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Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

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The association and dissociation of proteins and ligands are crucial in biophysics for potential drug development [Baron and McCammon, Annu. Rev. Phys. Chem. 64, 151-175 (2013)]. However, identifying and characterizing the reaction pathways for these rare events has been a long-standing challenge. Molecular dynamics (MD) simulations are limited in exploring biophysical processes on experimental timescales, so ligand transport processes through complex transient tunnels formed by proteins during dynamics are often simulated using enhanced sampling MD [Rydzewski and Nowak, Phys. Life Rev. 22-23, 58-74 (2017)]. Erroneously identified ligand binding pathways can affect thermodynamic and kinetic characteristics calculated from MD trajectories. A system that has the potential to be a therapeutic target for neurodegenerative diseases is prolyl oligopeptidase (PREP). This is due to its involvement in promoting protein aggregation and disrupting cellular function through affecting protein-protein interactions (PPI). The recent discovery of a new type of PREP inhibitor that targets PPI raises important questions about the diversity of ligand binding pathways in PREP and their impact on protein dynamics [Pätsi et al., J. Med. Chem. 67, 5421-5436 (2024); Kilpeläinen et al., J. Med. Chem. 66, 7475-7496 (2023); and Walczewska-Szewc et al., Phys. Chem. Chem. Phys. 24, 4366-4373 (2022)]. In this article, using results from enhanced sampling MD, we visually present how the binding process in PREP depends on subtle changes in inhibitors, which could be crucial in treating neurodegenerative disorders.

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The long-time behavior of many complex molecular systems can often be described by Markovian dynamics in a slow subspace spanned by a few reaction coordinates referred to as collective variables (CVs). However, determining CVs poses a fundamental challenge in chemical physics. Depending on intuition or trial and error to construct CVs can lead to non-Markovian dynamics with long memory effects, hindering analysis. To address this problem, we continue to develop a recently introduced deep-learning technique called spectral map [J. Rydzewski, J. Phys. Chem. Lett. 14, 5216-5220 (2023)]. Spectral map learns slow CVs by maximizing a spectral gap of a Markov transition matrix describing anisotropic diffusion. Here, to represent heterogeneous and multiscale free-energy landscapes with spectral map, we implement an adaptive algorithm to estimate transition probabilities. Through a Markov state model analysis, we validate that spectral map learns slow CVs related to the dominant relaxation timescales and discerns between long-lived metastable states.

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Diseases spread by mosquitoes lead to the death of 700,000 people each year. The main way to reduce transmission is vector control by biting prevention with chemicals. However, the most commonly used insecticides lose efficacy due to the growing resistance. Voltage-gated sodium channels (VGSCs), membrane proteins responsible for the depolarizing phase of an action potential, are targeted by a broad range of neurotoxins, including pyrethroids and sodium channel blocker insecticides (SCBIs). Reduced sensitivity of the target protein due to the point mutations threatened malaria control with pyrethroids. Although SCBIs-indoxacarb (a pre-insecticide bioactivated to DCJW in insects) and metaflumizone-are used in agriculture only, they emerge as promising candidates in mosquito control. Therefore, a thorough understanding of molecular mechanisms of SCBIs action is urgently needed to break the resistance and stop disease transmission. In this study, by performing an extensive combination of equilibrium and enhanced sampling molecular dynamics simulations (3.2 µs in total), we found the DIII-DIV fenestration to be the most probable entry route of DCJW to the central cavity of mosquito VGSC. Our study revealed that F1852 is crucial in limiting SCBI access to their binding site. Our results explain the role of the F1852T mutation found in resistant insects and the increased toxicity of DCJW compared to its bulkier parent compound, indoxacarb. We also delineated residues that contribute to both SCBIs and non-ester pyrethroid etofenprox binding and thus could be involved in the target site cross-resistance.

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The dynamics of physical systems that require high-dimensional representation can often be captured in a few meaningful degrees of freedom called collective variables (CVs). However, identifying CVs is challenging and constitutes a fundamental problem in physical chemistry. This problem is even more pronounced when CVs need to provide information about slow kinetics related to rare transitions between long-lived metastable states. To address this issue, we propose an unsupervised deep-learning method called spectral map. Our method constructs slow CVs by maximizing the spectral gap between slow and fast eigenvalues of a transition matrix estimated by an anisotropic diffusion kernel. We demonstrate our method in several high-dimensional reversible folding processes.

