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The computational task of protein-structure prediction is believed to require exponential time, but previous arguments as to its intractability have taken into account only the size of a protein's conformational space. Such arguments do not rule out the possible existence of an algorithm, more selective than exhaustive search, that is efficient and exact. (An efficient algorithm is one that is guaranteed, for all possible inputs, to run in time bounded by a function polynomial in the problem size. An intractable problem is one for which no efficient algorithm exists.) Questions regarding the possible intractability of problems are often best answered using the theory of NP-completeness. In this treatment we show the NP-hardness of two typical mathematical statements of empirical potential-energy-function minimization for macromolecules. Unless all NP-complete problems can be solved efficiently, these results imply that a function-minimization algorithm can be efficient for protein-structure prediction only if it exploits protein-specific properties that prohibit the simple geometric constructions that we use in our proofs. Analysis of further mathematical statements of molecular-structure prediction could constitute a systematic methodology for identifying sources of complexity in protein folding, and for guiding development of predictive algorithms.

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