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Lecture 11:Selection on the

Environmental varianceBruce Walsh lecture notes

Synbreed courseversion 4 July 2013

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Scales of Environmentalsensitivity

• Clones (pure lines) typically showdifferential (relative) performance acrossmacroenvironments,– evidence of G x E

• Genotypes can also show variation in theirsensitivity to microenvironmentaldifferences,– Fluctuating asymmetry

– Developmental noise

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Environmental vs. genetic canalization

• Waddington (1942) suggested thedevelopment systems become bufferedover time to be somewhat robust toenvironmental and/or genetic noise

• Environmental canalization– A genotype has robust performance over a set

of microenvironments (G x E)

• Genetic canalization– A particular genotype (one or several loci) has

robust performance when randomized overgenetic backgrounds (epistasis)

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Fraser & Schadt’s experiment

• Using variation within and between a series ofinbred lines, one can distinguish between geneticand environmental robustness (canalization)

• Fraser & Schadt examined mRNA levels atthousands of genes for their traits over a seriesof inbred lines of mice

• Environmental robustness means that the traitwill have reduced within-line variation

• Genetic robustness implies that the mean traitvalue over lines (the between-line variance) isreduced

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Their design: eQTLswere mapped and thenexamined both within(down a row) and acrosslines (across rows)

Environmental robustness(ER) QTLs: within-linevariation, no between-line

Genetic robustness (GR) QTLs: Between-linevariation, no within-line

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•Figure 2. Genetic robustness QTLs in mouse.

Fraser HB, Schadt EE (2010) The Quantitative Genetics of Phenotypic Robustness. PLoS ONE 5(1): e8635.

GR QTLs tend to bemainly trans-acting,modest sex-specificoverlap.When present, due toCis-acting factors

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•Figure 3. Environmental robustness QTLs in mouse.

Fraser HB, Schadt EE (2010) The Quantitative Genetics of Phenotypic Robustness. PLoS ONE 5(1): e8635.

ER QTLs trans-acting, No sex-specific overlap.

No overlap between ER, GR QTLs,Hence, appears to bedifferent pathways forgeneticvs environmentalrobustness

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Evidence for geneticvariation in ER

• Inbred lines differ in Var(E)

• Sire differences (in cattle) for variation intheir offspring– Could also simply be due to major genes

segregating in that sire

• Specific QTL mapping– Early QTL experiments showed marker effects

on variance

– Direct mapping of such vQTLs in more recentstudies

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Selection on the environmentalvariance

• Genotypes can differ in their environmentalvariances– Example: Different inbred lines show different

variances

– Different QTL genotypes can show different traitvariances

• Can get response to selection on the environmentalvariance– Response for increased cannalization (greater left-right

symmetry in traits like bristle number in flies)

• Hence, selection on a trait can also potentiallychange its environmental variance

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Modeling breeding values ofVar(E)

• Assuming that the environmental variationis like any other complex trait, it has abreeding value (passed from parent ofoffspring) and residual values that are not

• Several different ways to model thegenetic structure of the environmentalvariance

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Models of heritable environmentalvariation

• Base model is P = G + E, where E now variesover genotypes and hence we can speak ofa breeding value for E

• Gavrilets-Hastings multiplicative model– E = !i*e, where e ~ (0, "e

2)

– A genotype-specific effect ! (sensitivity) +noise (e)

– Hence, Var(E | G, !) = !2 * "e2

– Simplest approach ! = Av, the BV for variance• Hence, Var(E | G, Av) = Av

2* "e2

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Multiplicative Model (cont)

• Total environmental variance (forpopulation)– "E

2 = (µ!2 + "!2)*"e

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• Selecting µ! to zero minimizes popenvironmental variance

• When ! = Av + Dv, (dominance inenvironmental sensitivity), then– "E

2 = (µAv2 + "Av

2)*"e2 + "Dv

2*"e2

– selection can decrease mean of Av and also itsvariance (by generating negative LD), but can’timpact the dominance term

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The exponential model

• E = exp(Av/2)*e– Where e ~ N(0, "e

2), Av ~ N(µAv, "Av2),

• Also called the log-additive model, as– Ln ("2(E | Av) = ln("e

2) + Av

• Environmental variation for the population– "E

2 = "e2 exp( µAv + "2

Av/2)

– Under this model, decreasing the mean (tonegative values) continues to decrease thevariance

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The additive model

• While the multiplicative and exponentialmodels ensure a non-zero variance, theycan be hard to work with

• The additive model is easier to use butdoes not ensure a positive environmentalvariance

– Here, "E2 = "e

2 + µAv

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Under the additive model, we can define a heritability in Var(E) as

Under normalityassumptions,

Average heritability is small (0.038), but potential forresponse is significant, its evolability CV(A) = 0.41. Thus,while the actual response can be small, the percentageincrease in the trait is not

here, CV(A)2 = 0.16 is the expected change in the traitmean given a standard unit of selection

Breeding values for trait mean, trait Var(E)tends to be negatively correlated (r ~ -0.24)

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Selection response

• Assume the additive model for Var(E) andconsider truncation selection with theupper p saved. Hill & Zhang (2004) showedthat the probability P(a,b) that a genotypewith mean value µ + a and variance effect"2 + b is selected is

• P(a,b)/p ~ 1 + a (i/"z) + (b/2) i x[1-p] / "z2

• Where i is the selection intensity and x[1-p]

is defined by P(unit normal > x[1-p] ) = p.

• Hence, direction selection favors higherVar(E)

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Selection response (additive model)

Hence strong directional selection (by selecting outliers)can result in a significant increase in Var(E), reducing h2.

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Artificial selection

• An increase in the environmental varianceis often seen following several generationsof directional selection

• As the models show, this is expected whenthere is heritable variation in Var(E) and isexpected to be more dramatic thestronger the directional selection

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Stabilizing selection

• Selection to minimize deviations fromsome optimal value can occur throughthree pathways

– Change the mean to the optimal value

– Reduce the additive genetic variation(generation of negative LD)

– Reduce the environmental variance

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Stabilizing selection (cont)Assume Multiplicative model

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Direct selection on Var(E)using repeated records

• While direct and stabilizing selection applyindirect selection on Var(E), direct selection canbe applied for traits with repeated records

• Construct an index I for each individual whichamounts to their sample variance in the records,then selection on I directly

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Assuming the exponential model for the variance