Transcript Hardy Weinberg Equiibrium with more than 2 alleles

Chapter 7 Quantitative Genetics
 Read Chapter 7 sections 7.1 and 7.2. [You should read
7.3 and 7.4 to deepen your understanding of the topic,
but I will not cover these topics in lecture].
Quantitative Genetics
 Traits such as flower color in peas produce distinct
phenotypes.
 Such discrete traits, which are determined by a single
gene, are relatively rare.
 Most traits are determined by the effects of multiple
genes (polygenic traits) and these show continuous
variation in trait values.
Complex traits vary continuously
Continuous variation
 For example, grain color in winter wheat is determined
by three genes at three loci each with two alleles.
Additive effects of genes
 Genes affecting color of winter wheat interact in a
straightforward way.
 They have additive genetic effects.
 Thus the phenotype is obtained by summing the effects of
individual alleles.
 The more alleles for being dark (prev. slide) or large (next
slide) an individual has, the darker/taller it will be. A
continuous distribution results.
Continuous variation
 Examples of humans traits that show continuous
variation:
 height,
 intelligence,
 athletic ability,
 skin color.
Quantitative traits
 For continuous traits we cannot assign individuals to
discrete categories (e.g. tall or short). Instead we must
measure them.
 Therefore, characters with continuously distributed
phenotypes are called quantitative traits.
 Quantitative genetics is the study of the genetic
basis of quantitative traits.
Value of quantitative traits
 Quantitative traits determined by influence of (1)
genes and (2) environment.
 The value of a quantitative trait such as height is
determined by the organism’s genes operating within
their environment.
 Both the genes it inherited from its parents and the
conditions under which it grows up affect an
individual’s height.
Value of quantitative traits
 For a given individual the value of its phenotype (P)
(e.g., a person’s height in cm) can be considered to
consist of two parts -- the part due to genotype (G) and
the part due to environment (E)
 P = G + E.
 G is the expected value of P for individuals with that
genotype. Any difference between P and G is
attributed to environmental effects.
Genetic and environmental influences
create continuous distribution
Measuring Heritable Variation
 Quantitative genetics takes a population view and
tracks variation in phenotype and whether this
variation has a genetic basis.
 Variation in a sample is measured using a statistic
called the variance.
Measuring Heritable Variation
 We want to distinguish between heritable and
nonheritable factors affecting the variation in
phenotype.
 It turns out that the variance of a sum of independent
variables is equal to the sum of their individual
variances.
 Because P = G +E
 Then Vp = Vg + Ve
 where Vp is phenotypic variance, Vg is variance due to
genotypic effects and Ve is variance due to
environmental effects.
Measuring Heritable Variation
 Heritability is defined as the fraction of total
phenotypic variation that is due to variation in genes
Measuring Heritable Variation
 Heritability = Vg/Vp
 Heritability = Vg/Vg+Ve
 This is broad-sense heritability (H2). It defines the
fraction of the total variance that is due to genetic
causes.
 (Heritability is always a value between 0 and 1.)
Measuring Heritable Variation
 The genetic component of inheritance (Vg) includes the
effect of all genes in the genotype.
 If all gene effects combined additively an individual’s
genotypic value G could be represented as the sum of
individual gene effects.
Measuring Heritable Variation
 However, interactions between alleles
(dominance effects) and interactions between
different genes (epistatic effects) can affect the
phenotype and these effects are non-additive.
Measuring Heritable Variation
 To account for dominance and epistasis we break down
the equation for P (value of the phenotype)
 P = G +E
 Comonent G (genetic effects) is the sum of three
subcomponents – A [additive component], D
[dominance component] and I [epistatic or
interaction component].
G=A+D+I
 So therefore P = A + D + I + E
Measuring Heritable Variation
 Similarly, assuming all components of the equation P =
A + D + I + E are independent of each other then the
variance of this sum is equal to sum of the individual
variances.
 Vp = Va + Vd + Vi + Ve
Measuring Heritable Variation
 Only Va is directly operated on by natural selection.
 The effects of Vd and Vi are strongly context dependent i.e.,
their effects depend on what other alleles and genes are
present (the genetic background).
Measuring Heritable Variation
 Va, however exerts the same effect regardless of the
genetic background. Therefore, its effects are always
visible to selection.
Measuring Heritable Variation
 We defined broad sense heritability (H2) as the
proportion of total variance due to any form of genetic
variation
 H2 = Vg/Vg+Ve
 We similarly define narrow sense heritability h2 as
the proportion of variance due to additive genetic
variance
 h2 = Va/(Va + Vd + Vi + Ve)
Measuring Heritable Variation
 Because narrow sense heritability is a measure of
what fraction of the variation is visible to
selection, it plays an important role in predicting how
phenotypes will change over time as a result of natural
selection.
