Perspective: Identification of collective variables and metastable states of protein dynamics
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.
MELD-Path Efficiently Computes Conformational Transitions, Including Multiple and Diverse Paths

Machine Learning of Biomolecular Reaction Coordinates

Principal component analysis on a torus: Theory and application to protein dynamics
A dimensionality reduction method for high-dimensional circular data is developed, which is based on a principal component analysis (PCA) of data points on a torus. Adopting a geometrical view of PCA, various distance measures on a torus are introduced and the associated problem of projecting data onto the principal subspaces is discussed. The main idea is that the (periodicity-induced) projection error can be minimized by transforming the data such that the maximal gap of the sampling is shifted to the periodic boundary. In a second step, the covariance matrix and its eigendecomposition can be computed in a standard manner. Adopting molecular dynamics simulations of two well-established biomolecular systems (Aib9 and villin headpiece), the potential of the method to analyze the dynamics of backbone dihedral angles is demonstrated. The new approach allows for a robust and well-defined construction of metastable states and provides low-dimensional reaction coordinates that accurately describe the free energy landscape. Moreover, it offers a direct interpretation of covariances and principal components in terms of the angular variables. Apart from its application to PCA, the method of maximal gap shifting is general and can be applied to any other dimensionality reduction method for circular data.
Time-resolved observation of protein allosteric communication

Robust Density-Based Clustering To Identify Metastable Conformational States of Proteins

Contact- and distance-based principal component analysis of protein dynamics
To interpret molecular dynamics simulations of complex systems, systematic dimensionality reduction methods such as principal component analysis (PCA) represent a well-established and popular approach. Apart from Cartesian coordinates, internal coordinates, e.g., backbone dihedral angles or various kinds of distances, may be used as input data in a PCA. Adopting two well-known model problems, folding of villin headpiece and the functional dynamics of BPTI, a systematic study of PCA using distance-based measures is presented which employs distances between Cα-atoms as well as distances between inter-residue contacts including side chains. While this approach seems prohibitive for larger systems due to the quadratic scaling of the number of distances with the size of the molecule, it is shown that it is sufficient (and sometimes even better) to include only relatively few selected distances in the analysis. The quality of the PCA is assessed by considering the resolution of the resulting free energy landscape (to identify metastable conformational states and barriers) and the decay behavior of the corresponding autocorrelation functions (to test the time scale separation of the PCA). By comparing results obtained with distance-based, dihedral angle, and Cartesian coordinates, the study shows that the choice of input variables may drastically influence the outcome of a PCA.
Principal component analysis of molecular dynamics: On the use of Cartesian vs. internal coordinates

Computing Velocities and Accelerations from a Pose Time Sequence in Three-dimensional Space
In recent years, small flying robots have become a popular platform in robotics due to their low cost and versatile use. In the context of autonomous navigation, low-cost robots are often equipped with imprecise sensors and actuators and require a proper calibration and carefully designed and learned models. External systems like motion capture cameras usually provide accurate pose estimates for such devices. However, they do not provide the translational and rotational velocities and accelerations of the object. In this paper, we present an algorithm for accurate calculations of the six-dimensional velocity and the six-dimensional acceleration from a possibly noisy pose time sequence. We compute the velocities and accelerations in a regression using Newton's equation of motion as the model function. Thereby, we efficiently decouple the six individual dimensions and account for fictitious forces in the non-inertial body-fixed frame of reference. In simulation and experiments with a real inertial measurement unit (IMU), we show that our algorithm provides accurate velocity and acceleration estimates compared to the reference data.