A kernel-based approach to molecular conformation analysis

Learning transfer operators by kernel density estimation

Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The conventional description, known as the Ulam-Galerkin method, involves projecting onto basis functions represented as characteristic functions supported over a fine grid of rectangles. From this perspective, the Ulam-Galerkin approach can be interpreted as density estimation using the histogram method. In this study, we recast the problem within the framework of statistical density estimation. This alternative perspective allows for an explicit and rigorous analysis of bias and variance, thereby facilitating a discussion on the mean square error. Through comprehensive examples utilizing the logistic map and a Markov map, we demonstrate the validity and effectiveness of this approach in estimating the eigenvectors of the Frobenius-Perron operator. We compare the performance of histogram density estimation (HDE) and kernel density estimation (KDE) methods and find that KDE generally outperforms HDE in terms of accuracy. However, it is important to note that KDE exhibits limitations around boundary points and jumps. Based on our research findings, we suggest the possibility of incorporating other density estimation methods into this field and propose future investigations into the application of KDE-based estimation for high-dimensional maps. These findings provide valuable insights for researchers and practitioners working on estimating the Frobenius-Perron operator and highlight the potential of density estimation techniques in this area of study.

TYROSINE KINASES: COMPLEX MOLECULAR SYSTEMS CHALLENGING COMPUTATIONAL METHODOLOGIES

Classical molecular dynamics (MD) simulations based on atomic models play an increasingly important role in a wide range of applications in physics, biology, and chemistry. Nonetheless, generating genuine knowledge about biological systems using MD simulations remains challenging. Protein tyrosine kinases are important cellular signaling enzymes that regulate cell growth, proliferation, metabolism, differentiation, and migration. Due to the large conformational changes and long timescales involved in their function, these kinases present particularly challenging problems to modern computational and theoretical frameworks aimed at elucidating the dynamics of complex biomolecular systems. Markov state models have achieved limited success in tackling the broader conformational ensemble and biased methods are often employed to examine specific long timescale events. Recent advances in machine learning continue to push the limitations of current methodologies and provide notable improvements when integrated with the existing frameworks. A broad perspective is drawn from a critical review of recent studies.

Comprehensive characterization of oligosaccharide conformational ensembles with conformer classification by free-energy landscape via reproductive kernel Hilbert space

Oligosaccharides play versatile roles in various biological systems but are difficult to characterize from a structural viewpoint due to their remarkable degrees of freedom in internal motion. Therefore, molecular dynamics simulations have been widely used to delineate the dynamic conformations of oligosaccharides. However, hardly any methods have thus far been available for the comprehensive characterization of simulation-derived conformational ensembles of oligosaccharides. In this research, we attempted to develop a non-linear multivariate analysis by employing a kernel method using two homologous high-mannose-type oligosaccharides composed of ten and eleven residues as model molecules. These oligosaccharides' conformers derived from simulations were mapped into reproductive kernel Hilbert space with a positive definite function in which all required non-redundant variables for describing the oligosaccharide conformations can be treated in a non-biased manner. By applying Gaussian mixture model clustering, the oligosaccharide conformers were successfully classified by different funnels in the free-energy landscape, enabling a systematic comparison of conformational ensembles of the homologous oligosaccharides. The results shed light on the contributions of intraresidue conformational factors such as the hydroxyl group orientation and/or ring puckering state to their global conformational dynamics. Our methodology will open opportunities to explore oligosaccharides' conformational spaces, and more generally, molecules with high degrees of motional freedom.

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

Kernel methods for detecting coherent structures in dynamical data

We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely, the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition called coherent mode decomposition. Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.