set.seed(123) out=impute(x) SLU, Sweden Computations with Markers 8/20

9 Continue... Examples #Note that marker 167 and 179 are #monomorphic and should be excluded from analysis out$monomorphic #Remove monomorphic markers, #At this point no more missing values are present X=out$X[,-out$monomorphic] #compute p phat=colmeans(x)/2 MAF=ifelse(phat<0.5,phat,1-0.5) phat=maf hist(maf,main="") SLU, Sweden Computations with Markers 9/20

10 Continue... Examples Frequency MAF Figure 2: Distribution of allele frequencies. SLU, Sweden Computations with Markers 10/20

11 Examples Computations: three ways #Computing the genomic relationship matrix G1=tcrossprod(X) X2=scale(X,center=TRUE,scale=FALSE) k=2*sum(phat*(1-phat)) G2=tcrossprod(X2)/k X3=scale(X,center=TRUE,scale=TRUE) G3=tcrossprod(X3)/ncol(X3) heatmap(g3) hist(diag(g3),main="") SLU, Sweden Computations with Markers 11/20

12 Exercise Examples 1 Load the weath dataset that we were using yesterday. 2 Compute the Genomic relationship matrix using equation 1. 3 Compare the entries of G and A. SLU, Sweden Computations with Markers 12/20

13 Examples Continue Figure 3: Heatmap of G matrix. SLU, Sweden Computations with Markers 13/20

14 Continue... Examples Frequency diag(g3) Figure 4: Histogram of the diagonal elements of the G matrix. SLU, Sweden Computations with Markers 14/20

15 Distance matrix Examples The distance matrix, also appears naturally in RKHS models. We will review them in the next days, d ij = x i x j 2 = k (x ik x jk ) 2 Example: D=as.matrix(dist(X)) SLU, Sweden Computations with Markers 15/20

16 Big Data! Big Data! The computation of the genomic relationship matrix is straight forward if the matrix X is small. There are application where the number of markers can be very big, SLU, Sweden Computations with Markers 16/20

17 Big Data! Ober s prediction problem Ober et al. (2012) predicts starvation stress resistance and starle resistance in Drosophila using p = 2.5 millions SNPs and n = 192 D. melanogaster inbreed lines derived by 20 generations of full sib mating from wild-caught females from the Raleigh, North Carolina population. SLU, Sweden Computations with Markers 17/20

Continue... Big Data! Prediction in D. melanogaster Using Sequence Data Genomic relationship matrix for Ober s data. Figure 2. Heatmap of the genomic relationship matrix G.

18 Continue... Big Data! Prediction in D. melanogaster Using Sequence Data Genomic relationship matrix for Ober s data. Figure 2. Heatmap of the genomic relationship matrix G. The genomic relationship matrix G was calculated according to [8] using 157 lines and 2.5 million SNPs. The S after the line-id indicates that the line belongs to the set of lines for which phenotypic records for startle response were also available (in addition to the phenotypic records of starvation resistance). doi: /journal.pgen g002 NeLf SLU, Sweden Computations with Markers 18/20

19 Solution Big Data! Fortunately the computation of the G matrix can be fully paralleled in modern CPU processors, G ij = k (x ik 2p k )(x jk 2p k )/c When computing G ij only the genotypes of individuals (i, j) are needed. SLU, Sweden Computations with Markers 19/20

20 Continue... Big Data! SLU, Sweden Computations with Markers 20/20

General aspects of genome-wide association studies

General aspects of genome-wide association studies Abstract number 20201 Session 04 Correctly reporting statistical genetics results in the genomic era Pekka Uimari University of Helsinki Dept. of Agricultural