Markov State Models Research Articles

Crystallization in Microgravity and the Atomic-Resolution Structure of Uridine Phosphorylase from Vibrio cholerae

Uridine phosphorylases are known as key targets for the development of new anticancer and antiparasitic agents. Crystals of uridine phosphorylase from the pathogenic bacterium Vibrio cholerae were grown in microgravity by the capillary counter-diffusion method on board of the International Space Station. The three-dimensional structure of this enzyme was determined at atomic (1.04 Å) resolution (RCSB PDB ID: 6Z9Z). Alternative conformations of long fragments (β-strands and adjacent loops) of the protein molecule were found for the first time in the three-dimensional structure of uridine phosphorylase in the absence of specific bound ligands. Apparently, these alternative conformations are related to the enzyme function. Conformational analysis with Markov state models demonstrated that conformational rearrangements can occur in the ligand-free state of the enzyme.

Nonparametric Analysis of Nonequilibrium Simulations.

We extend the nonparametric framework of reaction coordinate optimization to nonequilibrium ensembles of (short) trajectories. For example, we show how, starting from such an ensemble, one can obtain an equilibrium free-energy profile along the committor, which can be used to determine important properties of the dynamics exactly. A new adaptive sampling approach, the transition-state ensemble enrichment, is suggested, which samples the configuration space by "growing" committor segments toward each other starting from the boundary states. This framework is suggested as a general tool, alternative to the Markov state models, for a rigorous and accurate analysis of simulations of large biomolecular systems, as it has the following attractive properties. It is immune to the curse of dimensionality, does not require system-specific information, can approximate arbitrary reaction coordinates with high accuracy, and has sensitive and rigorous criteria to test optimality and convergence. The approaches are illustrated on a 50-dimensional model system and a realistic protein folding trajectory.

Temperature-dependent kinetic pathways of heterogeneous ice nucleation competing between classical and non-classical nucleation

Ice nucleation on the surface plays a vital role in diverse areas, ranging from physics and cryobiology to atmospheric science. Compared to ice nucleation in the bulk, the water-surface interactions present in heterogeneous ice nucleation complicate the nucleation process, making heterogeneous ice nucleation less comprehended, especially the relationship between the kinetics and the structures of the critical ice nucleus. Here we combine Markov State Models and transition path theory to elucidate the ensemble pathways of heterogeneous ice nucleation. Our Markov State Models reveal that the classical one-step and non-classical two-step nucleation pathways can surprisingly co-exist with comparable fluxes at T = 230 K. Interestingly, we find that the disordered mixing of rhombic and hexagonal ice leads to a favorable configurational entropy that stabilizes the critical nucleus, facilitating the non-classical pathway. In contrast, the favorable energetics promotes the formation of hexagonal ice, resulting in the classical pathway. Furthermore, we discover that, at elevated temperatures, the nucleation process prefers to proceed via the classical pathway, as opposed to the non-classical pathway, since the potential energy contributions override the configurational entropy compensation. This study provides insights into the mechanisms of heterogeneous ice nucleation and sheds light on the rational designs to control crystallization processes.

Millisecond functional dynamics of RNA polymerases elucidated by Markov state models

Millisecond functional dynamics of RNA polymerases elucidated by Markov state models

N-Glycosylation Enhances Conformational Flexibility of Protein Disulfide Isomerase Revealed by Microsecond Molecular Dynamics and Markov State Modeling.

Secreted proteins of eukaryotes are decorated with branched carbohydrate oligomers called glycans. This fact is only starting to be considered for in silico investigations of protein dynamics. Using all-atom molecular dynamics (MD) simulations and Markov state modeling (MSM), we unveil the influence of glycans on the conformational flexibility of the multidomain protein disulfide isomerase (PDI), which is a ubiquitous chaperone in the endoplasmic reticulum (ER). Yeast PDI (yPDI) from Saccharomyces cerevisiae is glycosylated at asparagine side chains and the knowledge of its five modified sites enables a realistic computational modeling. We compare simulations of glycosylated and unglycosylated yPDI and find that the presence of glycan-glycan and glycan-protein interactions influences the flexibility of PDI in different ways. For example, glycosylation reduces interdomain interactions, shifting the conformational ensemble toward more open, extended structures. In addition, we compare our results on yPDI with structural information of homologous proteins such as human PDI (hPDI), which is natively unglycosylated. Interestingly, hPDI lacks a surface recess that is present in yPDI. We find that glycosylation of yPDI facilitates its catalytic site to reach close to this surface recess. Hence, this might point to a possible functional relevance of glycosylation in yeast to act on substrates, while glycosylation seems redundant for the human homologous protein. We conclude that glycosylation is fundamental for protein dynamics, making it a necessity for a truthful representation of the flexibility and function in in silico studies of glycoproteins.

