Markov models and dynamical fingerprints: Unraveling the complexity of molecular kinetics

Mechanism of β-hairpin formation in AzoChignolin and Chignolin

AzoChignolin is a photoswitchable variant of the mini-protein Chignolin with an azobenzene (AMPP) replacing the central loop. AzoChignolin is unfolded with AMPP in the trans-isomer. Transition to the cis-isomer causes β-hairpin folding similar to Chignolin. The AzoChignolin system is excellently suited for comprehensive analysis of folding nucleation kinetics. Utilizing multiple long-time MD simulations of AzoChignolin and Chignolin in MeOH and water, we estimated Markov models to examine folding kinetics of both peptides. We show that while AzoChignolin mimics Chignolin's structure well, the folding kinetics are quite different. Not only folding times but also intermediate states differ, particularly Chignolin is able to fold in MeOH into an α-helical intermediate which is impossible to form in AzoChignolin. The Markov models demonstrate that AzoChignolin's kinetics are generally faster, specifically when comparing the two main microfolding processes of hydrophobic collapse and turn formation. Photoswitchable loops are used frequently to understand the kinetics of elementary protein folding nucleation. However, our results indicate that intermediates and folding kinetics may differ between natural loops and photoswitchable variants.

Identifying the leading dynamics of ubiquitin: A comparison between the tICA and the LE4PD slow fluctuations in amino acids' position

Molecular Dynamics (MD) simulations of proteins implicitly contain the information connecting the atomistic molecular structure and proteins' biologically relevant motion, where large-scale fluctuations are deemed to guide folding and function. In the complex multiscale processes described by MD trajectories, it is difficult to identify, separate, and study those large-scale fluctuations. This problem can be formulated as the need to identify a small number of collective variables that guide the slow kinetic processes. The most promising method among the ones used to study the slow leading processes in proteins' dynamics is the time-structure based on time-lagged independent component analysis (tICA), which identifies the dominant components in a noisy signal. Recently, we developed an anisotropic Langevin approach for the dynamics of proteins, called the anisotropic Langevin Equation for Protein Dynamics or LE4PD-XYZ. This approach partitions the protein's MD dynamics into mostly uncorrelated, wavelength-dependent, diffusive modes. It associates with each mode a free-energy map, where one measures the spatial extension and the time evolution of the mode-dependent, slow dynamical fluctuations. Here, we compare the tICA modes' predictions with the collective LE4PD-XYZ modes. We observe that the two methods consistently identify the nature and extension of the slowest fluctuation processes. The tICA separates the leading processes in a smaller number of slow modes than the LE4PD does. The LE4PD provides time-dependent information at short times and a formal connection to the physics of the kinetic processes that are missing in the pure statistical analysis of tICA.

Progress in deep Markov state modeling: Coarse graining and experimental data restraints

Recent advances in deep learning frameworks have established valuable tools for analyzing the long-timescale behavior of complex systems, such as proteins. In particular, the inclusion of physical constraints, e.g., time-reversibility, was a crucial step to make the methods applicable to biophysical systems. Furthermore, we advance the method by incorporating experimental observables into the model estimation showing that biases in simulation data can be compensated for. We further develop a new neural network layer in order to build a hierarchical model allowing for different levels of details to be studied. Finally, we propose an attention mechanism, which highlights important residues for the classification into different states. We demonstrate the new methodology on an ultralong molecular dynamics simulation of the Villin headpiece miniprotein.

X-Entropy: A Parallelized Kernel Density Estimator with Automated Bandwidth Selection to Calculate Entropy

X-Entropy is a Python package used to calculate the entropy of a given distribution, in this case, based on the distribution of dihedral angles. The dihedral entropy facilitates an alignment-independent measure of local protein flexibility. The key feature of our approach is a Gaussian kernel density estimation (KDE) using a plug-in bandwidth selection, which is fully implemented in a C++ backend and parallelized with OpenMP. We further provide a Python frontend, with predefined wrapper functions for classical coordinate-based dihedral entropy calculations, using a 1D approximation. This makes the package very straightforward to include in any Python-based analysis workflow. Furthermore, the frontend allows full access to the C++ backend, so that the KDE can be used on any binnable one-dimensional input data. In this application note, we discuss implementation and usage details and illustrate potential applications. In particular, we benchmark the performance of our module in calculating the entropy of samples drawn from a Gaussian distribution and the analytical solution thereof. Further, we analyze the computational performance of this module compared to well-established python libraries that perform KDE analyses. X-Entropy is available free of charge on GitHub (https://github.com/liedllab/X-Entropy).

