A Comparison of Collective Coordinates for Analyzing Protein Dynamics

Abstract

The internal dynamics of a protein is commonly analyzed by examining the fluctuations along the protein's alpha-carbons. However, a vast majority of an N-residue protein's internal dynamics can be described successfully using less than the complete (3N-6)-dimensional space spanned by the protein's internal fluctuations. This dimensionality reduction is achieved by transforming from the lab-frame coordinates to a set of judiciously-selected collective coordinates. A common technique, called principal component analysis (PCA), is to select the collective coordinates along which the variance of the original (3N-6)-dimensional coordinates is maximized. These statistically-independent, collective coordinates are termed principal modes (PMs) and are ordered in terms of descending variance, with the first PM possessing the maximum variance. However, these PMs contain no inherent information regarding the timescale over which the motion described by the PM occurs. We present an alternative set of independent, collective coordinates generated using the Langevin Equation for Protein Dynamics (LE4PD), a method for analyzing the internal fluctuations of proteins along its alpha-carbon backbone. The LE4PD accounts for the presence of hydrodynamic interactions between each residue and barriers on each mode's free-energy surface. In contrast to PMs, the LE4PD normal modes are ordered by timescales instead of variance; thus, the first normal mode from the LE4PD describes longest timescale internal motion. The fluctuations predicted by the corresponding PMs and LE4PD normal modes are in good agreement without including hydrodynamics or barriers in the LE4PD, but inclusion of these phenomena have a significant impact on the location and magnitude of the predicted fluctuations along the protein's backbone. The results suggest the LE4PD is an effective method for describing the internal fluctuations of proteins along collective coordinates, and the inclusion of hydrodynamic interactions and free-energy barriers significantly alters the fluctuations predicted along each normal mode.