We’ve now covered breeding concepts and basic genetics, and a good amount of that discussion implied improving animals as individuals.
But breeding is not about improving an individual animal per se — once it has been born you can’t refine it further by tinkering with its genome. Breeding is really about improving populations of animals, such that those individuals that are born are collective improvements on their ancestors. Those individuals, collectively, then provide the genetic base that will improve their descendants, and so on.
As breeding is about improving populations, it makes sense to take our understanding of Mendelian inheritance at an individual level and apply that knowledge to a population. A scaling-up if you will! Population genetics is thus the study of the frequency of genes and genotypes within a population, and how these may change or be maintained through generations.
The latter part of the Background Information section of this blog touched briefly on selection and mating processes. The Selection Process determines which animals get to become parents at all, so as to pass their genes on to the next generation. The Mating Process determines which selected males cover which selected females for any of several end-goals in mind.
The ultimate aim in breeding is to improve a population of animals by taking what makes them ‘best’ in their generation, and amplifying that in the next generation, such that each successive generation contains a greater number each time of animals that are ‘best’.
For numbers of ‘best’ to increase, the frequency of the genes within a population that make it ‘best’ must also increase. Individual animals with the greater number of ‘best’ genes, and which are able to pass those genes on, have a better breeding value as they can improve the population better than the ones that don’t.
Understanding population genetics leads to informed and improved selection and mating systems — the very foundation of good animal breeding. The following posts in this Population Genetics section will discuss gene and genotypic frequencies, and how selection and mating systems can affect those frequencies.
Describing the genotype of some simply-inherited traits in an individual animal can be straightforward — horned cattle are always ‘pp’, and looking at an Andalusian chicken will tell you if it is ‘BB’ (black), ‘Bb’ (slate-blue) or ‘bb’ (white). We know the exact alleles carried, and in which proportions.
When it comes to describing the genotype of a population however, we refer to gene and genotypic frequencies instead.
Gene Frequency
A gene (or allelic) frequency is a measure of how common a particular allele is at a particular locus in a population. For example, how frequently ‘B’ and ‘b’ appears in an Andalusian flock.
An allele that doesn’t appear in a population at all has a gene frequency of 0 (0%). An allele that is the sole allele at that locus in a population has a gene frequency of 1 (100%). And an allele that makes up 50% of the genes at that locus has a gene frequency of 0.5 (50%).
Gene frequencies are denoted by lowercase letters. The frequency of the dominant allele at a locus is represented by p and that of the recessive one by q. If neither allele is dominant, p and q are assigned arbitarily. A third allele at that locus would have a gene frequency r.
The sum of all frequencies within a population always adds up to 1 (or 100%). From above, p + q = 1, or p + q + r = 1.
Let’s work an example with a flock of 100 Andalusian chickens, of which 42 are black (’BB’), 33 are slate-blue (’Bb’) and 25 are white (’bb’). There are 200 total alleles for feather colour in this population.
Each black chicken has two ‘B’ alleles, thus 42 × 2 = 84 ‘B’ alleles. Each slate-blue chicken has one ‘B’ each, thus 33 ‘B’ alleles. The sum total of ‘B’ alleles is 84 + 33, or 117.
Each white chicken has two ‘b’ alleles, or 25 × 2 = 50 ‘b’ alleles total. Each slate-blue chicken has one ‘b’ each, thus another 33 alleles to make 83 ‘b’ alleles total.
The sum of all alleles, ‘B’ and ‘b’, is 117 + 83, or 200 total.
The gene frequency of the ‘B’ allele is therefore p = 117 ÷ 200 = 0.585 (or 58.5%) The gene frequency of the ‘b’ allele is therefore q = 83 ÷ 200 = 0.415 (or 41.5%)
And p + q = 1.
Genotypic Frequency
A genotypic frequency is a measure of how common a particular one-locus genotype is in a population. For example, how frequently the homozygous dominant, heterozygous dominant, and homozygous recessive genotypes occur.
Genotypic frequencies are denoted by uppercase letters. The frequency of the homozygous dominant genotype is represented by P, that of the heterozygous dominant genotype by H, and that of the homozygous recessive genotype by Q. Again, these designations are arbitrary if there is no clearly dominant genotype.
In our flock of 100 Andalusians above: the ‘BB’ genotype has a genotypic frequency of P = 42 ÷ 100 = 0.42 (or 42%) the ‘Bb’ genotype has a genotypic frequency of H = 33 ÷ 100 = 0.33 (or 33%) the ‘bb’ genotype has a genotypic frequency of Q = 25 ÷ 100 = 0.25 (or 25%)
And P + H + Q = 1.
How do you express genotypes with more than two alleles at a locus? Let’s take a locus B, with the alleles ‘B’, ‘b’ and ‘b′’. The possible one-locus genotypes with possible genotypic frequency representatives could be:
‘BB’
P
‘bb’
Q
‘b′b′’
R
‘Bb’
H(’Bb’)
‘Bb′’
H(’Bb′’)
‘bb′’
H(’bb′’)
where P + Q + R + H(’Bb’) + H(’Bb′’) + H(’bb′’) = 1.