Catalytic Site pKa Values of Aspartic, Cysteine, and Serine Proteases: Constant pH MD Simulations

Enzymatic function and activity of proteases is closely controlled by the pH value. The protonation states of titratable residues in the active site react to changes in the pH value, according to their pK_a, and thereby determine the functionality of the enzyme. Knowledge of the titration behavior of these residues is crucial for the development of drugs targeting the active site residues. However, experimental pK_a data are scarce, since the systems’ size and complexity make determination of these pK_a values inherently difficult. In this study, we use single pH constant pH MD simulations as a fast and robust tool to estimate the active site pK_a values of a set of aspartic, cysteine, and serine proteases. We capture characteristic pK_a shifts of the active site residues, which dictate the experimentally determined activity profiles of the respective protease family. We find clear differences of active site pK_a values within the respective families, which closely match the experimentally determined pH preferences of the respective proteases. These shifts are caused by a distinct network of electrostatic interactions characteristic for each protease family. While we find convincing agreement with experimental data for serine and aspartic proteases, we observe clear deficiencies in the description of the titration behavior of cysteines within the constant pH MD framework and highlight opportunities for improvement. Consequently, with this work, we provide a concise set of active site pK_a values of aspartic and serine proteases, which could serve as reference for future theoretical as well as experimental studies.

Dynamical matrix propagator scheme for large-scale proton dynamics simulations

We derive a matrix formalism for the simulation of long range proton dynamics for extended systems and timescales. On the basis of an ab initio molecular dynamics simulation, we construct a Markov chain, which allows us to store the entire proton dynamics in an M × M transition matrix (where M is the number of oxygen atoms). In this article, we start from common topology features of the hydrogen bond network of good proton conductors and utilize them as constituent constraints of our dynamic model. We present a thorough mathematical derivation of our approach and verify its uniqueness and correct asymptotic behavior. We propagate the proton distribution by means of transition matrices, which contain kinetic data from both ultra-short (sub-ps) and intermediate (ps) timescales. This concept allows us to keep the most relevant features from the microscopic level while effectively reaching larger time and length scales. We demonstrate the applicability of the transition matrices for the description of proton conduction trends in proton exchange membrane materials.

pH-Dependent Protonation of the Phl p 6 Pollen Allergen Studied by NMR and cpH-aMD

We use state-of-the-art NMR experiments to measure apparent pK_a values in the native protein environment and employ a cutting-edge combination of enhanced sampling and constant pH molecular dynamics (MD) simulations to rationalize strong pK_a shifts. The major timothy grass pollen allergen Phl p 6 serves as an ideal model system for both methods due to its high number of titratable residues despite its comparably small size. We present a proton transition analysis as intuitive tool to depict the captured protonation state ensemble in atomistic detail. Combining microscopic structural details from MD simulations and macroscopic ensemble averages from NMR shifts leads to a comprehensive view on pH dependencies of protonation states and tautomers. Overall, we find striking agreement between simulation-based pK_a predictions and experiment. However, our analyses suggest subtle differences in the underlying molecular origin of the observed pK_a shifts. From accelerated constant pH MD simulations, we identify immediate proximity of opposite charges, followed by vicinity of equal charges as major driving forces for pK_a shifts. NMR experiments on the other hand, suggest only a weak relation of pK_a shifts and close contacts to charged residues, while the strongest influence derives from the dipolar character of α helices. The presented study hence pinpoints opportunities for improvements concerning the theoretical description of protonation state and tautomer probabilities. However, the coherence in the resulting apparent pK_a values from simulations and experiment affirms cpH-aMD as a reliable tool to study allergen dynamics at varying pH levels.

Molecular Dynamics Simulations of 2-Aminopurine-Labeled Dinucleoside Monophosphates Reveal Multiscale Stacking Kinetics

Molecular dynamics (MD) simulations of 2-aminopurine (2Ap)-labeled DNA dinucleoside monophosphates (DNMPs) were performed to investigate the hypothesis that base stacking dynamics occur on timescales sufficiently rapid to influence the emission signals measured in time-resolved fluorescence experiments. Analysis of multiple microsecond-length trajectories shows that the DNMPs sample all four coplanar stacking motifs. In addition, three metastable unstacked conformations are detected. A hidden Markov-state model (HMSM) was applied to the simulations to estimate transition rates between the stacked and unstacked states. Transitions between different stacked states generally occur at higher rates when the number of nucleobase faces requiring desolvation is minimized. Time constants for structural relaxation range between 1.6 and 25 ns, suggesting that emission from photoexcited 2Ap, which has an excited-state lifetime of 10 ns, is sensitive to base stacking kinetics. A master equation model for the excited-state population of 2Ap predicts multiexponential emission decays that reproduce the sub-10 ns emission decay lifetimes and amplitudes seen in experiments. Combining MD simulations with HMSM analysis is a powerful way to understand the dynamics that influence 2Ap excited-state relaxation and represents an important step toward using observed emission signals to validate MD simulations.

