We absolutely understand protein folding. The fact there isn't a closed form simple solution to tell us the shape of a protein shouldn't be surprising at all. It's just hard to calculate. It's like the 3 body problem, but there are 90. I don't know why you think there would be some simple solution to that that doesn't involve simulation and heavy computation. You'd need 8 dimensional 100x100 matrices or something to describe something like a protein in pure math, which is way more like programming than arithmetic anyway.
We do not understand protein folding. I'm an expert in this field with publications and what you just said makes no sense at all.
The simplified protein folding problem states that proteins fold to their energy minimum, and a succesful heuristic can find the "correct" fold by finding the structure argmin(energy). Although this is a real simplification that doesn't represent actual proteins, even that is not something you'd solve with matrices; combinatorial math.
More importantly, "protein folding" is the biophysical process that proteins experience; what DeepMind did was solve "structure prediction", which is another simpler problem, and they didn't do it by energy minimization, they did it by exploiting sequence similarity to provide structural constraints that massively reduced the search space.
I used matrix math as a hand wavey example of how one might purely numerically describe the angles in a structure. What don't we understand? We know the forces of all the atoms that act on each other, what is the mystery at this point other than calculating it is hard?
"We know the forces of all the atoms that act on each other, what is the mystery at this point other than calculating it is hard?"
might be true, but it's computationally intractable and even if you did find a way around it you'd just learn that your force field was inaccurate, or that proteins don't actually fold to their energy minimum.
Not just hard, but NP-hard. In fact, the "protein folding problem is NP hard" is misleading; in fact, the "static protein structure problem" is NP hard, and protein folding is "harder" than that (it's a superproblem of static structure prediction).
The way to describe angles for a protein is this: you use degrees or radians to describe the torsion angles of the backbone (2 torsions per amino acid, 360 possible positions per torsion, times the number of backbone atoms in the protein). Treat side chains with rotamers, that's already a solved problem. However, you can't compute self-collisions in angle space you need to embed the protein structure in cartesian coordinates, which is how they are normally represented (as a graph representing the bond topology, and an N*3 array of positions).
Right, my point is that it's as solved as it can really be as far as human understanding is concerned. There are too many variables at play to have any sort of elegant solution like Ohm's law or calculating an orbit. The best we can do is find different/more efficient/novel methods to calculate it. Am I wrong?
proteins don't fold to their energy minimum, they typically "collapse" to an intermediate state and then sample many different states (kinetically), rather than adopting the absolute thermodynamic minimum. In many proteins, the structure snapshot doesn't even correspond to a functionally active protein. The "protein folding problem is NP-hard is an entirely inaccurate view of the biophysics of folding", and if solved, would not address any useful question about folding.
What DeepMind solved was a far simpler problem: reproduce the structures that get produced by some experimental method. That's a well posed problem but ultimately a less interesting one, even if it's immediately "useful".
