Extensions of the Langevin Equation for Protein Dynamics for Modelling Equilibrium Fluctuations of Proteins

Abstract

Proteins are not static structures; they must undergo conformational fluctuations about their folded state to function. Typically, the slow, near-equilibrium conformational dynamics of proteins encode the functional motions; an accurate description of these dynamics is useful for elucidating the functional motions of proteins. Use of molecular dynamics (MD) simulations gives a physical model of proteins' motions, but the dynamics are too high dimensional and coupled to determine the functional motions purely from observation of the MD trajectory; thus, methods to effciently extract the slow conformational dynamics of proteins from atomistic models are valuable. This dissertation advances the Langevin equation for protein dynamics (LE4PD), a diffusive, coarse-grained equation of motion for modeling protein dynamics adapted from the field of polymer physics. The LE4PD is solved by an eigenvalue decomposition into a set of normal mode coordinates, each of which encodes dynamics on a specific time- and lengthscale. A discrete-state master equation approach, Markov state modeling, is used to precisely determine the dynamics and kinetics of conformational dynamics described by the slow LE4PD modes by analyzing a 1- microsecond, folded simulation of the protein ubiquitin. The approach is able to extract slow dynamics in important binding regions of ubiquitin. In chapter III, Markov state models are used to determine the contributions of metastable states to the circular dichroism spectrum of a dinucleotide system. Because protein dynamics is inherently anisotropic, we develop an anisotropic version of the LE4PD. When both hydrodynamic effects and free-energy barriers are neglected, the model reduces to a principal component analysis of the alpha-carbon coordinates; including both these effects are important for quantitatively modelling the decay of simulated autocorrelation functions. Finally, we compare the LE4PD predictions from the ubiquitin simulation to the slow modes extracted by a time-lagged independent component analysis of the trajectory. We nd both methods are able to extract the slow dynamics of the protein, but the tICA compresses the information into a smaller number of modes; however, for ubiquitin, the tICA modes cannot model the simulated autocorrelation functions as effectively as the anisotropic LE4PD model. This dissertation includes previously published and unpublished co-authored material.