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