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Lecture 5 Artificial Selection. R = h 2 S. Applications of Artificial Selection. Applications in agriculture and forestry Creation of model systems of human diseases and disorders Construction of genetically divergent lines for QTL mapping and gene expression (microarray) analysis

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applications of artificial selection
Applications of Artificial Selection
  • Applications in agriculture and forestry
  • Creation of model systems of human diseases and disorders
  • Construction of genetically divergent lines for QTL mapping and gene expression (microarray) analysis
  • Inferences about numbers of loci, effects and frequencies
  • Evolutionary inferences: correlated characters, effects on fitness, long-term response, effect of mutations
response to selection
Response to Selection
  • Selection can change the distribution of phenotypes, and we typically measure this by changes in mean
    • This is a within-generation change
  • Selection can also change the distribution of breeding values
    • This is the response to selection, the change in the trait in the next generation (the between-generation change)
the selection differential and the response to selection
The Selection Differential and the Response to Selection
  • The selection differential S measures the within-generation change in the mean
    • S = m* - m
  • The response R is the between-generation change in the mean
    • R(t) = m(t+1) - m(t)

Truncation selection

uppermost fraction

p chosen





the breeders equation translating s into r



Likewise, averaging over the regression gives

E[ yo - m ] = h2 ( m* - m ) = h2 S

R = h2 S

The Breeders’ Equation

The Breeders’ Equation: Translating S into R

Recall the regression of offspring value on midparent value

Averaging over the selected midparents,

E[ (Pf + Pm)/2 ] = m*,

Since E[ yo - m ] is the change in the offspring mean, it

represents the response to selection, giving:

Note that no matter how strong S, if h2 is small, the response is small
  • S is a measure of selection, R the actual response. One can get lots of selection but no response
  • If offspring are asexual clones of their parents, the breeders’ equation becomes
    • R = H2 S
  • If males and females subjected to differing amounts of selection,
    • S = (Sf + Sm)/2
    • An Example: Selection on seed number in plants -- pollination (males) is random, so that S = Sf/2
response over multiple generations
Response over multiple generations
  • Strictly speaking, the breeders’ equation only holds for predicting a single generation of response from an unselected base population
  • Practically speaking, the breeders’ equation is usually pretty good for 5-10 generations
  • The validity for an initial h2 predicting response over several generations depends on:
    • The reliability of the initial h2 estimate
    • Absence of environmental change between generations
    • The absence of genetic change between the generation in which h2 was estimated and the generation in which selection is applied

50% selected

Vp = 4, S = 1.6

20% selected

Vp = 4, S = 2.8

20% selected

Vp = 1, S = 1.4

The selection differential is a function of both

the phenotypic variance and the fraction selected

the selection intensity i



The Selection Intensity, i

As the previous example shows, populations with the

same selection differential (S) may experience very

different amounts of selection

The selection intensity i provided a suitable measure

for comparisons between populations,

One important use of i is that for a normally-distributed

trait under truncation selection, the fraction saved p

determines i,

the realized heritability

Cumulative selection response = sum of all responses

The Realized Heritability

Since R = h2 S, this suggests h2 = R/S, so that

the ratio of the observed response over the

observed differential provides an estimate of

the heritability, the realized heritability

Obvious definition for a single generation of response.

What about for multiple generations of response?


Cumulative selection differential = sum of the S’s

(1) The Ratio Estimator for realized heritability

= total response/total differential,

Regression passes through the

origin (R=0 when S=0). Slope =

(2) The Regression Estimator --- the slope of the

Regression of cumulative response on cumulative differential


Ratio estimator

= 17.4/56.9 = 0.292


Slope = 0.270

= Regression



Cumulative Response


Note x axis is differential,

NOT generations






Cumulative Differential

gene frequency changes under selection


In finite population, genetic drift can overpower

selection. In particular, when

drift overpowers the effects of selection

Gene frequency changes under selection

Additive fitnesses

Let q = freq(A2). The change in q from one

generation of selection is:

strength of selection on a qtl
Strength of selection on a QTL

Have to translate from the effects on a trait under

selection to fitnesses on an underlying locus (or QTL)

Suppose the contributions to the trait are additive:

For a trait under selection (with intensity i) and

phenotypic variance sP2, the induced fitnesses

are additive with s = i (a /sP )

Thus, drift overpowers

selection on the QTL when

more generally

Change in allele frequency:


Selection coefficients for a QTL

s = i (a /sP )

h = k

More generally
changes in the variance under selection
Changes in the Variance under Selection

The infinitesimal model --- each locus has a very small

effect on the trait.

Under the infinitesimal, require many generations

for significant change in allele frequencies

However, can have significant change in genetic

variances due to selection creating linkage disequilibrium

Under linkage equilibrium,

freq(AB gamete) = freq(A)freq(B)

With negativve linkage disequilibrium, f(AB) < f(A)f(B),

so that AB gametes are less frequent

With positive linkage disequilibrium, f(AB) > f(A)f(B),

so that AB gametes are more frequent



Additive genetic variance in the

unselected base population. Often

called the additive genic variance

Additive genetic variance after

one generation of selection


Changes in VA changes the phenotypic variance

The amount diseqilibrium generated by a single generation

of selection is

Within-generation change

in the variance

Changes in VA with disequilibrium

Under the infinitesimal model, disequilibrium only

changes the additive variance.

Starting from an unselected base population, a single

generation of selection generates a disequilibrium

contribution d to the additive variance

Changes in VA and VP change the heritability

An increase in the variance

generates d > 0 and hence

positive disequilibrium

A decrease in the variance

generates d < 0 and hence

negative disequilibrium



change in the variance

-- akin to S

New disequilibrium generated by

selection that is passed onto the

sext generation

d(t+1) - d(t) measures the response in selection

on the variance (akin to R measuring the mean)

Decay in previous disequilibrium from recombination

Many forms of selection (e.g., truncation) satisfy

A “Breeders’ Equation” for Changes in Variance

d(0) = 0 (starting with an unselected base population)

k < 0. Within-generation

increase in variance.

positive disequilibrium, d > 0

k > 0. Within-generation

reduction in variance.

negative disequilibrium, d < 0