Recovering Hidden Degrees of Freedom Using Gaussian Processes

TL;DR

This work tackles the limitation of traditional MD dimensionality reduction methods that ignore temporal structure by introducing a Gaussian Process Variational Autoencoder (GP-VAE) with a time-conditioned latent prior $p(\mathbf{z}|t)$. By employing a Matérn kernel $k_{\nu,\ell}(t,t')$ in the latent space, the method encodes temporal correlations and preserves Markovian dynamics in the reduced representation. The authors demonstrate, first with a 3D toy model and then on a $50\,\mu$s MD trajectory of T4 lysozyme, that GP-VAE can separate dynamically distinct states that are geometrically indistinguishable and reveal functional couplings between structural subunits. This time-aware framework improves the reliability and interpretability of subsequent Markov state model analyses and offers a general approach for uncovering hidden degrees of freedom in complex biomolecular systems.

Abstract

Dimensionality reduction represents a crucial step in extracting meaningful insights from Molecular Dynamics (MD) simulations. Conventional approaches, including linear methods such as principal component analysis as well as various autoencoder architectures, typically operate under the assumption of independent and identically distributed data, disregarding the sequential nature of MD simulations. Here, we introduce a physics-informed representation learning framework that leverages Gaussian Processes combined with variational autoencoders to exploit the temporal dependencies inherent in MD data. Time-dependent kernel functions--such as the Matérn kernel--directly impose the temporal correlation structure of the input coordinates onto a low-dimensional space, preserving Markovianity in the reduced representation while faithfully capturing the essential dynamics. Using a three-dimensional toy model, we demonstrate that this approach can successfully identify and separate dynamically distinct states that are geometrically indistinguishable due to hidden degrees of freedom. Applying the framework to a $50\,μ$s-long MD trajectory of T4 lysozyme, we uncover dynamically distinct conformational substates that previous analyses failed to resolve, revealing functional relationships that become apparent only when temporal correlations are taken into account. This time-aware perspective provides a promising framework for understanding complex biomolecular systems, in which conventional collective variables fail to capture the full dynamical picture.

Figures (5)

• Figure 1: Time trace obtained from a Langevin simulation of the potential $\Phi(x,y,z)$ described by Eq. \ref{['eq:toypotential']}. The simulation was carried out for $10^6$ simulation steps using a time step of $\Delta t = 5 \cdot 10^{-3}$, a friction coefficient of $\gamma=1$ and a temperature of $T=1$ (in dimension-less units).
• Figure 2: (a) Three-dimensional representation of the toy potential showing four distinct basins, labeled as state 1 (blue), 2 (cyan), 3 (yellow), and 4 (red), respectively. The contour lines in the $xy-$plane indicate the potential depth. (b) When projected onto the $xy$-plane, states 3 and 4 overlap, making them indistinguishable without additional information. (c) Using only the $xy$-plane data combined with time information, the GP-VAE is capable of distinguishing the original state 3 and 4 in the latent embedding $(z_1,z_2)$, where $z_1$ and $z_2$ denote latent coordinates (distinct from the physical Euclidean $z$-coordinate), effectively recovering the Markovian dynamics. Lower panels (d-i) show column-wise MSM analyis results for each scenario above: (d,g) correspond to the 3D case (a), (e,h) to the $xy$-projection (b), and (f,i) to the GP-VAE embedding in (c). Shown are the corresponding implied timescales, as well as the eigenvectors $\bm{v}$ and the stationary distribution $\bm{\mu}$.
• Figure 3: Sankey diagram illustrating the overlap between the states in the original 3D data (left), those obtained by clustering the latent embedding of the GP-VAE (center) and the $xy-$plane (right). The width of each band indicate the fraction of frames in which both corresponding states temporally coincide. In the GP-VAE embedding, all four states are largely preserved with only minor differences in the transition region of state 1 (which might stem from dynamical coring).
• Figure 4: Structure of T4L, indicating MoSAIC clusters of inter-residue contacts shown as red (C1), blue (C2), yellow (C3) and green (C4) lines. (a) Cluster C1 accounts for the open$\leftrightarrow$closed motion of T4L, spanning from the hinge region (H) to the mouth (M) region, with the most important coordinates, $d_{4,60}$ and $d_{22,137}$ highlighted in blue. (b) The remaining three clusters, C2-C4, describe different processes that are not directly linked to the open$\leftrightarrow$closed motion.
• Figure 5: Two-dimensional free energy landscapes of T4L, with marginal probability distributions shown along each axis. (a) Energy landscape constructed from the two key distances, $d_{4,60}$ and $d_{20,145}$, showing a clear two-state behavior. (b) The landscape of the corresponding GP-VAE embedding reveals a second "open" state ($2_\text{GP}$) that is dynamically disconnected from the open$\leftrightarrow$closed region. (c) Energy landscape of MoSAIC cluster C3, obtained from a principal component analyses of the coordinates of this cluster. The first two components $(x_1^{(3)}$ and $x_2^{(3)})$ are found to clearly separate the two GP states $1_\text{GP}$ and $2_\text{GP}$. Red crosses indicate the points of the the open$\leftrightarrow$closed transitions. (d) Probability distributions of states $1_\text{GP}$ and $2_\text{GP}$ projected onto $(x_1^{(3)}, x_2^{(3)})$ space.