Lecture 7 Correlated Characters Genetic vs Phenotypic correlations

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Lecture 7: Correlated Characters

Lecture 7: Correlated Characters

Genetic vs. Phenotypic correlations • Within an individual, trait values can be positively or

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, – r. P denotes the phenotypic correlation • Phenotypic correlations arise because genetic and/or environmental values within an individual are correlated.

The phenotypic values between traits x and y within an individual are correlated Correlations

The phenotypic values between traits x and y within an individual are correlated Correlations between the breeding values of Likewise, the environmental values for the x and y within the individual can generate a two traits within the individual could phenotypic correlation also be correlated

Genetic & Environmental Correlations • r. A = correlation in breeding values (the genetic

Genetic & Environmental Correlations • r. A = correlation in breeding values (the genetic correlation) can arise from – pleiotropic effects of loci on both traits – linkage disequilibrium, which decays over time • r. E = 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

The relative contributions of genetic and environmental correlations to the phenotypic correlation If heritability values are high for both traits, then the correlation in breeding values dominates the phenotypic corrrelation If heritability values in EITHER trait are low, then the correlation in environmental values dominates the phenotypic corrrelation In practice, phenotypic and genetic correlations often have the same sign and are of similar magnitude, but this is not always the case

Estimating Genetic Correlations Similarly, wewe can estimate. VVAAfrom (x, y), the regression covariance of

Estimating Genetic Correlations Similarly, wewe can estimate. VVAAfrom (x, y), the regression covariance of in the Recall that estimated breeding for traits x and by the regression of trait x invalues the parent on trait x iny, the offspring, trait x in the parent and trait y in the offspring Trait x y in offspring Slope = (x)/VP(x) (1/2) VA(x, y)/V P(x) = =2 2*slope VA(x, y) *slope* *VPV(x) P(x) Trait x in parent

Thus, one estimator of VA(x, y) is 2 *by|x * VP(x) + 2 *bx|y

Thus, one estimator of VA(x, y) is 2 *by|x * VP(x) + 2 *bx|y * VP(y) VA(x, y) == by|x VP(x) + bx|y VP(y) 2 Put another way, Cov(x. O, y. P) = Cov(y. O, x. P) = (1/2)Cov(Ax, Ay) Cov(x. O, x. P) = (1/2) VA (x) = (1/2)Cov(Ax, Ax) Cov(y. O, y. P) = (1/2) VA (y) = (1/2)Cov(Ay, Ay) Likewise, for half-sibs, Cov(x. HS, y. HS) = (1/4) Cov(Ax, Ay) Cov(x. HS, x. HS) = (1/4) Cov(Ax, Ax) = (1/4) VA (x) Cov(y. HS, y. HS) = (1/4) Cov(Ay, Ay) = (1/4) VA (y)

G X E and Genetic Correlations One way to deal with G x E

G X E and Genetic Correlations One way to deal with G x E is to treat the same trait measured in two (or more) different environments as correlated characters. If no G x E is present, the genetic correlation should be 1 Example: 94 half-sib families of seed beetles were placed in petri dishes which either contained (Environment 1) or lacked seeds (Environment 2) Total number of eggs laid and longevity in days were measured and their breeding values estimated

Negative genetic correlations Positive genetic Seeds Present. Seeds Absent. 30 30 20 20 Fecundity

Negative genetic correlations Positive genetic Seeds Present. Seeds Absent. 30 30 20 20 Fecundity 10 10 0 0 -10 -20 -30 -2 -1 0 1 2 -4 -2 0 2 4 Longevity in Days Fecundity. Longevity. Seeds Absent 30 6 20 4 10 2 0 0 -10 -2 -4 -20 -30 -20 -10 0 10 20 30 -2 Seeds Present -1 0 1 2

Correlated Response to Selection Direct selection of a character can cause a withingeneration change

Correlated Response to Selection Direct selection of a character can cause a withingeneration change in the mean of a phenotypically correlated character. Direct selection on x also changes the mean of y

Phenotypic correlations induce within-generation changes Phenotypic values Trait y Sy Sx Trait x For

Phenotypic correlations induce within-generation changes Phenotypic values Trait y Sy Sx Trait x For there to be a between-generation change, the breeding values must be correlated. Such a change is called a correlated response to selection

Breeding values Trait y Ry = 0 Trait x Rx

Breeding values Trait y Ry = 0 Trait x Rx

Predicting the correlated response The change in character y in response to selection on

