Here, I'll go through all the terminology and concepts you need to know to understand the rest of the site. I hope this will be reasonably comprehensive without being too overwhelming.
Dominant-recessive inheritance is a relatively simple type to understand, and we'll see a lot of it. This describes the relationship between two alleles on the same locus. In a dominant-recessive relationship, a heterozygous cat with one of each allele will only show the phenotype of the dominant allele.
In the case of the B gene, a cat with one black allele and one chocolate allele - Bb, is black, making black dominant over chocolate, and chocolate recessive to black. Dominance does not always mean a trait is more common or more likely - it only means that when there are two different alleles, the more dominant one is the one that shows.
Generally, when there is a pair of alleles with the dominant-recessive relationship, we represent the dominant one with a capital letter (B), and the recessive one with a lowercase letter (b). If there are more alleles, they will typically be represented by adding superscripts onto the upper or lowercase letters - most on this website are written as non-superscripts for readability.
Remember, dominant and recessive are relative terms. The B locus actually has a third allele in cats - b1, which is cinnamon. Cinnamon is recessive to both black and chocolate. This makes chocolate recessive to black, but dominant to cinnamon.
Incomplete dominance is when, instead of one allele dominating, a heterozygous organism displays the intermediate phenotype of their two alleles.
This can be seen in cats with the mink pattern. Cats that are cscs have the colorpoint or Siamese pattern, which is a light body with dark extremities. Burmese cats, who are cbcb, display a similar pattern, but the body is nearly as dark as the extremities. Mink cats are cscb and have mid-toned bodies.
In codominance, a heterozygous displays the phenotypes of both alleles in patches. This is not a common pattern of inheritance in mammals, but we can see it in tortoiseshell cats, who have one orange allele and on non-orange allele (Oo). This gene is also sex-linked, which is explained more on the page for sex-linked red. In both incomplete dominance and codominance, one allele may be said to be more dominant and they may be assigned capital/lowercase letters, but this is ultimately arbitrary.
Epistasis is when the effects of one gene overrules the effects of another. For example, sex-linked red is epistatic to the brown gene - all cats with the alleles for sex-linked red are red, regardless of whether the brown gene says they should be black, chocolate, or cinnamon. Conversely, the brown gene is hypostatic to sex-linked red.
Polygenic inheritance is when a trait is affected by many different genes, each playing a small role. The hallmark of polygenic inheritance is that the traits fall along a continuous spectrum. Skin color in humans is a good example - there are many different genes that each play a small part in melanin production, which allows a practically infinite range of skin tones.
When looking at polygenic inheritance, the rule of thumb is that the offspring's phenotype will usually fall somewhere in the middle of the parents' phenotypes. It is possible for offspring to have phenotypes outside of that range, but the farther from the midpoint you go, the less likely it is.
Finally, I'd like to clearly define the concept of "carrying". Usually, a trait is said to be carried when it is a recessive trait which the individual has one allele for. So, a Bb cat would appear black, but would carry chocolate. However, sometimes people may say that a red cat who is bb is "carrying" chocolate. I find this confusing, so will only use be using the term in the first sense.
Let's say we have two cats, and we want to know what their kittens will look like. Or, even better, what the chances are that their kittens will have certain appearances. How can we figure this out?
We can make some predictions based on just their appearance, but we need the genotypes of the cats to be accurate. These can be determined by examining pedigrees or by direct genetic testing. Let's say we have two black cats we know are Bb. Can any of their kittens be chocolate?
When cells divide to make gametes (egg or sperm cells), the two alleles from each gene separate, and each one goes into a different gamete. This means that if a cat is Bb, 50% of their gametes will randomly get the B allele, and 50% will randomly get the b allele. One of those gametes can then meet a random gamete from the other cat.
We can organise these chances into a Punnett square, where each box has a 1 in 4 chance of occurring. This means that each kitten has a one in four chance of being chocolate, or, if these cats had many litters, you'd expect about one in four kittens to be chocolate.
If you want to make predictions based on more than one gene, you can make a bigger Punnett square...but that can get messy quick, and I'm sure you can easily find other sources on that (the excellent doggenetics.co.uk being one of them). When I am doing calculations, I instead use the rule of that the probability of two things occurring together is (probability of thing 1) x (probability of thing 2).
So, let's say these two Bb cats are also Dd. The dilution gene, D, turns black to blue and chocolate to lilac when a cat is dd. I'll first do separate Punnett squares for each gene. The black/chocolate one, as before, outputs a 3/4 chance of black and a 1/4 chance of chocolate. The dilution one outputs a 3/4 chance of non-dilute color and a 1/4 chance of dilute color.
I can now use the multiplication rule to find the chances for each color:Black = black x non-dilute = 3/4 x 3/4 = 9/16
Unlike Punnett squares, this is easy to scale to include more genes. These calculations are the basis for all my calculators.
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