Probing molecular kinetics with Markov models: metastable states, transition pathways and spectroscopic observables

Information thermodynamics of transition paths between multiple mesostates

A central concern across the natural sciences is a quantitative understanding of the mechanism governing rare transitions between two metastable states. Recent research has uncovered a fundamental equality between the time-reversal asymmetry of the ensemble of such transition paths and the informativeness of system dynamics about the reactivity of a given trajectory, immediately leading to quantitative criteria for judging the importance of distinct system coordinates for the transition. Here we generalize this framework to multiple mesostates. We find that the main system-wide and coordinate-specific results generalize intuitively, while the combinatorial diversity of pairwise transitions raises new questions and points to new concepts. This work increases the previous framework's generality and applicability and forges connections to enhanced-sampling and coarse-grained dynamical approaches such as milestoning and Markov-state models.

Analysis and interpretation of first passage time distributions featuring rare events

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

Unraveling the allosteric inhibition mechanism of PARP-1 CAT and the D766/770A mutation effects via Gaussian accelerated molecular dynamics and Markov state model

PARP-1 (Poly (ADP-ribose) polymerase 1) is a nuclear enzyme and plays a key role in many cellular functions, such as DNA repair, modulation of chromatin structure, and recombination. Developing the PARP-1 inhibitors has emerged as an effective therapeutic strategy for a growing list of cancers. The catalytic structural domain (CAT) of PARP-1 upon binding the inhibitor allosterically regulates the conformational changes of helix domain (HD), affecting its identification with the damaged DNA. The typical type I (EB47) and III (veliparib) inhibitors were able to lengthening or shortening the retention time of this enzyme on DNA damage and thus regulating the cytotoxicity. Nonetheless, the basis underlying allosteric inhibition is unclear, which limits the development of novel PARP-1 inhibitors. Here, to investigate the distinct allosteric changes of EB47 and veliparib against PARP-1 CAT, each complex was simulated via classical and Gaussian accelerated molecular dynamics (cMD and GaMD). To study the reverse allosteric basis and mutation effects, the complexes PARP-1 with UKTT15 and PARP-1 D766/770A mutant with EB47 were also simulated. Importantly, the markov state models were built to identify the transition pathways of crucial substates of allosteric communication and the induction basis of PARP-1 reverse allostery. The conformational change differences of PARP-1 CAT regulated by allosteric inhibitors were concerned with to their interaction at the active site. Energy calculations suggested the energy advantage of EB47 in inhibiting the wild-type PARP-1, compared with D766/770A PARP-1. Secondary structure results showed the change of two key loops (αB-αD and αE-αF) in different systems. This work reported the basis of PARP-1 allostery from both thermodynamic and kinetic views, providing the guidance for the discovery and design of more innovative PARP-1 allosteric inhibitors.

AlF-AlF Reaction Dynamics between 200 K and 1000 K: Reaction Mechanisms and Intermediate Complex Characterization

AlF is a relevant molecule in astrochemistry as a tracer of F-bearing molecules. Additionally, AlF presents diagonal Franck-Condon factors and can be created very efficiently in the lab, which makes it a prototypical molecular for laser cooling. However, very little is known about the reaction dynamics of AlF. In this work, we report on the reaction dynamics of AlF-AlF between 200 and 1000 K using ab initio molecular dynamics and a highly efficient active learning approach for the potential energy surface, including all degrees of freedom. As a result, we identify the main reaction mechanisms and the lifetime of the intermediate complex AlF-AlF relevant to astrochemistry environments and regions in buffer gas cells.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Elucidating Interfacial Dynamics of Ti-Al Systems Using Molecular Dynamics Simulation and Markov State Modeling

Due to their remarkable mechanical and chemical properties, Ti-Al-based materials are attracting considerable interest in numerous fields of engineering, such as automotive, aerospace, and defense. With their low density, high strength, and resistance to corrosion and oxidation, these intermetallic alloys and metal-compound composites have found diverse applications. However, additive manufacturing and heat treatment of Ti-Al alloys frequently lead to brittleness and severe formation of defects. The present study delves into the interfacial dynamics of these Ti-Al systems, particularly focusing on the behavior of Ti and Al atoms in the presence of TiAl3 grain boundaries under experimental heat treatment conditions. Using a combination of molecular dynamics and Markov state modeling, we scrutinize the kinetic processes involved in the formation of TiAl3. The molecular dynamics simulation indicates that at the early stage of heat treatment, the predominating process is the diffusion of Al atoms toward the Ti surface through the TiAl3 grain boundaries. Markov state modeling identifies three distinct dynamic states of Al atoms within the Ti/Al mixture that forms during the process, each exhibiting a unique spatial distribution. Using transition time scales as a qualitative measure of the rapidness of the dynamics, it is observed that the Al dynamics is significantly less rapid near the Ti surface compared to the Al surface. Put together, the results offer a comprehensive understanding of the interfacial dynamics and reveal a three-stage diffusion mechanism. The process initiates with the premelting of Al, proceeds with the prevalent diffusion of Al atoms toward the Ti surface, and eventually ceases as the Ti concentration within the mixture progressively increases. The insights gained from this study could contribute significantly to the control and optimization of manufacturing processes for these high-performing Ti-Al-based materials.

