Collective Variable Approaches for Single Molecule Flexible Fitting and Enhanced Sampling

Abstract

It is now well established that proteins and nucleic acids undergo local and global conformational fluctuations to perform a variety of cellular functions such as signal transduction, transport, and catalysis.1 Many experimental techniques such as X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, small-angle X-ray scattering (SAXS), and single-particle electron microscopy (EM) have indeed provided structural evidence at different resolutions to support the view of multiple functional states of biomolecules. Yet it remains difficult to characterize the vast conformational repertoire of biomolecules via experimental methods alone. Therefore, biophysical theory, modeling, and simulation techniques rooted in statistical mechanics are often useful for a detailed molecular understanding of biomolecular structures.2−5 Despite the limitations of molecular mechanics interaction potentials, computational methods can now be combined with low-resolution structural data to generate experimentally consistent conformational ensembles as well as to probe underlying mechanistic questions. Analyses of structural data for different functional states of biomolecules have revealed large-scale conformational rearrangements on the scales of entire domains. This means that a large group of atoms collectively move in a concerted way to facilitate functional movements. Traditionally, one relatively less expensive computational method to analyze collective motions in proteins has been to carry out normal-mode analysis (NMA) of equilibrium structures, because low-frequency modes are typically indicative of high-amplitude/large-scale motions.6−8 Such global and collective modes are robust, independent of sequence detail, and are intrinsically accessible to each biomolecule because they are encoded in their global shape.9−13 Given that the total number of degrees-of-freedom (DOF) in biomolecules is very large, NMA provides an efficient way to describe biomolecular dynamics in a reduced number of variables. As was originally pointed out by Hayward and Go,6 this reduction in dimensionality has led to the concept of an important subspace of variables, “collective variables (CVs)”, that are well-suited to characterize the dynamics of biomolecules. Interestingly, the concept of CVs as reaction coordinates has been recently extended to atomistic molecular dynamics (MD) simulations,14 which has significantly increased their capability in capturing long time-scale motions. This is chiefly possible because sampling in these CVs can be carried out more extensively in comparison to all possible DOF. Such methods are typically referred to as enhanced sampling techniques because they increase the likelihood of observation of a rare biomolecular event. The range of studies in which NMA and MD simulations have played a central role is immense, and it is not possible to do justice to all such studies in this focused Review. However, we refer the reader to pertinent comprehensive literature on those subjects along the way. Therefore, we have limited the scope of this Review to some specific applications of NMA, and enhanced sampling via temperature-acceleration in the context of flexible fitting to low-resolution EM data on macromolecular complexes. Particularly, we concentrate on two methods in this Review: (a) normal mode flexible fitting (NMFF)15,16 for structural refinement into EM maps; and (b) temperature-accelerated molecular dynamics (TAMD)17,18 for conformational exploration and flexible fitting. We first discuss theoretical underpinnings of all-atom and coarse-grained NMA of protein structures, which is followed by highlights of various successful applications. The theory and applications of NMA for flexible fitting of protein structures into EM maps are described thereafter. This is immediately followed by a discussion of the importance of enhanced sampling in biomolecular simulations, and how dynamics in these systems can be explored by evolving CVs via temperature acceleration, for example. Specifically, we discuss in detail various aspects of TAMD. During these discussions, we further highlight a few of the many cases where NMA and TAMD have alleviated difficulties faced by other methods in understanding large-scale functional excursions in biomolecules. This Review concludes with a brief overview and future outlook for these methods.