Time-lagged Independent Component Analysis Research Articles

Leveraging Machine Learning-Guided Molecular Simulations Coupled with Experimental Data to Decipher Membrane Binding Mechanisms of Aminosterols.

Understanding the molecular mechanisms of the interactions between specific compounds and cellular membranes is essential for numerous biotechnological applications, including targeted drug delivery, elucidation of the drug mechanism of action, pathogen identification, and novel antibiotic development. However, estimation of the free energy landscape associated with solute binding to realistic biological systems is still a challenging task. In this work, we leverage the Time-lagged Independent Component Analysis (TICA) in combination with neural networks (NN) through the Deep-TICA approach for determining the free energy associated with the membrane insertion processes of two natural aminosterol compounds, trodusquemine (TRO), and squalamine (SQ). These compounds are particularly noteworthy because they interact with the outer layer of neuron membranes, protecting them from the toxic action of misfolded proteins involved in neurodegenerative disorders, in both their monomeric and oligomeric forms. We demonstrate how this strategy could be used to generate an effective collective variable for describing solute absorption in the membrane and for estimating free energy landscape of translocation via on-the-fly probability enhanced sampling (OPES) method. In this context, the computational protocol allowed an exhaustive characterization of the aminosterol entry pathway into a neuron-like lipid bilayer. Furthermore, it provided accurate prediction of membrane binding affinities, in close agreement with the experimental binding data obtained by using fluorescently labeled aminosterols and large unilamellar vesicles (LUVs). The findings contribute significantly to our understanding of aminosterol entry pathways and aminosterol-lipid membrane interactions. Finally, the computational methods deployed in this study further demonstrate considerable potential for investigating membrane binding processes.

Cooperative Membrane Binding of HIV-1 Matrix Proteins.

The HIV-1 assembly process begins with a newly synthesized Gag polyprotein being targeted to the inner leaflet of the plasma membrane of the infected cells to form immature viral particles. Gag-membrane interactions are mediated through the myristoylated (Myr) N-terminal matrix (MA) domain of Gag, which eventually multimerize on the membrane to form trimers and higher order oligomers. The study of the structure and dynamics of peripheral membrane proteins like MA has been challenging for both experimental and computational studies due to the complex transient dynamics of protein-membrane interactions. Although the roles of anionic phospholipids (PIP2, PS) and the Myr group in the membrane targeting and stable membrane binding of MA are now well-established, the cooperative interactions between the MA monomers and MA-membrane remain elusive in the context of viral assembly and release. Our present study focuses on the membrane binding dynamics of a higher order oligomeric structure of MA protein (a dimer of trimers), which has not been explored before. Employing time-lagged independent component analysis (tICA) to our microsecond-long trajectories, we investigate conformational changes of the matrix protein induced by membrane binding. Interestingly, the Myr switch of an MA monomer correlates with the conformational switch of adjacent monomers in the same trimer. Together, our findings suggest complex protein dynamics during the formation of the immature HIV-1 lattice; while MA trimerization facilitates Myr insertion, MA trimer-trimer interactions in the immature lattice can hinder the same.

Machine Learning Deciphered Molecular Mechanistics with Accurate Kinetic and Thermodynamic Prediction.

Time-lagged independent component analysis (tICA) and the Markov state model (MSM) have been extensively employed for extracting conformational dynamics and kinetic community networks from unbiased trajectory ensembles. However, these techniques may not be the optimal choice for elucidating transition mechanisms within low-dimensional representations, especially for intricate biosystems. Unraveling the association mechanism in such complex systems always necessitates permutations of several essential independent components or collective variables, a process that is inherently obscure and may require empirical knowledge for selection. To address these challenges, we have implemented an integrated unsupervised dimension reduction model: uniform manifold approximation and projection (UMAP) with hierarchy density-based spatial clustering of applications with noise (HDBSCAN). This approach effectively generates low-dimensional configurational embeddings. The hierarchical application of this architecture, in conjunction with MSM, reveals global kinetic connectivity while identifying local conformational states. Consequently, our methodology establishes a multiscale mechanistic elucidation framework. Leveraging the benefits of the uniform sample distribution and a denoising approach, our model demonstrates robustness in preserving global and local data structures compared to traditional dimension reduction methods in the field of MD analysis area. The interpretability of hyperparameter selection and compatibility with downstream tasks are cross-validated across various simulation data sets, utilizing both computational evaluation metrics and experimental kinetic observables. Furthermore, the predicted Mcl1-BH3 association kinetics (0.76 s-1) is in close agreement with surface plasmon resonance experiments (0.12 s-1), affirming the plausibility of the identified pathway composed of representative conformations. We anticipate that the devised workflow will serve as a foundational framework for studying recognition patterns in complex biological systems. Its contributions extend to the exploration of protein functional dynamics and rational drug design, offering a potent avenue for advancing research in these domains.

