In my headcanon when they passed the stargate of the city, the AI emits a beam who change the DNA of the one having the ATA, John begins to dream about dragons, dreams that he flies as a dragon until he knows that it's not a dream.
the others too.

Rodney has the ATA therapy before they figure out that they don't dream and begins to shift too.

The AI changed them to fight the wraiths, the shifting didn't took with the ancients because they were the first evolution....

But you don't want to know about that, right ?