So, as I've alluded to elsewhere, the thing to remember about P vs NP is that you can solve NP problems very quickly, but it takes NP time to verify that the so

So, as I've alluded to elsewhere, the thing to remember about P vs NP is that you can solve NP problems very quickly, but it takes NP time to verify that the solution is correct. In other words, you can walk a traveling salesman to all the houses in the time it takes to walk to all the houses (that's a solution), but in order to verify that the path you walked was the shortest, you'd have walk all the possible paths (or at least, walk them until they were longer than the original path you walked, which is part of how heuristic methods of solving the problem work). The way this works with proteins is assuming that proteins fold to a conformation that is a global minimum. For any given combination of atoms, you would have to consider all possible conformations (which, as you point out, can more easily be expressed in terms of pair-wise and higher-order interactions). This is the classic NP-complete, traveling-salesman type problem. Now, not all proteins fold to a global energy minimum, but many or most do. So really, whether protein folding is P or NP depends on what you mean by the "protein folding problem". If, by "protein folding problem", you mean determining the global energy minimum for any given chain of amino acids, then yes, this is NP. However, what Levinthal realized (with his paradox) was that if this was how proteins actually folded, it would take more than the age of the universe for a single protein to fold. This is even more paradoxical when you consider that most proteins fold on the millisecond to microsecond time scale. In other words, nature isn't solving an NP problem. If, by "protein folding problem", you mean figuring out how a protein reaches its final conformation, then this is where the funnel comes in. Think of a protein on a surface in conformation space. Each spatial arrangement of atoms represents a different point in space. The surface represents the region of space that is accessible to the protein at a given energy. So, at high energy, the protein can wiggle around pretty much at will. All conformation space is open to it. However, as you lower the energy, fewer and fewer conformations are energetically allowed (that is, the protein no longer has enough energy to break certain hydrogen bonds or ionic interactions). Eventually, you lower the energy enough, and you find the global energy minimum. Where this gets tricky is that there could be "traps" in the funnel; local energy minima that the protein can't escape from. If the protein gets caught in one of these traps, then it hasn't solved the NP problem of finding the global energy minimum. What's remarkable about proteins, however, is that the path they do follow (the one that's kinetically favorable) almost always avoids these traps. So, the protein is solving a P problem (finding the steepest path down the funnel), but in doing so also solves an NP problem (finding the global minimum). If you think about it then, it's actually the funnel that has solved the NP problem (or rather, turned it into a P problem). This funnel is constructed by the laws of physics as they pertain to proteins, and it seems as though the unique properties of the 20 amino acids that make up proteins are the key to this NP to P conversion. Now, do I think this means that P = NP? No. The protein folding funnel is a heuristic, just like any other. Just because proteins usually find the global energy minimum, doesn't mean they always do. To absolutely, positively verify that they are at their global minimum, you still have to solve an NP hard problem.