Coupling of conformational dynamics and inhibitor binding in the phosphodiesterase-5 family.

Phosphodiesterase-5 (PDE5) is responsible for regulating the concentration of the second messenger molecule cGMP by hydrolyzing it into 5'-GMP. PDE5 is implicated in erectile dysfunction and cardiovascular diseases. The substrate binding site in the catalytic domain of PDE5 is surrounded by several dynamic structural motifs (including the α14 helix, M-loop, and H-loop) that are known to switch between inactive and active conformational states via currently unresolved structural intermediates. We evaluated the conformational dynamics of these structural motifs in the apo state and upon binding of an allosteric inhibitor (evodiamine) or avanafil, a competitive inhibitor. We employed enhanced sampling-based replica exchange solute scaling (REST2) method, principal component analysis (PCA), time-lagged independent component analysis (tICA), molecular dynamics (MD) simulations, and well-tempered metadynamics simulations to probe the conformational changes in these structural motifs. Our results support a regulatory mechanism for PDE5, where the α14 helix alternates between an inward (lower activity) conformation and an outward (higher activity) conformation that is accompanied by the folding/unfolding of the α8' and α8″ helices of the H-loop. When the allosteric inhibitor evodiamine is bound to PDE5, the inward (inactive) state of the α14 helix is preferred, thus preventing substrate access to the catalytic site. In contrast, competitive inhibitors of PDE5 block catalysis by occupying the active site accompanied by stabilization of the outward conformation of the α14 helix. Defining the conformational dynamics underlying regulation of PDE5 activation will be helpful in rational design of next-generation small molecules modulators of PDE5 activity.

Predicting residue cooperativity during protein folding: A combined, molecular dynamics and unsupervised learning approach.

Allostery in proteins involves, broadly speaking, ligand-induced conformational transitions that modulate function at active sites distal to where the ligand binds. In contrast, the concept of cooperativity (in the sense used in phase transition theory) is often invoked to understand protein folding and, therefore, function. The modern view on allostery is one based on dynamics and hinges on the time-dependent interactions between key residues in a complex network, interactions that determine the free-energy profile for the reaction at the distal site. Here, we merge allostery and cooperativity, and we discuss a joint model with features of both. In our model, the active-site reaction is replaced by the reaction pathway that leads to protein folding, and the presence or absence of the effector is replaced by mutant-vs-wild type changes in key residues. To this end, we employ our recently introduced time-lagged independent component analysis (tICA) correlation approach [Ray et al. Proc. Natl. Acad. Sci. 118(43) (2021), e2100943118] to identify the allosteric role of distant residues in the folded-state dynamics of a large protein. In this work, we apply the technique to identify key residues that have a significant role in the folding of a small, fast folding-protein, chignolin. Using extensive enhanced sampling simulations, we critically evaluate the accuracy of the predictions by mutating each residue one at a time and studying how the mutations change the underlying free energy landscape of the folding process. We observe that mutations in those residues whose associated backbone torsion angles have a high correlation score can indeed lead to loss of stability of the folded configuration. We also provide a rationale based on interaction energies between individual residues with the rest of the protein to explain this effect. From these observations, we conclude that the tICA correlation score metric is a useful tool for predicting the role of individual residues in the correlated dynamics of proteins and can find application to the problem of identifying regions of protein that are either most vulnerable to mutations or-mutatis mutandis-to binding events that affect their functionality.

A variational approach to assess reaction coordinates for two-step crystallization.