Functionalized Fullerene Potentially Inhibits SARS-CoV-2 Infection by Modulating Spike Protein Conformational Changes

The disease of SARS-CoV-2 has caused considerable morbidity and mortality globally. Spike proteins on the surface of SARS-CoV-2 allow it to bind with human cells, leading to infection. Fullerenes and their derivatives are promising SARS-CoV-2 inhibitors and drug-delivery vehicles. In this study, Gaussian accelerated molecular dynamics simulations and the Markov state model were employed to delve into the inhibitory mechanism of Fullerene-linear-polyglycerol-b-amine sulfate (F-LGPS) on spike proteins. During the study, it was discovered that fullerene derivatives can operate at the interface of the receptor-binding domain (RBD) and the N-terminal domain (NTD), keeping structural domains in a downward conformation. It was also observed that F-LGPS demonstrated superior inhibitory effects on the XBB variant in comparison to the wild-type variant. This study yielded invaluable insights for the potential development of efficient therapeutics targeting the spike protein of SARS-CoV-2.

Probing the structure of human tRNA3Lys in the presence of ligands using docking, MD simulations and MSM analysis

The tRNA3Lys, which acts as a primer for human immunodeficiency virus type 1 (HIV-1) reverse transcription, undergoes structural changes required for the formation of a primer-template complex. Small molecules have been targeted against tRNA3Lys to inhibit the primer-template complex formation. The present study aims to understand the kinetics of the conformational landscape spanned by tRNA3Lys in apo form using molecular dynamics simulations and Markov state modeling. The study is taken further to investigate the effect of small molecules like 1,4T and 1,5T on structural conformations and kinetics of tRNA3Lys, and comparative analysis is presented. Markov state modeling of tRNA3Lys apo resulted in three metastable states where the conformations have shown the non-canonical structures of the anticodon loop. Based on analyses of ligand-tRNA3Lys interactions, crucial ion and water mediated H-bonds and free energy calculations, it was observed that the 1,4-triazole more strongly binds to the tRNA3Lys compared to 1,5-triazole. However, the MSM analysis suggest that the 1,5-triazole binding to tRNA3Lys has brought rigidity not only in the binding pocket (TΨC arm, D-TΨC loop) but also in the whole structure of tRNA3Lys. This may affect the easy opening of primer tRNA3Lys required for HIV-1 reverse transcription.

Transcending Markov: non-Markovian rate processes of thermosensitive TRP ion channels

The Markov state model (MSM) is a popular theoretical tool for describing the hierarchy of time scales involved in the function of many proteins especially ion channel gating. An MSM is a particular case of the general non-Markovian model, where the rate of transition from one state to another does not depend on the history of state occupancy within the system, i.e. it only includes reversible, non-dissipative processes. However, an MSM requires knowledge of the precise conformational state of the protein and is not predictive when those details are not known. In the case of ion channels, this simple description fails in real (non-equilibrium) situations, for example when local temperature changes, or when energy losses occur during channel gating. Here, we show it is possible to use non-Markovian equations (i.e. offer a general description that includes the MSM as a particular case) to develop a relatively simple analytical model that describes the non-equilibrium behaviour of the temperature-sensitive transient receptor potential (TRP) ion channels, TRPV1 and TRPM8. This model accurately predicts asymmetrical opening and closing rates, infinite processes and the creation of new states, as well as the effect of temperature changes throughout the process. This approach therefore overcomes the limitations of the MSM and allows us to go beyond a mere phenomenological description of the dynamics of ion channel gating towards a better understanding of the physics underlying these processes.

Target Recognition in Tandem WW Domains: Complex Structures for Parallel and Antiparallel Ligand Orientation in h-FBP21 Tandem WW

Protein-protein interactions often rely on specialized recognition domains, such as WW domains, which bind to specific proline-rich sequences. The specificity of these protein-protein interactions can be increased by tandem repeats, i.e., two WW domains connected by a linker. With a flexible linker, the WW domains can move freely with respect to each other. Additionally, the tandem WW domains can bind in two different orientations to their target sequences. This makes the elucidation of complex structures of tandem WW domains extremely challenging. Here, we identify and characterize two complex structures of the tandem WW domain of human formin-binding protein 21 and a peptide sequence from its natural binding partner, the core-splicing protein SmB/B'. The two structures differ in the ligand orientation and, consequently, also in the relative orientation of the two WW domains. We analyze and probe the interactions in the complexes by molecular simulations and NMR experiments. The workflow to identify the complex structures uses molecular simulations, density-based clustering, and peptide docking. It is designed to systematically generate possible complex structures for repeats of recognition domains. These structures will help us to understand the synergistic and multivalency effects that generate the astonishing versatility and specificity of protein-protein interactions.

Multiscale Modeling of Hepatitis B Virus Capsid Assembly and Its Dimorphism

Hepatitis B virus (HBV) is an endemic, chronic virus that leads to 800000 deaths per year. Central to the HBV lifecycle, the viral core has a protein capsid assembled from many copies of a single protein. The capsid protein adopts different (quasi-equivalent) conformations to form icosahedral capsids containing 180 or 240 proteins: T = 3 or T = 4, respectively, in Caspar-Klug nomenclature. HBV capsid assembly has become an important target for recently developed antivirals; nonetheless, the assembly pathways and mechanisms that control HBV dimorphism remain unclear. We describe computer simulations of the HBV assembly, using a coarse-grained model that has parameters learned from all-atom molecular dynamics simulations of a complete HBV capsid and yet is computationally tractable. Dynamical simulations with the resulting model reproduce experimental observations of HBV assembly pathways and products. By constructing Markov state models and employing transition path theory, we identify pathways leading to T = 3, T = 4, and other experimentally observed capsid morphologies. The analysis shows that capsid polymorphism is promoted by the low HBV capsid bending modulus, where the key factors controlling polymorphism are the conformational energy landscape and protein-protein binding affinities.