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Constructing reduced representations of high-dimensional systems is a fundamental problem in physical chemistry. Many unsupervised machine learning methods can automatically find such low-dimensional representations. However, an often overlooked problem is what high-dimensional representation should be used to describe systems before dimensionality reduction. Here, we address this issue using a recently developed method called the reweighted diffusion map [J. Chem. Theory Comput. 2022, 18, 7179-7192]. We show how high-dimensional representations can be quantitatively selected by exploring the spectral decomposition of Markov transition matrices built from data obtained from standard or enhanced sampling atomistic simulations. We demonstrate the performance of the method in several high-dimensional examples.

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Enhanced sampling methods are indispensable in computational chemistry and physics, where atomistic simulations cannot exhaustively sample the high-dimensional configuration space of dynamical systems due to the sampling problem. A class of such enhanced sampling methods works by identifying a few slow degrees of freedom, termed collective variables (CVs), and enhancing the sampling along these CVs. Selecting CVs to analyze and drive the sampling is not trivial and often relies on chemical intuition. Despite routinely circumventing this issue using manifold learning to estimate CVs directly from standard simulations, such methods cannot provide mappings to a low-dimensional manifold from enhanced sampling simulations, as the geometry and density of the learned manifold are biased. Here, we address this crucial issue and provide a general reweighting framework based on anisotropic diffusion maps for manifold learning that takes into account that the learning data set is sampled from a biased probability distribution. We consider manifold learning methods based on constructing a Markov chain describing transition probabilities between high-dimensional samples. We show that our framework reverts the biasing effect, yielding CVs that correctly describe the equilibrium density. This advancement enables the construction of low-dimensional CVs using manifold learning directly from the data generated by enhanced sampling simulations. We call our framework reweighted manifold learning. We show that it can be used in many manifold learning techniques on data from both standard and enhanced sampling simulations.

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The ability of phytochromes to act as photoswitches in plants and microorganisms depends on interactions between a bilin-like chromophore and a host protein. The interconversion occurs between the spectrally distinct red (Pr) and far-red (Pfr) conformers. This conformational change is triggered by the photoisomerization of the chromophore D-ring pyrrole. In this study, as a representative example of a phytochrome-bilin system, we consider biliverdin IXα (BV) bound to bacteriophytochrome (BphP) from Deinococcus radiodurans. In the absence of light, we use an enhanced sampling molecular dynamics (MD) method to overcome the photoisomerization energy barrier. We find that the calculated free energy (FE) barriers between essential metastable states agree with spectroscopic results. We show that the enhanced dynamics of the BV chromophore in BphP contributes to triggering nanometer-scale conformational movements that propagate by two experimentally determined signal transduction pathways. Most importantly, we describe how the metastable states enable a thermal transition known as the dark reversion between Pfr and Pr, through a previously unknown intermediate state of Pfr. We present the heterogeneity of temperature-dependent Pfr states at the atomistic level. This work paves a way toward understanding the complete mechanism of the photoisomerization of a bilin-like chromophore in phytochromes.

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The formation of protein aggregates is one of the leading causes of neuronal malfunction and subsequent brain damage in many neurodegenerative diseases. In Parkinson's disease, α-synucleins are involved in the accumulation of aggregates. The origin of aggregation is unknown, but there is convincing evidence that it can be reduced by prolyl oligopeptidase (PREP) inhibition. This effect cannot simply be related to the inhibition of the enzyme's catalytic function since not all PREP inhibitors stop α-synuclein aggregation. Finding differences in the dynamics of the enzyme inhibited by different compounds would allow us to identify the protein regions involved in the interaction between PREP and α-synuclein. Here, we investigate the effects of three PREP inhibitors, each of which affects α-synuclein aggregation to a different extent. We use molecular dynamics modelling to identify the molecular mechanisms underlying PREP inhibition and find structural differences between inhibitor-PREP systems. We suggest that even subtle variations in enzyme dynamics affect its interactions with α-synucleins. Our identification of these regions may therefore be biologically relevant in preventing α-synuclein aggregate formation.

