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Selection is defined as any act that restricts the random
mating of individuals in a population, sometimes just
called non-random mating.
Consider a locus with two alleles with frequencies, *p* and *q*.
The population is in Hardy-Weinberg equilibrium.
Now suppose that a fraction, *u*, of the
*A*_{1}*A*_{1} genotypes,
a fraction, *v*, of the
*A*_{1}*A*_{2} genotypes, and a fraction, *s*,
of the
*A*_{2}*A*_{2} genotypes have been culled from this population.
The resulting parent population can be summarized as follows:

Genotype | Initial | Fitness | After |

Frequency | Culling | ||

A_{1}A_{1} |
p^{2} |
1-u |
p^{2}(1-u) |

A_{1}A_{2} |
2pq |
1-v |
2pq(1-v) |

A_{2}A_{2} |
q^{2} |
1-s |
q^{2}(1-s) |

1 |
1-p^{2}u-2pqv-q^{2}s |

Assuming that culling has been applied equally to both sexes, then
the new frequency of the *A*_{1} allele in the parent generation is

The remaining animals, to be used for mating, are not in Hardy-Weinberg equilibrium. The remaining animals are mated randomly to produce the progeny generation. The progeny generation will be in Hardy-Weinberg equilibrium with

Culling could have been applied differentially to males and females, so that the allele frequencies in the remaining males and females could be different. Their progeny would not be in Hardy-Weinberg equilibrium.

The genotypic variance in the initial population is given by

The genotypic variance after culling will be different, and the genotypic variance in the progeny generation after random mating of the selected parents will be

If

**1. Assortative Mating**

Consider an initial population in Hardy-Weinberg equilibrium, as before.
Below is a mating diagram in which the frequency of the *A*_{1}allele can be different in the male and female population.
The frequencies in the table are the expected frequencies of
each type of mating - in a random mating population.

Female | Male Genotypes | |||

Genotypes | Frequencies | Frequencies | ||

A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |
||

p_{m}^{2} |
2
p_{m}q_{m} |
q_{m}^{2} |
||

A_{1}A_{1} |
p_{f}^{2} |
p_{f}^{2}p_{m}^{2} |
p_{f}^{2}2p_{m}q_{m} |
p_{f}^{2}q_{m}^{2} |

A_{1}A_{2} |
2p_{f}q_{f} |
2p_{f}q_{f}p_{m}^{2} |
4p_{f}q_{f}p_{m}q_{m} |
2p_{f}q_{f}q_{m}^{2} |

A_{2}A_{2} |
q_{f}^{2} |
q_{f}^{2}p_{m}^{2} |
q_{f}^{2}2p_{m}q_{m} |
q_{f}^{2}q_{m}^{2} |

Non-random mating means that the frequencies in the above
table are altered from their expected values. An example
is **assortative mating**, which would give a mating
diagram as follows:

Female | Male Genotypes | |||

Genotypes | Frequencies | Frequencies | ||

A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |
||

p_{m}^{2} |
2
p_{m}q_{m} |
q_{m}^{2} |
||

A_{1}A_{1} |
p_{f}^{2} |
p_{f}^{2}p_{m}^{2} |
0 | 0 |

A_{1}A_{2} |
2p_{f}q_{f} |
0 |
4p_{f}q_{f}p_{m}q_{m} |
0 |

A_{2}A_{2} |
q_{f}^{2} |
0 | 0 |
q_{f}^{2}q_{m}^{2} |

Note that the resulting frequencies do not sum to one.
Assuming that
*p*_{m}=*p*_{f} to simplify the discussion, then
the sum of the non-zero elements is

The progeny genotypic frequencies would be as follows:

Parent | Progeny Genotypic | |||

Genotypes | Frequencies | |||

Males | Females |
A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

A_{1}A_{1} |
A_{1}A_{1} |
p^{4}/S |
0 | 0 |

A_{1}A_{2} |
A_{1}A_{2} |
p^{2}q^{2}/S |
2p^{2}q^{2}/S |
p^{2}q^{2}/S |

A_{2}A_{2} |
A_{2}A_{2} |
0 | 0 | q^{4}/S |

The frequency of the *A*_{1} allele in the progeny is

The progeny genotypes are not in Hardy-Weinberg equilibrium. However, if the progeny generation is mated randomly, then the next generation will be in Hardy-Weinberg equilibrium again.

To illustrate the above, let *p*=.4 and let the genotypic values
be *a*=2 and *d*=1. The genotypic mean and variance in the
original random mating population would be

and the genetic variance would be

The mating diagram for assortative mating would be

Female | Male Genotypes | |||

Genotypes | Frequencies | Frequencies | ||

A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |
||

.16 | .48 | .36 | ||

A_{1}A_{1} |
.16 | .0256 | 0 | 0 |

A_{1}A_{2} |
.48 | 0 | .2304 | 0 |

A_{2}A_{2} |
.36 | 0 | 0 | .1296 |

Note that *S*=.3856.
The progeny figures would be

Parent | Progeny Genotypic | ||||

Genotypes | Frequencies | ||||

Males | Females | Freq. |
A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

A_{1}A_{1} |
A_{1}A_{1} |
.0664 | .0664 | 0 | 0 |

A_{1}A_{2} |
A_{1}A_{2} |
.5976 | .1494 | .2988 | .1494 |

A_{2}A_{2} |
A_{2}A_{2} |
.3360 | 0 | 0 | .3360 |

1.000 | .2158 | .2988 | .4854 |

The new genetic mean and variance in the progeny generation is
then

and

Thus, the mean has decreased and the variance has increased. The new frequency of the