Activation pathway of a G protein-coupled receptor uncovers conformational intermediates as targets for allosteric drug design

G protein-coupled receptors (GPCRs) are the most common proteins targeted by approved drugs. A complete mechanistic elucidation of large-scale conformational transitions underlying the activation mechanisms of GPCRs is of critical importance for therapeutic drug development. Here, we apply a combined computational and experimental framework integrating extensive molecular dynamics simulations, Markov state models, site-directed mutagenesis, and conformational biosensors to investigate the conformational landscape of the angiotensin II (AngII) type 1 receptor (AT1 receptor) — a prototypical class A GPCR—activation. Our findings suggest a synergistic transition mechanism for AT1 receptor activation. A key intermediate state is identified in the activation pathway, which possesses a cryptic binding site within the intracellular region of the receptor. Mutation of this cryptic site prevents activation of the downstream G protein signaling and β-arrestin-mediated pathways by the endogenous AngII octapeptide agonist, suggesting an allosteric regulatory mechanism. Together, these findings provide a deeper understanding of AT1 receptor activation at an atomic level and suggest avenues for the design of allosteric AT1 receptor modulators with a broad range of applications in GPCR biology, biophysics, and medicinal chemistry.

Markov State Models to Study the Functional Dynamics of Proteins in the Wake of Machine Learning.

Markov state models (MSMs) based on molecular dynamics (MD) simulations are routinely employed to study protein folding, however, their application to functional conformational changes of biomolecules is still limited. In the past few years, the field of computational chemistry has experienced a surge of advancements stemming from machine learning algorithms, and MSMs have not been left out. Unlike global processes, such as protein folding, the application of MSMs to functional conformational changes is challenging because they mostly consist of localized structural transitions. Therefore, it is critical to properly select a subset of structural features that can describe the slowest dynamics of these functional conformational changes. To address this challenge, we recommend several automatic feature selection methods such as Spectral-OASIS. To identify states in MSMs, the chosen features can be subject to dimensionality reduction methods such as TICA or deep learning based VAMPNets to project MD conformations onto a few collective variables for subsequent clustering. Another challenge for the application of MSMs to the study of functional conformational changes is the ability to comprehend their biophysical mechanisms, as MSMs built for these processes often require a large number of states. We recommend the recently developed quasi-MSMs (qMSMs) to address this issue. Compared to MSMs, qMSMs encode the non-Markovian dynamics via the generalized master equation and can significantly reduce the number of states. As a result, qMSMs can be built with a handful of states to facilitate the interpretation of functional conformational changes. In the wake of machine learning, we believe that the rapid advancement in the MSM methodology will lead to their wider application in studying functional conformational changes of biomolecules.

Markov state models from hierarchical density-based assignment

Markov state models (MSMs) have become one of the preferred methods for the analysis and interpretation of molecular dynamics (MD) simulations of conformational transitions in biopolymers. While there is great variation in terms of implementation, a well-defined workflow involving multiple steps is often adopted. Typically, molecular coordinates are first subjected to dimensionality reduction and then clustered into small "microstates," which are subsequently lumped into "macrostates" using the information from the slowest eigenmodes. However, the microstate dynamics is often non-Markovian, and long lag times are required to converge the relevant slow dynamics in the MSM. Here, we propose a variation on this typical workflow, taking advantage of hierarchical density-based clustering. When applied to simulation data, this type of clustering separates high population regions of conformational space from others that are rarely visited. In this way, density-based clustering naturally implements assignment of the data based on transitions between metastable states, resulting in a core-set MSM. As a result, the state definition becomes more consistent with the assumption of Markovianity, and the timescales of the slow dynamics of the system are recovered more effectively. We present results of this simplified workflow for a model potential and MD simulations of the alanine dipeptide and the FiP35 WW domain.

Impact of Increased Membrane Realism on Conformational Sampling of Proteins.