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Machine learning methods provide a general framework for automatically finding and representing the essential characteristics of simulation data. This task is particularly crucial in enhanced sampling simulations. There we seek a few generalized degrees of freedom, referred to as collective variables (CVs), to represent and drive the sampling of the free energy landscape. In theory, these CVs should separate different metastable states and correspond to the slow degrees of freedom of the studied physical process. To this aim, we propose a new method that we call multiscale reweighted stochastic embedding (MRSE). Our work builds upon a parametric version of stochastic neighbor embedding. The technique automatically learns CVs that map a high-dimensional feature space to a low-dimensional latent space via a deep neural network. We introduce several new advancements to stochastic neighbor embedding methods that make MRSE especially suitable for enhanced sampling simulations: (1) weight-tempered random sampling as a landmark selection scheme to obtain training data sets that strike a balance between equilibrium representation and capturing important metastable states lying higher in free energy; (2) a multiscale representation of the high-dimensional feature space via a Gaussian mixture probability model; and (3) a reweighting procedure to account for training data from a biased probability distribution. We show that MRSE constructs low-dimensional CVs that can correctly characterize the different metastable states in three model systems: the Müller-Brown potential, alanine dipeptide, and alanine tetrapeptide.

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The catalytic effects of complex minerals or meteorites are often mentioned as important factors for the origins of life. To assess the possible role of nanoconfinement within a catalyst consisting of montmorillonite (MMT) and the impact of local electric field on the formation efficiency of the simple hypothetical precursors of nucleic acid bases or amino acids, we performed ab initio Car-Parrinello molecular dynamics simulations. We prepared four condensed-phase systems corresponding to previously suggested prototypes of a primordial soup. We monitored possible chemical reactions occurring within gas-like bulk and MMT-confined four simulation boxes on a 20-ps time scale at 1 atm and 300 K, 400 K, and 600 K. Elevated temperatures did not affect the reactivity of the elementary components of the gas-like boxes considerably; however, the presence of the MMT nanoclay substantially increased the formation probability of new molecules. Approximately 20 different new compounds were found in boxes containing carbon monoxide or formaldehyde molecules. This observation and an analysis of the atom-atom radial distribution functions indicated that the presence of Ca2+ ions at the surface of the internal MMT cavities may be an important factor in the initial steps of the formation of complex molecules at the early stages of the Earth's history.

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Searching for reaction pathways describing rare events in large systems presents a long-standing challenge in chemistry and physics. Incorrectly computed reaction pathways result in the degeneracy of microscopic configurations and inability to sample hidden energy barriers. To this aim, we present a general enhanced sampling method to find multiple diverse reaction pathways of ligand unbinding through nonconvex optimization of a loss function describing ligand-protein interactions. The method successfully overcomes large energy barriers using an adaptive bias potential and constructs possible reaction pathways along transient tunnels without the initial guesses of intermediate or final states, requiring crystallographic information only. We examine the method on the T4 lysozyme L99A mutant which is often used as a model system to study ligand binding to proteins, provide a previously unknown reaction pathway, and show that by using the bias potential and the tunnel widths, it is possible to capture heterogeneity of the unbinding mechanisms between the found transient protein tunnels.

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The protein structure refinement using conformational sampling is important in hitherto protein studies. In this paper, we examined the protein structure refinement by means of potential energy minimization using immune computing as a method of sampling conformations. The method was tested on the x-ray structure and 30 decoys of the mutant of [Leu]Enkephalin, a paradigmatic example of the biomolecular multiple-minima problem. In order to score the refined conformations, we used a standard potential energy function with the OPLSAA force field. The effectiveness of the search was assessed using a variety of methods. The robustness of sampling was checked by the energy yield function which measures quantitatively the number of the peptide decoys residing in an energetic funnel. Furthermore, the potential energy-dependent Pareto fronts were calculated to elucidate dissimilarities between peptide conformations and the native state as observed by x-ray crystallography. Our results showed that the probed potential energy landscape of [Leu]Enkephalin is self-similar on different metric scales and that the local potential energy minima of the peptide decoys are metastable, thus they can be refined to conformations whose potential energy is decreased by approximately 250 kJ/mol.

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The brain activity is to a large extent determined by states of neural cortex microcircuits. Unfortunately, accuracy of results from neural circuits׳ mathematical models is often biased by the presence of uncertainties in underlying experimental data. Moreover, due to problems with uncertainties identification in a multidimensional parameters space, it is almost impossible to classify states of the neural cortex, which correspond to a particular set of the parameters. Here, we develop a complete methodology for determining uncertainties and the novel protocol for classifying all states in any neuroinformatic model. Further, we test this protocol on the mathematical, nonlinear model of such a microcircuit developed by Giugliano et al. (2008) and applied in the experimental data analysis of Huntington׳s disease. Up to now, the link between parameter domains in the mathematical model of Huntington׳s disease and the pathological states in cortical microcircuits has remained unclear. In this paper we precisely identify all the uncertainties, the most crucial input parameters and domains that drive the system into an unhealthy state. The scheme proposed here is general and can be easily applied to other mathematical models of biological phenomena.

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