Transcript 2002-11-14: Quantitative Traits IV

Lecture 24: Quantitative Traits IV
Date: 11/14/02
 Sources of genetic variation
additive
 dominance
 epistatic

Where We Are
 We have studied single marker locus QTL analysis
in controlled crosses (F2, BC).
 We have yet to study QTL analysis using interval
and multipoint mapping (more than one locus) and
QTL analysis for natural populations. We will
revisit these topics after Thanksgiving.
 QTL analysis is essentially the future of
quantitative genetics. It depends on the presence of
many markers and high density marker maps.
Where We Are
 QTL analysis not always possible:


Few markers available.
Complex trait which is not yet known to have a genetic
component.
 For these cases, we estimate and analyze variances,
trying to determine whether significant levels of
trait variation are actually genetic in nature.
 Knowing there is a genetic component and later
obtaining markers allows us to proceed with QTL
analysis.
Genotypic Value
 Therefore, one of the prime goals of quantitative
genetics is to distinguish the genetic and nongenetic determinants of a continuous phenotype.
 To this end, we write the phenotypic value of an
individual z as a sum:
z GE
 We defined the genotypic value G to be the sum
effect of all loci influencing a trait.
 We define the environmental effect through an
environmental deviation E.
Partitioning Genotypic Value
 We showed previously that the genotypic value
could be partitioned into two components:
G  Gˆ  d
 where Ĝ is the expected genotypic values predicted
by regressing genotypic value on number of alleles.
 and d is the deviation from linearity (additivity)
resulting from dominance.
Partitioning Genetic Variance
 The partition of genotypic value leads to a partition
of genetic variance:
   
2
G
2
A
2
D
 Where  A2 is the amount of the variance in
genotypic value that is explained by regression on
allele counts.
 And  D2 is the residual variance of the regression,
the variance associated with dominance effects.
Complications of Multilocus
Traits
 Can the genotypic values from multiple loci simply
be added to explain the observed phenotype? If
not, there are significant nonlinear interactions
exist.
 Is the inheritance and distribution of genes at one
contributing locus independent of another
contributing locus? How might this not be true?
 Does gene expression vary with environmental
context?
Resolution of Complications of
Multilocus Traits
 There really is no way to resolve all these
complicating determinates.
 The solution is to look at variances and
determine which sources of variation
contribute significantly to the overall
variance.
Epistasis
 Just as the dominance effect is a measure of
the nonadditivity of alleles at a single locus,
epistasis is a measure of the nonadditivity of
effects between loci.
 Metabolic networks, signal transduction,
basically any kind of protein-protein
interaction could potentially result in
epistasis.
Epistasis – A Model
 Just as we did for a single locus, we can
propose a model for epistasis. Consider only
two loci, the first with alleles indexed by i
and j and the second with alleles indexed by
k and l:
Gijkl   G   i   j  d ij    k   l  d kl    ijkl
 The actual number of possible interactions
gets overwhelming very fast.
Epistasis – Interactions Possible
 additive1  additive2 ()
 additive1  dominance2 or additive2 
dominance1 (d)
 dominance  dominance (dd)
 As the number of loci you consider increases,
the number of possible interactions
skyrockets.
Epistasis - Example
Domestic maize
Wild maize
Genetically
engineered
wild maize
AM
BM
AW
BW
AM
BM
Epistasis - Example
AMAM
AMAW
AWAW
a
k
BMBM
18.0
40.9
61.1
21.6
0.06
BMBW
54.6
47.6
66.5
6.0
-2.17
BWBW
47.8
83.6
101.7
27.0
0.33
a
14.9
21.4
20.3
k
1.46
-0.69
-0.73
Model – Additive Effects
 Assume random mating population.
 The additive effect of an allele was shown to
be the difference between the mean genotype
of individuals with that allele and the mean
genotypic value in the full population:
 i  Gi  G
Model – Dominance Effects
 The dominance effects at the first locus are
found by calculating the mean genotypic
value for all individuals with genotype ij at
the first locus:
d ij  Gij..   G   i   j
Model – Additive x Additive
 Similarly, consider the mean phenotype of
individuals with gene i at locus 1 and gene k
at locus 2, and call this Gi.k.. Then additive
by additive effects are modeled with:
 ik  Gi.k .  G   i   k
Model – Additive x Dominance
 Dominance by additive effects measure the
interaction between an allele at one locus
with a genotype at another locus:
d ikl  Gi.kl  G   i   k   l
 d kl   ik   il
Model – Dominance x
Dominance
 Dominance by dominance interactions are
modeled by subtracting everything else from
the genotypic value:
dd ijkl  Gijkl   G   i   j   k   l  d ij  d kl
  ik   il    jk    jl
 d ikl  d  jkl  d ijk  d ijl
Summary of Partitioning
 The additive effects of alleles are considered
first and are defined to explain as much of
the variation as possible (by least squares).
 Subsequent higher order effects are added,
each time explaining as much as the variance
as possible.
 We can recover the total genotypic values by
summing all the pieces back together again:
Final Genotypic Value Partition

