Normal Mode Analysis of Biomolecular Structures: Functional Mechanisms of Membrane Proteins

1.1. Protein Dynamics and Allostery 1.1.1. Dynamic Equilibrium between Pre-existing Conformations The ability of macromolecules to sample an ensemble of conformations has been evident for decades, starting from the statistical mechanical theory and simulations of polymers.1–3 A polymer chain of N atoms enjoys 3N – 6 internal degrees of freedom, which gives rise to infinitely many conformations. Even a simple model of N = 100 atoms where bond lengths and bond angles are fixed, and dihedral angles are restricted to discrete isomeric states—say three states per bond—has access to 3N–3 = 1.9 × 1046 conformations. Proteins, too, are polymers, and have access to ensembles of conformations. The main structural difference between proteins and other chain molecules is that, under physiological conditions, proteins sample a significantly narrower distribution of conformations compared to disordered polymers. Their conformational variations are confined to the neighborhood of a global energy minimum that defines their “native state”. While the native state has been traditionally viewed as a “unique structure” selected or encoded by the particular amino acid sequence, it is now established by theory, computations, and experiments, after the work of pioneering scientists in the field,4–15 that the native state actually represents an ensemble of microstates: these microstates maintain the overall “fold” and usually share common secondary structure, but they differ in their detailed atomic coordinates. Differences are manifested by variations in bond lengths, bond angles, dihedral angles, loop conformations, substructure packing, or even entire domain or subunit positions and orientations. Importantly, these microstates are not static: there is a dynamic equilibrium which allows for continual interconversions between them while maintaining their probability distribution. These “jigglings and wigglings of atoms” as expressed by Feynman,16 and clearly observed in molecular dynamics (MD) simulations, were originally viewed as random events, or stochastic properties, hardly relevant to biological function. They essentially account for local relaxation phenomena in the nanoseconds regime, which may facilitate, for example, the diffusion of oxygen into the heme cavity of myoglobin17 or the permeation of ions across selectivity filters in ion channels.18–20 However, recent studies indicate that these thermal fluctuations may not only passively facilitate but also actively drive concerted domain movements and/or allosteric interactions, such as those required for substrate binding, ion channel gating, or catalytic function.15,21–34 Figure 1 provides an overview of the broad range of equilibrium motions accessible under native state conditions, ranging from bond length vibrations, of the order of femtoseconds, to coupled movements of multimeric substructures, of the order of milliseconds or seconds. Figure 1 Equilibrium motions of proteins. Motions accessible near native state conditions range from femtoseconds (bond length vibrations) to milliseconds or slower (concerted movements of multiple subunits; passages between equilibrium substates). X-ray crystallographic ... 1.1.2. Functional Significance of Collective Motions In the last two decades, there has been a surge in the number of studies based on principal components analysis (PCA)36 of biomolecular structures and dynamics. These studies have proven useful in unraveling the collective modes, and in particular those at the low frequency end of the mode spectrum, that underlie the equilibrium dynamics of proteins.37 Normal mode analysis (NMA) of equilibrium structures,38,39 essential dynamics analysis (EDA) of the covariance matrices retrieved from MD runs,40 and singular value decomposition (SVD) of MD or Monte Carlo (MC) trajectories41–43 all fall in this category of PCA-based methods. Recently, a server has been developed to efficiently perform such calculations using a variety of input structures.44 PCA-based studies provide increasing support to the view that the apparently random fluctuations of proteins under native state conditions conceal contributions from highly cooperative movements (e.g., concerted opening and closing of domains) that are directly relevant to biological function. Functional movements indeed involve passages between collections of microstates or substates that coexist in a dynamic equilibrium (Figure 2). The most cooperative motions usually occur at the low frequency end of the mode spectrum. These modes engage large substructures, if not the entire structure, hence their designation as global or essential modes. They are intrinsically accessible to biomolecules, arising solely from structure. In a sense, in the same way as sequence encodes structure, structure encodes the equilibrium dynamics. We refer to these global movements that are collectively encoded by the 3-dimensional (3D) structure as intrinsic motions of the examined protein, intrinsic to the protein fold or topology of native contacts. Biomolecular structures conceivably evolved to favor the global modes that help them achieve their biological or allosteric functions.21 Briefly, the emerging paradigm is structure-encodes-dynamics-encodes-function, and an evolutionary pressure originating from functional dynamics requirements may have selected the relatively small space of functional structures. Figure 2 Energy profile of the native state modeled at different resolutions. N denotes the native state, modeled at a coarse-grained scale as a single energy minimum. A more detailed examination of the structure and energetics reveals two or more substates (S1, ... The predisposition of proteins to undergo functional changes in structure is now supported by numerous experimental and computational studies, and an increasing amount of data demonstrates that allosteric responses are driven by intrinsically accessible motions.15,23,24,45–51 These studies have brought a new understanding to the role of collective dynamics in protein functions, demonstrating in particular how the functions of membrane proteins such as signal transduction, pore opening, ion gating, or substrate translocation are enabled by the cooperative movements of symmetrically arranged subunits. These findings are in support of the original Monod–Wyman–Changeux (MWC) view of allosteric effects,52,53 the main tenets of which are predisposition of the structure to access alternative conformations via cooperative changes in structure (simultaneously engaging all subunits) and selection from this pool of accessible conformation to achieve biological function in the presence of ligand/substrate binding. Recent findings on the relevance of global modes to functional dynamics are presented below for select, widely studied membrane proteins. The goal here is to review NMA-based computational methods and their applications to membrane proteins. We will also discuss recent developments for improving the methodology and its implementation in structure refinement and drug discovery methods.