And that’s how gene and gene frequencies are calculated! Just remember that lowercase letters (eg p, q, r) denote gene frequencies and uppercase letters (eg P, Q, R, H) denote genotypic frequencies as we’ll be using these a bit.
Next week we’ll see the effect selection has on changing gene and genotypic frequencies.
Selecting animals for breeding is a process by which those deemed ‘best’ are allowed to be parents, and those deemed not, aren’t. The next generation is similarly assessed, and the next, and the next, with the population expected to improve incrementally each time.
This gradual improvement over time is due to the frequency of desirable genes increasing in the population and the frequency of undesirable genes decreasing in the population. This results in a group of animals with increased breeding value, as they have a higher concentration of ‘best’ genes more likely to be passed onto the next generation. That next generation, with its higher concentration of ‘best’ genes will perform* at a higher level than earlier generations did.
(* ‘Performance’ here is a breeding term that doesn’t necessarily refer to athletic performance such as speed. Rather, it refers to the resulting phenotype, as determined by the genotype. ‘Performance’ could be how fine a sheep’s wool is, for example.)
Gene frequencies, breeding values and performance are all intertwined. Increasing breeding values and performance in a population increases the frequencies of desirable genes. Increasing the frequencies of desirable genes increases breeding values and performance.
The following graph shows the interrelations between gene and genotypic frequencies at the A locus for the ‘A’ and ‘a’ alleles and ‘AA’, ‘Aa’ and ‘aa’ genotypes.
The x-axis value is q, the frequency of the ‘a’ allele. The y-axis value is the genotypic frequency of each of the genotypes P (the ‘AA’ genotype), H (the ‘Aa’ genotype) and Q (the ‘aa’ genotype). Please refer to Gene and Genotypic Frequencies for a brush-up on these terms if needed.
As the frequency of ‘a’ (q, the green line) increases and approaches 1, the frequency of ‘A’ (p, the blue line) simultaneously decreases and approaches 0, as p + q = 1.
The higher the frequency of ‘a’, the lower the frequency of ‘A’. With more and more ‘a’ alleles and less and less ‘A’ alleles in the population as a result, it follows that the ‘aa’ genotype (Q) increases as the ‘Aa’ genotype (H) decreases and the ‘AA’ genotype (P) decreases even more.
If the ‘A’ allele is favourable and selected for, its frequency (p) will increase, as will the ’AA’ (P) genotype, while the ‘aa’ genotype (Q) and the frequency of ‘a’ (q) simultaneously decrease.
The ‘Aa’ (H) heterozygous genotype becomes more common as the ‘A’ allele increases in frequency, but as the ‘AA’ genotype becomes more and more common, ‘Aa’ drops off in frequency.
Eventually p may increase such that there are no more ‘a’ alleles left in the population. The ‘A’ allele by default has become fixed at locus A (it has reached fixation). As the only possible genotype is ‘AA’, P = 1 and H = Q = 0.
Please note that selecting for a particular allelic expression (ie selecting for ‘best’ phenotype) can change the gene frequency significantly in a population, but the genotypic frequencies change as a consequence of this. Selection does not directly change genotypic frequencies — they ‘follow along’ instead.
The next post will cover how mating systems can shift gene and genotypic frequencies in a population — but this time it’s the genotypic frequencies that are changed directly.
Once animals have been selected for breeding, the next step is to decide which are mated to which via a mating system. Some examples of mating systems, each with different intentions, were briefly covered here.
Last week we saw how selecting for a particular allelic expression (ie selecting for ‘best’ phenotype) can change the gene frequency significantly in a population. Genotypic frequencies change indirectly as a result — they ‘tag along’.
Mating systems do however change genotypic frequencies directly, with gene frequencies less affected. These systems fall into two general categories: inbreeding systems and outbreeding systems.
Inbreeding is a mating system which increases homozygosity. Outbreeding is a mating system which increases heterozygosity.
We’ll cover inbreeding here, and outbreeding in the next post.
Inbreeding increases homozygosity by mating related animals. Related animals are more likely to have many alleles in common, and mating two related animals increases the likelihood of progeny becoming homozygous for a desirable allele. Inbreeding helps to fix the ‘AA’ or ‘aa’ genotype (whichever is desirable) in place.
Consider the following pedigree tree, which is a typical example of inbreeding, or linebreeding. The same animal A appears both as the paternal and maternal grandsire of animal X:
A pedigree tree shows the relationships of all ancestors, and the same animal may appear in several places, as A does. An arrow diagram differs in that it shows the flow of genetic material from ancestors to descendants. A particular animal will appear just once, while ancestors that do not contribute to inbreeding are excluded. (B and C were kept in this arrow diagram so as to make comparisons of the two diagrams clearer.)
