extensions of the breeder’s equation: permanent versus transient response response to selection on...
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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 +S
æ2z
µæ2
A A
2+
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 +æ2
AA
2æ2z
∂
R(1+ø) = Sµ
h2 +(1 ° c)ø æ2AA
2æ2z
∂
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)
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RAA has a limitingvalue given by
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Time to equilibrium afunction of c
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What about response with higher-order epistasis?
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Fixed incremental differencethat decays when selection
stops
Maternal Effects:Falconer’s dilution model
z = G + m zdam + e
Direct genetic effect on characterG = A + D + I. E[A] = (Asire + Adam)/2
Maternal effect passed from dam to offspring is justA 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) =Adam
2+
Asire
2+mzdam
Regression of BV on phenotype
A = 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 = Sdam
µh2
2 ° m+m
∂+ Ssire
h2
2 ° m
The response thus becomes
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Generation
Cu
mu
lati
ve R
esp
onse
to
Sel
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on
(i
n t
erm
s of
S)
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)
20151050
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Cu
mu
lati
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(in
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its
of S
)
m = -0.25
m = -0.5
m = -0.75
h2 = 0.35
Selection occurs for 10 generations and then stops
Ancestral Regressions
When regressions on relatives are linear, wecan think of the response as the sum over allprevious contributions QuickTime™ and a
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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
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2T-t = number of relatives fromgen. t for offspring from gen T
T,t = cov(zT,zt)
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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
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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
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Selection-induced changes in d change 2A, 2
z , h2
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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,
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Dynamics of d: Computing the effect of selection in generating d
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Consider the parent-offspring regression
Taking the variance of the offspring given theselected parents gives
Change in variance from selection
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Change in d = change from recombination pluschange from selection
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Recombination
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Selection
+ =
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This is the Bulmer Equation (Michael Bulmer), and it isakin to a breeder’s equation for the change in variance
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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= 34.8
Var(z) should increase to 55.9, giving h2 = 0.62
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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 families
When 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
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Proportional reduction model:constant fraction k of
variance removed
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Bulmer equation simplifiesto
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Closed-form solutionto equilibrium h2
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.
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
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Directional truncation selection
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Changes in the variance = changes in h2
and even S (under truncation selection)
R(t) = h2(t) S(t)
In Class Problem
You 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
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Selection can also focus entirely on thevariance (stabilizing & disruptive selection)
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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)