Structural compliance: A new metric for protein flexibility

Abstract

Proteins are the active players in performing essential molecular activities throughout biology, and their dynamics has been broadly demonstrated to relate to their mechanisms. The intrinsic fluctuations have often been used to represent their dynamics and then compared to the experimental B-factors. However, proteins do not move in a vacuum and their motions are modulated by solvent that can impose forces on the structure. In this paper, we introduce a new structural concept, which has been called the structural compliance, for the evaluation of the global and local deformability of the protein structure in response to intramolecular and solvent forces. Based on the application of pairwise pulling forces to a protein elastic network, this structural quantity has been computed and sometimes is even found to yield an improved correlation with the experimental B-factors, meaning that it may serve as a better metric for protein flexibility. The inverse of structural compliance, namely the structural stiffness, has also been defined, which shows a clear anticorrelation with the experimental data. Although the present applications are made to proteins, this approach can also be applied to other biomolecular structures such as RNA. This present study considers only elastic network models, but the approach could be applied further to conventional atomic molecular dynamics.

Low-Frequency Harmonic Perturbations Drive Protein Conformational Changes

“…Remarks need to be given also regarding the physical meaning associated with the adopted perturbation scheme (i.e., harmonic excitations with a random direction in the space-domain but with a well-defined frequency content in the time-domain). In the previous literature, different force application patterns have been applied in a static fashion to probe protein flexibility [ 41 , 42 ] and protein conformational changes [ 44 , 45 , 47 ]. The approach proposed here (i.e., applying random forces to the protein ANM in a dynamic fashion) is supposed to simulate the external perturbations to the protein structure mainly due to Brownian motions of the surrounding particles [ 70 ].…”

“…As mentioned in the Introduction, the ANM has also been used to predict protein flexibility and conformational changes upon the application of external perturbations to the protein network [ 41 , 42 , 44 , 45 , 47 ]. This is usually done in a static fashion, meaning that dynamic effects are neglected.…”

“…The choice of the force vector { F } depends on the specific application. For example, in [ 42 ] pairwise pulling forces are applied for each couple of residues i and j , whereas in the PRS technique a point force is usually applied at a single node [ 44 , 45 ] or in a localized region [ 46 , 47 ] with a random direction. In any case, the protein response { u } is computed by inverting Equation (7) as follows: where [ H −1 ] is the pseudo-inverse Hessian matrix, which can be computed from the 3 N – 6 eigenvalues λ n and eigenvectors { δ n } of the Hessian matrix, as follows: …”

“…This analysis was able to explain the anisotropy of the mechanical resistance observed from single-molecule manipulation techniques, such as atomic force microscopy (AFM). More recently, we made use of two structural metrics, which are well-known in the field of structural mechanics (i.e., compliance and stiffness), to study the flexibility of protein structures under pairwise force application [ 42 ]. These metrics enabled to predict the distribution of protein flexibility and rigidity throughout the protein chain and were verified against the experimental B-factors.…”

Entropies Derived from the Packing Geometries within a Single Protein Structure

“… 36 Therefore, we believe that protein packing can yield insights into protein stability as characterized by entropies. Over the past several years, we have investigated protein packing 37 as an important consideration for dynamics, 38 , 39 where we have demonstrated that hinges in proteins can be identified immediately from the static structure to be localized within the least densely packed regions. The present study aims to examine protein packing as the basis for protein entropy.…”