Perspective: Identification of collective variables and metastable states of protein dynamics

Abstract

The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables (CVs) {xi} = x that in some sense describe the essential dynamics of the system. Considering the distribution P(x) of the CVs, the primal goal of a statistical analysis is to detect the characteristic features of P(x), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape ΔG(x) = −kBT ln P(x), and therefore amount to the metastable states and transition regions of the system. In this perspective, we outline a systematic strategy to identify CVs and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally, we briefly discuss a machine learning approach to dimensionality reduction, which highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms.

Automatic mutual information noise omission (AMINO): generating order parameters for molecular systems

“…These OPs serve as internal coordinates that are useful as a basis set for the description of the processes of interest. 2 Through one of many available methods [3][4][5][6][7] , they can then also be mixed into an even lower-dimensional reaction coordinate (RC). By then enhancing fluctuations along this RC, enhanced sampling methods such as metadynamics 8 and umbrella sampling 9 can tackle the second challenge mentioned above, i.e.…”

Principal component analysis of nonequilibrium molecular dynamics simulations

“…Here we employ (φ i , ψ i ) backbone dihedral angles, which have been shown to be well suited to describe the dynamics of peptides and small proteins. 7,9,55,57 To take the periodicity of the dihedral angles into account, we shift the periodic boundary of the circular data to the region of the lowest point density. This "maximal gap shifting" approach was incorporated into the new version of the dihedral angle principal component analysis (dPCA+), 58 which represents a significant improvement to the previously advocated sine/cosine-transformed variables used in dPCA.…”

On the dynamics of reaction coordinates in classical, time-dependent, many-body processes

“…When studying a complex many-body system, it is often a pragmatic choice to describe a process of interest in terms of the evolution of a small set of relevant observables ("reaction coordinates"), which capture the main features of the process. In molecular biophysics this view is commonly adopted: for example, protein folding, amyloid fiber formation or DNA mechanics are described in terms of the size, shape and structure of the complex molecules involved, while the degrees of freedom of the surrounding water as well as detailed information on the molecular structure are "averaged over" [1][2][3]. Similarly, chemical reactions and phase transitions are often described in terms of reaction coordinates.…”