**2. Disassortative Mating**

*Disassortative* mating is the mating of unlike genotypes.
The mating diagram would look like

Female | Male Genotypes | |||

Genotypes | Frequencies | Frequencies | ||

A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |
||

p_{m}^{2} |
2
p_{m}q_{m} |
q_{m}^{2} |
||

A_{1}A_{1} |
p_{f}^{2} |
0 | 0 |
p_{f}^{2}q_{m}^{2} |

A_{1}A_{2} |
2p_{f}q_{f} |
0 |
4p_{f}q_{f}p_{m}q_{m} |
0 |

A_{2}A_{2} |
q_{f}^{2} |
q_{f}^{2}p_{m}^{2} |
0 | 0 |

Note that the mating of
*A*_{1}*A*_{2} genotypes appears in both
assortative and disassortative mating. Another possibility
in both cases would be to remove these matings as well.
The sum of the frequencies of the matings that were made is

assuming the male and female allele frequencies were equal. The progeny genotypic frequencies would be

Parent | Progeny Genotypic | ||||

Genotypes | Frequencies | ||||

Males | Females | Freq. |
A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

A_{1}A_{1} |
A_{2}A_{2} |
p^{2}q^{2}/S |
0 | 1/6 | 0 |

A_{1}A_{2} |
A_{1}A_{2} |
4p^{2}q^{2}/S |
1/6 | 1/3 | 1/6 |

A_{2}A_{2} |
A_{1}A_{1} |
p^{2}q^{2}/S |
0 | 1/6 | 0 |

The new frequency of *A*_{1} is 0.5, regardless of the starting
frequency, and the frequency will remain at 0.5 in future generations
whether doing disassortative matings of the same type or random matings.
The frequency of homozygous genotypes will diminish in future
generations of disassortative matings.

**3. Reality**

Both types of selection most likely occur simultaneously in the real world, plus other types of selection. Some animals are culled, perhaps for reasons other than their genotype or phenotype, but the reasons may be associated with the genotypes resulting in a change in the parental allele frequencies. After culling, matings may not be random, which will alter the progeny allele frequencies. If these processes occur each generation, then it is unlikely that Hardy-Weinberg equilibrium is ever achieved. Also, genotypic variation could be altered by many different factors. If selection is on phenotypes (because genotypes are masked), then the effects of selection may be reduced if the residual variation is large relative to genotypic variation. However, the effects of selection will still influence the genotypic variance.

Suppose the bottom 25% are culled, and that this affects males
and females equally. Let *p*=0.4 as before, then the new
frequencies of parental genotypes would be

Genotype | Initial | After | Re-scaled |

Frequency | Culling | ||

A_{1}A_{1} |
.16 | .16 | .21333 |

A_{1}A_{2} |
.48 | .48 | .64000 |

A_{2}A_{2} |
.36 | .11 | .14667 |

1 | .75 | 1.00000 |

The new progeny distribution after random mating of the selected parents would be

Parent | Progeny Genotypic | ||||

Genotypes | Frequencies | ||||

Males | Females | Freq. |
A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

A_{1}A_{1} |
A_{1}A_{1} |
.045511 | .045511 | 0 | 0 |

A_{1}A_{1} |
A_{1}A_{2} |
.136533 | .068267 | .068267 | 0 |

A_{1}A_{1} |
A_{2}A_{2} |
.031289 | 0 | .031289 | 0 |

A_{1}A_{2} |
A_{1}A_{1} |
.136533 | .068267 | .068267 | 0 |

A_{1}A_{2} |
A_{1}A_{2} |
.409600 | .102400 | .204800 | .102400 |

A_{1}A_{2} |
A_{2}A_{2} |
.093867 | 0 | .046933 | .046933 |

A_{2}A_{2} |
A_{1}A_{1} |
.031289 | 0 | .031289 | 0 |

A_{2}A_{2} |
A_{1}A_{2} |
.093867 | 0 | .046933 | .046933 |

A_{2}A_{2} |
A_{2}A_{2} |
.021511 | 0 | 0 | .021511 |

1.000000 | .284444 | .497778 | .217778 |

If another 25% were culled from the bottom, then the new frequencies of the parent genotypes would be

Genotype | Initial | After | Re-scaled |

Frequency | Culling | ||

A_{1}A_{1} |
.284444 | .284444 | .379259 |

A_{1}A_{2} |
.497778 | .465556 | .620741 |

A_{2}A_{2} |
.217778 | .000000 | .000000 |

1 | .75 | 1.00000 |

All of the
*A*_{2}*A*_{2} genotypes would be culled and part of
the heterozygotes. Eventually fixation to the *A*_{1} allele
would occur with continued selection.

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