Quantitative Genetics. February 16, 2010

Transcription

1 Quantitative Genetics February 16, 2010

2 A General Model y ij z ij y ij y ij = y i(j) = g j + ε ij 2 1 [zij =j]f ij ( complete likelihood ) j=0 2 ω ij f ij j=0 ( likelihood ) ω i0 = Pr[z ij = j] = q 2 = p 2 a ω i1 = Pr[z ij = j] = 2pq = 2p A p a ω i2 = Pr[z ij = j] = p 2 = p 2 A g 0 = µ a g 1 = µ + d g 2 = µ + a ε ij ( 0, σj 2 ) ( N 0, σ 2 ) February 16, 2010 Marker Data 2 / 16

3 Data Simulation Function: data.sim1 Inputs: n: sample size p: allele frequency mu: mean effect a: additive effect d: dominant effect sig: environmental variance Outputs: load("quantgenfun.r") set.seed(1) pop<-data.sim1(n=20000,p=.5,mu=2,a=1,d=.5,sig=1) str(pop) marker: marker data (coded as 0=aa, 1=Aa, 2=AA) y: observed phenotypic values param: parameters used in generating the data List of 3 $ marker: num [1:20000] $ y : num [1:20000] $ param : Named num [1:6] 2e+04 5e-01 2e+00 1e+00 5e-01 1e attr(*, "names")= chr [1:6] "n" "p" "mu" "a"... February 16, 2010 Marker Data 3 / 16

4 Data Simulation II Function: mate2 Inputs: n: sample size pop: population object output from data.sim1 pop2<-mate2(n=2000,pop) str(pop2) Outputs: marker: progeny marker data y: observed progeny phenotypic values midparent: average phenotype variation of the parents List of 3 $ marker : num [1:2000] $ y : num [1:2000] $ midparent: num [1:2000] February 16, 2010 Marker Data 4 / 16

5 Heritability p<-pop$param[["p"]] q<-1-p mu<-pop$param[["mu"]] a<-pop$param[["a"]] d<-pop$param[["d"]] alpha<-a+(q-p)*d sig.a<-2*p*q*alpha var.add<-2*p*q*alpha^2 var.dom<-(2*p*q*d)^2 H2; h2 [1] 0.36 [1] 0.32 lm(pop2$y~pop2$midparent) Call: lm(formula = pop2$y ~ pop2$midparent) var.gen<-var.add+var.dom var.env<-pop$param[["sig"]]^2 Coefficients: (Intercept) pop2$midparent H2<-var.gen/(var.gen+var.env) h2<-var.add/(var.gen+var.env) February 16, 2010 Marker Data 5 / 16

6 Heritability Simulation H 2 = σ2 g σg 2 + σ 2 = σ 2 = σ2 g(1 H 2 ) H 2 Function: data.sim2 Inputs: n: sample size p: allele frequency mu: mean effect a: additive effect d: dominant effect H2: broad-sense heritability Outputs: marker: marker data (coded as 0=aa, 1=Aa, 2=AA) y: observed phenotypic values param: parameters used in generating the data set.seed(2) pop<-data.sim2(n=2000,p=.5,mu=2,a=1,d=.5,h2=.1) pop$param[["sig"]] [1] 2.25 February 16, 2010 Marker Data 6 / 16

7 Simulated Cross: Marker Data + Phenotype Data library(qtl) R Library: qtl sim.map: constructs a genetic map sim.cross: simulates marker, QTL, and phenotype data pull.geno: extracts marker data pull.pheno: extracts phenotype data n<-100 map<-sim.map(len=100,n.mar=11,include.x=false,eq.spacing=true) cross<-sim.cross(map,model = c(1,45,2), type="bc",n.ind=n) markers<-pull.geno(cross) y<-pull.pheno(cross) The residual phenotypic variation is normally distributed with variance 1. For a backcross, the effect of a QTL corresponds to the difference between the homozygote and the heterozygote. February 16, 2010 Marker Data 7 / 16

8 F 2 Simulated Cross map; summary(cross) D1M1 D1M2 D1M3 D1M4 D1M5 D1M6 D1M7 D1M8 D1M9 D1M10 D1M Backcross No. individuals: 100 No. phenotypes: 1 Percent phenotyped: 100 No. chromosomes: 1 Autosomes: 1 Total markers: 11 No. markers: 11 Percent genotyped: 100 Genotypes (%): AA:53.1 AB:46.9 February 16, 2010 Marker Data 8 / 16

9 Simulated Backcross Marker Data head(markers) D1M1 D1M2 D1M3 D1M4 D1M5 D1M6 D1M7 D1M8 D1M9 D1M10 D1M11 [1,] [2,] [3,] [4,] [5,] [6,] mean(markers==1) [1] fit<-lm(y~i(markers-1)) summary(fit) February 16, 2010 Marker Data 9 / 16

10 Backcross Marker Regression Call: lm(formula = y ~ I(2 * markers - 3)) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-13 *** I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M ** I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M I(2 * markers - 3)D1M February , 2010 Marker Data 10 / 16

11 F 2 Simulated Cross n<-200 map<-sim.map(len=100,n.mar=11,include.x=false,eq.spacing=true) cross<-sim.cross(map,model = c(1,45,3,-2), type="f2",n.ind=n) markers<-pull.geno(cross) y<-pull.pheno(cross) For an intercross, the effect of a QTL is a pair of numbers, (a,d), where a is the additive effect (half the difference between the homozygotes) and d is the dominance deviation (the difference between the heterozygote and the midpoint between the homozygotes). head(markers) D1M1 D1M2 D1M3 D1M4 D1M5 D1M6 D1M7 D1M8 D1M9 D1M10 D1M11 [1,] [2,] [3,] [4,] [5,] [6,] February 16, 2010 Marker Data 11 / 16

12 Regression Coding variable AA Aa aa test X ij +1-1 backcross A ij F2: additive D ij F2: dominance A<-(markers-2) D<-(markers==2)+0 fit<-lm(y~a+d) summary(fit) Y i = µ + βx ij + e i Y i = µ + αa ij + βd ij + e i backcross F2 February 16, 2010 Marker Data 12 / 16

13 F 2 Results Estimate Std. Error t value Pr(> t ) AD1M AD1M AD1M AD1M AD1M e-05 *** AD1M e-05 *** AD1M AD1M AD1M AD1M AD1M DD1M DD1M DD1M DD1M DD1M ** DD1M * DD1M DD1M DD1M DD1M DD1M

14 Interval Mapping Backcross n<-100 map<-sim.map(len=100,n.mar=11,include.x=false,eq.spacing=true) cross<-sim.cross(map,model = c(1,45,1), type="bc",n.ind=n) crossb<-calc.genoprob(cross,step=1) est<-scanone(crossb) plot(est,lwd=10,main="backcross") abline(v=45,col="red",lwd=4) axis(1,at=45) 7 Backcross 6 5 lod Map position (cm) February 16, 2010 Marker Data 14 / 16

15 Permutation Testing est2<-scanone(crossb,n.perm=1000) summary(est2, alpha=c(0.01, )) LOD thresholds (1000 permutations) lod 1% % 3.21 plot(est2,main="permutation Testing") abline(v=2.45,col="orange",lwd=3) abline(v=3.21,col="red",lwd=3) Permutation Testing Frequency maximum LOD score February 16, 2010 Marker Data 15 / 16

16 QTL Scan with Threshold plot(est,main="backcross") abline(h=2.45,col="orange",lwd=3) abline(h=3.21,col="red",lwd=3) 7 Backcross 6 5 lod Map position (cm) February 16, 2010 Marker Data 16 / 16