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Lecture 2: Basic Population and Quantitative Genetics. Allele and Genotype Frequencies. Given genotype frequencies, we can always compute allele frequencies, e.g.,. The converse is not true: given allele frequencies we cannot uniquely determine the genotype frequencies.

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allele and genotype frequencies
Allele and Genotype Frequencies

Given genotype frequencies, we can always compute allele

frequencies, e.g.,

The converse is not true: given allele frequencies we

cannot uniquely determine the genotype frequencies

For n alleles, there are n(n+1)/2 genotypes

If we are willing to assume random mating,



hardy weinberg

• Prediction of genotype frequencies from allele freqs

• Allele frequencies remain unchanged over generations,


• Infinite population size (no genetic drift)

• No mutation

• No selection

• No migration

• Under HW conditions, a single generation of random

mating gives genotype frequencies in Hardy-Weinberg

proportions, and they remain forever in these proportions

gametes and gamete frequencies
Gametes and Gamete Frequencies

When we consider two (or more) loci, we follow gametes

Under random mating, gametes combine at random, e.g.

Major complication: Even under HW conditions, gamete

frequencies can change over time










In the F1, 50% AB gametes

50 % ab gametes

If A and B are unlinked, the F2 gamete frequencies are

AB 25%

ab 25%

Ab 25%

aB 25%

Thus, even under HW conditions, gamete frequencies change

linkage disequilibrium
Linkage disequilibrium

Random mating and recombination eventually changes

gamete frequencies so that they are in linkage equilibrium (LE).

once in LE, gamete frequencies do not change (unless acted

on by other forces)

At LE, alleles in gametes are independent of each other:

When linkage disequilibrium (LD) present, alleles are no

longer independent --- knowing that one allele is in the

gamete provides information on alleles at other loci

The disequilibrium between alleles A and B is given by


Departure from


LE value

Initial LD value

The Decay of Linkage Disequilibrium

The frequency of the AB gamete is given by

If recombination frequency between the A and B loci

is c, the disequilibrium in generation t is

Note that D(t) -> zero, although the approach can be

slow when c is very small

contribution of a locus to a trait

Genotypic value

Phenotypic value -- we will occasionally

also use z for this value

Environmental value

Contribution of a locus to a trait

Basic model: P = G + E

G = average phenotypic value for that genotype

if we are able to replicate it over the universe

of environmental values

G - E covariance -- higher performing animals

may be disproportionately rewarded

G x E interaction --- G values are different

across environments. Basic model now

becomes P = G + E + GE

alternative parameterizations of genotypic values



C + a + d

C + a(1+k)

C + 2a

C + 2a

C -a

C + d

C + a

2a = G(Q2Q2) - G(Q1Q1)

Alternative parameterizations of Genotypic values




d measures dominance, with d = 0 if the heterozygote

is exactly intermediate to the two homozygotes

d = ak =G(Q1Q2 ) - [G(Q2Q2) + G(Q1Q1) ]/2

k = d/a is a scaled measure of the dominance

example booroola b gene
Example: Booroola (B) gene

2a = G(BB) - G(bb) = 2.66 -1.46 --> a = 0.59

ak =d = G(Bb) - [ G(BB)+G(bb)]/2 = 0.10

k = d/a = 0.17

fisher s decomposition of g

Dominance deviations --- the difference (for genotype

AiAj) between the genotypic value predicted from the

two single alleles and the actual genotypic value,

Average contribution to genotypic value for allele i

Mean value, with

The genotypic value predicted from the individual

allelic effects is thus

Fisher’s Decomposition of G

One of Fisher’s key insights was that the genotypic value

consists of a fraction that can be passed from parent to

offspring and a fraction that cannot.

Since parents pass along single alleles to their

offspring, the ai (the average effect of allele i)

represent these contributions


Residual error

Predicted value


Independent (predictor) variable N = # of Q2 alleles

Regression slope

Regression residual

Fisher’s decomposition is a Regression

A notational change clearly shows this is a regression,


Allele Q1 common, a2 > a1

Allele Q2 common, a1 > a2

Both Q1 and Q2 frequent, a1 = a2 = 0

Slope = a2 - a1











Allelic effects

Dominance deviations

Consider a diallelic locus, where p1 = freq(Q1)




Average effects and Breeding Values

The a values are the average effects of an allele

Breeders focus on breeding value (BV)

Why all the fuss over the BV?

Consider the offspring of a QxQy sire mated

to a random dam. What is the expected value

of the offspring?


For random w and z alleles, this has an

expected value of zero

For a random dam, these have expected value 0


The expected value of an offspring is the expected value of


We can thus estimate the BV for a sire by twice

the deviation of his offspring from the pop mean,

More generally, the expected value of an offspring

is the average breeding value of its parents,

genetic variances

As Cov(a,d) = 0

Dominance Genetic Variance

(or simply dominance variance)

Additive Genetic Variance

(or simply Additive Variance)

Genetic Variances

Q1Q1 Q1Q2 Q2Q2

0 a(1+k) 2a

When dominance present,

asymmetric function of allele


Dominance effects

additive variance

Equals zero if k = 0

This is a symmetric function of

allele frequencies

Since E[a] = 0,

Var(a) = E[(a -ma)2] = E[a2]

One locus, 2 alleles:

One locus, 2 alleles:



Allele frequency, p

Additive variance, VA, with no dominance (k = 0)


Complete dominance (k = 1)



Allele frequency, p


Zero additive


Overdominance (k = 2)



Allele frequency, p

Allele frequency, p


Additive x Additive interactions --

interactions between a single allele

at one locus with a single allele at another

Dominance x dominance interaction ---

the interaction between the dominance

deviation at one locus with the dominance

deviation at another.

Additive x Dominant interactions --

interactions between an allele at one

locus with the genotype at another, e.g.

allele Ai and genotype Bkj

Dominance value -- interaction

between the two alleles at a locus

Breeding value


These components are defined to be uncorrelated,

(or orthogonal), so that