Genotypic and Phenotypic Variances in Genetic Studies

Phenotypic and Genotypic variances OUTLINE Genotypic variance in terms of breeding values Additive variance Dominance deviation Does additive variance implies no dominance gene effect?

Variances and Covariances Covariance is a measure of the joint variation between two or more variables E[(X- X) (Y- Y)] If Z=X+Y then Var(Z)=Var(X)+Var(Y)+2Cov(X,Y) If Z=X+Y-W then Var(Z)=Var(X)+Var(Y)+Var(W)+2Cov(X,Y)- 2Cov(X,W)-2Cov(Y,W)

Phenotypic and Genotypic variance In the previous section we modeled the phenotypic value of one individual kthwith genotype AiAjas Pij(k)=Gij+eij(k) Pij(k)= +gij+eij(k) [gij=Gij- dev from the population mean] with gijbeing the genotypic effect ofAiAjand eij(k)the non-genetic deviation for individual k. Then VP=Vgij+V[eij(k)]=VG+VE (with gijand eij(k) being uncorrelated)

Phenotypic and Genotypic variance Also we have seeing in the previous section that Gij= +gijGenotypic value of AiAj Gij= + i+ j+ ijwith i+ jbeing the breeding value and ijthe dominance deviation. Then VG=V( i+ j)+V( ij)=VA+VD We can partition the genotypic effect of two loci into intralocus effect and epistasis effect

Phenotypic and Genotypic variance Suppose locus A and B then the genotypic value of AiAjBkBlis Gijkl= +( i+ j+ ij)+( k+ l+ kl)+Iijkl and the genetic variance across the two loci is V(G)=V( i+ j)+V( ij)+V( k+ l)+V( kl)+V(Iijkl) =VA+VD+VI Also V[eij(k)] can be partitioned in two components VGEvariance of genotype x environments Veerror variance

Phenotypic and genetic variances P=G+E VP=VG+VE VP=V( i+ j)+V( ij)+VE(genotype AiAj) VP=VA+VD+VE VP=V( i+ j)+V( ij)+V( k+ l)+V( kl)+V(Iijkl)+VE VP=VA+VD+VI+VE VP=VA+VD+VI+VGE+Ve VP=VA+VD+VI+VAE+VDE+VIE+Ve

Additive Variance The VA measures the variation due to the average effects of alleles Recall that iquantifies the effect of allele Aion the mean of random offspring that inherit that allele Then VA measures the variation in the effects that are transmitted from one generation to the next and thus plays a key role in predicting the change in the population mean due to selection

Genotypic values, Breeding values and Dominance deviation Genotypic Breeding Dominance value value MP+a 2 1=2q MP+d 1+ 2 =(q-p) MP-a 2 2=-2p Genotype Freq. A1A1 A1A2 A2A2 MP=[Z+(Z+2a)]/2=Z+a; 1= q[a+d(p-p)]=q = a+d(p-p); 2 =-p[a+d(p-p)]=-p Deviation -2q2d 2pqd -2p2d p2 2pq q2 VDE

Additive variance VA=2V( i)= 2 pi( i)2 The breeding values of 2 1=2q for A1A1, (q-p) for A1A2 and -2 2=2p for A2A2 . They have a mean of zero and their variance is the sum of the products of the genotype frequency and the squared breeding value VA = p2(2q )2 + 2pq(q-p)2 + q2(-2p )2 =2pq( )2 =2pq[a+d(q-p)]2 When d=0 VA =2pq(a)2 Also VA can be expressed in terms of the average effects of alleles. That is VA=V( i+ j) and since V( i) = V( j) then VA=2V( i) = 2E( i)2=2 pi( i)2 in other words VA=2 the variance of average effect of an allele VA/2= V( i)

Additive variance VA = p2(2q )2 + 2pq(q-p)2 + q2(-2p )2 =2pq( )2 =2pq[a+d(q-p)]2 The term additive is misleading as it may imply that the allele acts in pure additive fashion. That is, additive may imply that VA exists only at loci where dominance is absent. However 2pq[a+d(q-p)]2 indicates that that any segregating loci with d=0 (no dominance), partial dominance (0<d<a) or overdominance (d>a) can contribute to VA Therefore the presence of VA does not imply that the allele act in purely additive manner.

Dominance deviation variance The dominance deviation for A1A1 is -2q2d, for A1A2 is 2pqd and for A2A2 is -2p2d V(A) = p2(2q 2 d)2 + 2pq(2pqd)2 + (-2p 2 d)2 =4p 2 q 2d2 = (2pqd)2

Covariance between relatives OUTLINE Coefficient of coancestry Identity by descent Covariance between relatives due to their breeding values

Covariance between relatives Close relatives such a parents and offspring have higher degree of resemblance than more distance relatives such as uncle and nice. Non-genetic factor can contribute to degree of resamblance between relative. Members of a family are more alike because genetic and environmental factor.

Environmental factors among relatives are uncorrelated Non-genetic factors among relatives are assumed to be equaled to zero. In plants this assumption is met through the randomization procedure that are inherent in the experimental design used in plant breeding

Covariance between relatives and selection Covariance between relatives measures the degree of resemblance between related individuals in a population. By definition Covariance between unrelated individuals = 0 Progress from selection is directly proportional to the degree of resamblance between the selected individuals and the progeny to recombine them. Also the covariance between relatives plays a key role in estimating the variance components and prediction.

