Local-structural diversity and protein folding: application to all-beta off-lattice protein models

Global measures of structural diversity within a distribution of biopolymers, such as the radius of gyration and percent native contacts, have proven useful in the analysis of simulation data for protein folding. In this paper we describe a statistical-based methodology to quantify the local structural variability of a distribution of biopolymers, applied to 46- and 69-"residue" off-lattice, three-color model proteins. Each folds into beta-barrel structures. First we perform a principal component analysis of all interbead distance variables for a large number of independent, converged Boltzmann-distributed samples of conformations collected at each of a wide range of temperatures. Next, the principal component vectors are subjected to orthogonal (varimax) rotation. The results are displayed on so-called "squared-loading" plots. These provide a quantitative measure of the contribution to the sample variance of the position of each residue relative to the others. Dominant structural elements, those having the largest structural diversity within the sampled distribution, are responsible for peaks and shoulders observed in the specific heat versus temperature curves, generated using the weighted histogram analysis method. The loading plots indicate that the local-structural diversity of these systems changes gradually with temperature through the folding transition but radically changes near the collapse transition temperature. The analysis of the structural overlap order statistic suggests that the 46-mer thermodynamic folding transition involves the native state and at least three other nearly native intermediates. In the case of the 46-mer protein model, data are generated at sufficiently low temperatures that squared-loading plots, coupled with cluster analysis, provide a local and energetic description of its glassy state.