Derivation and testing of pair potentials for protein folding. When is the quasichemical approximation correct?

Many existing derivations of knowledge-based statistical pair potentials invoke the quasichemical approximation to estimate the expected side-chain contact frequency if there were no amino acid pair-specific interactions. At first glance, the quasichemical approximation that treats the residues in a protein as being disconnected and expresses the side-chain contact probability as being proportional to the product of the mole fractions of the pair of residues would appear to be rather severe. To investigate the validity of this approximation, we introduce two new reference states in which no specific pair interactions between amino acids are allowed, but in which the connectivity of the protein chain is retained. The first estimates the expected number of side-chain contracts by treating the protein as a Gaussian random coil polymer. The second, more realistic reference state includes the effects of chain connectivity, secondary structure, and chain compactness by estimating the expected side-chain contrast probability by placing the sequence of interest in each member of a library of structures of comparable compactness to the native conformation. The side-chain contact maps are not allowed to readjust to the sequence of interest, i.e., the side chains cannot repack. This situation would hold rigorously if all amino acids were the same size. Both reference states effectively permit the factorization of the side-chain contact probability into sequence-dependent and structure-dependent terms. Then, because the sequence distribution of amino acids in proteins is random, the quasichemical approximation to each of these reference states is shown to be excellent. Thus, the range of validity of the quasichemical approximation is determined by the magnitude of the side-chain repacking term, which is, at present, unknown. Finally, the performance of these two sets of pair interaction potentials as well as side-chain contact fraction-based interaction scales is assessed by inverse folding tests both without and with allowing for gaps.