Dynamic and structural analysis of isotropically distributed molecular ensembles

An efficient new method is presented for the characterization of motional correlations derived from a set of protein structures without requiring the separation of overall and internal motion. In this method, termed isotropically distributed ensemble (IDE) analysis, each structure is represented by an ensemble of isotropically distributed replicas corresponding to the situation found in an isotropic protein solution. This leads to a covariance matrix of the cartesian atomic positions with elements proportional to the ensemble average of scalar products of the position vectors with respect to the center of mass. Diagonalization of the covariance matrix yields eigenmodes and amplitudes that describe concerted motions of atoms, including overall rotational and intramolecular dynamics. It is demonstrated that this covariance matrix naturally distinguishes between "rigid" and "mobile" parts without necessitating a priori selection of a reference structure and an atom set for the orientational alignment process. The method was applied to the analysis of a 5-ns molecular dynamics trajectory of native ubiquitin and a 40-ns trajectory of a partially folded state of ubiquitin. The results were compared with essential dynamics analysis. By taking advantage of the spherical symmetry of the IDE covariance matrix, more than a 10-fold speed up is achieved for the computation of eigenmodes and mode amplitudes. IDE analysis is particularly suitable for studying the correlated dynamics of flexible and large molecules.