An Anisotropic Langevin Equation for Protein Dynamics

The analysis and description of protein motions is greatly facilitated by reducing the effective dimensionality of the system through coarse-graining and transforming to decoupled normal-mode coordinates. We have developed a coarse-grained, diffusive, Langevin equation to model protein dynamics, the Langevin Equation for Protein Dynamics (LE4PD), which accounts for hydrodynamic effects and mode-dependent free-energy barriers. Here, we extend the LE4PD to describe anisotropic, directional fluctuations of a protein’s residues, projected along the alpha-carbons. We compare the dynamics predicted by the LE4PD to a conventional method to model protein dynamics, principal component analysis (PCA), which does not account for free-energy barriers or possess an associated equation of motion. Testing the formalism on a molecular dynamics simulation of ubiquitin, coarse-grained at the alpha-carbon level, when both free-energy barriers and hydrodynamic effects are neglected, the normal modes predicted by both methods are identical. However, we find that including the barriers and hydrodynamic effects in the mode-dependent description can alter significantly the predicted kinetic and dynamic properties of the protein.