Lecture 7: Correlated Characters
Feb 22, 2016
Lecture 7:Correlated Characters
Genetic vs. Phenotypic correlations• Within an individual, trait values can be
positively or negatively correlated,– height and weight -- positively correlated– Weight and lifespan -- negatively correlated
• Such phenotypic correlations can be directly measured, – rP denotes the phenotypic correlation
• Phenotypic correlations arise because genetic and/or environmental values within an individual are correlated.
r P
Px Py
Ar
AxAy
The phenotypic values between traits x and ywithin an individual are correlated
Correlations between the breeding values ofx and y within the individual can generate aphenotypic correlation
Likewise, the environmental values for the two traits within the individual couldalso be correlated
y
rE
ExE
Genetic & Environmental Correlations
• rA = correlation in breeding values (the genetic correlation) can arise from– pleiotropic effects of loci on both traits– linkage disequilibrium, which decays over time
• rE = correlation in environmental values– includes non-additive genetic effects– arises from exposure of the two traits to the
same individual environment
The relative contributions of genetic and environmental correlations to the phenotypic
correlation
If heritability values are high for both traits, thenthe correlation in breeding values dominates thephenotypic corrrelation
If heritability values in EITHER trait are low, thenthe correlation in environmental values dominates thephenotypic corrrelationIn practice, phenotypic and genetic correlations often have the same sign and are of similar magnitude, but this is not always the case
rP=rAhXhY+rEq(1°h2x)(1°h2Y)
Estimating Genetic Correlations
Recall that we estimated VA from the regression oftrait x in the parent on trait x in the offspring,
Trait x in parent
Trait x inoffspring
Slope = (1/2) VA(x)/VP(x)
VA(x) = 2 *slope * VP(x)
Similarly, we can estimate VA(x,y), the covariance in thebreeding values for traits x and y, by the regression oftrait x in the parent and trait y in the offspring
Trait x in parent
Trait y inoffspring
Slope = (1/2) VA(x,y)/VP(x)
VA(x,y) = 2 *slope * VP(x)
Thus, one estimator of VA(x,y) is
VA(x,y) = by|x VP(x) + bx|y VP(y)2 *by|x * VP(x) + 2 *bx|y * VP(y)
2VA(x,y) =VA(x,y) =
Put another way, Cov(xO,yP) = Cov(yO,xP) = (1/2)Cov(Ax,Ay)
Cov(xO,xP) = (1/2) VA (x) = (1/2)Cov(Ax, Ax) Cov(yO,yP) = (1/2) VA (y) = (1/2)Cov(Ay, Ay)
Likewise, for half-sibs,Cov(xHS,yHS) = (1/4) Cov(Ax,Ay)Cov(xHS,xHS) = (1/4) Cov(Ax,Ax) = (1/4) VA (x) Cov(yHS,yHS) = (1/4) Cov(Ay,Ay) = (1/4) VA (y)
G X E and Genetic CorrelationsOne way to deal with G x E is to treat the same traitmeasured in two (or more) different environments ascorrelated characters.
If no G x E is present, the genetic correlation should be 1Example: 94 half-sib families of seed beetles were placedin petri dishes which either contained (Environment 1)or lacked seeds (Environment 2)
Total number of eggs laid and longevity in days weremeasured and their breeding values estimated
-2 -1 0 1 2
Fecundity
-30
-20
-10
0
10
20
30
Longevity in Days
-4 -2 0 2 4-20
-10
0
10
20
30
Seeds Present. Seeds Absent.
-30 -20 -10 0 10 20 30
Seeds A
bsent
-20
-10
0
10
20
30
Seeds Present
-2 -1 0 1 2-4
-2
0
2
4
6
Fecundity. Longevity.
Positive genetic correlationsNegative genetic correlations
Correlated Response to Selection
Direct selection of a character can cause a within-generation change in the mean of a phenotypicallycorrelated character.