Covariance between relatives as function of identity by descent and VG The covariance between relatives is a function of the identity by descent between alleles and of the different components of VG The genetic covariance between relatives is due to three components additive genetic variance (breeding values) dominance deviations genetic variance epistasis genetic variance

General method to express relations between relatives (one locus) First follow methodology of Kempthorne (1955a,b) for the covariance between relatives which was originally developed by Malecot (1948) under the assumption of no inbreeding and that loci are independent. Extensions of the previous formulas due to Cockerham (1954) to the case of arbitrary number of loci with epistasis and no linkage

General method to express relations between relatives (one locus) X Y (a b) (c d) The genotypic value of X and Y measured as deviation from the mean of the random mating population may be written as gx=?a + ?b + ?ab gy=?c + ?d + ?cd

General method to express relations between relatives Cov (X,Y)=E(?a+ ?b+?ab)(?c+ ?d+?cd)= E(?a?c) + E(?a?d) + E(?a?cd) + E(?b?c) + E(?b?d) + E(?b?cd) + E(?ab?c) + E(?ab?d) + E(?ab?cd) P(a b)=P(c d)=0 (no inbreeding)

General method to express relations between relatives Consider E( a cd). If genes c and a are statistically dependent then gene d must be independent of gene a as we assumed there is no inbreeding. Hence E(?a?cd)=0 and similarly with E(?b?cd), E(?ab?c) E(?ab?d) [no with E(?ab?cd)].

General method to express relations between relatives Thus Cov (X,Y) reduces to Cov (X,Y)= E(?a?c) + E(?a?d) + E(?b?c) + E(?b?d) + E(?ab?cd) Now consider each of these terms E(?a?c)=P(a c)E(?a2) = P(a c)(1/2VA) [E( i2)=1/2 VA] E(?a?d)=P(a d)E(?a2) = P(a d)(1/2VA) E(?b?c)=P(b c)E(?b2) = P(b c)(1/2VA) E(?b?d)=P(b d)E(?b2) = P(b d)(1/2VA) E(?ab?cd)=P(a c, b d)E(?ab2) + P(a d, b c)E(?ab2) =[P(a c, b d)+P(a d, b c)]VD [E(?ab2)=VD]

General method to express relations between relatives Hence Cov (X,Y)= [P(a c)+P(a d)+P(b c)+P(b d)[1/2VA] + [P(a c, b d)+P(a d, b c)]VD Using Kempthorne (1995a,b) this is Cov (X,Y)=KVA +LVD or AVA +DVD which is similar to Malecot (1948)

General method to express relations between relatives Cockerham (1954) extended to an arbitrary number of loci with arbitrary epistasis and no linkage Cov (X,Y)=(A)VA+(D)VD+ (A2)VAA+(AD)VAD+ (D2)VDD+ (A3)VAAA

Covariance between relatives due to their breeding values A B C D X Y (AiAj) (AkAl) The alleles in X and Y contribute to the covariance between relatives if they are identical by descent (IBD) If Ai in X is IBD to Ak in Y then the covariance due to this allele is E( i, k)=E( i2)=Var( i) otherwise covariance=0

Covariance between relatives due to their breeding values A B C D X Y (AiAj) (AkAl) Alleles in X and Y can be IBD through out four events -- Ai IBD to Ak -- Ai IBD to Al -- Aj IBD to Ak -- Aj IBD to Al and the probability of these events equals fXY the coefficient of coancestry between X and Y

Covariance between relatives due to their breeding values A B C D X Y (AiAj) (AkAl) Then Cov( )=P(Ai IBD to Ak) Cov( i, k) + P(Ai IBD to Al) Cov( i, l)+ P(Aj IBD to Ak) Cov( j, k) + P(Aj IBD to Al) Cov( j, k) Cov( )=4 fXYVar( i)[ where 2Var( i)=VA] Cov( )=2 fXYVA Cov( )=A VA Then Cov due to breeding values=variance of the breeding value multiplied by the numerical relationship matrix A

Covariance between relatives due to their dominance deviation A B C D X Y (AiAj) (AkAl) Dominance deviations associated with pairs of genes at the same locus contribute to the covariance between X and Y if Ai Ak Aj Al Ai Al Aj Ak

Covariance between relatives due to their dominance deviation A B C D X Y (AiAj) (AkAl) fAC=P(Ai Ak) fBD=P(Aj Al) fAD=P(Ai Al) fBC=P(Aj Ak)

Covariance between relatives due to their dominance deviation A B C D X Y (AiAj) (AkAl) Cov ? = P(Ai Ak,Aj Al) Cov (???, ???) = P(Ai Al,Aj Aj) Cov (???, ???) = (fAC fBD + fAD fBC) V(D) = XY V(D)

Covariance between relatives due to their epistasis Each component of V(I) has a different contribution to the covariance between relatives. V(AA)=V(??)ik+ V(??)il+ V(??)jk+ V(??)jl =4 V(??)ik

Covariance between relatives due to their epistasis The resulting contribution of V(AA) to the Covariance between relatives is Cov (??) =f2xyV(??)ik = (2fxy)2 V(AA) The covariance due to additive x dominance effects is obtained by multiplying the coefficient for V(A) and the coefficient for V(D) resulting in (2fxy) XY V(AD)

Covariance between relatives due to their breeding values, dominance deviation and epistasis Cov(XY)=2fxy V(A)+ XY V(D)+(2fxy)2 V(AA)+ (2fxy) XY V(AD)+( XY)2 V(DD)+)+ (2fxy)3 V(AAA) + (2fxy)2 XYV(AAD)+ .