Mutagenic Activation of Glutathione Peroxidase-4: Approaches toward Rational Design of Allosteric Drugs

Glutathione peroxidase 4 (GPX4) reduces lipid hydroperoxides in lipid membranes, effectively inhibiting iron-dependent cell death or ferroptosis. The upregulation of the enzyme by the mutations at residues D21 and D23 has been suggested to be associated with higher protein activity, which confers more protection against neurodegenerative diseases such as Alzheimer's, Parkinson's, and Huntington's diseases. Therefore, it has become an attractive target for treating and preventing neurodegenerative diseases. However, identifying means of mimicking the beneficial effects of these mutations distant from the active site constitutes a formidable challenge in moving toward therapeutics. In this study, we explore using molecular dynamics simulations to computationally map the conformational and energetic landscape of the wild-type GPX4 protein and three mutant variants to identify the allosteric networks of the enzyme. We present the conformational dynamic profile providing the desired signature behavior of the enzyme. We also discuss the implications of these findings for drug design efforts.

Investigating the Bioactive Conformation of Angiotensin II Using Markov State Modeling Revisited with Web-Scale Clustering

Molecular dynamics simulation is a powerful technique for studying the structure and dynamics of biomolecules in atomic-level detail by sampling their various conformations in real time. Because of the long timescales that need to be sampled to study biomolecular processes and the big and complex nature of the corresponding data, relevant analyses of important biophysical phenomena are challenging. Clustering and Markov state models (MSMs) are efficient computational techniques that can be used to extract dominant conformational states and to connect those with kinetic information. In this work, we perform Molecular Dynamics simulations to investigate the free energy landscape of Angiotensin II (AngII) in order to unravel its bioactive conformations using different clustering techniques and Markov state modeling. AngII is an octapeptide hormone, which binds to the AT1 transmembrane receptor, and plays a vital role in the regulation of blood pressure, conservation of total blood volume, and salt homeostasis. To mimic the water-membrane interface as AngII approaches the AT1 receptor and to compare our findings with available experimental results, the simulations were performed in water as well as in water-ethanol mixtures. Our results show that in the water-ethanol environment, AngII adopts more compact U-shaped (folded) conformations than in water, which resembles its structure when bound to the AT1 receptor. For clustering of the conformations, we validate the efficiency of an inverted-quantized k-means algorithm, as a fast approximate clustering technique for web-scale data (millions of points into thousands or millions of clusters) compared to k-means, on data from trajectories of molecular dynamics simulations with reasonable trade-offs between time and accuracy. Finally, we extract MSMs using various clustering techniques for the generation of microstates and macrostates and for the selection of the macrostate representatives.

Molecular simulations of proteins: From simplified physical interactions to complex biological phenomena

Molecular dynamics simulation is the most popular computational technique for investigating the structural and dynamical behaviour of proteins, in search of the molecular basis of their function. Far from being a completely settled field of research, simulations are still evolving to best capture the essential features of the atomic interactions that govern a protein's inner motions. Modern force fields are becoming increasingly accurate in providing a physical description adequate to this purpose, and allow us to model complex biological systems under fairly realistic conditions. Furthermore, the use of accelerated sampling techniques is improving our access to the observation of progressively larger molecular structures, longer time scales, and more hidden functional events. In this review, the basic principles of molecular dynamics simulations and a number of key applications in the area of protein science are summarized, and some of the most important results are discussed. Examples include the study of the structure, dynamics and binding properties of 'difficult' targets, such as intrinsically disordered proteins and membrane receptors, and the investigation of challenging phenomena like hydration-driven processes and protein aggregation. The findings described provide an overall picture of the current state of this research field, and indicate new perspectives on the road ahead to the upcoming future of molecular simulations.

Enlightening activation gating in P2X receptors

P2X receptors are trimeric nonselective cation channels gated by ATP. They assemble from seven distinct subunit isoforms as either homo- or heteromeric complexes and contain three extracellularly located binding sites for ATP. P2X receptors are expressed in nearly all tissues and are there involved in physiological processes like synaptic transmission, pain, and inflammation. Thus, they are a challenging pharmacological target. The determination of crystal and cryo-EM structures of several isoforms in the last decade in closed, open, and desensitized states has provided a firm basis for interpreting the huge amount of functional and biochemical data. Electrophysiological characterization in conjugation with optical approaches has generated significant insights into structure–function relationships of P2X receptors. This review focuses on novel optical and related approaches to better understand the conformational changes underlying the activation of these receptors.

Discovery of Novel GR Ligands toward Druggable GR Antagonist Conformations Identified by MD Simulations and Markov State Model Analysis

Binding of different ligands to glucocorticoid receptor (GR) may induce different conformational changes and even trigger completely opposite biological functions. To understand the allosteric communication within the GR ligand binding domain, the folding pathway of helix 12 (H12) induced by the binding of the agonist dexamethasone (DEX), antagonist RU486, and modulator AZD9567 are explored by molecular dynamics simulations and Markov state model analysis. The ligands can regulate the volume of the activation function-2 through the residues Phe737 and Gln738. Without ligand or with agonist binding, H12 swings from inward to outward to visit different folding positions. However, the binding of RU486 or AZD9567 perturbs the structural state, and the passive antagonist state appears more stable. Structure-based virtual screening and in vitro bioassays are used to discover novel GR ligands that bias the conformation equilibria toward the passive antagonist state. HP-19 exhibits the best anti-inflammatory activity (IC50 = 0.041 ± 0.011 µm) in nuclear factor-kappa B signaling pathway, which is comparable to that of DEX. HP-19 also does not induce adverse effect-related transactivation functions of GR. The novel ligands discovered here may serve as promising starting points for the development of GR modulators.

