Extensions of the Breeder’s Equation: Permanent Versus Transient Response Response to selection on the variance

Extensions of the Breeder’s Equation:

Permanent Versus Transient ResponseResponse to selection on the variance

Permanent Versus Transient Response

Considering epistasis and shared environmental values,the single-generation response follows from the midparent-offspring regression

R = h2 S + Sæ2z

µæ2A A2 +æ2A A A

4 +¢¢¢+æ(Esi re;Eo) +æ(Edam;Eo)∂

Breeder’s EquationResponse from epistasisResponse from shared environmental effects

Permanent component of responseTransient component of response --- contributesto short-term response. Decays away to zeroover the long-term

Response with Epistasis

R = Sµ

h2 +æ2AA2æ2z

∂

R(1+ø) = Sµ

h2 +(1 ° c)ø æ2AA2æ2z

∂

The response after one generation of selection froman unselected base population with A x A epistasis is

The contribution to response from this single generationafter generations of no selection is

c is the average (pairwise) recombination between lociinvolved in A x A

Contribution to response from epistasis decays to zero aslinkage disequilibrium decays to zero

Response from additive effects (h2 S) is due to changes in allele frequencies and hence is permanent. Contribution from A x A due to linkage disequilibrium

R(t+ø) = th2 S + (1 ° c)ø RAA(t)

Why unselected base population? If history of previousselection, linkage disequilibrium may be present andthe mean can change as the disequilibrium decays

More generally, for t generation of selection followed by generations of no selection (but recombination)

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

RAA has a limitingvalue given by


are needed to see this picture.Time to equilibrium a

function of c



QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

What about response with higher-order epistasis?




Fixed incremental differencethat decays when selection

stops

Maternal Effects:Falconer’s dilution model

z = G + m zdam + eDirect genetic effect on characterG = A + D + I. E[A] = (Asire + Adam)/2Maternal effect passed from dam to offspring is just

A fraction m of the dam’s phenotypic value

m can be negative --- results in the potential for a reversed response

The presence of the maternal effects means that responseis not necessarily linear and time lags can occur in response

Parent-offspring regression under the dilution model

In terms of parental breeding values,

E(zo j Adam; Asire;zdam) = Adam2 + Asire

2 +mzdam

Regression of BV on phenotypeA = πA +bAz (z ° πz ) +e

With no maternal effects, baz = h2

æA;M =mæ2A=(2 ° m)With maternal effects, a covariance between BV and maternal effect arises, withThe resulting slope becomes bAz = h2 2/(2-m)

¢πz = Sdamµ h2

2 ° m +m∂

+ Ssireh2

2 ° m

The response thus becomes

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Response to a single generation of selection

Reversed response in 1st generation largely due to negative maternal correlation masking genetic gain

Recovery of genetic response after initial maternal correlation decays

h2 = 0.11, m = -0.13 (litter size in mice)

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m = -0.25

m = -0.5

m = -0.75

h2 = 0.35

Selection occurs for 10 generations and then stops

Ancestral RegressionsWhen regressions on relatives are linear, wecan think of the response as the sum over allprevious contributions QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

For example, consider the response after 3 gens:

8 great-grand parents from generation zero Cov(offspring gen 3 on great-grandparents in gen 0)Selection differential on great-grandparentsQuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

2T-t = number of relatives fromgen. t for offspring from gen T

T,t = cov(zT,zt)



Changes in the Variance under Selection

The infinitesimal model --- each locus has a very smalleffect on the trait.

Under the infinitesimal, require many generations for significant change in allele frequencies

However, can have significant change in geneticvariances due to selection creating linkage disequilibrium

Under linkage equilibrium, freq(AB gamete) = freq(A)freq(B)

With positive linkage disequilibrium, f(AB) > f(A)f(B), so that AB gametes are more frequentWith negative linkage disequilibrium, f(AB) < f(A)f(B), so that AB gametes are less frequent




Additive variance with LD:Additive variance is the variance of the sum ofallelic effects,

Additive varianceGenic variance: value of Var(A)in the absence of disequilibriumfunction of allele frequencies

Disequilibrium contribution. Requirescovariances between allelic effects atdifferent loci

Key: Under the infinitesimal model, no (selection-induced) changes in genicvariance 2

a



Selection-induced changes in d change 2A, 2

z , h2


Dynamics of d: With unlinked loci, d loses half its value each generation (i.e, d in offspring is 1/2 d of their parents,



Dynamics of d: Computing the effect of selection in generating d



Consider the parent-offspring regression

Taking the variance of the offspring given theselected parents gives

Change in variance from selection



Change in d = change from recombination pluschange from selection


Recombination


Selection

+ =


are needed to see this picture.In terms of change in d,

This is the Bulmer Equation (Michael Bulmer), and it isakin to a breeder’s equation for the change in variance

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.At the selection-recombination equilibrium,

Application: Egg Weight in Ducks

Rendel (1943) observed that while the change mean weight weight (in all vs. hatched) asnegligible, but their was a significance decreasein the variance, suggesting stabilizing selection

Before selection, variance = 52.7, reducing to43.9 after selection. Heritability was h2 = 0.6

ed = eh4 e±(æ2z)=0.64 (43.9 - 52.7) = -3.2

Var(A) = 0.6*52.7= 31.6. If selection stops, Var(A)is expected to increase to 31.6+3.2= 34.8Var(z) should increase to 55.9, giving h2 = 0.62


are needed to see this picture. QuickTime™ and aTIFF (LZW) decompressor


Contribution of within- vs. between-familyeffects to Var(A)

The total additive variance arises from twosources: differences between the mean BVsof families and variation of BVs within familiesWhen no LD is present, both these sourcescontribute equally, Var(A)/2.What happens when LD present?

Consider parent-offspring regression in BA

The within-family (or mendelian segregation variance)is simply the genic variance and is a constant (if allelefrequencies not changing).

LD is a function the between-family variance in BV

When LD < 0, families are more similar than expected,When LD > 0, families are more dissimilar

Specific models of selection-inducedchanges in variances



Proportional reduction model:constant fraction k of

variance removed



Bulmer equation simplifiesto


are needed to see this picture.Closed-form solution

to equilibrium h2



.

0.2

0.3

0.4

0.5

0.6

0.7

0.8 h2 = 0.75

h2 = 0.50

h2 = 0.25

Fraction saved, p (in percentage)0 20 40 60 80 100

Equilibrium h2 under directiontruncation selection



Directional truncation selection







Changes in the variance = changes in h2

and even S (under truncation selection)

R(t) = h2(t) S(t)

In Class ProblemYou are selecting the upper 5% ofa trait with h2 = 0.75 and z

2 = 100initially in linkage equilibrium

• Compute the response over 3 populationsalso compute d(t), h2(t), z

2(t), and S(t)

• Compare the total 3 generations of responsewith the result from the standard breeder’sequation



Selection can also focus entirely on thevariance (stabilizing & disruptive selection)







Disruptive selection inflates the varianceafter selection, generating positive d

Stabilizing selection deflates the varianceafter selection, generating negative d

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In class problem #2

You have a trait with phenotypic variance100, heritability 0.4, d(0) = 0

Compare d, h2 and Var(A) after four generations of truncation selection (p=0.1)vs. double truncation selection (p=0.05 onboth tails)

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Extensions of the Breeder’s Equation: Permanent Versus Transient Response Response to selection on the variance