Collective motions in protein structures: Applications of elastic network models built from electron density distributions

Abstract

In this work, the Gaussian Network Model (GNM) and Anisotropic Network Model (ANM) approaches are applied to describe the dynamics of Pancreatic Trypsin Inhibitor protein graphs built from smoothed promolecular electron density (ED) distribution functions. A specific smoothing degree is selected which provides a clear partitioning of the protein structure into fragments located either on the protein backbone or side chains. A first set of analyses is carried out on results obtained from ED maxima calculated at that specific smoothing level. A second set is achieved for a protein ED network whose edges are weighted by ED overlap integral values. Results are compared with those obtained through GNM, ANM, and Normal Mode Analysis approaches, applied to the network of $C_{\alpha }$ atoms.

Introduction

Computational simulations of protein dynamics play an important role in deciphering protein functions, and usually require the knowledge of atomic coordinates. Coarse-grain protein models are often used to circumvent the problems of representing interactions of a protein structure at a full atomic description. The reduction of the number of degrees of freedom allows the reduction of the calculation time but also the smoothing of energy barriers, for example, during a Molecular Dynamics (MD) simulation. Like for all-atom interaction potentials, parametrization of coarse-grain potentials is an active field of research [1].

It has actually been shown that it is feasible to efficiently extract information about protein motions, at a reasonable degree of accuracy, without even knowing the detailed amino acid structure. Several years ago Bahar et al. [2], through the so-called Gaussian Network Model (GNM) approach, deepened the study of a single-parameter Hookean potential [3] which is an approximation based on a Gaussian distribution of interatomic distances around their equilibrium values. The authors showed that this simple potential, applied to the $C_{\alpha }$ atoms only of a protein in its folded state, provides a satisfactory description of the correlation between atomic fluctuations. This is justified by the agreement between calculated temperature factors B values and corresponding crystallographic experimental data. Since then, the GNM approach has often been applied to model protein dynamics [4] and was assessed vs. Normal Mode Analysis (NMA) and Molecular Dynamics (MD) techniques [5], [6]. Doruker et al. [6] extended the GNM algorithm in the so-called ANM approach in order to take account of the anisotropy of fluctuations through directional considerations.

GNM can be related to NMA approaches, which have proven to be useful in investigating the slowest motions in macromolecular systems [7], in that they express the dynamics of a chemical structure in terms of coordinates that involve the collective displacement of a large number of atoms. Hinsen et al. [8] actually showed that the global residue motion can be reproduced by a simple harmonic potential that contains two contributions: a vibrational term, described by a normal mode calculation, and a long-time diffusive term, described by a Brownian motion in a effective harmonic potential.

Rather than working with $C_{\alpha }$ coordinates, Ming et al. [9] applied the GNM and ANM theories to centroids of Voronoi cells of electron density (ED) maps in order to determine the motions of a protein structure without any knowledge of the atomic coordinates. For generating low-resolution density maps, the authors employed the Gaussian kernel convolution technique to electron microscopy (EM) data. Tama et al. [10] also computed elastic modes of proteins that can undergo large conformational changes from simulated experimental EM density maps. They later expanded their approach to the fitting of high-resolution structures in low-resolution [11].

The purpose of the present work is to suggest new applications in the field of ED analyses, with possible applications to experimental ED data. The interest lies in the perspective to predict the slow large amplitude dynamics of a macromolecular structure from experimental data without any knowledge of its atomic coordinates, and possibly to link various degrees of resolution. Following the work of Ming et al. [9], vertices of a protein graph were selected as the local maxima of the ED distribution function. We have additionally implemented a new way to calculate connectivity matrix elements in the GNM and ANM approaches, through the overlap integral values over smoothed ED distributions of protein fragments. Applications are presented for Pancreatic Trypsin Inhibitor (PTI), a small protein whose dynamics have already been largely studied by various methods such as NMA and MD.