Visualization of the nature of protein folding by a study of a distance constraint approach in two‐dimensional models

A distance constraint approach is applied to two‐dimensional models of proteins in order to visualize the nature of protein folding and to examine the relative roles of different ranges of interaction. Three different native structures (I, II, and III) are considered; they have two different kinds of residues, viz., hydrophobic and hydrophilic, and different sequences of these residues. We examine how the distance constraint approach functions in the prediction of protein folding when we know the sequence of the residues, the (fixed) bond lengths, the mean distances between residues i and i + 2, and i and i + 3, and the mean distances for hydrophobic–hydrophobic, hydrophobic–hydrophilic, and hydrophilic–hydrophilic contacts between residues i and i + j, where j ≥ 4. This approach involves optimization of an object function with respect to 98 variables and is not free of the multiple‐minimum problem. The optimization is always terminated if the chain is entangled and/or the segments (residues) are packed too compactly to move. In order to escape from such situations and to take the excluded‐volume effect into account, a Monte Carlo method is used after the optimization is trapped in local minima. Success in the prediction of folding is found to depend on the starting conformations and on the native conformations. Fair success is obtained in predicting the helix‐like structure in protein I and the overall structure of protein III, but not the β‐like structures of proteins I and II. Insofar as the prediction of the structure of protein III is reasonable, it appears that some sequences of residues produce greater constraints on their conformations than others, if one considers only the hydrophobic and hydrophilic nature of the residues. These results imply that, in the folding of real proteins in three dimensions, the competition for hydrophobic (and hydrophilic) residues for inside (outside) positions in the molecule probably constitutes a necessary but not a sufficient condition to form and stabilize the native structure. The failure to predict the structure of protein II, and part of that of protein I, suggests that there are two types of long‐range interactions. One (which we considered here) is nonspecific (i.e., is defined only in terms of contacts between residues of the same or different polarity) and acts at any stage of protein folding; the other (which we did not consider here) is a specific interaction between residues in pairs and contributes only when the residues in the specific pair take on the native conformation. Presumably, incorporation of such specific long‐range interactions, together with the nonspecific ones, is necessary for successful protein folding, using the distance constraint approach.