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.

Nearly reducible finite Markov chains: Theory and algorithms

Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications.

Estradiol induces allosteric coupling and partitioning of sex-hormone-binding globulin monomers among conformational states

Sex-hormone-binding globulin (SHBG) regulates the transport and bioavailability of estradiol. The dynamics of estradiol's binding to SHBG are incompletely understood, although it is believed that estradiol binds to each monomer of SHBG dimer with identical affinity (Kd ∼2 nM). Contrary to the prevalent view, we show that estradiol's binding to SHBG is nonlinear, and the "apparent" Kd changes with varying estradiol and SHBG concentrations. Estradiol's binding to each SHBG monomer influences residues in the ligand-binding pocket of both monomers and differentially alters the conformational and energy landscapes of both monomers. Monomers are not energetically or conformationally equivalent even in fully bound state. Estradiol's binding to SHBG involves bidirectional, inter-monomeric allostery that changes the distribution of both monomers among various energy and conformational states. Inter-monomeric allostery offers a mechanism to extend the binding range of SHBG and regulate hormone bioavailability as estradiol concentrations vary widely during life.

Transition pathway of hydrogen bond switching in supercooled water analyzed by the Markov state model

In this work, we examine hydrogen-bond (H-bond) switching by employing the Markov State Model (MSM). During the H-bond switching, a water hydrogen initially H-bonded with water oxygen becomes H-bonded to a different water oxygen. MSM analysis was applied to trajectories generated from molecular dynamics simulations of the TIP4P/2005 model from a room-temperature state to a supercooled state. We defined four basis states to characterize the configuration between two water molecules: H-bonded ("H"), unbound ("U"), weakly H-bonded ("w"), and alternative H-bonded ("a") states. A 16 × 16 MSM matrix was constructed, describing the transition probability between states composed of three water molecules. The mean first-passage time of the H-bond switching was estimated by calculating the total flux from the HU to UH states. It is demonstrated that the temperature dependence of the mean first-passage time is in accordance with that of the H-bond lifetime determined from the H-bond correlation function. Furthermore, the flux for the H-bond switching is decomposed into individual pathways that are characterized by different forms of H-bond configurations of trimers. The dominant pathway of the H-bond switching is found to be a direct one without passing through such intermediate states as "w" and "a," the existence of which becomes evident in supercooled water. The pathway through "w" indicates a large reorientation of the donor molecule. In contrast, the pathway through "a" utilizes the tetrahedral H-bond network, which is revealed by the further decomposition based on the H-bond number of the acceptor molecule.

Complex Reaction Network Thermodynamic and Kinetic Autoconstruction Based on Ab Initio Statistical Mechanics: A Case Study of O2 Activation on Ag4 Clusters

An approach based on ab initio statistical mechanics is demonstrated for autoconstructing complex reaction networks. Ab initio molecular dynamics combined with Markov state models are employed to study relevant transitions and corresponding thermodynamic and kinetic properties of a reaction. To explore the capability and flexibility of this approach, we present a study of oxygen activation on Ag4 as a model reaction. Specifically, with the same sampled trajectories, it is possible to study the structural effects and the reaction rate of the cited reaction. The results show that this approach is suitable for automatized construction of reaction networks, especially for non-well-studied reactions, which can benefit from this ab initio molecular dynamics based approach to construct comprehensive reaction networks with Markov state models without prior knowledge about the potential energy landscape.

The molecular basis for the pH-dependent calcium affinity of the pattern recognition receptor langerin

The C-type lectin receptor langerin plays a vital role in the mammalian defense against invading pathogens. Langerin requires a Ca²⁺ cofactor, the binding affinity of which is regulated by pH. Thus, Ca²⁺ is bound when langerin is on the membrane but released when langerin and its pathogen substrate traffic to the acidic endosome, allowing the substrate to be degraded. The change in pH is sensed by protonation of the allosteric pH sensor histidine H294. However, the mechanism by which Ca²⁺ is released from the buried binding site is not clear. We studied the structural consequences of protonating H294 by molecular dynamics simulations (total simulation time: about 120 μs) and Markov models. We discovered a relay mechanism in which a proton is moved into the vicinity of the Ca²⁺-binding site without transferring the initial proton from H294. Protonation of H294 unlocks a conformation in which a protonated lysine side chain forms a hydrogen bond with a Ca²⁺-coordinating aspartic acid. This destabilizes Ca²⁺ in the binding pocket, which we probed by steered molecular dynamics. After Ca²⁺ release, the proton is likely transferred to the aspartic acid and stabilized by a dyad with a nearby glutamic acid, triggering a conformational transition and thus preventing Ca²⁺ rebinding. These results show how pH regulation of a buried orthosteric binding site from a solvent-exposed allosteric pH sensor can be realized by information transfer through a specific chain of conformational arrangements.

What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models

Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.