Molecule- and particle-based simulations provide the tools to test, in microscopic detail, the validity of classical nucleation theory. In this endeavor, determining nucleation mechanisms and rates for phase separation requires an appropriately defined reaction coordinate to describe the transformation of an out-of-equilibrium parent phase for which myriad options are available to the simulator. In this article, we describe the application of the variational approach to Markov processes to quantify the suitability of reaction coordinates to study crystallization from supersaturated colloid suspensions. Our analysis indicates that collective variables (CVs) that correlate with the number of particles in the condensed phase, the system potential energy, and approximate configurational entropy often feature as the most appropriate order parameters to quantitatively describe the crystallization process. We apply time-lagged independent component analysis to reduce high-dimensional reaction coordinates constructed from these CVs to build Markov State Models (MSMs), which indicate that two barriers separate a supersaturated fluid phase from crystals in the simulated environment. The MSMs provide consistent estimates for crystal nucleation rates, regardless of the dimensionality of the order parameter space adopted; however, the two-step mechanism is only consistently evident from spectral clustering of the MSMs in higher dimensions. As the method is general and easily transferable, the variational approach we adopt could provide a useful framework to study controls for crystal nucleation.

Protein folding pathway and intermediates on the dimensionality reduced landscape with UMAP and native contact likelihood.

To understand protein folding mechanisms from molecular dynamics (MD) simulations, it is important to explore not only folded/unfolded states but also representative intermediate structures on the conformational landscape. Generally, such a conformational landscape is obtained by linear dimensionality reduction methods, principal component analysis (PCA) or time-lagged independent component analysis (tICA), for instance. However, it is often difficult to apply these linear methods to large-scale, nonlinear conformational changes such as protein folding. Here, we propose a novel approach to construct the landscape using the uniform manifold approximation and projection (UMAP) method, which nonlinearly reduces the dimensionality without losing data-point proximity. In the approach, native contact likelihood, a measure for native contact formation, is used as feature variables rather than the conventional Cartesian coordinates or dihedral angles of protein structures. We tested the performance of UMAP for coarse-grained MD simulation trajectories of B1 domain in protein G and observed on-pathway transient structures and other metastable states on the UMAP conformational landscape. In contrast, these structures were not clearly distinguished on the dimensionality reduced landscape using PCA or tICA. This approach is also useful to obtain dynamical information through Markov State Modeling and would be applicable to large-scale conformational changes in many other biomacromolecules.

Protein folding intermediates on the dimensionality reduced landscape with UMAP and native contact likelihood

To understand protein folding mechanisms from molecular dynamics (MD) simulations, it is important to explore not only folded/unfolded states but also representative intermediate structures on the conformational landscape. Here, we propose a novel approach to construct the landscape using the uniform manifold approximation and projection (UMAP) method, which reduces the dimensionality without losing data-point proximity. In the approach, native contact likelihood is used as feature variables rather than the conventional Cartesian coordinates or dihedral angles of protein structures. We tested the performance of UMAP for coarse-grained MD simulation trajectories of B1 domain in protein G and observed on-pathway transient structures and other metastable states on the UMAP conformational landscape. In contrast, these structures were not clearly distinguished on the dimensionality reduced landscape using principal component analysis or time-lagged independent component analysis. This approach is also useful to obtain dynamical information through Markov state modeling and would be applicable to large-scale conformational changes in many other biomacromolecules.

Correlation-Based Feature Selection to Identify Functional Dynamics in Proteins.

To interpret molecular dynamics simulations of biomolecular systems, systematic dimensionality reduction methods are commonly employed. Among others, this includes principal component analysis (PCA) and time-lagged independent component analysis (TICA), which aim to maximize the variance and the time scale of the first components, respectively. A crucial first step of such an analysis is the identification of suitable and relevant input coordinates (the so-called features), such as backbone dihedral angles and interresidue distances. As typically only a small subset of those coordinates is involved in a specific biomolecular process, it is important to discard the remaining uncorrelated motions or weakly correlated noise coordinates. This is because they may exhibit large amplitudes or long time scales and therefore will be erroneously considered important by PCA and TICA, respectively. To discriminate collective motions underlying functional dynamics from uncorrelated motions, the correlation matrix of the input coordinates is block-diagonalized by a clustering method. This strategy avoids possible bias due to presumed functional observables and conformational states or variation principles that maximize variance or time scales. Considering several linear and nonlinear correlation measures and various clustering algorithms, it is shown that the combination of linear correlation and the Leiden community detection algorithm yields excellent results for all considered model systems. These include the functional motion of T4 lysozyme to demonstrate the successful identification of collective motion, as well as the folding of the villin headpiece to highlight the physical interpretation of the correlated motions in terms of a functional mechanism.

