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The properties of the ammino acids. One key factor is whether the amino acid is hydrophilic or hydrophobic.Wiki, provided there are citations, is an excellent source for explaining things in lay terms.
[qutoe]What drives the folding in the particular way? How early in life was folding present?
How early in life was folding present? I’m not a paleontologist, but if I had to guess I’d say very early.
Now consider the combination of possible 3D folds.
A ‘scientific revolution’ is taking place, as researchers explore the genomic jungle:
“The science of life is undergoing changes so jolting that even its top researchers are feeling something akin to shell-shock. Just four years after scientists finished mapping the human genome - the full sequence of 3 billion DNA “letters” folded within every cell - they find themselves confronted by a biological jungle deeper, denser, and more difficult to penetrate than anyone imagined.”
boston.com/news/globe/health_science/articles/2007/09/24/dna_unraveled/?page=1
The Case Against a Darwinian Origin of Protein Folds - Podcast on peer reviewed paper