Markov models from the square root approximation of the Fokker-Planck equation: calculating the grid-dependent flux

Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to [Formula: see text] states can be obtained within seconds on a single high-performance compute server if regular grids are used.

Path probability ratios for Langevin dynamics-Exact and approximate

Path reweighting is a principally exact method to estimate dynamic properties from biased simulations-provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since molecular dynamics simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio M_L for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. This new path probability ratio allows for exact reweighting of Langevin dynamics propagated by this integrator. We also show that a previously derived approximate path probability ratio M_approx differs from the exact M_L only by O(ξ⁴Δt⁴) and thus yields highly accurate dynamic reweighting results. (Δt is the integration time step, and ξ is the collision rate.) The results are tested, and the efficiency of path reweighting is explored using butane as an example.

Analysis of COVID-19 epidemic and clinical risk factors of patients under epidemiological Markov model

OBJECTIVE

It aimed to analyze the epidemic situation of new coronary pneumonia (COVID-19) based on the epidemiological Markov model, and to study the clinical risk factors of the patients based on the patient's cardinal data and clinical symptoms.

METHODS

A total of 500 patients with COVID-19 diagnosed by nucleic acid testing in the X hospital from January 2020 to May 2020 were collected. According to the severity of the disease, they were classified into general group (200 cases) and acute critical group (300 cases). Markov model to predict the number of COVID-19 infections was constructed. Patient's general information, clinical characteristics, and prevention methods were analyzed.

RESULTS

According to Markov model statistics, the developmental expected stay time of patients infected with COVID-19 was 14 days. 2. The two groups of patients had statistically considerable differences in complications such as gender, age, hypertension, coronary heart disease, shortness of breath, myocardial damage, and thrombocytopenia (P < 0.05). 3. Logistic multivariate regression analysis showed that the clinical risk factors for patients with COVID-19 mainly included the patient's gender, age, whether they were associated with hypertension, coronary heart disease, shortness of breath, myocardial damage, and thrombocytopenia.

CONCLUSION

Markov model can be utilized to judge the time course of the COVID-19 in various development states. In addition, the COVID-19 spread rapidly and is extremely harmful. Clinically, through active prevention, the treatment effect can be improved, the patient's respiratory function, and the quality of life can also be improved.

Unravelling the intricate cooperativity of subunit gating in P2X2 ion channels

Ionotropic purinergic (P2X) receptors are trimeric channels that are activated by the binding of ATP. They are involved in multiple physiological functions, including synaptic transmission, pain and inflammation. The mechanism of activation is still elusive. Here we kinetically unraveled and quantified subunit activation in P2X2 receptors by an extensive global fit approach with four complex and intimately coupled kinetic schemes to currents obtained from wild type and mutated receptors using ATP and its fluorescent derivative 2-[DY-547P1]-AET-ATP (fATP). We show that the steep concentration-activation relationship in wild type channels is caused by a subunit flip reaction with strong positive cooperativity, overbalancing a pronounced negative cooperativity for the three ATP binding steps, that the net probability fluxes in the model generate a marked hysteresis in the activation-deactivation cycle, and that the predicted fATP binding matches the binding measured by fluorescence. Our results shed light into the intricate activation process of P2X channels.

Polymerization and editing modes of a high-fidelity DNA polymerase are linked by a well-defined path

Proofreading by replicative DNA polymerases is a fundamental mechanism ensuring DNA replication fidelity. In proofreading, mis-incorporated nucleotides are excised through the 3'-5' exonuclease activity of the DNA polymerase holoenzyme. The exonuclease site is distal from the polymerization site, imposing stringent structural and kinetic requirements for efficient primer strand transfer. Yet, the molecular mechanism of this transfer is not known. Here we employ molecular simulations using recent cryo-EM structures and biochemical analyses to delineate an optimal free energy path connecting the polymerization and exonuclease states of E. coli replicative DNA polymerase Pol III. We identify structures for all intermediates, in which the transitioning primer strand is stabilized by conserved Pol III residues along the fingers, thumb and exonuclease domains. We demonstrate switching kinetics on a tens of milliseconds timescale and unveil a complete pol-to-exo switching mechanism, validated by targeted mutational experiments.

Simple Model of Protein Energetics To Identify Ab Initio Folding Transitions from All-Atom MD Simulations of Proteins

A fundamental requirement to predict the native conformation, address questions of sequence design and optimization, and gain insights into the folding mechanisms of proteins lies in the definition of an unbiased reaction coordinate that reports on the folding state without the need to compare it to reference values, which might be unavailable for new (designed) sequences. Here, we introduce such a reaction coordinate, which does not depend on previous structural knowledge of the native state but relies solely on the energy partition within the protein: the spectral gap of the pair nonbonded energy matrix (ENergy Gap, ENG). This quantity can be simply calculated along unbiased MD trajectories. We show that upon folding the gap increases significantly, while its fluctuations are reduced to a minimum. This is consistently observed for a diverse set of systems and trajectories. Our approach allows one to promptly identify residues that belong to the folding core as well as residues involved in non-native contacts that need to be disrupted to guide polypeptides to the folded state. The energy gap and fluctuations criteria are then used to develop an automatic detection system which allows us to extract and analyze folding transitions from a generic MD trajectory. We speculate that our method can be used to detect conformational ensembles in dynamic and intrinsically disordered proteins, revealing potential preorganization for binding.