Employing Artificial Neural Networks to Identify Reaction Coordinates and Pathways for Self-Assembly.

Capturing the autonomous self-assembly of molecular building blocks in computer simulations is a persistent challenge, requiring to model complex interactions and to access long time scales. Advanced sampling methods allow to bridge these time scales but typically need accurate low-dimensional representations of the transition pathways. In this work, we demonstrate for the self-assembly of two single-stranded DNA fragments into a ring-like structure how autoencoder neural networks can be employed to reliably provide a suitable low-dimensional representation and to expose transition pathways: The assembly proceeds through a two-step process with two distinct half-bound states, which are correctly identified by the neural net. We exploit this latent space representation to construct a Markov state model for predicting the four molecular conformations and their transition rates. We present a detailed comparison with two other low-dimensional representations based on empirically determined order parameters and a time-lagged independent component analysis (TICA). Our work opens up new avenues for the computational modeling of multistep and hierarchical self-assembly, which has proven challenging so far.

Expanding the Accessible Chemical Space of SIRT2 Inhibitors through Exploration of Binding Pocket Dynamics.

Considerations of binding pocket dynamics are one of the crucial aspects of the rational design of binders. Identification of alternative conformational states or cryptic subpockets could lead to the discovery of completely novel groups of the ligands. However, experimental characterization of pocket dynamics, besides being expensive, may not be able to elucidate all of the conformational states relevant for drug discovery projects. In this study, we propose the protocol for computational simulations of sirtuin 2 (SIRT2) binding pocket dynamics and its integration into the structure-based virtual screening (SBVS) pipeline. Initially, unbiased molecular dynamics simulations of SIRT2:inhibitor complexes were performed using optimized force field parameters of SIRT2 inhibitors. Time-lagged independent component analysis (tICA) was used to design pocket-related collective variables (CVs) for enhanced sampling of SIRT2 pocket dynamics. Metadynamics simulations in the tICA eigenvector space revealed alternative conformational states of the SIRT2 binding pocket and the existence of a cryptic subpocket. Newly identified SIRT2 conformational states outperformed experimentally resolved states in retrospective SBVS validation. After performing prospective SBVS, compounds from the under-represented portions of the SIRT2 inhibitor chemical space were selected for in vitro evaluation. Two compounds, NDJ18 and NDJ85, were identified as potent and selective SIRT2 inhibitors, which validated the in silico protocol and opened up the possibility for generalization and broadening of its application. The anticancer effects of the most potent compound NDJ18 were examined on the triple-negative breast cancer cell line. Results indicated that NDJ18 represents a promising structure suitable for further evaluation.

Uncertainty in Markov state models for small macromolecules

Markov state models (MSMs) are a tool to describe and analyse protein dynamics. They are useful in particular to determine characteristic timescales of protein motion. For larger biomolecules such as proteins it is challenging to obtain sufficient sampling for the timescales to converge, which is thus a frequent concern. There are, however, also several other sources of uncertainty, such as choice of lag time and the number of dimensions in the dimension reduction preprocessing step (e.g. time-lagged independent component analysis), the number of Markov states, or choice of the MSM lag time.

