A Coarse-Grained Langevin Equation for Protein Dynamics: Global Anisotropy and a Mode Approach to Local Complexity

Abstract

We utilize a multi-scale approach where molecular dynamic simulations are performed to obtain quantitative structural averages used as input to a coarse-grained Langevin Equation for Protein Dynamics, which can be solved analytically. The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. The normal mode analytical solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole, and internal modes which describe local fluctuations about the folded structure. Complexity in the configurational free energy landscape around the folded state of the macromolecule leads to a renormalization of the internal modes, while the global modes provide a basis set in which the dipolar orientation and global anisotropy can be accounted for when comparing to experiments. Fundamental to this approach is the inclusion of internal dissipation which is absent in any rigid-body hydrodynamical modeling scheme. This simple approach predicts the dynamics of both global rotational diffusion and internal motion from the picosecond to the nanosecond regime, and is quantitative when compared to time correlation functions calculated from molecular dynamic simulations and in good agreement with Nuclear Magnetic Resonance relaxation experiments. Results for several well-characterized globular proteins are presented, suggesting our method describes the relevant dynamics around the global minimum well. Use of non-equilibrium simulation techniques such as metadynamics to sample the full free-energy landscape of the protein, and extension of the theoretical treatment to describe the dynamics into the biologically interesting microsecond to millisecond regime, will be discussed.