Here you can see how alleles at a particular locus will flow from A to S and D. If S and D both acquire the same allele from A, there is a chance that X will become homozygous for that allele. If homozygous, then that allele is fixed in place for X, and X will only ever pass that particular allele on to all of its progeny. X’s breeding value with respect to passing on the desirable phenotype has increased.
This sounds straightforward and simple to implement, but that chance of X being homozygous through a common ancestor is only one in eight. (We’ll cover the maths later when going into more depth on linebreeding.)
Nonetheless, linebreeding is a well-established and time-honoured practice for ‘fixing’ genes into a population as it increases homozygosity and decreases heterozygosity.
The next post will discuss gene frequencies with respect to outbreeding.
Outbreeding (also called crossbreeding) is the opposite of inbreeding, in that unrelated animals are mated with the effect of increasing heterozygosity.
Let’s step through an example by starting with two unrelated populations, 1 and 2.
Population 1 has gene frequencies at the A locus of p1 = 0.8 and q1 = 0.2. Population 2 has gene frequencies at the A locus of p2 = 0.1 and q2 = 0.9.
Right away we can tell the two are unrelated as the gene frequencies differ so much.
Now cross them to create an F1 generation. (Back in Explaining Mendel’s Results Visually we defined an F1 generation as one resulting from the crossing of two purebred populations. Here we are using it more broadly to include two unrelated, but not necessarily purebred, populations.)
Where do the numbers come from? The gene frequency of ‘A’ in population 1 is 0.1, and in population 2, 0.8. Thus the probability of the ‘AA’ genotype occurring is the probability of inheriting the ‘A’ allele from population 1 and the probability of inheriting the ‘A’ allele from population 2.
In statistics, where we’d say ‘and’, we’d write ‘×’.
Therefore the probability of the ‘AA’ genotype arising is 0.1 × 0.8 = 0.08.
Likewise the probability of the ‘aa’ genotype can be calculated as 0.9 × 0.2 = 0.18.
However, the ‘Aa’ genotype could come from the ‘Aa’ combination or the ‘aA’ combination. They’re the exact same genotype, but can be formed in either of two ways depending on which populations contribute the ‘A’ and the ‘a’ alleles.
In statistics, where we’d say ‘or’, we’d write ‘+’.
Therefore the probability of the ‘Aa’ genotype is 0.72 + 0.02 = 0.74.
From the Punnett square results, the genotypic frequencies are: PF1 = 0.08 HF1 = 0.74 QF1 = 0.18
(Note how these add up to 1.)
We can actually determine gene frequencies from genotypic frequencies. Let’s assume an F1 population of 100 animals. Statistically, 8%, or eight, are likely to be ‘AA’ genotype. 74% are likely to be ‘Aa’ genotype, and 18% of ‘aa’ genotype.
One hundred animals between them have 200 alleles at the A locus. Of these, eight will have two copies of the ‘A’ allele each, and 74 will have one copy each.
Therefore at the A locus there will be (8 × 2) + 74 = 90 individual ‘A’ alleles amongst the population, with a frequency of p = 90 ÷ 200 = 0.45.
As p + q must always add to one, the frequency of the ‘a’ allele, q, is therefore 1 - 0.45 = 0.55. But let’s work it out anyway:
At the A locus there will be (18 × 2) + 74 = 110 individual ‘a’ alleles amongst the population, with a frequency of p = 110 ÷ 200 = 0.55. pF1 = 0.45 qF1 = 0.55
From these calculations can you see the formulae for working gene frequencies from genotypic frequencies? p = P + ½H q = Q + ½H
Now let’s cross the F1 population with themselves to produce an F2 population:
You may be wondering why, if we are crossing an F1 population with ‘AA’, Aa’ and ‘aa’ genotypes, that this square shows only ‘A’ and ‘a’ combinations and not ‘A’ and ‘A’ and ‘a’ and ‘a’ combinations.
Please don’t confuse the squares here for the ones used back in Explaining Mendel’s Results Visually. There, we were determining the genotypes possible from the alleles carried by the parents. Here, we are determining the frequencies of genotypes possible in a population, based on the frequencies of the alleles in that population.
Thus we have F2 genotypic frequencies of: PF2 = 0.2025 HF2 = 0.4950 (from 0.2475 + 0.2475) QF2 = 0.3025
Interestingly, the gene frequencies have not changed between the F1 and F2 generations: p, the frequency of the ‘A’ allele, remains at 0.45 q, the frequency of the ‘a’ allele, remains at 0.55
If gene frequencies stay constant between generations, and if the genotypic frequencies in the offspring generation are a result of the gene frequencies in the parent generation, it follows that mating an F2 population amongst itself would produce an F3 generation with the same genotypic frequencies as the F2 one.
Or, if matings within a population are completely random, then gene and genotypic frequencies do not change from generation to generation.
This concept is known as the Hardy-Weinberg equilbrium (named after the two men who independently discovered it), which we’ll cover in the next post!