Distant residues modulate conformational opening in SARS-CoV-2 spike protein

Infection by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) involves the attachment of the receptor-binding domain (RBD) of its spike proteins to the ACE2 receptors on the peripheral membrane of host cells. Binding is initiated by a down-to-up conformational change in the spike protein, the change that presents the RBD to the receptor. To date, computational and experimental studies that search for therapeutics have concentrated, for good reason, on the RBD. However, the RBD region is highly prone to mutations, and is therefore a hotspot for drug resistance. In contrast, we here focus on the correlations between the RBD and residues distant to it in the spike protein. This allows for a deeper understanding of the underlying molecular recognition events and prediction of the highest-effect key mutations in distant, allosteric sites, with implications for therapeutics. Also, these sites can appear in emerging mutants with possibly higher transmissibility and virulence, and preidentifying them can give clues for designing pan-coronavirus vaccines against future outbreaks. Our model, based on time-lagged independent component analysis (tICA) and protein graph connectivity network, is able to identify multiple residues that exhibit long-distance coupling with the RBD opening. Residues involved in the most ubiquitous D614G mutation and the A570D mutation of the highly contagious UK SARS-CoV-2 variant are predicted ab initio from our model. Conversely, broad-spectrum therapeutics like drugs and monoclonal antibodies can target these key distant-but-conserved regions of the spike protein.

Time-Lagged Independent Component Analysis of Random Walks and Protein Dynamics.

Time-lagged independent component analysis (tICA) is a widely used dimension reduction method for the analysis of molecular dynamics (MD) trajectories and has proven particularly useful for the construction of protein dynamics Markov models. It identifies those “slow” collective degrees of freedom onto which the projections of a given trajectory show maximal autocorrelation for a given lag time. Here we ask how much information on the actual protein dynamics and, in particular, the free energy landscape that governs these dynamics the tICA-projections of MD-trajectories contain, as opposed to noise due to the inherently stochastic nature of each trajectory. To answer this question, we have analyzed the tICA-projections of high dimensional random walks using a combination of analytical and numerical methods. We find that the projections resemble cosine functions and strongly depend on the lag time, exhibiting strikingly complex behavior. In particular, and contrary to previous studies of principal component projections, the projections change noncontinuously with increasing lag time. The tICA-projections of selected 1 μs protein trajectories and those of random walks are strikingly similar, particularly for larger proteins, suggesting that these trajectories contain only little information on the energy landscape that governs the actual protein dynamics. Further the tICA-projections of random walks show clusters very similar to those observed for the protein trajectories, suggesting that clusters in the tICA-projections of protein trajectories do not necessarily reflect local minima in the free energy landscape. We also conclude that, in addition to the previous finding that certain ensemble properties of nonconverged protein trajectories resemble those of random walks; this is also true for their time correlations.

Integrated Variational Approach to Conformational Dynamics: A Robust Strategy for Identifying Eigenfunctions of Dynamical Operators.

One approach to analyzing the dynamics of a physical system is to search for long-lived patterns in its motions. This approach has been particularly successful for molecular dynamics data, where slowly decorrelating patterns can indicate large-scale conformational changes. Detecting such patterns is the central objective of the variational approach to conformational dynamics (VAC), as well as the related methods of time-lagged independent component analysis and Markov state modeling. In VAC, the search for slowly decorrelating patterns is formalized as a variational problem solved by the eigenfunctions of the system's transition operator. VAC computes solutions to this variational problem by optimizing a linear or nonlinear model of the eigenfunctions using time series data. Here, we build on VAC's success by addressing two practical limitations. First, VAC can give poor eigenfunction estimates when the lag time parameter is chosen poorly. Second, VAC can overfit when using flexible parametrizations such as artificial neural networks with insufficient regularization. To address these issues, we propose an extension that we call integrated VAC (IVAC). IVAC integrates over multiple lag times before solving the variational problem, making its results more robust and reproducible than VAC's.

Conformational and Dynamical Effects of Tyr32 Phosphorylation in K-Ras: Molecular Dynamics Simulation and Markov State Models Analysis.

