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Given a sequence of amino acids, the protein folding problem is to find a geometric structure of the amino acids that minimizes energy. Given that this problem is NP-Hard, one should be able to do the following:

  1. Convert an instance of an arbitrary problem to 3-SAT
  2. Use the reduction from 3-SAT to protein folding to generate an instance of the protein folding problem.
  3. Synthesize the amino acid sequence in a lab and allow the protein structure to fold.
  4. Observe the structure of the resultant protein.
  5. Finally, use the observed structure to find a solution to the original problem utilizing the aforementioned reduction.

This is almost certainly not possible at the moment as it would have some very serious implications. Here are my theories as to why

  1. The proof that the protein folding problem is NP-hard relies on the Hydrophobic-polar protein folding model, a simplified model of protein structure. Perhaps the simplifications that allow us to study the problem abstractly prevent us from applying this result to the physical problem of protein folding.
  2. While the problem of protein structure prediction is NP-hard in the worst case, perhaps proteins that are found in nature are a special case on which the protein folding problem is in P. So, if you were to generate an amino acid sequence corresponding to an instance of a supposedly hard problem, then the protein would fail to reliably fold to a minimal-energy state. My understanding is that proteins occasionally misfold to local minima in nature; maybe some proteins are more prone to this behavior than others.
  3. Synthesizing a protein in a lab given its amino acid sequence is not feasible.
  4. Observing the physical structure of a protein at the required resolution is not possible with today’s technology.

Number 1 does not seem entirely convincing since effects considered in the Hydrophobic-polar protein folding model should still be present in real life. Similarly, theories 3 and 4 are also likely incomplete explanations. If they are true, they are simply technological limitations as opposed to fundamental properties of the the problem. So, as technology progresses and we are able to synthesize and observe proteins, this just raises more questions such as “why can physical/chemical systems compute things that computers cannot?”

Kai Arakawa
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  • You want a quantum analog computer, that's what it is. Quantum computers are known to be able to solve NP-hard problems, so no surprise. This one looks somewhat like "IP protocol over pigeon mail" (RFC 1149) - I mean, compared to real, digital quantum computers, which are, er, well, not a thing yet. – Ivan Neretin Dec 29 '22 at 11:06
  • @IvanNeretin so what you are saying is that when proteins fold, certain quantum mechanical effects affect the resultant geometry? Also, it is not known if quantum computers can solve NP-hard problems. If they could, then they would be able to solve all problems which reduce to SAT. This is why some post-quantum cryptography schemes are based on NP-complete problems. – Kai Arakawa Dec 29 '22 at 23:21
  • I stand corrected on the second part. As for the first one, molecules are quantum things indeed, but the HP folding model abstracts that away! So now I'm split between your theories 1 and 2, while rejecting 3 and 4. Protein synthesis is pretty much a solved problem, and X-ray structural analysis... well, sometimes it is hard, but not NP-hard. – Ivan Neretin Dec 29 '22 at 23:39

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