Until now we only looked at genetics within one individual. This area of genetics is usually called Mendel genetics. A different area — population genetics — is divided into two parts:

population genetics (in the strict sense of the word) : studies the 'behaviour' of genes within a group of individuals

quantitative genetics : studies inheritance of different characteristics that are not determined by one or a few genes.

## Population genetics (in the strict sense of the word)

In all calculations and discussions the population (group of animals) is assumed to be a random mating population (every animals has an equal chance of mating any other animal (of the opposite sex of course)). In the world of dogs this is not a realistic situation. On one side the breeder selects a certain combination and on the other side there are geographical limitations (the chances of me using a dog from Australia are considerably less (almost zero) than those of a dog from the Netherlands).

In reality the results are however very close to the predictions that assume a random mating population. Always check whether this approximation is sensible in each case!

### Hardy and Weinberg's Law

These two investigators independently described the relationship between the frequency of certain alleles (allele frequency) and the frequency of certain genotypes in the population. Two properties of the population are required: random mating occurs and the population is large.

If we consider a gene with alleles A and a, and the allele frequency of A is represented by \({p}\) and the frequency of a is represented by \({q}\), then:

$$AA : Aa : aa = p^2 : 2pq : q^2$$

The importance of this formula can be illustrated by a hereditary disorder. When you can only detect homozygous recessive animals (sufferers), this formula helps to calculate the number of carriers. Let's say 1% of the population suffers from such a simple recessive, autosomal disorder. aa equals 0.01 and therefor q^{2} also equals 0.01 . This results in q being the square of that, \({q=0.1}\) . If 10% of the genes in the population is a, then 90% must be A; so \({p=0.9}\) . The ratio \({AA : Aa: aa = 0.9^2 : 2 \times 0.9 \times 0.1 : 0.1^2 = 0.81 : 0.18 : 0.01}\)

In other words: 81% is completely healthy, 18% is carrier and 1% is affected. You only want to breed with non-carriers, which means that 19% of the population cannot be used! And we only looked at one disorder!

If we change the number of affected dogs the other numbers will be (a few examples:)

Healthy | Carrier | Affected |
---|---|---|

74% | 24% | 2% |

68% | 29% | 3% |

64% | 32% | 4% |

60% | 35% | 5% |

### Mutation and selection

The frequency of different alleles can be influenced in various ways. A number of these influences will be discussed in brief.

#### Mutation

In the article on biological aspects we have learned that information in genes is coded by way of chemicals compounds. Sometimes this information changes because the 'wrong' chemical is inserted at a certain point. These changes can be spontaneous (for example an incorrect copy is made when dividing a cell) or artificial (e.g. because of radioactive radiation).

Since the genes contain coding for certain proteins, these proteins will also be slightly changed. In most cases these changes are adverse changes, but now and then a favourable change occurs, which makes the cell function better. When the mutation occurs in reproductive cells the entire animal that grows from those cells will be affected.

#### Selection

Selection can be caused by environmental factors (natural selection) and by human interference (artificial selection). Natural selection happens when a (hereditary) characteristic influences the functionality of the animal in his environment. If this characteristic is preferred by humans (and they only breed with individuals which have that characteristic) we call it artificial selection.

Let's see how artificial selection (that's the essence of breeding) influences the frequency of certain genes.

#### Selection against a recessive allele

Assume that we can't distinguish between AA and Aa animals, but that we can tell the aa animals from the rest. If we use selection we can only remove the aa individuals from the breeding stock. This is the situation for various kinds of hereditary diseases.

A quick calculation; Assume that 5% has aa (sufferers), Hardy and Weinberg's law tells us that the ratio \(AA : Aa : aa = 60\% : 35\% : 5\%\). If we exclude the aa individuals (5%) from 100 dogs we'll end up with 60 AA and 35 Aa. The frequency of the A-allele becomes \((60 + 35/2)/(60+35) = 82\%\). The frequency of the a-allele becomes 18%, which gives the ratio of the new generation: \(AA:Aa:aa=67\%:30\%:3\%\). The number of affected dogs has decreased to 3%. We can continue to select against aa, which results in numbers depicted in the graph. Note that it was not taken into account that the previous generation is still included in the breeding stock, so the actual percentages are slightly different. The graph only shows the percentages in the next generations.

#### The influence of a 'mistake'

In real life once in a while a stud dog comes along that is so wonderful that a lot of breeders decide to use him. Such a dog serves a relatively large part of the bitches. It is not so sure that this has only benefits for the population. Many times such a dog is found to carry a disease we try to remove from the population. The consequenses are clear: half of his offspring also carries this gene and with a part of the bitches even affected pups show up.

If we continue the way we were used to, our breeding program has been set back a number of years. The graph shows the consequenses for the breeding program.

#### Inbreeding

This subject is by many people considered to be a 'scary' concept, while in fact we are all participating in inbreeding! Inbreeding can be explained as: breeding individuals which are closer related than average. If we consider the dogs in the Netherlands as a population, we must conclude that breeding pedigree dogs is inbreeding; the dogs within a population are closer related than two random dogs. Inbreeding must not be seen as a black and white concept, but as a quantity.

#### Inbreeding coefficient

Inbreeding will result in less heterozygous individuals in the population and in less heterozygous genes within an individual. The amount of inbreeding can be expressed by the (expected) decrease of heterozygous genes or animals. The inbreeding coefficient depends on the relationship of both parents; it is in fact half this level of relationship. The inbreeding coefficient is now easy to calculate: determine the common ancestor in the pedigree, count the number of steps from one parent via the common ancestor to the other parent, multiply that number of times ½ with itself and divide by two (half brother and half sister: 1 step on each side, so \(0.5 \times 0.5 / 2 = 0.125 = 12.5\%\)). To makes things a little more complicated, the result must be multiplied by the inbreeding coefficient of the common ancestor. When more than one common ancestor can be found, the individually calculated coefficient must be added (e.g. brother and sister have a both their father and their mother as common ancestor!). For the mathematical minds among us, this calculation can be summarised in a beautiful formula:

$$F = \sum \left(\frac{1}{2}\right) ^ {(n_m + n_v + 1)} \cdot \left(1 + F\!_A \right) $$

\(n_m\) : number of generations on the dams side

\(n_v\) : number of generations of the sires side

\(F\!_A\) : inbreeding coeff. of the common ancestor

We've seen before that inbreeding can't be avoided when breeding pedigree dogs. The breeding goal of many breeds (=meet the breed standard) is such that a uniform look and thus more homozygous genes is wanted.

So, selection also increases inbreeding. Inbreeding in a limited amount is not harmful for a population. By registering in a stud-book the ancestors of many generations can be determined, which makes it possible to breed unrelated individuals. This measure (outcross) reduces the inbreeding coefficient by 50%!