Apart from protein folding there are other interfering folding mechanisms like chaperone proteins that also make it hard to phrase it as a straight "sequence -> structure" prediction problem, though they often only exist for a small number of the total proteins (in humans).
It's not clear to me that you could truly demonstrate substantially different folding due to codon usage like that, on an experimental level, to make a general statement about all proteins.
This would also seem to imply that "sequence -> structure" isn't quite the right problem.
That's a reason that evolution-based methods, which use statistics about families of related proteins to estimate distances between pairs of amino acids (in 3D space), are more effective- many times in biology we can use evolutionary relationships between proteins to infer things that would be hard to determine through experiments or rigorous, thorough simulations.
But it's important to appreciate there are a large number of proteins that don't fold to a single unique structure rapidly-- and there are many ways this can be the case and many different biologically relevant behaviors depend on these properties. The tools from CASP are much less useful for proteins that violate the assumptions of Anfinsen's dogma, although the evolutionary data is helpful there too, it can often be a lot more challenging to deconvolute the signal.
Ultimately, "what is the right problem"? THe one that makes the most money? Produces the most "useful" scientific result? Is accessible with today's technology? For now, there's plenty of value in these sorts of competitions.
Personally I think the "right problem" is: "given a collection of diseases, use experimentally derived data and clever math, to discover biological treatments that reduce the total suffering from those diseases, subject to monetary and ethical constraints". That's what pharma attempts to do, although not particularly well. Others might say simply solving interesting problems like protein folding is inherently valuable as the right problem.
When you're doing protein expression, it's a standard procedure to do a codon optimization in order to adapt the codons to your host expression system. When you're expressing a human protein in yeast or ecoli the codons will be quite different but the folding is the same. If translation wouldn't show this stability it would be difficult to use foreign expression systems at all.
Imagine software that evolved by trial and error rather than being designed - it's not 'clean'... there are no absolutes.
It's well known that codons effect translation rate and translation rate could/can effect the kinetic pathway of folding.
When a protein folds ( even away from the ribosome ) it isn't in isolation - it's in a solution - being bombarded by Brownian motion of other molecules.
It's amazing you are alive at all really - it also suggests most proteins would need the functional state to be strongly energetically preferred.
So there are potentially lots of things beyond just the protein sequence that could be important variables.
However some proteins will reversibly fold/unfold in simple solution all day long - so it makes sense to start there.
> sequence -> structure
I recall reading about a result a couple of years ago supposedly demonstrating that "synonymous" DNA codons were in fact not synonymous, because the ribosome took systematically different amounts of time to process them, and the difference in construction time resulted in different folding for the protein.
This would imply that the problem "sequence -> structure" is not well defined, at least if the sequence in question is the sequence of peptides making up the protein and not the sequence of codons making up the gene that codes for the protein.
Do you know anything about this? Am I just making it up?