Markov State Model Analysis of Haloperidol Binding to the D3 Dopamine Receptor

We have developed Markov state models (MSMs) and hidden Markov models (HMMs) that describe the binding of haloperidol to the D₃ dopamine receptor. Haloperidol is an antipsychotic drug that binds with nanomolar affinity to the D₃ dopamine receptor, where it functions as an inverse agonist. The models were constructed using an adaptive sampling approach from 519 individual molecular dynamics simulations totaling 122 μs of simulated time and encompass the entire drug binding process. They reveal short-lived metastable bound states and two distinct long-lived bound conformations that cannot be separated in affinity using our current methodology. This work extends the use of MSMs and HMMs to study ligand binding, which thus far has been limited to simpler systems.

Dynamic design: manipulation of millisecond timescale motions on the energy landscape of cyclophilin A

Proteins need to interconvert between many conformations in order to function, many of which are formed transiently, and sparsely populated. Particularly when the lifetimes of these states approach the millisecond timescale, identifying the relevant structures and the mechanism by which they interconvert remains a tremendous challenge. Here we introduce a novel combination of accelerated MD (aMD) simulations and Markov state modelling (MSM) to explore these 'excited' conformational states. Applying this to the highly dynamic protein CypA, a protein involved in immune response and associated with HIV infection, we identify five principally populated conformational states and the atomistic mechanism by which they interconvert. A rational design strategy predicted that the mutant D66A should stabilise the minor conformations and substantially alter the dynamics, whereas the similar mutant H70A should leave the landscape broadly unchanged. These predictions are confirmed using CPMG and R1ρ solution state NMR measurements. By efficiently exploring functionally relevant, but sparsely populated conformations with millisecond lifetimes in silico, our aMD/MSM method has tremendous promise for the design of dynamic protein free energy landscapes for both protein engineering and drug discovery.

Local and global perspectives on diffusion maps in the analysis of molecular systems

Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalizing the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).

Disclosing the Mechanism of Spontaneous Aggregation and Template-Induced Misfolding of the Key Hexapeptide (PHF6) of Tau Protein Based on Molecular Dynamics Simulation

The microtubule-associated protein tau is critical for the development and maintenance of the nervous system. Tau dysfunction is associated with a variety of neurodegenerative diseases called tauopathies, which are characterized by neurofibrillary tangles formed by abnormally aggregated tau protein. Studying the aggregation mechanism of tau protein is of great significance for elucidating the etiology of tauopathies. The hexapeptide ³⁰⁶VQIVYK³¹¹ (PHF6) of R3 has been shown to play a vital role in promoting tau aggregation. In this study, long-term all-atom molecular dynamics simulations in explicit solvent were performed to investigate the mechanisms of spontaneous aggregation and template-induced misfolding of PHF6, and the dimerization at the early stage of nucleation was further specifically analyzed by the Markov state model (MSM). Our results show that PHF6 can spontaneously aggregate to form multimers enriched with β-sheet structure and the β-sheets in multimers prefer to exist in a parallel way. It is observed that PHF6 monomer can be induced to form a β-sheet structure on either side of the template but in a different way. In detail, the β-sheet structure is easier to form on the left side but does not extend well, but on the right side, the monomer can form the extended β-sheet structure. Furthermore, MSM analysis shows that the formation of dimer mainly occurs in three steps. First, the separated monomers collide with each other at random orientations, and then a dimer with short β-sheet structure at the N-terminal forms; finally, β-sheets elongate to form an extended parallel β-sheet dimer. During these processes, multiple intermediate states are identified and multiple paths can form a parallel β-sheet dimer from the disordered coil structure. Moreover, the residues I308, V309, and Y310 play an essential role in the dimerization. In a word, our results uncover the aggregation and misfolding mechanism of PHF6 from the atomic level, which can provide useful theoretical guidance for rational design of effective therapeutic drugs against tauopathies.

Directed kinetic transition network model

Molecular dynamics simulations contain detailed kinetic information related to the functional states of proteins and macromolecules, but this information is obscured by the high dimensionality of configurational space. Markov state models and transition network models are widely applied to extract kinetic descriptors from equilibrium molecular dynamics simulations. In this study, we developed the Directed Kinetic Transition Network (DKTN)-a graph representation of a master equation which is appropriate for describing nonequilibrium kinetics. DKTN models the transition rate matrix among different states under detailed balance. Adopting the mixing time from the Markov chain, we use the half mixing time as the criterion to identify critical state transition regarding the protein conformational change. The similarity between the master equation and the Kolmogorov equation suggests that the DKTN model can be reformulated into the continuous-time Markov chain model, which is a general case of the Markov chain without a specific lag time. We selected a photo-sensitive protein, vivid, as a model system to illustrate the usage of the DKTN model. Overall, the DKTN model provides a graph representation of the master equation based on chemical kinetics to model the protein conformational change without the underlying assumption of the Markovian property.