The realism and accuracy of lipid bilayer simulations through molecular dynamics (MD) are heavily dependent on the lipid composition. While the field is pushing toward implementing more heterogeneous and realistic membrane compositions, a lack of high-resolution lipidomic data prevents some membrane protein systems from being modeled with the highest level of realism. Given the additional diversity of real-world cellular membranes and protein-lipid interactions, it is still not fully understood how altering membrane complexity affects modeled membrane protein functions or if it matters over long-timescale simulations. This is especially true for organisms whose membrane environments have little to no computational study, such as the plant plasma membrane. Tackling these issues in tandem, a generalized, realistic, and asymmetric plant plasma membrane with more than 10 different lipid species is constructed herein. Classical MD simulations of pure membrane constructs were performed to evaluate how altering the compositional complexity of the membrane impacted the plant membrane properties. The apo form of a plant sugar transporter, OsSWEET2b, was inserted into membrane models where lipid diversity was calculated in either a size-dependent or size-independent manner. An adaptive sampling simulation regime validated by Markov-state models was performed to capture the gating dynamics of OsSWEET2b in each of these membrane constructs. In comparison to previous OsSWEET2b simulations performed in a pure POPC bilayer, we confirm that simulations performed within a native-like membrane composition alter the stabilization of apo OsSWEET2b conformational states by ∼1 kcal/mol. The free-energy barriers of intermediate conformational states decrease when realistic membrane complexity is simplified, albeit roughly within sampling error, suggesting that protein-specific responses to membranes differ due to altered packing caused by compositional fluctuations. This work serves as a case study where a more realistic bilayer composition makes unbiased conformational sampling easier to achieve than with simplified bilayers.

Ion Conduction Mechanism as a Fingerprint of Potassium Channels.

K+-channels are membrane proteins that regulate the selective conduction of potassium ions across cell membranes. Although the atomic mechanisms of K+ permeation have been extensively investigated, previous work focused on characterizing the selectivity and occupancy of the binding sites, the role of water molecules in the conduction process, or the identification of the minimum energy pathways enabling permeation. Here, we exploit molecular dynamics simulations and the analytical power of Markov state models to perform a comparative study of ion conduction in three distinct channel models. Significant differences emerged in terms of permeation mechanisms and binding site occupancy by potassium ions and/or water molecules from 100 μs cumulative trajectories. We found that, at odds with the current paradigm, each system displays a characteristic permeation mechanism, and thus, there is not a unique way by which potassium ions move through K+-channels. The high functional diversity of K+-channels can be attributed in part to the differences in conduction features that have emerged from this work. This study provides crucial information and further inspiration for wet-lab chemists designing new synthetic strategies to produce versatile artificial ion channels that emulate membrane transport for their applications in diagnosis, sensors, the next generation of water treatment technologies, etc., as the ability of synthetic channels to transport molecular ions across a bilayer in a controlled way is usually governed through the choice of metal ions, their oxidation states, or their coordination geometries.

Independent Markov decomposition: Toward modeling kinetics of biomolecular complexes

To advance the mission of in silico cell biology, modeling the interactions of large and complex biological systems becomes increasingly relevant. The combination of molecular dynamics (MD) simulations and Markov state models (MSMs) has enabled the construction of simplified models of molecular kinetics on long timescales. Despite its success, this approach is inherently limited by the size of the molecular system. With increasing size of macromolecular complexes, the number of independent or weakly coupled subsystems increases, and the number of global system states increases exponentially, making the sampling of all distinct global states unfeasible. In this work, we present a technique called independent Markov decomposition (IMD) that leverages weak coupling between subsystems to compute a global kinetic model without requiring the sampling of all combinatorial states of subsystems. We give a theoretical basis for IMD and propose an approach for finding and validating such a decomposition. Using empirical few-state MSMs of ion channel models that are well established in electrophysiology, we demonstrate that IMD models can reproduce experimental conductance measurements with a major reduction in sampling compared with a standard MSM approach. We further show how to find the optimal partition of all-atom protein simulations into weakly coupled subunits.

The kinetic landscape of nucleosome assembly: A coarse-grained molecular dynamics study

The organization of nucleosomes along the Eukaryotic genome is maintained over time despite disruptive events such as replication. During this complex process, histones and DNA can form a variety of non-canonical nucleosome conformations, but their precise molecular details and roles during nucleosome assembly remain unclear. In this study, employing coarse-grained molecular dynamics simulations and Markov state modeling, we characterized the complete kinetics of nucleosome assembly. On the nucleosome-positioning 601 DNA sequence, we observe a rich transition network among various canonical and non-canonical tetrasome, hexasome, and nucleosome conformations. A low salt environment makes nucleosomes stable, but the kinetic landscape becomes more rugged, so that the system is more likely to be trapped in off-pathway partially assembled intermediates. Finally, we find that the co-operativity between DNA bending and histone association enables positioning sequence motifs to direct the assembly process, with potential implications for the dynamic organization of nucleosomes on real genomic sequences.

