The complexity and accuracy of discrete state models of protein structure

The prediction of protein structure depends on the quality of the models used. In this paper, we examine the relationship between the complexity and accuracy of representation of various models of protein alpha-carbon backbone structure. First, we develop an efficient algorithm for the near optimal fitting of arbitrary lattice and off-lattice models of polypeptide chains to their true X-ray structures. Using this, we show that the relationship between the complexity of a model, taken as the number of possible conformational states per residue, and the simplest measure of accuracy, the root-mean-square deviation from the X-ray structure, is approximately (Accuracy) varies; is directly proportional to (Complexity)-1/2. This relationship is insensitive to the particularities of individual models, i.e. lattice and off-lattice models of the same complexity tend to have similar average root-mean-square deviations, and this also implies that improvements in model accuracy with increasing complexity are very small. However, other measures of model accuracy, such as the preservation of X-ray residue-residue contacts and the alpha-helix, do distinguish among models. In addition, we show that low complexity models, which take into account the uneven distribution of residue conformations in real proteins, can represent X-ray structures as accurately as more complex models, which do not: a selected 6-state model can represent protein structures almost as accurately (1.7 A root-mean-square) as a 17-state lattice model (1.6 A root-mean-square). Finally, we use a novel optimization procedure to generate eight 4-state models, which fit native proteins to an average of 2.4 A, and preserve 85% of native residue-residue contacts. We discuss the implications of these findings for protein folding and the prediction of protein conformation.