Dynamic Neutron Scattering by Biological Systems

Dynamic neutron scattering directly probes motions in biological systems on femtosecond to microsecond timescales. When combined with molecular dynamics simulation and normal mode analysis, detailed descriptions of the forms and frequencies of motions can be derived. We examine vibrations in proteins, the temperature dependence of protein motions, and concepts describing the rich variety of motions detectable using neutrons in biological systems at physiological temperatures. New techniques for deriving information on collective motions using coherent scattering are also reviewed.

Modeling the mechanism of CLN025 beta-hairpin formation

Beta-hairpins are substructures found in proteins that can lend insight into more complex systems. Furthermore, the folding of beta-hairpins is a valuable test case for benchmarking experimental and theoretical methods. Here, we simulate the folding of CLN025, a miniprotein with a beta-hairpin structure, at its experimental melting temperature using a range of state-of-the-art protein force fields. We construct Markov state models in order to examine the thermodynamics, kinetics, mechanism, and rate-determining step of folding. Mechanistically, we find the folding process is rate-limited by the formation of the turn region hydrogen bonds, which occurs following the downhill hydrophobic collapse of the extended denatured protein. These results are presented in the context of established and contradictory theories of the beta-hairpin folding process. Furthermore, our analysis suggests that the AMBER-FB15 force field, at this temperature, best describes the characteristics of the full experimental CLN025 conformational ensemble, while the AMBER ff99SB-ILDN and CHARMM22* force fields display a tendency to overstabilize the native state.

Hierarchical Time-Lagged Independent Component Analysis: Computing Slow Modes and Reaction Coordinates for Large Molecular Systems

Analysis of molecular dynamics, for example using Markov models, often requires the identification of order parameters that are good indicators of the rare events, i.e. good reaction coordinates. Recently, it has been shown that the time-lagged independent component analysis (TICA) finds the linear combinations of input coordinates that optimally represent the slow kinetic modes and may serve in order to define reaction coordinates between the metastable states of the molecular system. A limitation of the method is that both computing time and memory requirements scale with the square of the number of input features. For large protein systems, this exacerbates the use of extensive feature sets such as the distances between all pairs of residues or even heavy atoms. Here we derive a hierarchical TICA (hTICA) method that approximates the full TICA solution by a hierarchical, divide-and-conquer calculation. By using hTICA on distances between heavy atoms we identify previously unknown relaxation processes in the bovine pancreatic trypsin inhibitor.

Quasielastic neutron scattering in biology: Theory and applications

Neutrons scatter quasielastically from stochastic, diffusive processes, such as overdamped vibrations, localized diffusion and transitions between energy minima. In biological systems, such as proteins and membranes, these relaxation processes are of considerable physical interest. We review here recent methodological advances and applications of quasielastic neutron scattering (QENS) in biology, concentrating on the role of molecular dynamics simulation in generating data with which neutron profiles can be unambiguously interpreted. We examine the use of massively-parallel computers in calculating scattering functions, and the application of Markov state modeling. The decomposition of MD-derived neutron dynamic susceptibilities is described, and the use of this in combination with NMR spectroscopy. We discuss dynamics at very long times, including approximations to the infinite time mean-square displacement and nonequilibrium aspects of single-protein dynamics. Finally, we examine how neutron scattering and MD can be combined to provide information on lipid nanodomains. This article is part of a Special Issue entitled "Science for Life" Guest Editor: Dr. Austen Angell, Dr. Salvatore Magazù and Dr. Federica Migliardo.

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Discovery of Small-Molecule Nonfluorescent Inhibitors of Fluorogen-Fluorogen Activating Protein Binding Pair

A new class of biosensors, fluorogen activating proteins (FAPs), has been successfully used to track receptor trafficking in live cells. Unlike the traditional fluorescent proteins (FPs), FAPs do not fluoresce unless bound to their specific small-molecule fluorogens, and thus FAP-based assays are highly sensitive. Application of the FAP-based assay for protein trafficking in high-throughput flow cytometry resulted in the discovery of a new class of compounds that interferes with the binding between fluorogens and FAP, thus blocking the fluorescence signal. These compounds are high-affinity, nonfluorescent analogs of fluorogens with little or no toxicity to the tested cells and no apparent interference with the normal function of FAP-tagged receptors. The most potent compound among these, N,4-dimethyl-N-(2-oxo-2-(4-(pyridin-2-yl)piperazin-1-yl)ethyl)benzenesulfonamide (ML342), has been investigated in detail. X-ray crystallographic analysis revealed that ML342 competes with the fluorogen, sulfonated thiazole orange coupled to diethylene glycol diamine (TO1-2p), for the same binding site on a FAP, AM2.2. Kinetic analysis shows that the FAP-fluorogen interaction is more complex than a homogeneous one-site binding process, with multiple conformational states of the fluorogen and/or the FAP, and possible dimerization of the FAP moiety involved in the process.

Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has, therefore, been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase-space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs/PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, a 1 ms Anton MD simulation of the bovine pancreatic trypsin inhibitor protein, and an optical tweezer force probe trajectory of an RNA hairpin.

Markov state models of biomolecular conformational dynamics

It has recently become practical to construct Markov state models (MSMs) that reproduce the long-time statistical conformational dynamics of biomolecules using data from molecular dynamics simulations. MSMs can predict both stationary and kinetic quantities on long timescales (e.g. milliseconds) using a set of atomistic molecular dynamics simulations that are individually much shorter, thus addressing the well-known sampling problem in molecular dynamics simulation. In addition to providing predictive quantitative models, MSMs greatly facilitate both the extraction of insight into biomolecular mechanism (such as folding and functional dynamics) and quantitative comparison with single-molecule and ensemble kinetics experiments. A variety of methodological advances and software packages now bring the construction of these models closer to routine practice. Here, we review recent progress in this field, considering theoretical and methodological advances, new software tools, and recent applications of these approaches in several domains of biochemistry and biophysics, commenting on remaining challenges.

Microsecond folding experiments and simulations: a match is made

For the past two decades, protein folding experiments have been speeding up from the second or millisecond time scale to the microsecond time scale, and full-atom simulations have been extended from the nanosecond to the microsecond and even millisecond time scale. Where the two meet, it is now possible to compare results directly, allowing force fields to be validated and refined, and allowing experimental data to be interpreted in atomistic detail. In this perspective we compare recent experiments and simulations on the microsecond time scale, pointing out the progress that has been made in determining native structures from physics-based simulations, refining experiments and simulations to provide more quantitative underlying mechanisms, and tackling the problems of multiple reaction coordinates, downhill folding, and complex underlying structure of unfolded or misfolded states.

Modulation of a ligand's energy landscape and kinetics by the chemical environment

Understanding how the chemical environment modulates the predominant conformations and kinetics of flexible molecules is a core interest of biochemistry and a prerequisite for the rational design of synthetic catalysts. This study combines molecular dynamics simulation and Markov state models (MSMs) to a systematic computational strategy for investigating the effect of the chemical environment of a molecule on its conformations and kinetics. MSMs allow quantities to be computed that are otherwise difficult to access, such as the metastable sets, their free energies, and the relaxation time scales related to the rare transitions between metastable states. Additionally, MSMs are useful to identify observables that may act as sensors for the conformational or binding state of the molecule, thus guiding the design of experiments. In the present study, the conformation dynamics of UDP-GlcNAc are studied in vacuum, water, water + Mg(2+), and in the protein UDP-GlcNAc 2-epimerase. It is found that addition of Mg(2+) significantly affects the conformational stability, thermodynamics, and kinetics of UDP-GlcNAc. In particular, the slowest structural process, puckering of the GlcNAc sugar, depends on the overall conformation of UDP-GlcNAc and may thus act as a sensor of whether Mg(2+) is bound or not. Interestingly, transferring the molecule from vacuum to water makes the protein-binding conformations UDP-GlcNAc first accessible, while adding Mg(2+) further stabilizes them by specifically associating to binding-competent conformations. While Mg(2+) is not cocrystallized in the UDP-GlcNAc 2-epimerase complex, the selectively stabilized Mg(2+)/UDP-GlcNAc complex may be a template for the bound state, and Mg(2+) may accompany the binding-competent ligand conformation to the binding pocket. This serves as a possible explanation of the enhanced epimerization rate in the presence of Mg(2+). This role of Mg(2+) has previously not been described and opens the question whether "binding co-factors" may be a concept of general relevance for protein-ligand binding.

EMMA: A Software Package for Markov Model Building and Analysis

The study of folding and conformational changes of macromolecules by molecular dynamics simulations often requires the generation of large amounts of simulation data that are difficult to analyze. Markov (state) models (MSMs) address this challenge by providing a systematic way to decompose the state space of the molecular system into substates and to estimate a transition matrix containing the transition probabilities between these substates. This transition matrix can be analyzed to reveal the metastable, i.e., long-living, states of the system, its slowest relaxation time scales, and transition pathways and rates, e.g., from unfolded to folded, or from dissociated to bound states. Markov models can also be used to calculate spectroscopic data and thus serve as a way to reconcile experimental and simulation data. To reduce the technical burden of constructing, validating, and analyzing such MSMs, we provide the software framework EMMA that is freely available at https://simtk.org/home/emma .