Selection Rules for Outliers in Outlier Flooding Method Regulate Its Conformational Sampling Efficiency

The outlier flooding method (OFLOOD) has been proposed as an enhanced conformational sampling method of proteins. In OFLOOD, rarely occurring states of proteins are detected as sparse conformational distributions (outliers) with a clustering algorithm. The detected outliers are intensively resampled with short-time molecular dynamics (MD) simulations. As a set of cycles, OFLOOD repeats selections of outliers and their conformational resampling. Herein, as an essential issue to be tackled to perform OFLOOD efficiently, a selection rule for outliers should be carefully specified. Generally, many outliers are detected from distributions on conformational subspaces with the clustering. Judging from its computational costs, it is unreasonable to select all the detected outliers upon the conformational resampling. Therefore, it is important to consider which outliers should be selected from the sparse distributions when restarting their short-time MD simulations with limited computational costs. In this sense, we investigated the conformational sampling efficiency of OFLOOD by changing the selection rules for outliers. To address the conformational sampling efficiency of OFLOOD depending on its selection rules, outliers to be resampled were selected by focusing their probability occurrences (populations of outliers). As a comparison, a random selection rule for outliers was also considered. Through the present assessment, the random selection of outliers showed the most efficient conformational sampling efficiency compared to the other OFLOOD trials using the biased selection rules, indicating that a variety of outliers should be selected and resampled during the OFLOOD cycles. In conclusion, the random outlier selection rule is the best strategy to perform OFLOOD efficiently.

Nonproductive Binding Modes as a Prominent Feature of Aβ40 Fiber Elongation: Insights from Molecular Dynamics Simulation

The formation of amyloid fibers has been implicated in a number of neurodegenerative diseases. The growth of amyloid fibers is strongly thermodynamically favorable, but kinetic traps exist where the incoming monomer binds in an incompatible conformation that blocks further elongation. Unfortunately, this process is difficult to follow experimentally at the atomic level. It is also too complex to simulate in full detail and to date has been explored either through coarse-grained simulations, which may miss many important interactions, or full atomic simulations, in which the incoming peptide is constrained to be near the ideal fiber geometry. Here we use an alternate approach starting from a docked complex in which the monomer is from an experimental NMR structure of one of the major conformations in the unbound ensemble, a largely unstructured peptide with the central hydrophobic region in a 3₁₀ helix. A 1000 ns full atomic simulation in explicit solvent shows the formation of a metastable intermediate by sequential, concerted movements of both the fiber and the monomer. A Markov state model shows that the unfolded monomer is trapped at the end of the fiber in a set of interconverting antiparallel β-hairpin conformations. The simulation here may serve as a model for the binding of other non-β-sheet conformations to amyloid fibers.

Complete Coupled Binding-Folding Pathway of the Intrinsically Disordered Transcription Factor Protein Brinker Revealed by Molecular Dynamics Simulations and Markov State Modeling

Intrinsically disordered proteins (IDPs) make up a large class of proteins that lack stable structures in solution, existing instead as dynamic conformational ensembles. To perform their biological functions, many IDPs bind to other proteins or nucleic acids. Although IDPs are unstructured in solution, when they interact with binding partners, they fold into defined three-dimensional structures via coupled binding-folding processes. Because they frequently underlie IDP function, the mechanisms of this coupled binding-folding process are of great interest. However, given the flexibility inherent to IDPs and the sparse populations of intermediate states, it is difficult to reveal binding-folding pathways at atomic resolution using experimental methods. Computer simulations are another tool for studying these pathways at high resolution. Accordingly, we have applied 40 μs of unbiased molecular dynamics simulations and Markov state modeling to map the complete binding-folding pathway of a model IDP, the 59-residue C-terminal portion of the DNA binding domain of Drosophila melanogaster transcription factor Brinker (BrkDBD). Our modeling indicates that BrkDBD binds to its cognate DNA and folds in ∼50 μs by an induced fit mechanism, acquiring most of its stable secondary and tertiary structure only after it reaches the final binding site on the DNA. The protein follows numerous pathways en route to its bound and folded conformation, occasionally becoming stuck in kinetic traps. Each binding-folding pathway involves weakly bound, increasingly folded intermediate states located at different sites on the DNA surface. These findings agree with experimental data and provide additional insight into the BrkDBD folding mechanism and kinetics.

All-atom calculation of protein free-energy profiles

The Bias Functional (BF) approach is a variational method which enables one to efficiently generate ensembles of reactive trajectories for complex biomolecular transitions, using ordinary computer clusters. For example, this scheme was applied to simulate in atomistic detail the folding of proteins consisting of several hundreds of amino acids and with experimental folding time of several minutes. A drawback of the BF approach is that it produces trajectories which do not satisfy microscopic reversibility. Consequently, this method cannot be used to directly compute equilibrium observables, such as free energy landscapes or equilibrium constants. In this work, we develop a statistical analysis which permits us to compute the potential of mean-force (PMF) along an arbitrary collective coordinate, by exploiting the information contained in the reactive trajectories calculated with the BF approach. We assess the accuracy and computational efficiency of this scheme by comparing its results with the PMF obtained for a small protein by means of plain molecular dynamics.

Simulation Studies on the Lipid Interaction and Conformation of Novel Drug-Delivery Pseudopeptidic Polymers

Pseudopeptides based on poly(l-lysine isophthalamide) backbone have emerged as promising drug delivery candidates due to their pH-activated membrane disruption ability. To gain molecular understanding on these novel polymeric species, we have constructed force-field parameters and simulated the behaviors of polymers with and without phenylalanine grafted as side chains under conditions compatible with different pHs. The free energy changes upon polymer permeation through membrane were calculated using the umbrella sampling technique. We show that both polymers with and without grafts interact better with the membrane under conditions compatible with lower pH. The conformational states of the polymers were investigated in water and at a water-membrane interface. On the basis of Markov state modeling results, we propose a possible advantage of the grafted polymer over the ungrafted polymer for membrane rupture because of its quicker conformational rearrangement kinetics.

