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Lecture 7: Correlated Characters

This lecture discusses the concept of correlated characters and the genetic and phenotypic correlations between them. It explains how these correlations arise within individuals and how they can be measured. The lecture also explores the contributions of genetic and environmental factors to the phenotypic correlation and discusses the estimation of genetic correlations through selection.

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Lecture 7: Correlated Characters

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  1. Lecture 7:Correlated Characters

  2. 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, • rPdenotes the phenotypic correlation • Phenotypic correlations arise because genetic and/or environmental values within an individual are correlated.

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

  4. 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 (e.g., D, I) • arises from exposure of the two traits to the same individual environment

  5. 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

  6. Similarly, we can estimate VA(x,y), the covariance in the breeding values for traits x and y, by the regression of trait x in the parent and trait y in the offspring Recall that we estimated VA from the regression of trait x in the parent on trait x in the offspring, Slope = (1/2) VA(x)/VP(x) Slope = (1/2) VA(x,y)/VP(x) Trait x in offspring Trait y in offspring VA(x) = 2 *slope * VP(x) VA(x,y) = 2 *slope * VP(x) Trait x in parent Trait x in parent Estimating Genetic Correlations

  7. 2 *by|x * VP(x) + 2 *bx|y * VP(y) VA(x,y) = VA(x,y) = 2 Thus, one estimator of VA(x,y) is VA(x,y) = by|x VP(x) + bx|y VP(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)

  8. 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

  9. Negative genetic correlations Positive genetic correlations S e e d s P r e s e n t . S e e d s A b s e n t . 3 0 3 0 2 0 2 0 1 0 1 0 Fecundity 0 0 - 1 0 - 1 0 - 2 0 - 3 0 - 2 0 - 2 - 1 0 1 2 - 4 - 2 0 2 4 L o n g e v i t y i n D a y s F e c u n d i t y . L o n g e v i t y . 3 0 6 2 0 4 2 1 0 Seeds Absent 0 0 - 2 - 1 0 - 4 - 2 0 - 2 - 1 0 1 2 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 S e e d s P r e s e n t

  10. Select All Y * S Y + X S X Correlated Response to Selection Direct selection of a character can cause a within- generation change in the mean of a phenotypically correlated character. Direct selection on x also changes the mean of y

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

  12. Phenotypic values Breeding values Breeding values Breeding values Trait y Trait y Trait y Trait y Ry = 0 Trait x Trait x Trait x Trait x Rx

  13. 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 = bAy|Ax Ax where s(Ay) Cov(Ax,Ay) bAy|Ax = = rA Var(Ax) s(Ax) Predicting the correlated response If Rx denotes the direct response to selection on x, CRy denotes the correlated response in y, with CRy = bAy|Ax Rx

  14. Recall that ix = Sx/sP (x)is the selection intensity shows that hx hy rAin the corrected response plays the same role as hx2does in the direct response. As a result, hx hy rA is often called the co-heritability We can rewrite CRy = bAy|Ax Rxas follows First, note that Rx = h2xSx = ixhxsA (x) Since bAy|Ax = rAsA(x) / sA(y), We have CRy = bAy|Ax Rx =rAsA (y) hxix Substituting sA (y)= hysP (y)gives our final result: CRy = ix hx hy rAsP (y) Noting that we can also express the direct response as Rx = ixhx2sp (x)

  15. CRx CRy rA2 = Rx Ry 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 This is the realized genetic correlation, akin to the realized heritability, h2 = R/S

  16. 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 Example: A double selection experiment in bristle number in fruit flies Direct responses in white, correlated in blue RAB = 33.4-2.4 = 31.0, RST = 45.0-9.5 = 35.5 CRAB = 26.4-12.8 = 13.6, C RST = 22.2 - 11.1 = 11.1

  17. Direct vs. Indirect Response We can change the mean of x via a direct response Rx or an indirect response CRx due to selection on y Hence, indirect selection gives a large response when • 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. • Character y has a greater heritability than x, and the genetic correlation between x and y is high. This could occur if x is difficult to measure with precison but y is not.

  18. Matrix algebra

  19. µ ( ) ) ( ) ( µ ∂ a b A B i = C = c d j µ ∂ 1 0 I = 0 1 2 x 2 ( ) Matrices ∂ The identity matrix I,

  20. µ ∂ µ ∂ µ ∂ ) ) ( ) ( ( a b e f a e + b g a f + b h Matrix Multiplication A B = = c d g h c e + d g c f + d h ) µ ∂ ( ( µ ∂ ) a e + c f e b + d f a i + b j B A = A C = g a + c h g d + d h c i + d j The identity matrix serves the role of one in matrix multiplication: AI =A, IA = A

  21. - 1 - 1 A A = A A = I µ ∂ µ ∂ ) ( ) ( 1 d ° b a b For ° 1 A = A = c d ° c a a d ° b c If this quantity (the determinant) is zero, the inverse does not exist. The Inverse Matrix, A-1 For a square matrix A, define the Inverse of A, A-1, as the matrix satisfying

  22. 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

  23. 0 1 0 1 R S 1 1 R B C S B C 2 2 B C R = B C . S = . . @ A . @ A . . R ( ) n S n ( ) µ ∂ µ ∂ 2 2 æ ( P ) æ ( P ; P ) æ ( A ) æ ( A ; A ) 1 1 2 1 1 2 P = G = 2 2 æ ( P ; P ) æ ( P ) æ ( A ; A ) æ ( A ) 1 2 2 1 2 2 Multivariate trait selection Vector of responses Vector of selection differentials P = phenotypic covariance matrix. Pij = Cov(Pi,Pj) G = Genetic covariance matrix. Gij = Cov(Ai,Aj)

  24. Natural parallels with univariate breeders equation R= h2S = (VA/VP) S The multidimensional breeders' equation R = G P-1 S P-1 S = bis called theselection gradient and measures the amount of direct selection on a character The gradient version of the breeders’ Equation is R = G b

  25. X 2 S = æ ( P ) Ø + æ ( P ; P ) Ø j j j j i i i 6 j = X Response from direct selection on trait j Change in mean from direct selection on trait j Within-generation change in trait j Between-generation change in trait j 2 R = æ ( A ) Ø + æ ( A ; A ) Ø j Indirect response from genetically correlated characters under direct selection Change in mean from phenotypically correlated characters under direct selection j j j i i i 6 j = Sources of within-generation change in the mean Since b = P-1 S, S = P b,

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