Phosphorylation of tyrosine 32 in K-Ras has been shown to influence function by disrupting the GTPase cycle. To shed light on the underlying mechanism and atomic basis of this process, we carried out a comparative investigation of the oncogenic G12D K-Ras mutant and its phosphorylated variant (pTyr32) using all-atom molecular dynamics simulations and Markov state models. We show that, despite sharing a number of common features, G12D and pTyr32-G12D K-Ras exhibit some distinct conformational states and fluctuations. In addition to notable differences in conformation and dynamics of residues surrounding the GTP binding site, nonlocal changes were observed at a number of loops. Switch I is more flexible in pTyr32-G12D K-Ras while switch II is more flexible in G12D K-Ras. We also used time-lagged independent component analysis and k-means clustering to identify five metastable states for each system. We utilized transition path theory to calculate the transition probabilities for each state to build a Markov state model for each system. These models and other close inspections suggest that the phosphorylation of Tyr32 strongly affects protein dynamics and the active site conformation, especially with regards to the canonical switch conformations and dynamics.

Identification of kinetic order parameters for non-equilibrium dynamics.

A popular approach to analyze the dynamics of high-dimensional many-body systems, such as macromolecules, is to project the trajectories onto a space of slowly varying collective variables, where subsequent analyses are made, such as clustering or estimation of free energy profiles or Markov state models. However, existing "dynamical" dimension reduction methods, such as the time-lagged independent component analysis (TICA), are only valid if the dynamics obeys detailed balance (microscopic reversibility) and typically require long, equilibrated simulation trajectories. Here, we develop a dimension reduction method for non-equilibrium dynamics based on the recently developed Variational Approach for Markov Processes (VAMP) by Wu and Noé. VAMP is illustrated by obtaining a low-dimensional description of a single file ion diffusion model and by identifying long-lived states from molecular dynamics simulations of the KcsA channel protein in an external electrochemical potential. This analysis provides detailed insights into the coupling of conformational dynamics, the configuration of the selectivity filter, and the conductance of the channel. We recommend VAMP as a replacement for the less general TICA method.

Improving collective variables: The case of crystallization.

Several enhanced sampling methods, such as umbrella sampling or metadynamics, rely on the identification of an appropriate set of collective variables. Recently two methods have been proposed to alleviate the task of determining efficient collective variables. One is based on linear discriminant analysis; the other is based on a variational approach to conformational dynamics and uses time-lagged independent component analysis. In this paper, we compare the performance of these two approaches in the study of the homogeneous crystallization of two simple metals. We focus on Na and Al and search for the most efficient collective variables that can be expressed as a linear combination of X-ray diffraction peak intensities. We find that the performances of the two methods are very similar. Wherever the different metastable states are well-separated, the method based on linear discriminant analysis, based on its harmonic version, is to be preferred because simpler to implement and less computationally demanding. The variational approach, however, has the potential to discover the existence of different metastable states.

A kernel-based approach to molecular conformation analysis.

We present a novel machine learning approach to understand conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov state models, extended dynamic mode decomposition (EDMD), and time-lagged independent component analysis (TICA) can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular, the alanine dipeptide and the protein NTL9.

Mapping the Ligand Binding Landscape

The interaction between a ligand and a protein involves a multitude of conformational states. To achieve a particular deeply bound pose, the ligand must search across a rough free-energy landscape with many metastable minima. Creating maps of the ligand binding landscape is a great challenge, as binding and release events typically occur on timescales that are beyond the reach of molecular simulation. The WExplore enhanced sampling method is well suited to build these maps because it is designed to broadly explore free-energy landscapes and is capable of simulating ligand release pathways that occur on timescales as long as minutes. WExplore also uses only unbiased trajectory segments, allowing for the construction of Markov state models (MSMs) and conformation space networks that combine the results of multiple simulations. Here, we use WExplore to study two bromodomain-inhibitor systems using multiple docked starting poses (Brd4-MS436 and Baz2B-ICR7) and synthesize our results using a series of MSMs using time-lagged independent component analysis. Ranking the starting poses by exit rate agrees with the crystal structure pose in both cases. We also predict the most stable pose using the equilibrium populations from the MSM but find that the prediction is not robust as a function of MSM parameters. The simulated trajectories are synthesized into network models that visualize the entire binding landscape for each system, and we examine transition paths between deeply bound stable states. We find that, on average, transitions between deeply bound states convert through the unbound state 81% of the time, implying a trial-and-error approach to ligand binding. We conclude with a discussion of the implications of this result for both kinetics-based drug discovery and virtual screening pipelines that incorporate molecular dynamics.