Predicting the correlated response The change in character y in response to selection on x is the regression of the breeding value of y on the breeding value of x, Ay = b. Ay|Ax Ax where b. Ay|Ax = Cov(Ax, Ay) Var(Ax) = r. A s(Ay) s(Ax) If Rx denotes the direct response to selection on x, CRy denotes the correlated response in y, with CRy = b. Ay|Ax Rx

We can rewrite CRy = b. Ay|Ax Rx as follows First, note that Rx

We can rewrite CRy = b. Ay|Ax Rx as follows First, note that Rx = h 2 x. Sx = ixhx s. A (x) Since b. Ay|Ax = r. A s. A(x) / s. A(y), Recall that ix = Sx/s. P (x) is the selection intensity We have CRy = b. Ay|Ax Rx = r. A s. A (y) hxix Substituting s. A (y)= hy s. P (y) gives our final result: CRy = ix hx hy r. A s. P (y) Noting that we can also express the direct response as Rx = ixhx 2 sp (x) shows that hx hy r. A in the corrected response plays the same role as hx 2 does in the direct response. As a result, hx hy r. A is often called the co-heritability

Estimating the Genetic Correlation from Selection Response Suppose we have two experiments: Direct selection

Estimating the Genetic Correlation from Selection Response Suppose we have two experiments: Direct selection on x, record Rx, CRy Direct selection on y, record Ry, CRx Simple algebra shows that r. A = 2 CRx CRy Rx Ry This is the realized genetic correlation, akin to the realized heritability, h 2 = R/S

Example: A double selection experiment in bristle number in fruit flies In one line,

Example: A double selection experiment in bristle number in fruit flies In one line, direct selection on abdominal Mean Bristle Number (AB) bristles. The direct R AB and correlated CRST responses measured Selection Line abdominal sternopleural AB line, direct selection 33. 4 on sternopleural (ST)26. 4 In the. High other Bristle, with the direct RST and correlated CRAB responses measured Low AB 2. 4 12. 8 High ST 22. 2 45. 0 Low ST 11. 1 9. 5 Direct responses in red, correlated in blue RAB = 33. 4 -2. 4 = 31. 0, CRAB = 26. 4 -12. 8 = 13. 6, RST = 45. 0 -9. 5 = 35. 5 C RST = 22. 2 - 11. 1 = 11. 1

Direct vs. Indirect Response We can change the mean of x via a direct

Direct vs. Indirect Response We can change the mean of x via a direct response R x or an indirect response CRx due to selection on y Hence, indirect selection gives a large response when selection intensity is much greaterthan for yx, than x. This would be true • • The Character y has a greater heritability and the genetic if y were measurable both measurable one sexto. correlation between x inand y issexes high. but Thisx could occur ifinxonly is difficult measure with precison but x is not.

Matrices ( ) The identity matrix I ( ) ()

Matrices ( ) The identity matrix I ( ) ()

( )( ) ( ( ) ) ( ) The identity matrix serves the

( )( ) ( ( ) ) ( ) The identity matrix serves the role of one in matrix multiplication: AI =A, IA = A

The Inverse Matrix, -1 A For a square matrix A, define the Inverse of

The Inverse Matrix, -1 A For a square matrix A, define the Inverse of A, A-1, as the matrix satisfying For ( ) ( 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

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 Vector of selection differentials responses P = phenotypic covariance matrix. Pij

Multivariate trait selection Vector of selection differentials responses P = phenotypic covariance matrix. Pij = Cov(Pi, Pj) G = Genetic covariance matrix. Gij = Cov(Ai, Aj) ( )

The multidimensional breeders' equation R = G P-1 S Naturalgradient parallels P-1 R= S

The multidimensional breeders' equation R = G P-1 S Naturalgradient parallels P-1 R= S =h 2 b. S is called the selection = (VA/VP) S with univariate and measures the amount of direct selection breeders equation on a character The gradient version of the breeders’ Equation is R=Gb

Sources of within-generation change in the mean Since b = P-1 S, S =

Sources of within-generation change in the mean Since b = P-1 S, S = P b, Change in meanchange from direct selection on trait j Within-generation in trait j Change in mean from phenotypically correlated characters under direct selection Response fromchange direct selection Between-generation in trait j on trait j Indirect response from genetically correlated characters under direct selection