How orientational order governs collectivity of folded proteins

The past decade has witnessed the development and success of coarse-grained network models of proteins for predicting many equilibrium properties related to collective modes of motion. Curiously, the results are usually robust toward the different cutoff distances used for constructing the residue networks from the knowledge of the experimental coordinates. In this study, we present a systematical study of network construction and their effect on the predicted properties. Probing bond orientational order around each residue, we propose a natural partitioning of the interactions into an essential and a residual set. In this picture, the robustness originates from the way with which new contacts are added, so that an unusual local orientational order builds up. These residual interactions have a vanishingly small effect on the force vectors on each residue. The stability of the overall force balance then translates into the Hessian as small shifts in the slow modes of motion and an invariance of the corresponding eigenvectors. We introduce a rescaled version of the Hessian matrix and point out a link between the matrix Frobenius norm based on spectral stability arguments and orientational local order. A recipe for the optimal choice of partitioning the interactions into essential and residual components is prescribed. Implications for the study of biologically relevant properties of proteins are discussed with specific examples.