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 non-genetic determinants of a continuous phenotype.

To this end, we write the phenotypic value of an individual z as a sum:

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.

EGz

Partitioning Genotypic Value

We showed previously that the genotypic value could be partitioned into two components:

where is the expected genotypic values predicted by regressing genotypic value on number of alleles.

and is the deviation from linearity (additivity) resulting from dominance.

GG ˆG

Partitioning Genetic Variance

The partition of genotypic value leads to a partition of genetic variance:

Where is the amount of the variance in genotypic value that is explained by regression on allele counts.

And is the residual variance of the regression, the variance associated with dominance effects.

222DAG

2A

2D

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:

The actual number of possible interactions gets overwhelming very fast.

ijklkllkijjiGijklG

Epistasis – Interactions Possible

additive1 additive2 ()

additive1 dominance2 or additive2 dominance1 ()

dominance dominance () As the number of loci you consider

increases, the number of possible interactions skyrockets.

Epistasis - Example

Wild maizeAW BW

Domestic maizeAM BM

Genetically engineeredwild maize 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:

Gii 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:

jiGijij G ..

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:

kiGkiik G ..

Model – Additive x Dominance

Dominance by additive effects measure the interaction between an allele at one locus with a genotype at another locus:

ilikkl

lkiGkliikl G

.

Model – Dominance x Dominance

Dominance by dominance interactions are modeled by subtracting everything else from the genotypic value:

ijlijkjklikl

jljkilik

klijlkjiGijklijkl G

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

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).

ijklijlijkjklikl

jljkilik

klijlkjiGijklG

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).

Under the aforementioned conditions, the total variance is just the sum of the variances of each term in the model.

Define:

Model – Total Variance

ijklDD

ijlijkjkliklAD

jljkilikAA

klijD

lkjiA

22

22222

22222

222

22222

Model – Total Variance

Then, the total variance is given by:

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.`

222222DDADAADAG

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.

AMAM AMAW AWAWBMBM 1/16 1/8 1/16

BMBW 1/8 1/ 4 1/8

BWBW 1/16 1/8 1/16

Example – Mean Genotypic Value

The mean genotypic value in this population is:

8875.56

6.474

16.835.666.549.40

8

17.1018.471.6118

16

1

P,,,

lkji

ijkllkjiG GBBAA

Example – Additive Effects

We seek the conditional genotype means for each allele.

lkj

jklBMB MMGBiijklG

,,

P

4375.64

3375.49

6250.66

1500.47

5.664

1

2

16.47

2

1

2

16.54

4

1

2

11.61

4

1

2

19.40

2

1

2

10.18

4

1

2

1

W

M

W

M

A

A

B

B

G

G

G

G

Example – Additive Effects

Gii G

5500.7

5500.7

7375.9

7375.9

W

M

W

M

A

A

B

B

Convenient check:weight by allele frequencies and sumto 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.

225.401.614

19.40

2

10.18

4

1

,P,

lk

klBBMBB MMMMGBjiijklG

Example – Dominance Effects

jiGijij G ..

8125.2

8125.2

8125.2

WW

WM

MM

BB

BB

BB

Convenient check:weight by HWE genotype frequenciesand 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.

6428.303

2222

7375.92

1

7375.92

1

22222

22

22

WAMAWAMAA

WA

MA

AABB

B

B

Example – Variance Components

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

17%

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.