Enzyme-Mediated Directional Transport of a Small-Molecule Walker With Chemically Identical Feet

We describe a small-molecule "walker" that uses enzyme catalysis to discriminate between the relative positions of its "feet" on a track and thereby move with net directionality. The bipedal walker has identical carboxylic acid feet, and "steps" along an isotactic hydroxyl-group-derivatized polyether track by the formation/breakage of ester linkages. Lipase AS catalyzes the selective hydrolysis of the rear foot of macrocyclized walkers (an information ratchet mechanism), the rear foot producing an (R)-stereocenter at its point of attachment to the track. If the hydrolyzed foot reattaches to the track in front of the bound foot it forms an (S)-stereocenter, which is resistant to enzymatic hydrolysis. Only macrocyclic walker-track conjugates are efficiently hydrolyzed by the enzyme, leading to high processivity of the walker movement along the track. Conventional chemical reagents promote formation of the ester bonds between the walker and the track. Iterative macrocyclization and hydrolysis reactions lead to 68% of walkers taking two steps directionally along a three-foothold track.

Molecular Dynamics-Markov State Model of Protein Ligand Binding and Allostery in CRIB-PDZ: Conformational Selection and Induced Fit

Conformational selection and induced fit are well-known contributors to ligand binding and allosteric effects in proteins. Molecular dynamics (MD) simulations now enable the theoretical study of protein-ligand binding in terms of ensembles of interconverting microstates and the population shifts characteristic of "dynamical allostery." Here we investigate protein-ligand binding and allostery based on a Markov state model (MSM) with states and rates obtained from all-atom MD simulations. As an exemplary case, we consider the single domain protein par-6 PDZ with and without ligand and allosteric effector. This is one of the smallest proteins in which allostery has been experimentally observed. In spite of the increased complexity intrinsic to a statistical ensemble perspective, we find that conformational selection and induced fit mechanisms can be readily identified in the analysis. In the nonallosteric pathway, MD-MSM shows that PDZ binds ligand via conformational selection. However, the allosteric pathway requires an activation step that involves a conformational change induced by the allosteric effector Cdc42. Once in the allosterically activated state, we find that ligand binding can proceed by conformational selection. Our MD-MSM model predicts that allostery in this and possibly other systems involves both induced fit and conformational selection, not just one or the other.

Molecular simulations and Markov state modeling reveal the structural diversity and dynamics of a theophylline-binding RNA aptamer in its unbound state

RNA aptamers are oligonucleotides that bind with high specificity and affinity to target ligands. In the absence of bound ligand, secondary structures of RNA aptamers are generally stable, but single-stranded and loop regions, including ligand binding sites, lack defined structures and exist as ensembles of conformations. For example, the well-characterized theophylline-binding aptamer forms a highly stable binding site when bound to theophylline, but the binding site is unstable and disordered when theophylline is absent. Experimental methods have not revealed at atomic resolution the conformations that the theophylline aptamer explores in its unbound state. Consequently, in the present study we applied 21 microseconds of molecular dynamics simulations to structurally characterize the ensemble of conformations that the aptamer adopts in the absence of theophylline. Moreover, we apply Markov state modeling to predict the kinetics of transitions between unbound conformational states. Our simulation results agree with experimental observations that the theophylline binding site is found in many distinct binding-incompetent states and show that these states lack a binding pocket that can accommodate theophylline. The binding-incompetent states interconvert with binding-competent states through structural rearrangement of the binding site on the nanosecond to microsecond timescale. Moreover, we have simulated the complete theophylline binding pathway. Our binding simulations supplement prior experimental observations of slow theophylline binding kinetics by showing that the binding site must undergo a large conformational rearrangement after the aptamer and theophylline form an initial complex, most notably, a major rearrangement of the C27 base from a buried to solvent-exposed orientation. Theophylline appears to bind by a combination of conformational selection and induced fit mechanisms. Finally, our modeling indicates that when Mg²⁺ ions are present the population of binding-competent aptamer states increases more than twofold. This population change, rather than direct interactions between Mg²⁺ and theophylline, accounts for altered theophylline binding kinetics.

The folding mechanism and key metastable state identification of the PrP127–147 monomer studied by molecular dynamics simulations and Markov state model analysis

MD simulation combined with MSM analysis was employed to investigate the structural dynamics and the folding mechanism of the key fragment 127–147 monomer of prion protein.

Bridge helix bending promotes RNA polymerase II backtracking through a critical and conserved threonine residue

The dynamics of the RNA polymerase II (Pol II) backtracking process is poorly understood. We built a Markov State Model from extensive molecular dynamics simulations to identify metastable intermediate states and the dynamics of backtracking at atomistic detail. Our results reveal that Pol II backtracking occurs in a stepwise mode where two intermediate states are involved. We find that the continuous bending motion of the Bridge helix (BH) serves as a critical checkpoint, using the highly conserved BH residue T831 as a sensing probe for the 3′-terminal base paring of RNA:DNA hybrid. If the base pair is mismatched, BH bending can promote the RNA 3′-end nucleotide into a frayed state that further leads to the backtracked state. These computational observations are validated by site-directed mutagenesis and transcript cleavage assays, and provide insights into the key factors that regulate the preferences of the backward translocation.

RNA Polymerase II can detect and cleave mis-incorporated nucleotides by a proofreading mechanism that requires backtracking of the enzyme. Here the authors show that Pol II backtracking occurs in a stepwise mode that involves two intermediate states where the fraying of the terminal RNA nucleotide is a prerequisite.