Well, I had a giant poster of Gem Twist autographed by Greg Best on my wall as a girl, so while I'm not technically a geneticist, I'm obviously some kind of authority here :winkgrin: ...
Yes.
So the son of Gem Twist could have the SAME genotype, but due to the differences in pregnancies, you could have different phenotypes?[/QUOTE]
Yes.
And certainly early training or just clicking with a certain person can change a horse.[/QUOTE]
Yes.
But in every situation above, the horse is still the same genotype. It's like saying "Am I still the mother (or father) of my child now that I have a sun tan?"
A different phenotype is not the same as a different genotype. That's the whole point. I may have a genetic propensity to tan and suffer schizophrenia and my twin would be the same. But if I drop acid a lot and go to Burning Man, I will probably get a tan and schizophrenia. She may not ever express either trait. Our genotypes are still the same.
Heritance is really a third concept, but I think the question is, "Can there be changes to the genotype and how does that impact clones as sires?"
A clone is at least as genetically similar to the original as an identical twin, if not more so. (Right? Help me out scientists) In any case, from a heritance standpoint, the same. You are not exactly the same from year to year, things change, even in what you will pass on. But you are still always the father or mother of your children. Any difference is extremely minute and not well understood. Demonstrating performance actually demonstrates nothing to a breeder in this circumstance. Gemini will not be a better sire by learning to jump. His propensity to jump is exactly the same as Gem Twist's and however much of that was a result of Gem Twist's particular environment is irrelevant from a breeding perspective. There is no way to control Gemini's environment to make it like Gem Twists. For one thing, no matter what the kids wear the 80s are never coming back.
Or: you can only breed for the things you can breed for :D
BTW this is already years underway, with performance results, in the world of mule racing
http://www.cbsnews.com/2100-205_162-1680235.html