 Narrow sense heritability reflects the degree to which
offspring resemble their parents in a population.
Estimating heritability from parents
and offspring
 Narrow sense heritability is the slope of a linear
regression between the average phenotype of the two
parents and the phenotype of the offspring.
 Can assess the relationship using scatterplots.
 Plot midparent value (average of the two parents)
against offspring value.
 If offspring don’t resemble parents then best fit line
has a slope of approximately zero.
 Slope of zero indicates most variation in individuals
due to variation in environments.
If offspring strongly
resemble
parents then slope of
best fit line
will be close to 1.
Most traits in most
populations fall
somewhere in the middle
with offspring showing
moderate resemblance to
parents.
 When estimating heritability it’s important to
remember parents and offspring share an
environment.
 We need to make sure there is no correlation between
the environments experienced by parents and their
offspring. This requires cross-fostering experiments to
randomize environmental effects.
Smith and Dhondt (1980)
 Smith and Dhondt (1980) studied heritability of beak
size in Song Sparrows.
 Moved eggs and young to nests of foster parents.
Compared chicks beak dimensions to parents and
foster parents.
Smith and Dhondt estimated heritability of bill depth as about 0.98.
Evolutionary response to selection
 Once we know the sources of variation in a
quantitative trait we can study how it evolves.
 If selection favors certain values of a trait then we
expect the population to evolve in response.
 The effect on the distribution of the trait will depend
on which phenotypes are being favored (see next
slide).
Directional selection for oil content in corn
Disruptive selection for
bristle number in Drosophila
Evolutionary response to selection
 To quantify the amount and direction of change in a
trait value from one generation to the next (i.e., how a
trait evolves) we need to quantify heritability and the
effect of selection.
 To assess the effect of selection we have to measure
differences in survival and reproductive success
among individuals.
Measuring differences in
survival and reproduction
 Need to be able to quantify difference between
winners and losers in whatever trait we are interested
in. This is strength of selection.
Measuring differences in
survival and reproduction
 If some members of a population breed and others
don’t and you compare the mean values of some trait
(say mass) for the breeders and the whole population,
the difference between them (and one measure of the
strength of selection) is the selection differential
(S).
 This term is derived from selective breeding trials.
Selection Differential
Response to Selection
 We want to be able to measure the effect of selection
on a population.
 This is called the Response to Selection and is
defined as the difference between the mean trait value
for the offspring generation and the mean trait value
for the parental generation i.e. the change in trait value
from one generation to the next.
Evolutionary response to selection
 Knowing heritability and selection differential we
can predict evolutionary response to selection (R).
 Given by the simple formula: R=h2S
 R is predicted response to selection, h2 is
heritability, S is selection differential.
 R is the proportional (or if you multiply by 100, the
percentage) change in a trait value.
Effect of difference in heritability (h2) on a population’s response to
selection (R) with same selection differential (S). Plots of parent offspring
regressions for two populations. Intersection of axes is midpoint of
parental (x-axis) and offspring (y-axis) trait values.
Alpine skypilots and bumble bees
 Alpine skypilot perennial wildflower found in the
Rocky Mountains.
 Populations at timberline and tundra differed in
size. Tundra flowers about 12% larger in diameter.
 Timberline flowers pollinated by many insects, but
tundra only by bees. Bees known to be more
attracted to larger flowers.
Alpine skypilots and bumble bees
 Candace Galen (1996) wanted to know if selection by
bumblebees was responsible for larger size flowers in
tundra and, if so, how long it would take flowers to
increase in size by 12%.
Alpine skypilots and bumble bees
 First, Galen estimated heritability of flower size.
Measured plants flowers, planted their seeds and
(seven years later!) measured flowers of offspring.
 Concluded 20-100% of variation in flower size was
heritable (h2).
Alpine skypilots and bumble bees
 Next, she estimated strength of selection by
bumblebees by allowing bumblebees to pollinate a
caged population of plants, collected seeds and
grew plants from seed.
 Correlated number of surviving young with flower
size of parent. Estimated the selection differential
(S) to be 5% (successfully pollinated plants 5%
larger than population average).
Alpine skypilots and bumble bees
 Using her data Galen predicted response to selection
R.
 R=h2S
 R=0.2*0.05 = 0.01 (low end estimate)
 R=1.0*0.05 = 0.05 (high end estimate)
Alpine skypilots and bumble bees
 Thus, expect 1-5% increase in flower size per
generation.
 Difference between populations in flower size
plausibly due to bumblebee selection pressure.