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In "Variation under Domestication", Darwin makes several references to the concept of true breeding:

They believe that every race which breeds true, let the distinctive characters be ever so slight, has had its wild prototype.

And:

I crossed some white fantails, which breed very true, with some black barbs.

Before diving into the book, I never heard of the term, so I looked it up and found this definition in the Wikipedia article on Zygosity:

A cell is said to be homozygous for a particular gene when identical alleles of the gene are present on both homologous chromosomes. The cell or organism in question is called a homozygote. True breeding organisms are always homozygous for the traits that are to be held constant.

But this is rather technical and it's not the kind of explanation I would give to a child.

What is true breeding, in simple terms? When is breeding not considered true?

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In the context of Darwin's Variation under Domestication, "true breeding" is a phenotypic characteristic rather than a genetic one. True-breeding organisms produce offspring that are identical to themselves, concerning some trait -- i.e. white fantails, when bred with white fantails, produce characteristically white offspring. For diploid organisms, true-breeding typically implies that the parents are homozygous at the locus conferring the trait of interest.

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Case A: Not Breeding True

If you cross-breed two random dogs, the offspring are likely to only vaguely resemble either parent. If you could clone a random dog to obtain two genetically identical dogs with different sex, you would likely still only get offspring that vaguely resemble their parents, because the chromosomes of the parents would come in pairs composed of chromosomes from the grandparents; but which grandparent contributed each chromosome in the offspring would be random. There is of course a very small probability that an occasional puppy would be identical to its parents — about 1 in 278.

Case B: Breeding True

On the other hand, if both chromosomes in each pair were identical for both parents (a situation that can be approached via a long series of inbreedings and selection, or could probably be obtained artificially in the lab), all the randomness would be eliminated and the offspring would be identical to the parents, if we neglect the likelihood of mutations and other genetic novelties.

So, in fact "breeding true" is just one end of a spectrum that is determined both by the amount of similarity between the parents' genetics, and by the amount of homozygosity in each parent. This whole concept goes right over the head of the majority of adults, so getting it across to a child could be a serious challenge. So to simplify: "Breeding True" means that the parents are identical and all the kids are identical to the parents. It happens only very, very rarely.

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