 

            
 d   d   d   d   dd 
Gijkl   G   i   j   k   l  d ij  d kl
ikl
ik
il
jk
jl
jkl
ijk
ijl
ijkl
 There are many more terms to consider when there are more
than two loci considered.
 The mean value of each effect type is 0.
 Each effect depends on the population (allele frequencies,
etc).
Model – In Terms of Variances
 The preceding model is terribly complex
looking.
 Assume:


random mating
independent segregation of loci (no linkage, no
linkage disequilibrium)
 Then: genes found within and between loci
are independent (0 covariation).
Model – Total Variance
 Under the aforementioned conditions, the total
variance is just the sum of the variances of each
term in the model.
 Define:
 A2   2  i    2  j    2  k    2  l 
 D2   2 d ij    2 d kl 
2
 AA
  2  ik    2  il    2   jk   2   jl 
2
 AD
  2 d ikl    2 d  jkl    2 d ijk   2 d ijl 
2
 DD
  2 dd ijkl 
Model – Total Variance
 Then, the total variance is given by:
2
2
2
 G2   A2   D2   AA
  AD
  DD

 Epistatic interactions can inflate the additive or
dominance components of genetic variance.
 Even when epistatic components are relatively
small, there can still be strong epistatic effects
because of the population context.`
Example – Genotype
Frequencies
 Pretend that the previous data arose from a random mating
population and the entries are accurate estimates of the
genotypic values.
 We seek the probability of each genotype. Since it is
actually F2, the allele frequencies are all equal.
BMBM
BMBW
BWBW
AMAM
1/16
1/8
1/16
AMAW
1/8
1/4
1/8
AWAW
1/16
1/8
1/16
Example – Mean Genotypic
Value
 The mean genotypic value in this population
is:
   PA A B B G
G
i
j
k
l
ijkl
i , j , k ,l
1
18  61.1  47.8  101.7   1 40.9  54.6  66.5  83.6  1 47.6
16
8
4
 56.8875

Example – Additive Effects
 We seek the conditional genotype means for
each allele.
GBM  
 Pijkl i  B G
M
BM jkl
j , k ,l
11
11
11
11
11
11
18.0 
40.9 
61.1 
54.6 
47.6 
66.5
24
22
24
24
22
24
 47.1500
 66.6250
GBM  
GBW 
G AM   49.3375
G AW   64.4375
Example – Additive Effects
 i  Gi  G
 B  9.7375
M
 B  9.7375
W
 A  7.5500
M
 A  7.5500
W
Convenient check:
weight by allele
frequencies and sum
to get 0.
Example – Dominance Effects
 Need to calculate conditional genotypic
means. This calculation for a genotype at
locus 1 is done by averaging the three
genotypic values weighted by the HW
frequencies of the genotypes at the second
locus.
GBM BM    Pijkl i, j  BM GBM BM kl
k ,l
1
1
1
 18.0  40.9  61.1  40.225
4
2
4
Example – Dominance Effects
d ij  Gij..   G   i   j
dB
M BM
dB
M BW
dB
W BW
 2.8125
 2.8125
 2.8125
Convenient check:
weight by HWE
genotype frequencies
and sum to get 0
Example – Variance
Components
 Since means are always 0, it makes for
convenient calculation of variances: they are
just the mean squared effects weighted by the
frequency of that effect.
 A2 BM  
1
 9.73752
2
1
 A2 BW   9.73752
2
 A2  2 A2 BM   2 A2 BW   2 A2  AM   2 A2  AW 
 303.6428
Example – Variance
Components
17%
Variance Component
Value
Additive
303.6428
Dominance
11.7615
Additive by additive
1.8225
Additive by dominance
43.6191
Dominance by Dominance
20.3627
Genetic
381.2086
Linkage Disequilibrium
 If genes with a positive effect on the character tend
to be associated on some chromosomes, and vice
versa for those genes with negative effects, then the
genetic variation will be increased over expectation
when assuming linkage equilibrium.
 One could also get a decrease in genetic variance
below expectation.
 There are equations for genetic variance when
linkage disequilibrium between two loci is
considered that take a page to print.
A Biological Test for Linkage
Disequilibrium
 Expand and randomly mate a population
while minimizing selection.
 The variance after a few generations will
converge on the variance associated with
linkage equilibrium.
 The difference between start and end values
provide an estimate of the disequilibrium
covariance.