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1. Parent-Offspring
Progeny will inherit one-half the additive breeding value of the parent. Below is a table that summarizes the quantities that will be needed to calculate the covariance between parent and offspring.
Parent | Freq. | Parent | Offspring | |
Genotype | BV | BV | ||
A_{1}A_{1} | p^{2} | |||
A_{1}A_{2} | 2pq | |||
A_{2}A_{2} | q^{2} |
2. Half-Sibs
From the table above,
the covariance between half-sibs can be computed.
The
offspring column represents half-sibs, so that the variance of
offspring means gives the covariance between half-sibs.
3. Midparent - Offspring
The midparent value involves both parents, and the value of the offspring depends on the genotypes of the parents. So the offspring of each possible pair of genotype matings has to be considered.
Possible Parental | Frequency | Midparent | Offspring | |
Matings | Genotypic Value | Mean Value | ||
A_{1}A_{1} | A_{1}A_{1} | p^{4} | a | a |
A_{1}A_{1} | A_{1}A_{2} | 4p^{3}q | 0.5(a+d) | 0.5(a+d) |
A_{1}A_{1} | A_{2}A_{2} | 2p^{2}q^{2} | 0 | d |
A_{1}A_{2} | A_{1}A_{2} | 4p^{2}q^{2} | d | 0.5d |
A_{1}A_{2} | A_{2}A_{2} | 4pq^{3} | 0.5(d-a) | 0.5(d-a) |
A_{2}A_{2} | A_{2}A_{2} | q^{4} | -a | -a |
4. Full-Sibs
The table for Midparent-Offspring covariance can be
used to obtain the covariance between full-sibs. The offspring
within a mating pair are full-sibs, and the variance of offspring
means gives the covariance between full-sibs.
5. Phenotypic Values
The phenotype, what is visible, is composed of a mean plus the
genotypic value plus a random residual term, i.e.,
The phenotypic mean of the population will be
Suppose you have two individuals of genotype A_{1}A_{2} with the male having P = 3 + 0.5 + .621 = 4.121, and the female having P = 3 + 0.5 - .894 = 2.606. To generate a progeny record from mating these two individuals,
What would be the expected average phenotype of a large number
of progeny from this mating? This mating can produce all
three genotypes in the ratio 1:2:1. Those with genotype
A_{1}A_{1} would have a mean of 4, those with
A_{1}A_{2}would have a mean of 3.5, and those with
A_{2}A_{2} would
have a mean of 2. Weighting these by their expected
frequencies, then
6. Small Populations and Drift
In large populations gene frequencies remain constant from generation to generation under random mating, no selection, no mutations, and no migration. In small populations there are random changes in gene frequencies due to the sampling of gametes during matings. This change in gene frequency is known as random drift. Hardy-Weinberg equilibrium is lost in a small population.
Within the small population, due to random drift, there is
Assume a large population which is split into subpopulations (lines) of N individuals each. Within each line, assume individuals are randomly mated (including selfing), in discrete generations, and that population size remains constant. The large population is assumed to be in Hardy-Weinberg equilibrium. The allele frequencies in each line are initially assumed to be equal to the population frequencies, p_{0} and q_{0}.
If there were a large number of lines, then the average frequency
of the A_{2} allele over all lines at generation t is
q_{0} | t=1 | t=5 | t=10 | t=20 | |
.1 | .00045 | .00223 | .00440 | .00858 | .09 |
.2 | .00080 | .00396 | .00782 | .01526 | .16 |
.3 | .00105 | .00520 | .01027 | .02003 | .21 |
.4 | .00120 | .00594 | .01173 | .02283 | .24 |
.5 | .00125 | .00619 | .01222 | .02385 | .25 |
By definition,
Genotype | Frequency | |
A_{1}A_{1} | ||
A_{1}A_{2} | ||
A_{2}A_{2} |
7. Inbreeding
Inbreeding and drift are related phenomena in small populations. Inbreeding results from the mating of related individuals. In a small population the degree of relationship among individuals increases with generation number and depends on the size of the population. Let F represent the coefficient of inbreeding, which is defined as the probability that two genes at a locus are identical by descent.
Consider a population of size N where every parent contributes
equally to the next generation. Within the N parents there are
2N possible gametes. If two gametes are chosen at random from
the population of 2N gametes, then the probability that the two
gametes are identical and from the same parent is
.
The probability that the two gametes are not from the same
parent is therefore,
.
Even though the two gametes are not from the same parent, it is
possible that alleles are equal, for example, both could be
A_{1}. The probability that two alleles at a locus are
identical by descent is 1 in the first case which occurs with
probability
.
The probability that two alleles at
a locus are identical by descent in the second case is F_{t-1} in
the parent generation, which occurs with probability
,
so that the coefficient of inbreeding in the
progeny generation is
In the previous section, drift variance was equal
to
Genotype | Frequency | |
A_{1}A_{1} | p_{0}^{2} + p_{0}q_{0}F_{t} | |
A_{1}A_{2} | 2p_{0}q_{0} (1 - F_{t}) | |
A_{2}A_{2} | q_{0}^{2} + p_{0}q_{0} F_{t} |
8. Genetic Variance
Assume an additive model for a single locus, so that d=0, then
the genetic variance in the base population is
The genetic variance between lines is calculated by
computing the variance of means of each line. The mean of
line i is
The total genetic variance is the sum of the within line genetic variance and the between lines genetic variance.
Source | |
Within Lines | |
Between Lines | |
Total |
With a dominance model, ,
then Weir and Cockerham (1977)
have shown that the total genetic variance, between and within, is
This LaTeX document is available as postscript or asAdobe PDF.
Larry Schaeffer