Determining sequence-dependent DNA oligonucleotide hybridization and dehybridization mechanisms using coarse-grained molecular simulation, Markov state models, and infrared spectroscopy

Determining sequence-dependent DNA oligonucleotide hybridization and dehybridization mechanisms using coarse-grained molecular simulation, Markov state models, and infrared spectroscopy

Determining sequence-dependent DNA oligonucleotide hybridization and dehybridization mechanisms using coarse-grained molecular simulation, Markov state models, and infrared spectroscopy

Determining sequence-dependent DNA oligonucleotide hybridization and dehybridization mechanisms using coarse-grained molecular simulation, Markov state models, and infrared spectroscopy

Computational modeling of protein conformational changes - Application to the opening SARS-CoV-2 spike

We present a new approach to compute and analyze the dynamical electro-geometric properties of proteins undergoing conformational changes. The molecular trajectory is obtained from Markov state models, and the electrostatic potential is calculated using the continuum Poisson-Boltzmann equation. The numerical electric potential is constructed using a parallel sharp numerical solver implemented on adaptive Octree grids. We introduce novel a posteriori error estimates to quantify the solution's accuracy on the molecular surface. To illustrate the approach, we consider the opening of the SARS-CoV-2 spike protein using the recent molecular trajectory simulated through the Folding@home initiative. We analyze our results, focusing on the characteristics of the receptor-binding domain and its vicinity. This work lays the foundation for a new class of hybrid computational approaches, producing high-fidelity dynamical computational measurements serving as a basis for protein bio-mechanism investigations.

Modulation of adenosine A2a receptor oligomerization by receptor activation and PIP2 interactions

SummaryGPCRs have been shown to form oligomers, which generate distinctive signaling outcomes. However, the structural nature of the oligomerization process remains uncertain. We have characterized oligomeric configurations of the adenosine A2a receptor (A2aR) by combining large-scale molecular dynamics simulations with Markov state models. These oligomeric structures may also serve as templates for studying oligomerization of other class A GPCRs. Our simulation data revealed that receptor activation results in enhanced oligomerization, more diverse oligomer populations, and a more connected oligomerization network. The active state conformation of the A2aR shifts protein-protein association interfaces to those involving intracellular loop ICL3 and transmembrane helix TM6. Binding of PIP2 to A2aR stabilizes protein-protein interactions via PIP2-mediated association interfaces. These results indicate that A2aR oligomerization is responsive to the local membrane lipid environment. This, in turn, suggests a modulatory effect on A2aR whereby a given oligomerization profile favors the dynamic formation of specific supramolecular signaling complexes.

Physics-based protein structure refinement in the era of artificial intelligence.

Protein structure refinement is the last step in protein structure prediction pipelines. Physics-based refinement via molecular dynamics (MD) simulations has made significant progress during recent years. During CASP14, we tested a new refinement protocol based on an improved sampling strategy via MD simulations. MD simulations were carried out at an elevated temperature (360 K). An optimized use of biasing restraints and the use of multiple starting models led to enhanced sampling. The new protocol generally improved the model quality. In comparison with our previous protocols, the CASP14 protocol showed clear improvements. Our approach was successful with most initial models, many based on deep learning methods. However, we found that our approach was not able to refine machine-learning models from the AlphaFold2 group, often decreasing already high initial qualities. To better understand the role of refinement given new types of models based on machine-learning, a detailed analysis via MD simulations and Markov state modeling is presented here. We continue to find that MD-based refinement has the potential to improve AI predictions. We also identified several practical issues that make it difficult to realize that potential. Increasingly important is the consideration of inter-domain and oligomeric contacts in simulations; the presence of large kinetic barriers in refinement pathways also continues to present challenges. Finally, we provide a perspective on how physics-based refinement could continue to play a role in the future for improving initial predictions based on machine learning-based methods.

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.

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.

Dynamic Profiling of β-Coronavirus 3CL Mpro Protease Ligand-Binding Sites.

β-coronavirus (CoVs) alone has been responsible for three major global outbreaks in the 21st century. The current crisis has led to an urgent requirement to develop therapeutics. Even though a number of vaccines are available, alternative strategies targeting essential viral components are required as a backup against the emergence of lethal viral variants. One such target is the main protease (Mpro) that plays an indispensable role in viral replication. The availability of over 270 Mpro X-ray structures in complex with inhibitors provides unique insights into ligand–protein interactions. Herein, we provide a comprehensive comparison of all nonredundant ligand-binding sites available for SARS-CoV2, SARS-CoV, and MERS-CoV Mpro. Extensive adaptive sampling has been used to investigate structural conservation of ligand-binding sites using Markov state models (MSMs) and compare conformational dynamics employing convolutional variational auto-encoder-based deep learning. Our results indicate that not all ligand-binding sites are dynamically conserved despite high sequence and structural conservation across β-CoV homologs. This highlights the complexity in targeting all three Mpro enzymes with a single pan inhibitor.