Direct selection onx also changes themean of y
*
+
Select All
X
Y
SX
SY
Phenotypic correlations induce within-generationchanges
For there to be a between-generation change, thebreeding values must be correlated. Such a changeis called a correlated response to selection
Trait y
Trait x
Phenotypic values
Sy
Sx
Trait y
Trait x
Phenotypic values
Trait y
Trait x
Breeding values
Trait y
Trait x
Breeding values
Trait y
Trait x
Breeding values
Rx
Ry = 0
Predicting the correlated response
bAy|Ax =Cov(Ax,Ay)Var(Ax)
= rAAx)Ay)
The change in character y in response to selectionon x is the regression of the breeding value of y on the breeding value of x,
Ay = bAy|Ax Ax
where
If Rx denotes the direct response to selection on x,CRy denotes the correlated response in y, with
CRy = bAy|Ax Rx
We can rewrite CRy = bAy|Ax Rx as follows
First, note that Rx = h2xSx = ixhx A (x)
Recall that ix = Sx/P (x) is the selection intensitySince bAy|Ax = rA A(x) / A(y),
We have CRy = bAy|Ax Rx = rA A (y) hxix Substituting A (y)= hy P (y) gives our final result:
CRy = ix hx hy rA P (y)
Noting that we can also express the direct response as Rx = ixhx
2 p (x)shows that hx hy rA in the corrected response plays thesame role as hx
2 does in the direct response. As a result,hx hy rA is often called the co-heritability
Estimating the Genetic Correlation from Selection Response
Suppose we have two experiments:Direct selection on x, record Rx, CRyDirect selection on y, record Ry, CRx
Simple algebra shows that
rA2 = CRx CRy
Rx Ry
This is the realized genetic correlation, akin to the realized heritability, h2 = R/S
Example: A double selection experiment in bristle number in fruit flies
Mean Bristle Number
Selection Line abdominal sternopleural
High AB 33.4 26.4
Low AB 2.4 12.8
High ST 22.2 45.0
Low ST 11.1 9.5
Direct responses in white, correlated in blue
In one line, direct selection on abdominal (AB) bristles. The direct RAB and correlated CRST responses measured
In the other line, direct selection on sternopleural (ST)Bristle, with the direct RST and correlated CRAB responses measured
RAB = 33.4-2.4 = 31.0, RST = 45.0-9.5 = 35.5CRAB = 26.4-12.8 = 13.6, C RST = 22.2 - 11.1 = 11.1rA=rCRABRABCRSTRST=r11:13113:635:3=0:37
Direct vs. Indirect ResponseWe can change the mean of x via a direct response Rxor an indirect response CRx due to selection on y
Hence, indirect selection gives a large response when
• Character y has a greater heritability than x, and the geneticcorrelation between x and y is high. This could occur if x is difficult tomeasure with precison but x is not.
• The selection intensity is much greater for y than x. This would be true if y were measurable in both sexes but x measurable in only one sex.
iYrAhY>iXhXCRXRX=iYrAæAXhYiXhXæAX=iYrAhYiXhX
Matrix algebra
Matrices
The identity matrix I,
∂=
µ ij
∂( )C( )=µefgh∂B( )=µ a b
c dA
)(=µ1 0
0 1∂
I2 x 2
AB =µa b
c d∂ µe f
g h∂
=µ ae+bg af +bh
ce+dg cf +dh∂
( ( () ) )
BA =µae+cf eb+df
ga+ch gd+dh∂
( ) )(A C =µ ai +bj
ci +dj∂
The identity matrix serves the role of one in matrix multiplication: AI =A, IA = A
Matrix Multiplication
The Inverse Matrix, A-1
∂µA = a b
c d( )For
For a square matrix A, define the Inverse of A, A-1, asthe matrix satisfying
A -1 A = A A -1 = I
A °1 = 1ad ° bc
µ d °b°c a
∂
If this quantity (the determinant)is zero, the inverse does not exist.
The inverse serves the role of division in matrix multiplication
Suppose we are trying to solve the system Ax = c for x.
A-1 Ax = A-1 c. Note that A-1 Ax = Ix = x, giving x = A-1 c
Multivariate trait selection
S =
0BB
S1S2...Sn
1CC@ A R @ A=
0BB
R1R2...Rn
1CCVector of selection differentialsVector of
responses
P = phenotypic covariance matrix. Pij = Cov(Pi,Pj)
P =µ æ2(P1) æ(P1; P2)
æ(P1;P2) æ2(P2)∂
G = Genetic covariance matrix. Gij = Cov(Ai,Aj)
G =µ æ2(A1) æ(A1;A2)æ(A1; A2) æ2(A2)
∂
The multidimensional breeders' equation
R = G P-1 S
R= h2S = (VA/VP) SNatural parallelswith univariatebreeders equation
P-1 S = is called the selection gradientand measures the amount of direct selectionon a character
The gradient version of the breeders’ Equation is R = G
Sources of within-generation change in the mean
Since = P-1 S, S = P
Response from direct selection on trait jBetween-generation change in trait j
Change in mean from phenotypicallycorrelated characters under direct selection
Within-generation change in trait j
Indirect response from geneticallycorrelated characters under direct selection
Change in mean from direct selection on trait j
Sj =æ2(P j )Øj + Xi6= j
æ(P j ; Pi)Øi
R j =æ2(A j )Øj +Xi6=j
æ(A j ; A i)Øi