Quantifying viscous coupling between global diffusion and conformational dynamics from simulated protein trajectories

In the simulation of protein folding and binding, it is convenient to analyze the results in a body-fixed reference system.The transition from a lab frame to the protein’s body-fixed frame kinematically isolates the protein’s internal fluctuations. However, it also reveals dynamical couplings between those fluctuations and global diffusion. We present a Langevin equation of motion in the body-fixed frame which explicitly quantifies the instantaneous coupling, via inertial and viscous forces, between the molecule’s global diffusion and its internal conformational fluctuations. Couplings due to inertial effects are expected to have little relevance at low Reynolds numbers. Couplings due to viscous forces, however, cannot be dismissed in principle and require careful examination. We hypothesize that these coupling terms could facilitate slow fluctuations necessary for the protein’s function. Our formalism helps to quantify the extent of the global-internal couplings directly from a Molecular Dynamics simulation trajectory, thus identifying the conditions in which each coupling term actively participates in the dynamics. Additionally, in the quasi-rigid limit of tightly folded proteins, we show how the equations can be linearized and subsequently parameterized from fluctuation statistics obtained via Molecular Dynamics simulation. This leads to coarse-grained equations of motion for the molecule and a description of ballistic and diffusive modes of motion and their timescales. We demonstrate the approach by studying the coupling contributions and linearizations of some model systems.