Dynamics of essential collective motions in proteins: Theory

Abstract

A general theoretical background is introduced for characterization of conformational motions in protein molecules, and for building reduced coarse-grained models of proteins, based on the statistical analysis of their phase trajectories. Using the projection operator technique, a system of coupled generalized Langevin equations is derived for essential collective coordinates, which are generated by principal component analysis of molecular dynamic trajectories. The number of essential degrees of freedom is not limited in the theory. An explicit analytic relation is established between the generalized Langevin equation for essential collective coordinates and that for the all-atom phase trajectory projected onto the subspace of essential collective degrees of freedom. The theory introduced is applied to identify correlated dynamic domains in a macromolecule and to construct coarse-grained models representing the conformational motions in a protein through a few interacting domains embedded in a dissipative medium. A rigorous theoretical background is provided for identification of dynamic correlated domains in a macromolecule. Examples of domain identification in protein G are given and employed to interpret NMR experiments. Challenges and potential outcomes of the theory are discussed.

Comparative analysis of essential collective dynamics and NMR-derived flexibility profiles in evolutionarily diverse prion proteins

“…It also allows one to visualize them in terms of collective subunit motions (dynamic domains) and determine local conformational flexibility from a single theoretical framework. 47,48 In contrast to the conventional empirical use of principal component analysis (PCA), the present methodology relies on a robust statistical-mechanical background, 47,49 which allows for a dynamically consistent definition of protein domains, as well as a clearer description of the local bond flexibility within the same theoretical framework. Using this technique, the essential collective coordinates of a protein can be determined via PCA using small portions of standard MD trajectories thereby providing a characterization of stable structural properties that persist over a longer period (Potapov et al, submitted).…”

“…[43][44][45] Recently, a novel theoretical formalism has been developed in our group to analyze collective coarse-grained dynamics in proteins. 47 This approach allows one to readily identify long term collective motions in a large molecule. It also allows one to visualize them in terms of collective subunit motions (dynamic domains) and determine local conformational flexibility from a single theoretical framework.…”

“…This formalism-denoted here as essential collective dynamics (ECD)-has been validated in a number of ways. In particular it has been used to compare the numerical predictions for protein G 47 as well as human and chicken prion proteins 48 with NMR-derived structural data such as the model-free order parameters 47 and random coil indices. [50][51][52] These studies have shown excellent agreement between the computational (ECD) and the experimental (NMR) results.…”

Algorithmic dimensionality reduction for molecular structure analysis

“…The difficulty arises from two factors: ͑1͒ the intrinsic dimensionality for most problems is unknown, and ͑2͒ there is debate for some problems as to whether simulation approaches can provide sufficient sampling of the phase space to facilitate an accurate analysis of dimensionality reduction. 3,13,24,25,27,42 In this paper, we investigate the ability of well-known nonlinear dimensionality reduction algorithms to identify accurate, low-dimensional substructures in the conformation space for an eight-membered ring. We chose this particular molecule for several reasons.…”

“…3,12,18,19 In addition to providing reduced complexity, an accurate representation for correlated molecular motions can aid in the interpretation of NMR and x-ray studies. [20][21][22][23][24][25][26] It has been shown that dimensionality reduction can be used to extend the time scales of MD, and a theoretical framework for low-dimensional simulation with Langevin MD or metadynamics is a topic of current investigation. 25,27,28 In addition to improving the efficiency of simulation, a logical extension is to utilize lowdimensional surrogate spaces for problems in optimization that occur in molecular recognition and self-assembly.…”