Maximizing Kinetic Information Gain of Markov State Models for Optimal Design of Spectroscopy Experiments

Markov State Models: To Optimize or Not to Optimize

Markov state models (MSM) are a popular statistical method for analyzing the conformational dynamics of proteins including protein folding. With all statistical and machine learning (ML) models, choices must be made about the modeling pipeline that cannot be directly learned from the data. These choices, or hyperparameters, are often evaluated by expert judgment or, in the case of MSMs, by maximizing variational scores such as the VAMP-2 score. Modern ML and statistical pipelines often use automatic hyperparameter selection techniques ranging from the simple, choosing the best score from a random selection of hyperparameters, to the complex, optimization via, e.g., Bayesian optimization. In this work, we ask whether it is possible to automatically select MSM models this way by estimating and analyzing over 16,000,000 observations from over 280,000 estimated MSMs. We find that differences in hyperparameters can change the physical interpretation of the optimization objective, making automatic selection difficult. In addition, we find that enforcing conditions of equilibrium in the VAMP scores can result in inconsistent model selection. However, other parameters that specify the VAMP-2 score (lag time and number of relaxation processes scored) have only a negligible influence on model selection. We suggest that model observables and variational scores should be only a guide to model selection and that a full investigation of the MSM properties should be undertaken when selecting hyperparameters.

Learned Reconstruction of Protein Folding Trajectories from Noisy Single-Molecule Time Series

Single-molecule Förster resonance energy transfer (smFRET) is an experimental methodology to track the real-time dynamics of molecules using fluorescent probes to follow one or more intramolecular distances. These distances provide a low-dimensional representation of the full atomistic dynamics. Under mild technical conditions, Takens' Delay Embedding Theorem guarantees that the full three-dimensional atomistic dynamics of a system are diffeomorphic (i.e., related by a smooth and invertible transformation) to a time-delayed embedding of one or more scalar observables. Appealing to these theoretical guarantees, we employ manifold learning, artificial neural networks, and statistical mechanics to learn from molecular simulation training data the a priori unknown transformation between the atomic coordinates and delay-embedded intramolecular distances accessible to smFRET. This learned transformation may then be used to reconstruct atomistic coordinates from smFRET time series data. We term this approach Single-molecule TAkens Reconstruction (STAR). We have previously applied STAR to reconstruct molecular configurations of a C24H50 polymer chain and the mini-protein Chignolin with accuracies better than 0.2 nm from simulated smFRET data under noise free and high time resolution conditions. In the present work, we investigate the role of signal-to-noise ratio, data volume, and time resolution in simulated smFRET data to assess the performance of STAR under conditions more representative of experimental realities. We show that STAR can reconstruct the Chignolin and Villin mini-proteins to accuracies of 0.12 and 0.42 nm, respectively, and place bounds on these conditions for accurate reconstructions. These results demonstrate that it is possible to reconstruct dynamical trajectories of protein folding from time series in noisy, time binned, experimentally measurable observables and lay the foundations for the application of STAR to real experimental data.

Reconciling membrane protein simulations with experimental DEER spectroscopy data

Spectroscopy experiments are crucial to study membrane proteins for which traditional structure determination methods still prove challenging. Double electron-electron resonance (DEER) spectroscopy experiments provide protein residue-pair distance distributions that are indicative of their conformational heterogeneity. Atomistic molecular dynamics (MD) simulations are another tool that have been proven to be vital to study the structural dynamics of membrane proteins such as to identify inward-open, occluded, and outward-open conformations of transporter membrane proteins, among other partially open or closed states of the protein. Yet, studies have reported that there is no direct consensus between the distributional data from DEER experiments and MD simulations, which has challenged validation of structures obtained from long-timescale simulations and using simulations to design experiments. Current coping strategies for comparisons rely on heuristics, such as mapping the nearest matching peaks between two ensembles or biased simulations. Here we examine the differences in residue-pair distance distributions arising due to the choice of membranes around the protein and covalent modification of a pair of residues to nitroxide spin labels in DEER experiments. Through comparing MD simulations of two proteins, PepT_So and LeuT-both of which have been characterized using DEER experiments previously-we show that the proteins' dynamics are similar despite the choice of the detergent micelle as a membrane mimetic in DEER experiments. On the other hand, covalently modified residues show slight local differences in their dynamics and a huge divergence when the oxygen atom pair distances between spin labeled residues are measured rather than protein backbone distances. Given the computational expense associated with pairwise MTSSL labeled MD simulations, we examine the use of biased simulations to explore the conformational dynamics of the spin labels only to reveal that such simulations alter the underlying protein dynamics. Our study identifies the main cause for the mismatch between DEER experiments and MD simulations and will accelerate the development of potential mitigation strategies to improve the match.

Mechanistic Origin of Partial Agonism of Tetrahydrocannabinol for Cannabinoid Receptors

Cannabinoid receptor 1 (CB₁) is a therapeutically relevant drug target for controlling pain, obesity, and other central nervous system disorders. However, full agonists and antagonists of CB₁ have been reported to cause serious side effects in patients. Therefore, partial agonists have emerged as a viable alternative as they can mitigate overstimulation and side effects. One of the key bottlenecks in the design of partial agonists, however, is the lack of understanding of the molecular mechanism of partial agonism itself. In this study, we examine two mechanistic hypotheses for the origin of partial agonism in cannabinoid receptors and predict the mechanistic basis of partial agonism exhibited by Δ⁹-Tetrahydrocannabinol (THC) against CB₁. In particular, we inspect whether partial agonism emerges from the ability of THC to bind in both agonist and antagonist-binding poses or from its ability to only partially activate the receptor. We used extensive molecular dynamics simulations and Markov state modeling to capture the THC binding in both antagonist and agonist-binding poses in the CB₁ receptor. Furthermore, we predict that binding of THC in the agonist-binding pose leads to rotation of toggle switch residues and causes partial outward movement of intracellular transmembrane helix 6 (TM6). Our simulations also suggest that the alkyl side chain of THC plays a crucial role in determining partial agonism by stabilizing the ligand in the agonist and antagonist-like poses within the pocket. Taken together, this study provides important insights into the mechanistic origin of the partial agonism of THC.

How do antiporters exchange substrates across the cell membrane? An atomic-level description of the complete exchange cycle in NarK

Major facilitator superfamily (MFS) proteins operate via three different mechanisms: uniport, symport, and antiport. Despite extensive investigations, the molecular understanding of antiporters is less advanced than that of other transporters due to the complex coupling between two substrates and the lack of distinct structures. We employ extensive all-atom molecular dynamics simulations to dissect the complete substrate exchange cycle of the bacterial NO₃^-/NO₂^- antiporter, NarK. We show that paired basic residues in the binding site prevent the closure of unbound protein and ensure the exchange of two substrates. Conformational transition occurs only in the presence of substrate, which weakens the electrostatic repulsion and stabilizes the transporter. Furthermore, we propose a state-dependent substrate exchange model, in which the relative spacing between the paired basic residues determines whether NO₃^- and NO₂^- bind simultaneously or sequentially. Overall, this work presents a general working model for the antiport mechanism within the MFS.